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\begin{flushright}
\begin{tabular}{r}
CERN-TH.6736/92 \\
FTUAM-38/92
\end{tabular}
\end{flushright}

\vspace{4cm}

\begin{center}

{\Large \bf Topics in String Theory and Quantum Gravity}
\footnote{Based  on the lectures given by L. Alvarez-Gaum\'e at Les
Houches
Summer School on Gravitation and Quantization, July $6-31$, $1992$.}

\vspace{1.5cm}

L. Alvarez-Gaum\'e \\
{\it Theory Division CERN \\ CH-1211 Geneva 23 \\ Switzerland}
\vspace{0.4cm}

and

\vspace{0.4cm}

M.A. V\'azquez-Mozo \\
{\it Departamento de F\'{\i}sica Te\'orica C-XI \\
Universidad Aut\'onoma de Madrid \\
E-28049 Madrid, Spain}

\end{center}

\vspace{7cm}

\begin{tabular}{l}

CERN-TH.6736/92 \\
November 1992

\end{tabular}


\newpage

\tableofcontents

\section*{General Introduction}

The goal of this Les Houches school is to bring together the
practitioners of different research lines concerned with the
quantization
of gravity (\cite{Oxford-symp} and references therein). In this
collection, the present lectures represent in part the point of view of
string theorist (for details and references see for example
\cite{Green-Schwarz-Witten,Kaku,Kaku2,Lust-Theisen}). The most ambitious
approach to the problem is certainly embodied by Superstring Theories
where one obtains together with the gravitational field many other
interactions with enough richness to account for many features of the
Standard Model (SM). This is not an accident. One can consider String
Theory as the culmination of several decades of effort dedicated to
incorporate within the same framework gravity together with the other
known interactions. After nearly a decade of renewed interest in String
Theory it seems reasonable to recapitulate how this theory has come to
be one of the leading candidates of the unification of all known
interactions. We still do not know the basic physical or geometrical
principles underlying String Theory. There has been some progress in the
formulation of a String Field Theory (see the lecture by B. Zwiebach,
\cite{SFT} and references therein) but we are still far from having a
complete theory of string fields. Progress has been made by following a
number of consistency requirements which worked quite well in ordinary
field theories, and in particular for the SM. These requirements when
applied to the unification of gravity with the SM lead to very strong
constraints which for the time being seem to be fulfilled only by String
Theory.

One of the basic properties of the SM which supports many of its
successes is its renormalizability, or equivalently, the fact that after
a finite number of parameters are given, we have a rather successful
machinery to calculate and explain a large number of high and low energy
phenomena. This machinery is renormalizable Quantum Field Theory (QFT).
This renormalizability, or rather, predictability of the SM is one of
the desired features we would like to have in a quantum theory of
gravity.

There are also several features of the SM that should be explained in
any theory trying to go beyond. Namely, the chiral nature of the
families of quarks and leptons with respect to the SM gauge group
$SU(3)\times SU(2)\times U(1)$, the origin of the gauge interaction, the
origin of the symmetry breaking, the vanishing of the cosmological
constant etc. One of the more popular ``Beyond the SM''avenues is the
study of theories incorporating supersymmetry \cite{SUSY}. Supersymmetry
is a central property of Superstring Theory. Unfortunately there is no
evidence in present accelerators that supersymmetry is realized in
Nature. However, if present or future accelerators would find evidence
of supersymmetry partners of the known elementary particles, the
temptation to extrapolate and to believe that Superstring Theory should
play a central	r\^{o}le in the unification of the known interactions
would be nearly irresistible.

When one tries to look for a predictive theory containing Quantum
Gravity together with the chiral structure of the low energy degrees of
freedom, one seems to be left only with String Theory. These two
requirements alone pose very stringent constraints on candidate
Theories. As we will see in the next chapter various proposals including
combinations of Supergravity \cite{SUGRA} and Kaluza-Klein Theories
\cite{KK}
do not satisfy any of these requirements. If one consider only the
quantization of the gravitational field independently of other
interactions, there has been some substantial progress in the recent
past which is reviewed in the lectures by A. Ashtekar.

The outline of these lectures is as follows. In Chapter 1 we present
many of the approaches used to quantize gravity and to unify it with
other interactions in the framework of Quantum Field Theory. We briefly
analyze the difficulties one encounters when one tries to apply standard
field theory techniques to perturbative computations including graviton
loops. We also sumarize the approaches including Supergravity and
Kaluza-Klein Theories. These theories are not only afflicted with
uncontrollable infinities, but also by the fact that low energy chiral
fermions are hard to obtain. We also present a rather brief study of the
conceptual problems encountered in the quantization of fields in the
presence of external gravitational backgrounds. We conclude the chapter
with a collection of general remarks concerning the Euclidean and
Hamiltonian approaches to the quantization of gravity. If taking
seriously, the na\"{\i}ve approach of summing over all possible
topologies
and geometries meets with formidable mathematical difficulties.
Furthermore, the use of semiclassical methods based on instantons, which
could in principle provide some insights into quantum-gravitationally
induced processes is also presented (and criticized). This chapter gives
a rather negative portrait of the field theoretic attempts to understand
Quantum Gravity, and needless to say, this view was not shared by many
of the others lecturers.

In Chapter two we come to study the constraints imposed by the
requirement of having consistent gauge and gravitational interactions
between chiral fermions. This brings us to the analysis of Anomalies,
both local and global. We present the general conceptual features of the
computation of local anomalies in diverse dimensions; we present some
efficient computational techniques, and apply them to few interesting
examples, in particular the Green-Schwarz anomaly cancelation mechanism
\cite{Green-Schwarz} which opened the way to the formulation of the
Heterotic String \cite{Pricenton-quartet}. We also present briefly the
analysis of global anomalies, and in particular Witten's $SU(2)$ anomaly
\cite{WittenSU(2),Witten-anom}. In this chapter we want to exhibit how
difficult it is to have anomaly-free chiral theories with gravitational
and gauge interactions. In the Green-Schwarz mechanism for example we
are left only with the possible gauge groups $SO(32)$, $E_{8}\times
E_{8}$ and $E_{8}\times U(1)^{248}$. Thus, the spectrum of low energy
excitations (massless states) is severely restricted by the requirement
of absence of anomalies. Global anomalies (anomalies with respect to
diffeomorphisms or gauge transformations not in the identity component)
play a central r\^{o}le in determining the spectrum of possible
Superstring
Theories. We will analyze in detail in chapter four how the absence of
global world-sheet diffeomorphism anomalies (modular invariance), a
purely two-dimensional condition, restricts the space-time spectrum in
String Theory.

In Chapter three we begin our study of String Theory. In this chapter we
restrict for simplicity to the bosonic string. We present briefly the
quantization of the bosonic string, the difference between critical and
non-critical strings, and the appearance of a tachyon and a graviton in
the spectrum in the critical dimension ($d=26$). These two states are
ubiquitous in critical bosonic string theories. We then present a
general analysis of String Perturbation Theory. We use the operator
formalism presented in \cite{Alvarez-Gaume-Gomez-Moore-Vafa}, (other
approaches can
be found in \cite{Op-form-2-1,Op-form-2-2}). This approach emphasizes
the geometrical
nature of first quantized String Theory. String scattering amplitudes
and physical state conditions are beautifully reflected in the
properties of the moduli space of Riemann surfaces. Using this geometric
formulation of string amplitudes provides us with a neat understanding
of where infinities in string processes might come from. In particular,
for the bosonic string it will become clear that the source of
infinities is the presence of a tachyon in the spectrum. We have also
included a brief section on space-duality to exhibit a property of
strings not shared by field theories. This symmetry renders further
evidence that in String Theory there is a fundamental length (even
though the string interactions are purely local, something that one
cannot have in field theory), and ultimately this fundamental length may
be the basic reason for the finiteness of superstrings amplitudes. We
also analyze how the Einstein equations follow from a two-dimensional
requirement, conformal invariance, which provides the classical
equations of motion.

In Chapter four we study fermionic and supersymmetric strings. We see
how the tachyon is projected out using the GSO projection \cite{GSO},
and exhibit the geometric interpretation of this projection. This
will make clear the interplay between modular invariance and the
spectrum of the theory. The most remarkable example is the Heterotic
String, which will be sumarized with special attention devoted to the
r\^{o}le of modular invariance. We close this chapter with two topics:
the
first is a summary of the finite temperature of fermionic strings. Here
again the modular group plays an interesting r\^{o}le. We present the
issue
of temperature duality (the canonical partition function at temperature
$T$ and $const/T$
is the same) and some of the riddles associated with this property.
Second we give a brief status report on the finiteness of superstring
perturbation theory and what is known about high order behavior of
perturbation theory.

In Chapter five we summarize some of the actives areas of current
research in String Theory and present the conclusions. In particular we
summarize some of the present research on string or string inspired
black hole physics.

There are obviously many aspect of String Theory which have been left
out in these lectures. Most notably the subject of classical solutions
to String Theory, which started with the Calabi-Yau \cite{Calabi-Yau}
and orbifold solutions \cite{orbifolds}, and String Phenomenology
\cite{phenomenology} which have bloomed into rather large subjects
have not been included. The aim of the school was centered upon the
issues of Quantum Gravity, and it seems more reasonable to leave this
subject to other ocassions (and to a more competent lecturer). Our aim
was to show how far we have gone with a rather simple set of
requirements. A lot of work is still necessary to clarify and explore
many aspects of String Theory. Perhaps the most outstanding open
problem in String Theory (as in any other theory of gravity) is the
vanishing of the cosmological constant. String Theory does not provide
us with any clue on this issue.

Needless to say, many attendants to the school did not share the
moderately optimistic opinion of this lecturer. This provides a constant
source of useful (and often entertaining) discussions. We still believe
that many of the ingredients and properties of String Theory will be
contained in the correct theory of Quantum Gravity. It is too early to
tell whether this belief carries any truth.

\underline{Acknowledgements:} It is a pleasure to thank the organizers
of the school B. Julia and J. Zinn-Justin for the opportunity to present
these lectures and for creating a very productive and inspiring
enviroment. In preparing these lectures we have benefitted from
discussions with many colleagues. We would like to thank E. Alvarez,
J.L.F. Barb\'on,  E. Br\'ezin, A. Connes, C. Gomez, L. Iba\~nez, S.
Jain, W. Lerche, J. Louis and M.A.R. Osorio for many conversations
about String Theory. The reviews by E. Alvarez
\cite{EAlvarez1,EAlvarez2} were
particularly useful in the preparation of chapter one.


\section{Field-theoretical approach to Quantum Gravity}

General Relativity (GR) and Quantum Field Theory (QFT)
are two of the most impressive achievements in XXth century Science.
One of the first predictions of GR was the centennial precession of the
orbit of Mercury. The relativistic contribution is (see for example
\cite{Weinberg})
\begin{equation}
\Delta\phi^{th}=43.03 \hspace{0.5cm} seconds/century
\end{equation}
while the observed value is
\begin{equation}
\Delta\phi^{obs}=43.11\pm 0.45 \hspace{0.5cm} seconds/century
\end{equation}
Even gravitational radiation, which is also a prediction of GR, has been
indirectly detected by the observation of the decay of the orbit
of the binary pulsar PSR $1913+16$ \cite{Taylor-pulsar}.

On the other hand QFT allow us to calculate observables at microscopic
scales, often with a very high precision. To give an example, the
predicted
value in Quantum Electrodynamics for the anomalous magnetic moment of
the electron is \cite{Itzykson-Zuber}
\begin{equation}
a_{e}^{th}=1.159652359(282) \times 10^{-12}
\end{equation}
where uncertainities are given by those in the value of the fine
structure constant. Experimentally it is found that:
\begin{equation}
a_{e}^{exp}=1.159652410(200) \times 10^{-12}
\end{equation}

To obtain a more complete fundamental picture of the physical world we
would like to put together QFT and GR. There are still many obtacles
to the achievement of this goal. It is also reasonably clear that such a
synthesis should provide profound insights into many of the riddles of
the Standard Model of Strong, Weak and Electromagnetic interactions.
Most notably, the origin of the chiral nature of the families of quarks
and leptons and possibly the origin of mass. When we study the running
of the coupling constants for the $SU(3)\times SU(2)\times U(1)$ gauge
group of the SM ($\alpha_{3},\alpha_{2},\alpha_{1}$
respectively), it is found that, extrapolating their present
experimental value and making some assumption about the degrees of
freedom between present experimental energies and the Planck scale (the
desert hypothesis), they converge to a common value at a scale between
$10^{16}-10^{17}$ GeV where quantum gravitational effects can certainly
not be neglected. Unification of GR and QFT is the most coveted Holy
Grail in Theoretical Physics in the last quarter of the XXth century.


\subsection{Linearized gravity}

QFT has been succesfully applied to both strong and electroweak
interaction, which are described in terms of local gauge field
theories. The
local gauge symmetry in the SM ensures that the theory is
perturbatively
renormalizable \cite{'tHooft1}; this means that there is an effective
decoupling of the high energy degrees of freedom from the low energy
predictions, so the infinities that appear in the perturbative expansion
can be absorbed in redefinitions of the parameters of the theory that
are experimentally measurable.

Gauge invariance+renormalizability seem to be two of the cornerstones of
our
understanding of present high energy physics. One then may wonder why
not try the same program with GR. We begin with the
Einstein-Hilbert action
\begin{equation}
S_{E-H}=-\frac{1}{2\kappa^{2}}\int d^{4}x\sqrt{-g} R
\end{equation}
and make a decomposition of the metric into a background part
$\hat{g}_{\mu \nu}$ and a dynamical part $h_{\mu \nu}$
\begin{equation}
g_{\mu \nu}=\hat{g}_{\mu \nu}+ \kappa h_{\mu \nu}
\end{equation}
By doing so, 't Hooft and Veltman \cite{'tHooft-Veltman} showed that
four-dimensional pure gravity has a finite one-loop $S$-matrix
although off-shell it is not one-loop renormalizable.
Considering gravity coupled to a scalar field, the one-loop counterterm
of the lagrangian is the sum of two terms, the first one is
proportional to the equation of motion and the second one is given by
\begin{equation}
{\cal L}_{c}^{1}=\frac{1}{\epsilon}\sqrt{-\hat{g}}\frac{203}{80}
\hat{R}^{2}
\end{equation}
where $\epsilon=4-d$ is the usual parameter in dimensional
regularization. In the case of pure gravity we should have $\hat{R}=0$
hence this counterterm vanishes and the theory is finite at one-loop
level.
However when considering the coupling to matter the counterterm is
divergent (since now we have $\hat{R}\neq 0$).

The finiteness of the one-loop $S$-matrix for pure gravity was a rather
surprising result, since Einstein theory is non-renormalizable by
na\"{\i}ve power counting. At two loops the situation changes radically.
Goroff and Sagnotti \cite{Goroff-Sagnotti} were
able to prove that Einstein pure gravity diverges at two loop level, and
that the divergent part of the on-shell effective action is
\begin{equation}
\Gamma^{(2)}=\frac{209}{2880(4\pi)^{4}}\frac{1}{\epsilon}
\int d^{4}x\sqrt{-g} R^{\alpha \beta}_{\;\;\;\;\gamma \delta} R^{\gamma
\delta}_{\;\;\;\;\rho \sigma} R^{\rho \sigma}_{\;\;\;\;\alpha \beta}
\end{equation}
so the program that allowed us to construct a quantum field theory for
both the strong and the electroweak interactions fails when applied to
GR.

If we continue to maintain the notion of renormalizability (or
predictability) should hold in the presence of gravitational
interactions, this result imposes strong constraints on
the possible quantum theories describing gravity.
If this theory includes other fields, then
there must be subtle cancellations between the different contributions.
Following this line of thought the first proposed candidate was
Supergravity (SUGRA) \cite{SUGRA}.

\subsection{Supergravity}

In SUGRA theories supersymmetry is a local symmetry. Since the
commutator of two supercurrents generates the energy-momentum tensor,
once supersymmetry is gauged we are inevitably led to a theory
containing gravity. As in ordinary supersymmetry, in N=1 supergravity
\cite{N=1}
each particle of mass $m$ and spin $s$ has associated with it another
particle with the same mass and spin $s\pm\frac{1}{2}$. Thus the
graviton which has spin $2$ comes accompanied by the gravitino field
with $s=\frac{3}{2}$. In principle one can also construct theories with
extended supergravity, $N>1$. Since it is not known how to consistently
quantize an interacting theory containing a massless spin $\frac{5}{2}$
field, the requirement that the highest helicity state be the graviton
imposed the constraint that $N\leq 8$ \cite{N<9}.

The ultraviolet properties of perturbation theory improve with respect
to those of pure gravity due to the cancelation of divergences between
bosons and fermions in the same supermultiplet. Although SUGRA is again
non-renormalizable by power counting, it is legitimate to ask whether
for some $1\leq N\leq 8$ the cancelation of infinities will render the
$S$-matrix finite without the need to renormalize. Thus, even if the
off-shell Green functions may be plaged with infinities, a presumably
weaker requirement that renormalizability is to impose finiteness of the
scattering amplitudes.

As an example, let us consider $N=1$, $d=4$ SUGRA described by the
lagrangian \cite{N=1}
\begin{equation}
{\cal L}=-\frac{1}{2\kappa^{2}}eR-
\frac{1}{2}\epsilon^{\mu\nu\sigma\lambda}
\bar{\psi}_{\mu}\gamma_{5}\gamma_{\nu}D_{\sigma}\psi_{\lambda}\;,
\end{equation}
where $\psi_{\mu}$ is the gravitino field, $e=det(e_{\mu}^{\;\;a})$ the
determinant of the vierbein and $D_{\mu}$ is the covariant derivative
\begin{equation}
D_{\mu}=\partial_{\mu}+\frac{1}{2}\omega_{\mu}^{\;\;mn}\sigma_{mn}\;,
\end{equation}
with $\omega_{\mu}^{\;\;mn}$ the spin connection and
$\sigma_{mn}=\frac{1}{4}[\gamma_{m},\gamma_{n}]$. The
lagrangian ${\cal L}$ is invariant under the local supersymmetry
transformations
\begin{eqnarray}
\delta
e_{\mu}^{\;\;m}
&=&\frac{1}{2}\kappa\bar{\epsilon}(x)\gamma^{m}\psi_{\mu}(x)
\nonumber \\
\delta\omega_{\mu}^{\;\;mn}&=&0
\nonumber \\
\delta \psi_{\mu}&=&D_{\mu}\epsilon(x)
\end{eqnarray}
In the quantum theory, besides the graviton and the gravitino
we have to include a coordinate spin-$1$ ghost and its
corresponding partner, a spin-$\frac{1}{2}$ ghost field
\cite{Freedman-Das}.

It can be shown that on-shell amplitudes in $N=1$ SUGRA are finite up to
two-loops order \cite{N=1-finite}, since the possible counterterms in
the action are
zero on-shell. However at three-loop level there is a counterterm which
does not vanish even on-shell \cite{N=1-3-l} (see also the second
reference in \cite{N=1-finite}). Unless the coefficient
multiplying this counterterm is zero, $N=1$ SUGRA would be three-loop
divergent. This is still an open question. The computation to settle it
is a formidable task that nobody has undertaken. However, since there
exists a possible counterterm, one would tend to be rather pesimist
about its outcome.

Since $N=1$ SUGRA is expected to diverge at three-loops, one has to
examine the case for
a finite theory in $N$-extended supergravity theories. For
some time $N=8$ SUGRA in four dimensions was a candidate for a theory of
``everything''. This theory can be formulated as $N=1$ SUGRA in
eleven dimensions \cite{N=1-d=11} which, after dimensional reduction,
becomes
$N=8$ SUGRA in four dimensions \cite{Cremmer-Julia}. In the
eleven-dimensional theory we have an {\it elf-bein} $e_{\mu}^{\;\;m}$
together with the gravitino $\psi_{\mu}$ and an antisymmetric tensor of
rank $3$, $A_{\mu\nu\sigma}$ (it is easy to see that the number of
bosonic and fermionic degrees of freedom is the same). The
mechanism of dimensional reduction consists of making the fields
depend only on the four-dimensional coordinates
$x^{1},\ldots,x^{4}$ and not on the eleven-dimensional
ones $x^{1},\ldots,x^{11}$. At the end we have the following spectrum
in $d=4$
\begin{eqnarray}
\begin{array}{ccc}
\# \; \mbox{of fields}	    & \mbox{spin}  &	 \\
\\
1  & 2	& \mbox{graviton} \\
8  & \frac{3}{2}  & \mbox{gravitinos}  \\
28 & 1	  &  \\
56 & \frac{1}{2}  &  \\
70 & 0	 &  \\
\end{array}
\nonumber
\end{eqnarray}
It is not possible to construct SUGRA theories in dimensions $d>11$
\cite{Nahm} since there is no possibility of matching bosonic and
fermionic degrees of freedom.
Moreover, could it be possible to construct simple SUGRA in $d>11$ then
by dimensional reduction we would be left with $N>8$ SUGRA in $d=4$,
violating so the upper bound for the number of supersymmetric partners
(see above).

Although, by symmetry arguments, $N=8$ SUGRA is expected to be finite up
to seven loops, for higher loops there are possible
on-shell countertems \cite{Kallosh}. This means that the theory could be
divergent at that
order if there is no a miraculous cancellation of the coefficients in
front of the counterterms. It seems na\"{\i}vely that supergravity is not the
correct place to look for a finite quantum theory of gravity. In fact
for $N$-extended SUGRA there are possible non-zero on-shell counterterms
beyond
$(N-1)$-loop order for $N\geq 3$ and beyond $2$-loops order for $N=1,2$.

\subsection{Kaluza-Klein theories}

During the seventies and early eighties there was a good deal of
activity in
Kaluza-Klein theories. The main idea goes
back to the 1920's when Kaluza and Klein \cite{Kaluza-Klein} tried to
unify Einstein's General Relativity and Maxwell theory by formulating
GR in $M_{4}\times S^{1}$, where $M_{4}$ is the ordinary
$4$-dimensional space-time and $S^{1}$ is a circle of very small radius
(typically of the order of the Planck lenght $l_{P}\sim 1.6\times
10^{-33}$
cm). By introducing coordinates $(x^{\mu},\phi)$ we make the
Kaluza-Klein ansatz for
the metric tensor in five dimensions
\begin{equation}
g_{AB}(x,\phi)=\left(
\begin{array}{cc}
g_{\mu \nu}(x) &   A_{\mu}(x)  \\
A_{\mu}(x)  & \sigma(x)
\end{array}
\right)\;,
\label{U(1)KK}
\end{equation}
where we have retained only the zero-modes in the fifth coordinate
$\phi$ (this is equivalent to restricting ourselves to massless models
in $M_{4}$). Substituting (\ref{U(1)KK}) in the five-dimensional
Einstein-Hilbert action we
obtain the gravitational action in four dimensions plus the Maxwell
kinetic term $(F_{\mu \nu})^{2}$ for the 4-dimensional vector field
$A_{\mu}$ and the kinetic term for $\sigma$.

This scheme can be generalized to non-abelian gauge theories
\cite{Cho-Freund}. To do so we consider the Einstein-Hilbert action in a
$4+n$-dimensional space-time  $M_{4}\times B$ where now $B$ is a compact
internal manifold having $G$ as its group of isometries. Considering
coordinates
$(x^{\mu},\phi_{i})$ ($\mu=0,1,2,3$ and $i=1,\ldots,n$) the Kaluza-Klein
ansatz for the metric tensor in $M_{4}\times B$ is
\begin{equation}
g_{AB}(x,\phi)=\left(
\begin{array}{cc}
g_{\mu \nu}(x)	  &  \sum_{a=1}^{dim G} A_{\mu}^{a}(x)\xi_{i}^{a}(\phi)
\\
\sum_{a=1}^{dim G} A_{\mu}^{a}(x)\xi_{i}^{a}(\phi) &  \gamma_{ij}(\phi)
\end{array}
\right)
\label{KK-n}
\end{equation}
where $\xi_{i}^{a}$ ($a=1,\ldots,dim G$) are the Killing vectors
associated with the symmetry group $G$
\begin{equation}
T^{a}:\phi_{i} \longrightarrow \phi_{i}+\xi_{i}^{a}(\phi)\epsilon^{a}
\end{equation}
Substituting (\ref{KK-n}) in
\begin{equation}
S_{E-H}=\frac{-1}{2\kappa^{2}}\int d^{4+n}x \sqrt{-g} R^{(4+n)}
\end{equation}
we obtain in particular the Einstein-Hilbert action in four dimensions
as well as the
Yang-Mills action $\sum_{a}(F_{\mu \nu}^{a})^{2}$, invariant under the
gauge transformation
\begin{eqnarray}
(x^{\mu},\phi_{i}) &  \longrightarrow  &
\left( x^{\mu},\phi_{i}+\sum_{a}\epsilon^{a}(x)\xi_{i}^{a}(\phi) \right)
\\
A_{\mu}^{a}(x) &  \longrightarrow & A_{\mu}^{a}(x)+D_{\mu}\epsilon^{a}(x)
\end{eqnarray}
where $\epsilon^{a}(x)$ are the gauge parameters and $D_{\mu}$ is the
covariant derivative with respect to the gauge field $A_{\mu}^{a}$.

So the main idea of Kaluza-Klein theories is to represent gauge
invariance in four-dimensional space-time as resulting from the
group of isometries of an internal $n$-dimensional manifold, a
beautiful geometrical idea. Of course
the problem now is to construct realistic Kaluza-Klein theories , i.e.,
theories with gauge group $G$ containing the SM gauge group
$SU(3)\times SU(2) \times U(1)$. This requirement imposes a lower bound
on the dimension of the internal space \cite{K-K-Witten1}. The space of
lowest dimension with symmetry group $G$ is always a homogeneous
space $G/H$ with $H\subset G$ a maximal subgroup of $G$. In our case
$G=SU(3)\times SU(2) \times U(1)$ and the largest subgroup of suitable
dimension
is $H=SU(2)\times U(1)\times U(1)$. This means that the minimal
dimension of a manifold with symmetry group  $SU(3)\times SU(2) \times
U(1)$ is $12-5=7$, so the minimal dimension of the space-time is $11$,
exactly equal to the maximal dimension for supergravity (see above).

Nevertheless, Kaluza-Klein theories have serious problems. One of them
is, being essentially Einstein gravity in
dimensions higher than four, non-renormalizability. Perhaps their
main difficulty is that they are unable to reproduce the chiral nature
of the four-dimensional world. Consider a massless Dirac
fermion in $4+n$ dimensions. It obeys the Dirac equation
\begin{equation}
\Dsl \psi=\sum_{i=1}^{4+n}\Gamma^{i}D_{i}\psi=0,
\end{equation}
which may be rewritten as
\begin{equation}
\Dsl ^{(4)}\psi+\Dsl ^{(int)}\psi=0
\label{4+int}
\end{equation}
where $\Dsl ^{(4)}=\sum_{i=1}^{4} \Gamma^{i}D_{i}$ and
$\Dsl ^{(int)}=\sum_{i=5}^{4+n} \Gamma^{i}D_{i}$. It is easy to see
from (\ref{4+int}) that the eigenvalues of $\Dsl ^{(int)}$ are the
masses of the fermions in four dimensions. Since non-zero eigenvalues
will be of the order of $1/R$ with $R$ the typical length of the
internal manifold ($\sim$ Planck length) we see that the
phenomenologically relevant four-dimensional fermions will be the
zero-modes of $\Dsl ^{(int)}$. But in order to reproduce the
SM we need these fermions to be chiral in four dimensions,
i.e., the left and right handed representations of the gauge group
should be different. This
cannot be achieved by using Dirac fermions in $4+n$ dimensions, as can
be seen by the following argument due to Lichnerowicz
\cite{Lichnerowicz,K-K-Witten2}: if we square
the Dirac operator $i\Dsl ^{(int)}$ in the internal manifold we have
(dropping the label {\it int} for simplicity)
\begin{equation}
(i\Dsl )^{2}=-\sum_{i} D_{i}D^{i} + \frac{1}{4}R
\end{equation}
where R is the scalar curvature. If $R>0$ everywhere on the manifold
$B$, and since $-\sum_{i} D_{i}D^{i}$ is positive defined,
the Dirac operator cannot have zero-modes. Furthermore,
a theorem by Lawson and Yau \cite{Lawson-Yau} states that on any compact
space $B$ with
non-abelian symmetry group $G$, there is always a $G-$invariant metric
of positive curvature. Thus, we cannot obtain massless fermions in four
dimensions from Dirac
fermions in $4+n$ dimensions. What can be said about Rarita-Schwinger
fields?
After all in $N=8$ SUGRA we start not with Dirac spinors
but with
Rarita-Schwinger fields in eleven dimensions, which after dimensional
reduction give raise to four-dimensional Dirac spinors. The problem of
obtaining four-dimensional chiral $\frac{1}{2}$ fermions from
Rarita-Schwinger
fields in higher dimension has been studied by Witten who also carried
out the general analysis of whether chirality could be generated from
Kaluza-Klein theories. We are essentially summarizing his arguments
\cite{K-K-Witten2}.
He showed by using Atiyah-Hirzebruch theorem
\cite{Atiyah-Hirzebruch} that the zero-modes of
the Rarita-Schwinger operator on any homogeneous space $G/H$ form a real
representation leading to vector-like quantum numbers in four
dimensions. It is then imposible to reproduce a chiral low-energy
field theory in four dimensions using the Kaluza-Klein program.

The only way to avoid Witten's result without renouncing to
Kaluza-Klein ideas is to consider internal
manifolds that are neither coset spaces nor compact
\cite{Wetterich}. There is besides another way to bypass the problem of
chiral fermions by including gauge fields in the higher-dimensional
theory \cite{gauge-d>4}. This last proposal, however, spoils
the most relevant
aim of Kaluza-Klein theories, namely, the unification of gravity with
Yang-Mills theories in a geometrical setting. One needs other principles
to justify the presence of gauge symmetries in higher dimensions. String
theory is the only candidate which generates at the same time extra
gauge symmetries, chirality, and a likely consistent quantum theory of
the gravitational field.

\subsection{Quantum field theory and classical gravity}

Instead of trying to be so ambitious as to try to start from the
beginning with Quantum Gravity, we can learn many lessons from the study
of QFT in the presence of a classical gravitational
background. This subject is reviewed in the
lectures by R. Wald, and
we will limit our considerations to some general remarks. QFT in this
setting presents many interesting features in its own right
\cite{Birrel-Davies}. Many
properties of QFT which hold in Minkowski space either do not apply, or
change radically on a curved backgroud. For example, the
vacuum state in Minkowski QFT has several well-known properties:
\begin{itemize}
\item{It is unique.}
\item{There is a well defined concept of localized excitations
(particles).}
\item{It determines the symmetries of the world and their realizations.}
\end{itemize}
In the presence of a gravitational field the vacuum state loses the
absolute meaning it has in Minkowski space. The reason is that strong
or rapidly varying gravitational fields can produce
particles. It is easy to see that this makes the concept of
particle ambiguous (observer-dependent); let us consider a region of
space-time in which we have a very strong gravitational field (for
example near the horizon of a black hole). A static observer will see,
because of the strong gravity, that particles are created in pairs. On
the other hand a free-falling observer, by the equivalence principle,
will not observe (locally) any gravitational field, so for him there is
no creation of particles at all.
This means that what the free-falling observer sees as the
vacuum state of his QFT is not a vacuum for the static observer.

There are a number of elements that we need in the construction of a QFT
in a fixed background \cite{Gibbons}:
\begin{itemize}
\item{A Hilbert space of states ${\cal H}$.}
\item{A classical space-time $({\cal M},g_{\mu\nu})$.}
\item{Fields operators acting on ${\cal H}$.}
\item{A set of canonical commutation relations obeyed by these fields.}
\item{Wave equations.}
\item{Rules for constructing ``particles'' observables and Fock basis of
${\cal H}$.}
\item{Regularization and renormalization schemes to make expectation
values like $\langle T_{\mu\nu} \rangle$ finite.}
\end{itemize}

In constructing a QFT in curved background we face some difficult
problems. First of all we are performing the quantization of the theory
in a given background metric $g_{\mu\nu}$. It is natural
to ask about the back-reaction of the QFT over the space-time geometry.
To be more concrete, when solving the quantum theory we usually impose
the consistency condition (in units in which $G=1$)
\begin{equation}
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}(R-2\Lambda)=8\pi\langle
T_{\mu\nu}\rangle
\label{Einstein}
\end{equation}
In general our energy-momentum tensor will not verify (\ref{Einstein}),
so it is necessary to proceed {\it \`a la} Hartree-Fock: we quantize the
theory in the presence of a ``test'' background metric
$g_{\mu\nu}^{(0)}$, and we evaluate the vacuum expectation value of
the energy-momentum tensor $\langle T_{\mu\nu} \rangle$. Then we
solve (\ref{Einstein}) with $\langle T_{\mu\nu} \rangle$ and so we
obtain a first-corrected metric $g_{\mu\nu}^{(1)}$. Repeating the
process we get a series of metrics $\{g_{\mu\nu}^{(0)},g_{\mu\nu}^{(1)},
\ldots,g_{\mu\nu}^{(N)},\ldots\}$ which, hopefully, should eventually
converges to a self-consistent solution $g_{\mu\nu}$.

A second problem is that, in order to solve wave equations, we need our
background space-time to be globally hyperbolic \cite{Hawking-Ellis}
(this means that there exist a global Cauchy hypersurface $\Sigma$). In
the case in which we have singularities or Cauchy horizons we would need
certain (unclear) boundary conditions.

In ordinary QFT the divergent vacuum energies of the fields can be
substracted without any further problem since the substration amounts
to a redefinition of the origin of energies.
In the presence of gravity
vacuum energies contributes to the value of the
cosmological constant. As soon as we have symmetry breaking, the
quantum contributions are
unacceptably large. In view of this, the smallness of the cosmological
constant is probably the most misterious fact which receives no
explanation from any theory of Quantum Gravity, including String Theory.

Let us now consider the fields in a curved background. Since we have no
invariant notion of time, it seems natural to work in the Heisenberg
picture in which the states do not depend on time. The existence of a
Cauchy hypersurface $\Sigma$ allow us to define an inner product in the
space of fields; for two classical fields $\phi_{1}$, $\phi_{2}$ we
define their inner product $(\phi_{1},\phi_{2})$ as
\begin{equation}
(\phi_{1},\phi_{2})=-\int_{\Sigma}
d\Sigma^{\mu}J_{\mu}(\phi_{1},\phi_{2})
\label{product}
\end{equation}
where $J_{\mu}(\phi_{1},\phi_{2})$ is given by
\begin{equation}
J_{\mu}(\phi_{1},\phi_{2})=i(\bar{\phi}_{1}\nabla_{\mu}\phi_{2}-
\bar{\phi}_{2}\nabla_{\mu}\phi_{1})
\end{equation}
for two scalar fields and
\begin{equation}
J_{\mu}(\psi_{1},\psi_{2})=\bar{\psi}_{1}\gamma_{\mu}\psi_{2}
\end{equation}
if $\psi_{1}$, $\psi_{2}$ are spinor fields. By using the equations
of motion it is easy to check that both currents are conserved
($\nabla_{\mu}J^{\mu}=0$) so the product (\ref{product}) does not depend
on the particular choice of the Cauchy hypersurface (as long as
the fields vanish at infinity).

For simplicity we are going to consider the case of a real massless
scalar
field $\phi(x)$ \cite{DeWitt}. Let $\{u_{i}(x)\}$ be a complete set of
solution of its wave
equation $g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi(x)=0$ whose
general solution is
\begin{equation}
\phi(x)=\sum_{i}u_{i}(x)a_{i}+\bar{u}_{i}(x)\bar{a}_{i}
\end{equation}
We assume that $\{u_{i}(x)\}$ is an orthonormal set with
respect to the inner product (\ref{product}). If $\xi^{\mu}(x)$ is a
Killing field we can construct the quantity
\begin{equation}
H(\xi)=\int_{\Sigma} d\Sigma^{\nu}T_{\mu\nu}\xi^{\mu},
\label{hamiltonian}
\end{equation}
where $\Sigma$ is again a Cauchy hypersurface and $T_{\mu\nu}$ is the
energy-momentum tensor for the scalar field. It is easy to evaluate the
Poisson bracket of (\ref{hamiltonian}) with the field $\phi(x)$:
\begin{equation}
\{H(\xi),\phi(x)\}={\cal L}_{\xi}\phi(x),
\end{equation}
${\cal L}_{\xi}$ is the Lie derivative with respect to the Killing
vector $\xi$. Since ${\cal L}_{\xi}$ is an antihermitian operator with
respect
to the inner product, we will choose the $u_{i}(x)$'s to be
eigenstates of this operator
\begin{equation}
{\cal L}_{\xi}u_{i}(x)=-i\kappa u_{i}(x),
\label{eigenstates}
\end{equation}
with real $\kappa$. Then we can classify the modes $u_{i}(x)$ into two
classes: if $\kappa>0$ we will say that the associated mode is a
positive frequency solution and a negative one if $\kappa<0$. From
(\ref{eigenstates}) it is easy to see that $u_{i}(x)$ and
$\bar{u}_{i}(x)$
both have the same frequencies but with opposite sign. Upon
quantization
we associate the creation operators with positive frequency
solutions, thus
\begin{equation}
\hat{\phi}(x)=\sum_{i} u_{i}(x)a_{i}+\bar{u}_{i}(x)a_{i}^{\dagger},
\end{equation}
with the vacuum state defined by $a_{i}|0\rangle=0$ for every $a_{i}$.
It is important to note that the classification of the modes into
positive and negative frequency solutions, and the definition of
the vacuum state, was made with respect to a given Killing field
$\xi^{\mu}(x)$. Once the choice of the Killing field is made one has
introduced the notion of particle. A specially interesting case is
when we consider so-called ``sandwitch'' space-times in which we have
three regions: two of them $M^{-}$ and $M^{+}$ are static and correspond
respectively to the past and to the future. Between them we have the
third region $M^{0}$ in which we have a time varying gravitational
field. In this case we have different notions of particle
appropiate to particular measurement processes at different times. Then
the concept of particle will be observer dependent.

The ambiguity in the concept of particle steems from its global nature.
We may speculate that since we always have (by the equivalence
principle) free falling observers it should be possible to define
particles only by the fact that they produce localized effects and by
the dispersion relation $p^{2}=m^{2}$. This might lead to a ``bundle''
of ground states with may not have global section in the presence of
horizons.

We are going to illustrate the ambiguity in the concept of particle with
a very well known example due to Unruh \cite{Unruh}. We
consider a particle detector consisting of a quantum system which in
absence of interaction with an external field is in its ground state.
When it interacts the system get excited and
it is no longer in the fundamental state. We can detect the
presence of particles by observing if our detector is in an excited
state.

Imagine the detector moving along a world line
$x^{\mu}(\tau)$, where $\tau$ is the proper time. Suppose
that it interacts with the scalar field
$\phi(x)$ through the lagrangian
\begin{equation}
{\cal L}_{int}=c m(\tau)\phi[x^{\mu}(\tau)]
\end{equation}
where $m(\tau)$ is the monopole moment of the detector. Furthermore,
assume
that we are in Minkowski space and that the field $\phi(x)$ is in its
ground state $|0_{M}\rangle$. For a general trayectory the detector will
not remain in its ground state with energy $E_{0}$, but will go to
an excited state with $E>E_{0}$ while the field will be also in an
excited state $|\psi\rangle$. For $|c|<<1$ the amplitude for this
process to ocur is
\begin{equation}
{\cal A}=ic\langle
E,\psi|\int_{-\infty}^{+\infty}d\tau\,m(\tau)\phi[x(\tau)]
|0_{M},E_{0}\rangle
\end{equation}
$m(\tau)$ can be rewritten as:
\begin{equation}
m(\tau)=e^{iH_{0}\tau}m(0)e^{-iH_{0}\tau},
\end{equation}
with $H_{0}$ the hamiltonian of the
detector. Since $H_{0}|E\rangle=E|E\rangle$.
The amplitude ${\cal A}$ becomes,
\begin{equation}
{\cal A}=
ic\langle E|m(0)|E_{0}\rangle\int_{-\infty}^{+\infty}d\tau\,
e^{i(E-E_{0})\tau} \langle\psi|\phi[x(\tau)]|0_{M}\rangle.
\end{equation}
If $|\psi\rangle$ describes a state with one particle with
momentum $k$, $|k\rangle$, using the decomposition of
$\phi(x)$ in terms of Minkowski modes
\begin{equation}
\phi(x)=\int\frac{d^{3}k^{'}}{\sqrt{16\pi^{3}\omega^{'}}}\left[
a_{k}^{\dagger}e^{-i\vec{k}^{'}\cdot\vec{x}+i\omega^{'}t}+ h.c.\right],
\end{equation}
with $\omega=\sqrt{k^{2}+m^{2}}$ we obtain
\begin{equation}
\langle k|\phi(x)|0\rangle=\frac{1}{\sqrt{2(2\pi)^{3}\omega}}
e^{-i\vec{k}\cdot\vec{x}+i\omega t}
\end{equation}
For an inertial observer, we have ${\bf x}={\bf x}_{0}+
{\bf v}\tau(1-v^{2})^{-1/2}$, and performing the integration we obtain
\begin{equation}
{\cal A}=\frac{1}{\sqrt{4\pi\omega}}e^{-i\vec{k}\cdot\vec{x}_{0}}
\delta\left[E-E_{0}+(\omega-{\bf k}\cdot{\bf v})(1-v^{2})^{-1/2}\right]
\end{equation}
but since $E>E_{0}$ and ${\bf k}\cdot{\bf v}\leq kv<\omega$, the
argument of the $\delta$-function is positive and the
amplitude vanishes. For a more complicated trayectory $x^{\mu}(\tau)$
the probability transition to all $E$ and all $\psi$ is simply
\begin{equation}
{\cal P}=\sum_{E}\left|\langle E|m(0)|E_{0}\rangle\right|^{2}
{\cal F}(E-E_{0})
\end{equation}
with
\begin{equation}
{\cal F}(E-E_{0})=\int_{-\infty}^{+\infty} d\tau
\int_{-\infty}^{+\infty} d\tau^{'}\,e^{-iE(\tau-\tau^{'})}
G^{+}[x(\tau),x(\tau^{'})],
\end{equation}
hence, the particle absortion per unit time is
\begin{equation}
{\cal W}=c^{2}\sum_{E}\left|\langle E|m(0)|E_{0}\rangle
\right|^{2}\int_{-\infty}^{+\infty} d(\Delta\tau)
e^{-i(E-E_{0})\Delta\tau}G^{+}(\Delta\tau).
\end{equation}
In the particular case of a massless scalar and a
uniformly accelerated detector, the world line is given by
\begin{eqnarray}
x&=&y=0 \nonumber \nonumber \\
z&=&\alpha\cosh{(\tau/\alpha)} \nonumber \\
t&=&\alpha \sinh{(\tau/\alpha)}
\end{eqnarray}
and the probability per unit time is
\begin{equation}
{\cal W}=\frac{c^{2}}{2\pi}\sum_{E}(E-E_{0})\frac{
\left|\langle E|m(0)|E_{0}\rangle \right|^{2}}{
e^{2\pi\alpha (E-E_{0})}-1}.
\end{equation}
Therefore a uniformly accelerated observer will see a flux of
particles as if he were in a thermal bath of scalar massless particles
at a temperature
\begin{equation}
T=\frac{1}{2\pi\alpha}
\end{equation}
while the inertial observer will see that the field $\phi(x)$ is in its
vacuum state $|0_{M}\rangle$ and consequently will not see any particle
at all. This shows that when we depart from Minkowski space the
concepts of vacuum and particles states become observer-dependent.

A general way to understand how the notion of no-particle state changes
in curved backgrounds is to use Bogoliubov transformations (for more
details and references see \cite{Gibbons}). We have seen above
that upon choosing a Killing field $\xi^{\mu}(x)$ we
can make a decomposition of any solution of the wave equation in terms
of a set of positive frequency solutions with respect to $\xi^{\mu}(x)$,
$u_{i}(x)$ and their complex conjugates $\bar{u}_{i}(x)$. In
many curved backgrounds there
are no timelike Killing fields or the Killing fields are not timelike
everywhere.
In the first case we usually have to deal with vector fields that are
asymptotically Killing fields and we have to use a $S$-matrix
formulation.

Let us consider two complete sets of solutions of the wave equation for
the massless scalar field $\{u_{i}(x)\}$ and $\{w_{i}(x)\}$.
Associated
with these sets we have the corresponding creation and annihilation
sets of operators $\{a_{i};a_{i}^{\dagger}\}$ and
$\{b_{i};b_{j}^{\dagger}\}$. Since both sets of modes are bases for
the solutions of the wave equation, we can expand one set in terms of
the other, namely
\begin{eqnarray}
w_{i}(x)&=&\sum_{j} \alpha_{ij}u_{j}(x)+\beta_{ij}\bar{u}_{j}(x)
\nonumber \\
\bar{w}_{i}(x)&=&\sum_{j}
\bar{\beta}_{ij}u_{j}(x)+\bar{\alpha}_{ij}\bar{u}_{j}(x)
\end{eqnarray}
By general covariance we can write
\begin{equation}
\phi(x)=\sum_{i}a_{i}u_{i}(x)+a_{i}^{\dagger}\bar{u}_{i}(x)=
\sum_{i}b_{i}w_{i}(x)+b_{i}^{\dagger}\bar{w}_{i}(x)
\end{equation}
So both sets of creation-annihilation operators are
related by the Bogoliubov transformations
\begin{eqnarray}
b_{i}&=&\sum_{j}\bar{\alpha}_{ij}a_{j}-\bar{\beta}_{ij}a_{j}^{\dagger}
\nonumber \\
b_{i}^{\dagger}&=&\sum_{j}-\beta_{ij}a_{j}+\alpha_{ij}a_{j}^{\dagger}.
\end{eqnarray}

We can apply these relations to the case with two
asymptotically flat regions: {\it in} and {\it out}. We have the
corresponding
set of modes $u_{i}^{in}(x)$ and $u_{i}^{out}(x)$ related by
\begin{equation}
u_{i}^{out}(x)=\sum_{j}\alpha_{ij}u_{j}^{in}(x)+
\beta_{ij}\bar{u}_{j}^{in},
\end{equation}
and by applying the previous result we obtain:
\begin{eqnarray}
a^{out}_{i}&=&\sum_{j}\bar{\alpha}_{ij}a^{in}_{j}-
\bar{\beta}_{ij}a_{j}^{in\dagger} \nonumber \\
b_{i}^{out \dagger}&=&
\sum_{j}-\beta_{ij}a_{j}^{in}+\alpha_{ij}a_{j}^{in \dagger}.
\label{in-out}
\end{eqnarray}
The {\it in} and {\it out} vacuum states are defined
respectively by
\begin{eqnarray}
a_{i}^{in}|0_{in}\rangle &=& 0 \hspace{1cm} \forall i \nonumber \\
a_{i}^{out}|0_{out}\rangle &=& 0 \hspace{1cm} \forall i \; ,
\end{eqnarray}
{}From (\ref{in-out}) we see that when $\beta\neq 0$, the {\it in} vacuum
is not a vacuum state for {\it out} particles:
\begin{equation}
n_{i}^{out}=\langle 0_{in}|a_{i}^{out \dagger}a_{i}^{out}| 0_{in}
\rangle= \sum_{i}|\beta_{ij}|^{2}
\end{equation}
Thus the total number of {\it out} particles in the {\it in} vacuum is
given by
\begin{equation}
N^{+}=\sum_{j} n_{j}^{out}=tr\,\beta\beta^{\dagger}
\end{equation}
This show that if we consider that our system is in the vacuum state in
the {\it in} region, in the {\it out} region an observer will see a
total number of particles
given by $tr\,\beta\beta^{\dagger}$. In practical cases the problem
is to find the Bogoliubov transformation between the two asymptotic
regions. Although in ordinary QFT we may also encounter Bogoliubov
transformations they are ubiquitous in the presence of interesting
gravitational backgrounds.

One of the most spectacular results which can be extracted from the
study of quantum fields in the presence of the gravitational interaction
was achieved by Hawking in $1975$ \cite{Hawking1}\footnote{For details
and references on Hawking radiation see the lectures by R. Wald.}.
He found that black holes emit
particles as if they were black bodies at a temperature
\begin{equation}
T_{Hawking}=\frac{1}{8\pi M}
\end{equation}
for the case of the Schwarzschild black hole with $M$ being its mass.
Since the temperature is inversely proportional to the mass and the
total amount of energy radiated per unit time and per unit area is
proportional to the fourth power of
the temperature we find that, asuming the
semiclassical approach all the way, the black hole will evaporate
completely in a time proportional to $M_{0}^{3}$ with $M_{0}$ the
initial mass of the black hole. It is also quite unintuitive
thermodynamically that as the black hole radiates it heats up.

Black hole evaporation gives rise to serious conceptual problems (for a
recent review see \cite{Preskill}). Hawking found that the radiation
emited by a black hole is thermal; if this would exactly correct, it
would mean that the radiation is in a mixed quantum state.
In classical GR no information from inside the horizon can escape to the
asymptotic region. It would seem natural in quantizing in the presence
of a black hole to always take a trace over the states inside the
horizon. This produces a mixed state. If we imagine a matter system in a
pure state which collapse to form a black hole, and if we consider the
complete evaporation process assuming the radiation to be exactly
thermal, at the end we are left with a mixed state.
In other words, although we know
the initial state of the system we cannot predict what the final state
would be. This clearly violates the laws of Quantum Mechanics.

Of course in the previous discussion it was assumed that the
semiclassical derivation of the emission of particles is valid during
the whole process of the black hole evaporation.
When the mass of the black hole becomes close to the Planck mass
the semiclassical description breaks down. The prediction of what
happens after that point is impossible without a reliable theory of
quantum gravity. One can imagine at least four possibilities
\cite{Hawking2}:
\begin{itemize}
\item{The evaporations produces a naked singularity of negative mass
which persists.}
\item{The evaporation slows down and stops leaving a remnant black hole
with a mass of the order of the Planck mass.}
\item{The black hole evaporates completely but all the information
about the initial state is encoded in the radiation that scapes to
infinity.}
\item{The black hole evaporates completely and takes with it all the
information about its initial state and conserved quantities, except
those coupled to long range fields (mass, charge and angular momentum).}
\end{itemize}

There are some arguments \cite{Hawking2,Preskill} in favour of
the fourth posibility. In this case we are left with the puzzles pointed
out before. Consider an asymptotic region in the past where
our system is described by a density operator $\rho_{-}$ defined in the
Fock space of a free field and an asymptotic region in the future in
which the system is described by a density
operator $\rho_{+}$. Hawking \cite{Hawking2} proposed to use a
superscattering operator $\$$ relating the density operators in the
two regions
\begin{equation}
\rho_{+\;\;B}^{\;\;A}=\$^{A\;\;\;\; D}_{\;\;BC}\rho_{-\;\;D}^{\;\;C}.
\end{equation}
We have to require the $\$$-operator to satisfy several conditions:
first it has to map initial density operators of unit trace into final
density operators which are positive, semi-definite and of unit trace.
This implies that
\begin{equation}
\$^{C\;\;\;\;B}_{\;\;CA}=\delta_{A}^{\;\;B}.
\end{equation}
Moreover, $\$$ has to map hermitian operators into hermitian operators,
\begin{equation}
\$^{A\;\;\;\;D}_{\;\;BC}=\$^{B\;\;\;\;D}_{\;\;AC}.
\end{equation}

In ordinary (Minkowski) QFT we have the $S$-operator which maps states
in the asymptotic past into states in the asymptotic future. In this
case the $\$$-operator can be factorized as the tensor product $S\otimes
S^{\dagger}$
\begin{equation}
\$^{A\;\;\;\;D}_{\;\;BC}=S^{A}_{\;\;C}\bar{S}_{B}^{\;\;D}
\label{factorization}
\end{equation}
With this form for the superscattering operator, if the system is in a
pure state, it will remain in a pure state, as it is required by the
laws
of Quantum Mechanics. This is not the case for a general
$\$$. In fact it is possible to show that
$\$$-operator has the form (\ref{factorization})
as long as the Green's functions of the theory verify
asymptotic completeness,
that is to say, if the asymptotic {\it in} and {\it out} states span the
Hilbert space of the theory. Hawking has argued that the axiom of
asymptotic completeness is not verified in quantum gravity; this
has the consequence that quantities that are not coupled to long range
fields are not conserved (for example global $U(1)$ charges
\cite{Hawking2}). Further analysis and criticism of these arguments
can be found in \cite{Banks-Peskin-Susskind}.

There has been some recent developments in the study of black hole
evaporation. On the one hand, the classical no-hair theorems have been
reanalyzed \cite{Kraus-Wilczek} and found to
be modified substancially by Quantum Fields. Furthermore, some simple
renormalizable theories of two-dimensional gravity coupled to dilatons,
and containing many of the interesting features of four-dimensional
black holes have recently been proposed \cite{Callan-Giddings}. In this
models one can study the collapse of matter to form a black hole and its
subsequent evaporation. Unfortunately, the final stages of this process
are not yet accesible to the approximations used in
\cite{Callan-Giddings}.

\subsection{Euclidean approach to Quantum Gravity}

In ordinary QFT we are faced with the computation of path integrals of
the form \begin{equation}
Z=\int {\cal D}\phi \,e^{iS[\phi]}
\label{field-fpi}
\end{equation}
with $S[\phi]$ the classical action. However, the oscillatory nature of
the exponential in (\ref{field-fpi}) makes
the expression ill-defined. In this context the problem can be
ameliorated by performing what is called a Wick rotation to euclidean
space; this means that we will make an analytic continuation of real
Minkowski time to pure imaginary values $t\rightarrow -it$ such that
we can rewrite (\ref{field-fpi}) as
\begin{equation}
Z_{E}=\int {\cal D}\phi \,e^{-S_{E}[\phi]}
\label{eucl-field}
\end{equation}
where $S_{E}[\phi]$ is the Euclidean action. Since in most cases
in QFT this euclidean action is positive definite
(\ref{eucl-field}) is now perfectly well defined because field
configurations with large values of $S_{E}[\phi]$ are damped by the
exponential factor. Once the path integral is evaluated in Euclidean
space we return to Minkowski space by analytical continuation.

For the procedure we outlined above to be justified some conditions has
to be met. In QFT the Wightman's axioms \cite{Glimm-Jaffe}
guarantee the possibility of analytic continuation.
Besides in QFT the
continuation to the Euclidean space has many other applications.
When we study quantum systems
at finite temperature we need to calculate the canonical partition
function $Z(\beta)$
\begin{equation}
Z(\beta)=tr\,e^{-\beta H}
\end{equation}
with $\beta=1/T$ and $H$ the hamiltonian of the system. This trace can
be represented as a path integral \cite{Ramond}.
Using standard techniques one obtains a path integral in Euclidean space
with time compactified to a circle whose length is the inverse
temperature. The thermal partition function is
\cite{Ramond}
\begin{equation}
Z(\beta)=\int_{{\bf R}^{3}\times S^{1}_{\beta}} {\cal D}\phi
\,e^{-S_{E}[\phi]}
\end{equation}
For bosons we have to asume periodic boundary conditions
in the compactified dimension. On the other hand if we have fermions
they have to be antiperiodic.
Euclidean field theory is also the arena in which
Yang-Mills instantons are formulated \cite{Eguchi-Gilkey-Hanson}.

Now we come to gravity. We may expect to find a similar formulation for
the
General Theory of Relativity, namely, to make an analytical continuation
from the physical signature to Euclidean signature in which a path
integral approach to quantum gravity would give better results than
the methods described in previous sections; after calculating
physical observables we would come back to the physical signature.
However this is not so easy because we find difficulties from the very
beginning: for curved spaces, even for
globally hyperbolic ones, there are no axioms ensuring the analytic
continuation in the time coordinate. Moreover, when we perform the
continuation we find that
singularities may disappear. This is the case for example of the
Schwarzshild metric \begin{equation}
ds^{2}=\left(1-\frac{2m}{r}\right)dt^{2}-
\left[\frac{dr^{2}}{1-\frac{2m}{r}}+
r^{2}(d\theta^{2}+\sin{\theta}d\phi^{2})\right]
\end{equation}
When we perform the analytical continuation into Euclidean space we miss
the region inside the horizon $r<2m$ and we are left only with the
exterior part $r>2m$ and the horizon in which the Killing vector
$\partial_{t}$ vanishes.

Even if we do not worry about this problem and continue ahead, we
will soon be in trouble. Unlike the case of Yang-Mills theories,
the gravitational action is unbounded from below; this is because the
integrand is linear in the scalar curvature, so the action can be made
arbitrarily negative. Let us consider the action for the
Euclidean gravitational
field coupled to matter in a four-dimensional riemannian manifold $M$
with boundary $\partial M$ (see \cite{Hawking-cent} and references
therein)
\begin{eqnarray}
S&=&-\frac{1}{16\pi G}\int_{M}d^{4}x\sqrt{g}(R-2\Lambda)-
\frac{1}{8\pi G}\int_{\partial M}d^{3}x\sqrt{h}(K-K^{0}) \nonumber \\
&-&\int_{M}d^{4}x\sqrt{g}\,{\cal L}_{matt},
\label{gravity-action}
\end{eqnarray}
where now $K$ is the trace of the second fundamental form on $\partial
M$ (i.e., the extrinsic curvature) and $h_{ab}$ is the metric on the
three-dimensional boundary. The third term in the action represents the
action for the matter fields. Let us write the metric $g_{ab}$ in the
form
\begin{equation}
\tilde{g}_{ab}=\Omega^{2}g_{ab}
\end{equation}
with $\Omega(x)$ a positive function. The scalar curvature changes
according to
\begin{equation}
\tilde{R}= \Omega^{-2}R-6\Omega^{-3}\Box \Omega \; ,
\end{equation}
and for the extrinsic curvature
\begin{equation}
\tilde{K}=\Omega^{-1}K+3\Omega^{-2}n^{a}\nabla_{a}\Omega \; ,
\end{equation}
with $n^{a}$ the unitary normal vector on $\partial M$. With this
decompositions we can rewrite the gravitational part of
(\ref{gravity-action}) as
\begin{eqnarray}
S&=&-\frac{1}{16\pi
G}\int_{M}d^{4}x\sqrt{g}\left[\Omega^{2}R+6g^{ab}\nabla_{a}\Omega
\nabla_{b}\Omega-2\Lambda\Omega^{4}\right] \nonumber\\
&-&\frac{1}{8\pi G}\int_{\partial M} d^{3}x\sqrt{h}\Omega^{2}(K-K^{0})
\end{eqnarray}
It is easy to see that by considering a rapidly varying conformal
factor $\Omega(x)$ we can make the action arbitrarily negative. This
implies that the path integral over $\Omega(x)$ diverges since, in
principle, we have to integrate over all possible conformal factors.
Gibbons, Hawking and Perry \cite{Gibbons-Hawking-Perry} gave a
prescription for the evaluation of the functional integral over the
conformal factor: we separate all possible four-dimensional metrics into
conformal
classes and then we pick in each class one for which the scalar
curvature
vanishes. This could be accomplished for a given metric $g_{ab}$
by solving
\begin{equation}
(\Box-\frac{1}{6}R)\Omega(x)=0
\end{equation}
so $\Omega^{2}g_{ab}$ has zero scalar curvature.
Once this is achieved, we integrate over the conformal factor and over
all conformal classes, and a particular analytic continuation of the
integral over $\Omega$ is given to render it well defined. Since in
this prescription we have to consider
metrics with zero scalar curvature, the gravitational action is enterely
governed by the surface term, and then by the boundary conditions over
$\partial M$. Physical gravitational fields vanish asymptotically at
infinity, so in physical cases we deal with metrics that are
asymptotically flat. This fact led Gibbons, Hawking and Perry to
formulate the positive action conjecture \cite{Gibbons-Hawking-Perry}
according to which the gravitational action is non-negative for any
asymptotically euclidean and positive defined metric with $R=0$. This
conjecture was finally proved in $1979$ by Schoen and Yau
\cite{Schoen-Yau}.

Assume now that we have succesfully dealt with
the unboundness of the gravitational action. Then, we can
think about applying Polyakov's prescription, which has given such good
results in two-dimensional gravity \cite{Polyakov}. When we
calculate the partition function (or any correlator) we have to sum over
all topologies, which in the case of two-dimensional orientable surfaces
are classified symply by their genus, hence
\begin{equation}
Z_{d=2}=\sum_{g=0}^{\infty} \int_{{\cal I}_{g}} {\cal D}g
\,e^{-S_{E}[g]},
\end{equation}
where the path integral is performed at fixed topology. Then we would
try to make a similar expansion in the case of four-dimensional gravity
\begin{equation}
Z_{d=4}=\sum_{topologies} \int_{fix.\, top.} {\cal D}g \,e^{-S_{E}[g]}.
\end{equation}
However in trying to carry out this program in four dimensions lead us
to an essential
difficulty: there is no algorithmic way of deciding when any two
four-dimensional manifolds are homeomorphic. We will briefly review
the set of theorems which lead to this result (see \cite{EAlvarez1}
and references therein).

The starting point is the fact (Markov's theorem) that any
finitely generated group $G$ can be the fundamental group of a
four-dimensional, smooth, compact, and connected manifold. That the
group
\begin{equation}
G=\langle a_{1},\ldots,a_{n} |R_{1},\ldots,R_{k}\rangle,
\end{equation}
is finitely
generated means simply that it is generated
by a finite number of elements $\{a_{i}\}$ which are subject to a finite
number of relations $\{R_{j}\}$. The second important ingredient we need
is a result first posed by Dehn and
proved by Novikov known in group theory as the word problem:
given a family of finitely generated groups $\{G_{k}\}$ with
$k=1,2,\ldots$ there is no algorithmic way of distinguishing
any trivial group in that family.

It is now easy to show the impossibility of classifying topologies in
four dimensions. Let us construct for the family $G_{k}$ a set of
manifolds $M_{k}$ such that $\pi_{1}(M_{k})=G_{k}$ (which is
possible by Markov's theorem).
If we were able
to say whether a manifold are homeomophic to one of the manifolds in
the family we could say that
their fundamental groups are equal. Thus by seeing what are the elements
of $\{M_{k}\}$ which are homeomorphic to a given manifold $M_{0}$
with trivial fundamental group, we would be able to
say what elements of $\{G_{k}\}$ are trivial; but that cannot be done.

Although the previous result tells us that it is impossible to classify
four-dimensional topologies in general, we can try to classify the
topologies in some subclass of four-dimensional manifolds. For example
we can impose the restriction to the sector with trivial fundamental
group
$\pi_{1}(M)=0$. Let us begin by defining the Kirby-Siebenman invariant
for a manifold $M$:  $\alpha(M)\in {\bf Z}_{2}$ such that
$\alpha(M)=0$ if the manifold
$M\times S^{1}$ admits a smooth structure and $\alpha(M)=1$ if it does
not. A theorem by M. Freedman states that if $M$ is a compact connected
manifold  with trivial fundamental group then it is classified by
$H_{2}(M,{\bf Z})$, the second homology group and the Kirby-Siebenman
invariant $\alpha(M)$.

Given a four-dimensional manifold we can define the intersection form
$\omega$ as a symmetric bilinear form $\omega:H^{2}(M,{\bf
Z})\rightarrow {\bf Z}$ such that
\begin{equation}
\omega(\beta_{1},\beta_{2})=\int_{M}\beta_{1}\wedge \beta_{2}
\end{equation}
This intersection form can be used to classify simply connected
($\pi_{1}(M)=0$) compact
four-dimensional manifolds into homeomorphism classes. Two of such
manifolds are homotopy equivalent iff their intersection forms are
isomorphic (Whitehead's theorem). We introduce the concept of signature
$\sigma(\omega)$ for the intersection form $\omega$ as the number of
its positive eigenvalues
minus the number of its negative ones. It can be shown that whenever
$\omega$ is even it is automatically divisible by eight. The concept of
signature is important because there is a one-to-one correspondence
between the set of simply-connected compact, four-dimensional manifolds
and the pairs $\langle \omega,\alpha \rangle$, where $\alpha$ is the
Kirby-Siebenman invariant defined earlier and if $\omega$ is even then
$\omega/8\equiv\alpha\,\,(mod\,2)$ is satisfied.
Thus, the pairs $\langle \omega,\alpha
\rangle$ constitute the space of parameters for four-dimensional
compact topological manifolds in the sector with $\pi_{1}(M)=0$.

GR is invariant under diffeomorphisms;
when performing the path integral at fixed
topology we have to integrate over all diffeomorphism classes without
counting each class more than once. So we have to classify
four-dimensional manifolds into diffeomorphism classes in order to
carry out the path integration. This can be done by using a theorem by
Donaldson which states that given a closed four-dimensional, smooth and
simply connected
manifold with an intersection form $\omega$ which is positive definite,
$\omega$ over the integers is equivalent to $1+\ldots+1$, i.e.,
the set of manifolds with intersection form
$\omega$ does not exists smoothly for any non-zero positive definite
$\omega$. This implies the existence of different classes of
differentiable structures which are classified in part by Donaldson's
invariants. For example, there are exotic
differentiable
structures in ${\bf R}^{4}$, the so called ${\bf R}_{fake}^{4}$.

After this description of the problems we encountered in trying to
use the Polyakov approach in four-dimensional gravity we have to
conclude that it seems imposible to carry out this program completely.
Nevertheless, one could, based in some yet unknown physical principle,
impose some kind of  restriction in the set of manifolds we integrate
over; for example to integrate over simply connected manifolds
($\pi_{1}(M)=0$). The previous arguments are too na\"{\i}ve. They are
based
on a semiclassical analysis of the problem of quantizing gravity. In the
functional integral we will certainly have contributions coming from
singular metrics. If we relax the conditions that the manifolds and
geometries should be continuous and smooth, the objections raised above
lose much of their strenght. What is not clear is the understanding of
what physical conditions would imply the consideration of more general
manifolds.

\subsection{Canonical quantization of gravity}

In this approach we restrict {\it ab initio} to space-times
with
a topology $\Sigma_{3}\times {\bf R}$, where $\Sigma_{3}$ is a compact
three-manifold; these hypersurfaces $\Sigma_{3}$ are interpreted as
surfaces of constant time which ``folliate'' the whole space-time.
Introducting this privileged time
coordinate we can construct a hamiltonian and proceed to the canonical
quantization of the space-time \footnote{A thorough presentation of
canonical gravity is found in the lectures by A. Ashtekar in this
volume.}. The corresponding Fock space would not
be
equivalent to that obtained from path-integral quantization, because in
the second case we integrate also over four-dimensional manifolds which
are not topologically $\Sigma_{3}\times {\bf R}$. When canonically
quantizing we are more interested on the classification of
three-dimensional manifolds. Now the determination of the classical
phase space is ``simpler'' as long as one believes Thurston's
geometrization program \cite{Thurston}. In fact we can include every
three-dimensional manifold in a list with a
countable infinite number of parameters without leaving any
three-manifold out of the list. However we cannot be completely happy
because we may have the problem of an infinite overcounting.

There are several steps in the classification of three-dimensional
manifolds. First we can consider genus $g$ Heegard splitting
(for an application of this technique see \cite{Dijkgraaf-Witten})
which consist of decomposing a three-manifold into two ``handle
bodies'' of genus $g$ by cutting along a Riemman surface $\Sigma_{g}$.
After that the boundaries of the two parts are identified by a
diffeomorphism not connected with the identity.
A problem,
nevertheless, arises when considering the fundamental group
$\pi_{1}(M)$
of a three-dimensional manifold because although it is finitely
generated there is no algorithmic way of knowing whether a given group
is the fundamental group of a three-manifold.

We would like to find a set of topological invariants
we could use to classify three-manifolds along the same lines as
in two dimensions in where
compact orientable manifolds are classified according to their genus.
The search of these invariants in the
three-dimensional case is a matter of current investigation
(Jones-Witten
invariants, Casson invariants...). An attempt to classify
three-dimensional manifolds has been made by Thurston
\cite{Thurston} based in the
geometrization conjecture: every compact, orientable three-manifold can
be cut by disjoint embedded two-spheres and tori into pieces which,
after gluing three-balls to all boundary spheres, admit geometric
structures. That the manifold admits a geometric structure simply means
that for every pair of points $x,y \in M$ there are two isometric
neighborhoods $U_{x},U_{y} \subset M$.

One way in which we could get rid of these problems it to consider that
at short distances we have a topological field theory
\cite{Witten-Topological}. Then, below some critical lenght, the
theory is in a symmetrical phase in which it only depends on some well
defined topological invariants so the phase space is perfectly well
characterized. In the low energy phase the invariance under
diffeomorphisms is spontaneously broken by the vacuum, which is only
Lorentz invariant, and the
graviton is the corresponding Goldstone boson. This idea is an old one
and can be traced back to the
early seventies \cite{Isham-Salam-Strathdee}.

The group of $C^{\infty}$-diffeomorphisms of a smooth compact manifold
$M$, $\mbox{Diff}(M)$, has some peculiar properties \cite{Milnor}. It is
a Lie
group with its Lie algebra given by the set $\mbox{Vect}(M)$ of smooth
vector fields on $M$ which close under Lie brackets.
This group is very different from ordinary Lie groups because it is not
possible in general to get any element of
the connected component of $\mbox{Diff}(M)$ as the image by the
exponential
map of an element of $\mbox{Vect}(M)$. However there always exist a set
$v_{1},\ldots ,v_{k} \in \mbox{Vect}(M)$ such that any element $f\in
\mbox{Diff}(M)$ can be written as
\begin{equation}
f=EXP(v_{1})\circ EXP(v_{2}) \circ \ldots \circ EXP(v_{k})
\end{equation}

We are interested in studying the topological structure of the set of
all metrics modulo the group of diffeomorphism. The topology of this
space is intimately connected to the topology of $\mbox{Diff}(M)$. For
instance
\begin{equation}
\pi_{1}[\mbox{Metrics}(M)/\mbox{Diff}(M)] \cong \pi_{0}[\mbox{Diff}(M)],
\end{equation}
that is to say, the symply connectness of
$\mbox{Metrics}(M)/\mbox{Diff}(M)$ is given
by the connectness of $\mbox{Diff}(M)$ and similarly for higher
homotopy groups. Considering the group $\mbox{Diff}(S^{n})$ Smale proved
that \cite{Smale}
\begin{equation}
\pi_{k}[\mbox{Diff}(S^{n})] \cong \pi_{k}[O(n+1)]
\end{equation}
for $n=1,2$ and for all $k$. Smale himself conjectured that this
relation also holds for
$n=3$, namely
\begin{equation}
\pi_{k}[\mbox{Diff}(S^{3})] \cong \pi_{k}[O(4)]
\end{equation}
This conjecture was finally proved by Hatcher \cite{Hatcher}.

As a final speculation, one can think that
perhaps in the right theory one is allowed to make deformations wilder
than homeomorphisms, so that there is some kind of quantum equivalence
of otherwise distinct classical geometrical structures. This happens for
example in String Theory in connection with duality relations
\cite{duality} and mirror symmetries \cite{mirror}. For example a string
moving on a circle of radius $R$ is equivalent to a string on a circle
of radius $1/R$ (see sec. 3.6). One also finds that
topologically very different Calabi-Yau manifolds (Ricci-flat compact
K\"{a}hler manifolds) lead to the same physical theory (mirror symmetry).
This is properly known to hold only for string theory. This short
section has hopefully convinced the reader that a na\"{\i}ve analysis
of the
problem of canonical quantization of gravity is going to meet with
rather severe mathematical difficulties if we insists on having smooth
manifolds and geometries. Very important progress towards a proper
understanding of canonical Quantum Gravity has been achieved with the
work initiated by Ashtekar and coworkers \cite{Ashtekar}.

\subsection{Gravitational instantons}

In Yang-Mills theory, instantons appear as classical solutions of the
Euclidean field equations. From the
point of view of Minkowsky space these solutions are interpreted as
tunneling between vacuum states that are not topologically equivalent.
If Euclidean gravity makes any sense we could expect to find
gravitational instantons which would represent tunneling between
inequivalent vacua. Some properties of these gravitational instantons
are \cite{Eguchi-Hanson}:
\begin{itemize}
\item{They describe gravitational fields which are localized in
Euclidean space-time.}
\item{These solutions approach an asymptotically locally Euclidean
vacuum
metric at infinity. This means that although the
metric at infinity is locally flat, globally the space is not
topologically equivalent
to flat space-time. This is equivalent to the property of Yang-Mills
self-dual solutions of being pure-gauge at infinity.}
\item{They have non-trivial topological quantum numbers.}
\end{itemize}
All these properties are very similar to those of the instantons we find
in Yang-Mills theory \cite{Eguchi-Gilkey-Hanson}.
The reason to expect to find self-dual solutions in
four-dimensional Euclidean space is that the holonomy group
is $SU(2)\times SU(2)$. Requiring the curvature to take values in only
one of the $SU(2)$'s is the self-duality condition. If we look for
regular manifolds
with instanton-like solutions we find that the surface $K3$ is the only
compact, regular and simply connected manifold without boundary which
admits a non-trivial self-dual curvature. If we look for solutions with
some singular points, the number of possible candidates is much bigger.

Let us illustrate the subject of gravitational instantons by considering
some particular examples \cite{Eguchi-Hanson}: let us take the
following ansatz for the metric
\begin{equation}
ds^{2}=U({\bf x})^{-1}(d\tau^{2}+ {\bf \omega}\cdot d{\bf x})
+U({\bf x})d{\bf x}\cdot d{\bf x}
\label{instanton}
\end{equation}
which is self-dual as long as
\begin{equation}
\nabla \times {\bf \omega}=\nabla U({\bf x})
\end{equation}
The general solution can be written as
\begin{equation}
U({\bf x})=\epsilon+\sum_{i=1}^{k}\frac{2M_{i}}{|{\bf x}-{\bf x}_{i}|}
\end{equation}
with $\epsilon$ an integration constant. In order to make
(\ref{instanton}) well behaved and free of singularities, we have to
choose $M_{i}=M$ and to make $\tau$ periodic with period $8\pi M/k$.

Let us take now $\epsilon=1$; in this case we have that
the action is $S=4\pi kM^{2}$. For the case $k=1$ we have
Euclidean Schwarzschild solution while for $k>1$ it is not clear what
physical situation requires this kind of
instantons. They are known as multi-Taub-NUT metrics and they are
assymptotically flat in the spatial directions $|{\bf x}|\rightarrow
\infty$. Near spatial infinity these manifolds look like
$S^{2}\times(S^{1}/{\bf Z}_{k})$ where $S^{2}$ has a radius growing like
$|x|$ as $|x|\rightarrow\infty$ while $S^{1}/{\bf Z}_{k}$ approaches a
fixed radius. The fiber identification implied by the
${\bf Z}_{k}$-modding makes the physical interpretation of these
solutions very difficult. When $k=1$, we have the only case of a
gravitational instanton which has been used so far. It appears in the
nucleation rate of black holes on a thermal gas of gravitons at a given
temperature \cite{Gross-Perry-Yaffe}.

With the choice $\epsilon=0$ we have that the action is zero
for every value of $k$. For $k=1$ we recover flat space-time
and for $k=2$ we have the Eguchi-Hanson metric in both cases
modulo coordinate transformations. Asymptotically, these instantons look
like $S^{3}/{\bf Z}_{k}$. Again their physical interpretation is
unclear.

There is a very important difference between Yang-Mills and
gravitational instantons. In the Yang-Mills case, instanton mediated
amplitudes are suppresed by factors typically of the form
$\exp{(-4\pi^{2}/g^{2})}$ (the exponent is the instanton action). For
the asymptotically locally euclidean instantons (like
multi-Eguchi-Hanson instantons), their action vanishes, and therefore
thery will not be suppressed by exponential factors. They are supressed
only by powers due to the quantization of their zero modes and moduli.
Unfortunately no physical process has been described which would require
the use of these instantons.

This concludes our quick overview of some of the approaches used to
describe Quantum Gravity in the context of QFT.


\section{Consistency Conditions: Anomalies}

\subsection{Generalities about anomalies}

In classical field theory it is well known that the
presence of a continuous symmetry in the action leads to the existence
of
a conserved current associated with it (Noether's theorem). Nevertheless
when quantizing the theory, classical symmetries may not be conserved,
and their conservation equation may be modified by terms proportional to
$\hbar$
\begin{equation}
\nabla_{\mu}\langle j^{\mu}(x) \rangle = O(\hbar)
\end{equation}
The reason for this is simple. Currents $j^{\mu}(x)$ are composite
operators and therefore are naively ill-defined, and we need to
regularize them. The problem arises when
the regularization
methods are not compatible with the symmetries of the classical
theory. In this case, after renormalization, the quantum theory does not
necessarily recover the classical symmetries.

By definition an anomaly is a breakdown of a classical symmetry by
quantum corrections. There are well known examples of anomalies in
QFT, for
example the massless $\phi^{4}$ scalar field theory is invariant under
scale transformations at the classical level. However, in the process of
regularization and renormalization it is necessary
to introduce an energy scale $\mu$. As a result, not only the
renormalized coupling constant $\lambda_{R}$ is a function of this
scale, but also the fields adquire anomalous dimensions.

Perhaps the most celebrated example of an anomaly in QFT is the
Adler-Bell-Jackiw (ABJ) anomaly which arises when considering the
decay $\pi^{0}\rightarrow 2\gamma$ \cite{ABJ}. This comes from the
anomalous term
in the Ward identities associated with the triangle diagram with
two vector currents and one axial vector current.

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\put(48.00,115.11){\makebox(0,0)[cc]{$A$}}
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\put(70.00,97.78){\makebox(0,0)[cc]{$V$}}
\end{picture}

If one imposes both
Bose symmetry and the conservation of vector currents, the result is
that the axial vector current $A_{\mu}(x)=\bar{\psi}(x)\gamma_{5}
\gamma_{\mu} \psi(x)$ is not conserved for a massless fermion, but
instead we have
\begin{equation}
\partial_{\mu} A^{\mu}(x)=-\frac{\alpha\hbar}{16\pi^{2}} \epsilon_{\mu
\nu \sigma \lambda} F^{\mu \nu}(x)F^{\sigma \lambda}(x),
\label{anomaly1}
\end{equation}
$F^{\mu \nu}(x)$ is the electromagnetic field strength and
$\epsilon_{\mu \nu \sigma \lambda}$ the completely antisymmetric
Levi-Civita tensor. This anomaly is far from being dangerous, it affects
a local current and in fact our understanding of the decay
$\pi^{0}\rightarrow 2\gamma$ is based on its existence \cite{Pi-0}.

Much more critical is the existence of anomalies in local gauge
symmetries. The reason is that very important issues such as
renormalizability or unitarity are based upon the preservation of the
gauge symmetry at the quantum level. Thus the existence of gauge
anomalies
jeopardizes the consistency of the theory. The best example of this type
is the SM based in the gauge group $SU(3)\times SU(2)\times
U(1)$. As we will see below the first diagram in which an anomalous
contribution could arise is the triangle diagram with three $V-A$
currents in the vertices coupled to gauge fields \cite{V-A}.

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\end{picture}

If
after summing over all fermion species in the fermionic loop we obtain a
non-zero contribution coming from this diagram we will have an anomalous
Ward identity for $\langle (V-A)(V-A)(V-A)\rangle$. When applying
Feynman rules to calculate a diagram with one fermionic loop and
$N$ vertices coupled to gauge fields we obtain an overall group
theoretical factor of the form
\begin{equation}
\sum_{T_{L}}str(T_{L}^{a_{1}}\ldots T_{L}^{a_{N}})-
\sum_{T_{R}}str(T_{R}^{a_{1}}\ldots T_{R}^{a_{N}}),
\end{equation}
$T_{L}^{a}$ and $T_{R}^{a}$ are respectively the generators of the
representation carried by the left and right-handed fermions
and $str$ is the symmetrized trace (symmetrization is
needed because of Bose symmetry). For non-chiral theories in which left
and right-handed fermions transform under the same representation of the
gauge group this factor is always zero, and consequently there is no
anomaly. The problem arises when the theory is chiral (as the real world
is) because in this case, unless we have a cancellation when summing
over
all fermions in the loop, the contribution of the diagram in non-zero.

Consider the SM. The quantum numbers of a single
family with respect to $SU(3)\times SU(2)\times U(1)$ are
\begin{equation}
(3,2)_{1/6} \oplus (\bar{3},1)_{-2/3} \oplus (\bar{3},1)_{1/3} \oplus
(1,2)_{-1/2} \oplus (1,1)_{1}\; ,
\label{SM}
\end{equation}
where the first entry in the parenthesis indicates the representation of
$SU(3)$, the second one that of $SU(2)$ and the subindex the weak
hypercharge $Y$. All anomalies cancel for this representation; for
example, it is easy to see that the contribution
of the leptons to the anomaly is exactly canceled by the quarks
(note that all the fermions in (\ref{SM}) are left-handed)
\begin{eqnarray}
\sum_{leptons}Y^{3}_{L}&=&2\times
\left(-\frac{1}{2}\right)^{3}+1=\frac{3}{4} \nonumber \\
\sum_{quarks}Y^{3}_{L}&=&3\times\left[2\times\left(\frac{1}{6}\right)^{3}
+\left(-\frac{2}{3}\right)^{3}+\frac{1}{3}\right]=-\frac{3}{4},
\end{eqnarray}
All other combinations of the generators of $SU(3)\times SU(2)\times
U(1)$ also vanish (as can be easily checked) and then the theory is
anomaly free.

In general we expect gauge anomalies only in chiral theories (or
equivalently in parity violating theories). This is because for non
chiral theories it is always possible to construct a mass term in the
lagrangian in a gauge invariant way. The existence of this mass term
allows us to use of Pauli-Villars regularization, which does not break
gauge invariance and thus the renormalized theory will be anomaly free.

\subsection{Spinors in $2n$ dimensions}

Since the existence of anomalies is intimately related to the presence
chiral fermions it will be useful before plunging into the study of
anomalies to review briefly the properties of spinors in $2n$ dimensions
\footnote{Chiral theories exist only in even-dimensional
spaces.}\cite{LAG1}.
The starting point is the $2n$-dimensional Clifford algebra
\begin{equation}
\{\Gamma^{a},\Gamma^{b}\}=2\eta^{ab},
\label{Clifford}
\end{equation}
with $a,b=0,1,\ldots,2n-1$ and $\eta_{ab}$ the flat metric with
signature $(s,t)$
\begin{equation}
\eta_{ab}=diag\left((+1)^{s},(-1)^{t}\right).
\end{equation}
A Dirac spinor field is an object that under an infinitesimal
transformation of $SO(t,s)$ it transforms according to
\begin{equation}
\delta \psi=-\frac{1}{2}\epsilon_{mn}\Sigma^{mn} \psi\; ,
\end{equation}
$\epsilon_{mn}$ are the parameters of the tranformation, and
\begin{equation}
\Sigma^{ab}=-\frac{1}{4} [\Gamma^{m},\Gamma^{n}]
\label{spinorial}
\end{equation}
with
\begin{equation}
(\Gamma^{1})^{\dagger}=\Gamma^{1},\ldots,
(\Gamma^{t})^{\dagger}=\Gamma^{t},
(\Gamma^{t+1})^{\dagger}=-\Gamma^{t+1},\ldots,(\Gamma^{t+s})^{\dagger}=
-\Gamma^{t+s}
\end{equation}
It is easy to check by using (\ref{Clifford}) that $\Sigma^{ab}$
satisfies the commutation relations of the Lie algebra of $SO(t,s)$.

Since we are working in even dimension, we can define an element
$\bar{\Gamma}$ of the Clifford algebra (the analog of the
$4$-dimensional $\gamma_{5}$)
\begin{equation}
\bar{\Gamma}=\alpha\Gamma^{0}\Gamma^{1}\ldots\Gamma^{2n-1}
\end{equation}
satisfying the properties
\begin{equation}
\{\bar{\Gamma},\Gamma^{m}\}=0 \hspace{2cm} m=0,\ldots,2n-1
\label{gamma51}
\end{equation}
and
\begin{equation}
\bar{\Gamma}^{2}=1, \hspace{1cm} \bar{\Gamma}^{+}=\bar{\Gamma}
\label{gamma52}
\end{equation}
The last conditions can be translated into conditions over $\alpha$
\begin{equation}
\alpha^{2}=(-1)^{\frac{s-t}{2}} \hspace{1cm}
\alpha^{*}=(-1)^{\frac{s-t}{2}}\alpha
\end{equation}
{}From (\ref{gamma51}) we deduce that $[\bar{\Gamma},\Sigma^{ab}]=0$.
This, together with (\ref{gamma52}) suffices to construct the chiral
projectors
\begin{equation}
P_{\pm}=\frac{1}{2}(1\pm \bar{\Gamma})
\end{equation}
These two projectors $P_{\pm}$ split the representation $\Sigma^{ab}$
into two irreducible representations of $SO(t,s)$
\begin{equation}
\psi_{\pm} = P_{\pm}\psi
\end{equation}
where $\psi_{+}$ ($\psi_{-}$) is a positive (resp. negative) chirality
Weyl spinor.

It is a known fact that the Clifford algebra (\ref{Clifford}) has a
unique faithful representation of dimension $2^{n}$. Thus, the
representations supplied by
\begin{equation}
\Gamma^{m},(\Gamma^{m})^{*},-(\Gamma^{m})^{*},(\Gamma^{m})^{T},
-(\Gamma^{m})^{T},
\end{equation}
are all equivalent. In particular there exists a matrix $B$ such that
\begin{equation}
(\Sigma^{ab})^{*}=B\Sigma^{ab}B^{-1}.
\end{equation}
We define the charge conjugate spinor by means of the antilinear
operator $C$
\begin{equation}
\psi^{c}=C\psi\equiv B^{-1}\psi^{*}.
\end{equation}
It is easy to check that $\psi$ and $\psi^{c}$ transform in the same way
under $SO(s,t)$ transformations
\begin{equation}
\delta \psi^{c}=B^{-1} \delta \psi^{*}=
-\frac{1}{2}\epsilon_{ab}[B^{-1}(\Sigma^{ab})^{*}B]B^{-1}\psi^{*}=
-\frac{1}{2}\epsilon_{ab}\Sigma^{ab}\psi^{c}
\end{equation}
Note that if $\psi$ transforms under the gauge group (i.e. if it carries
some gauge index) with the representations $R(G)$, the charge conjugate
spinor transforms with the complex conjugate representation
$\bar{R}(G)$.

In the case in which $B$ can be found such that
$C^{2}=1$ it is possible to construct two Majorana projectors
\begin{equation}
\frac{1}{2}(1\pm C),
\end{equation}
such that
\begin{equation}
\psi_{M}=\frac{1}{2}(1+C),
\end{equation}
and
\begin{equation}
\psi_{\bar{M}}=\frac{1}{2}(1-C),
\end{equation}
are Majorana and anti-Majorana fields.

Let us now see under what circunstances we can have Weyl and Majorana
conditions at the same time. We calculate the
commutator of $\bar{\Gamma}$ and $C$, the result being
\begin{equation}
C\bar{\Gamma}=(-1)^{\frac{s-t}{2}}\bar{\Gamma}C
\end{equation}
This means that for $(s-t)/2$ odd the charge conjugate spinor has the
opposite chirality of the original one, while for $(s-t)/2$ even we have
the same chirality for both the spinor and its charge conjugated. If we
consider now the case of Minkowski space, $s=2n-1$, $t=1$, it is easy to
see that chirality and helicity are equivalent concepts. Indeed, the
massless Dirac equation in momentum space can be written as
\begin{equation}
\Gamma^{0}\psi(p)=\Gamma^{2n-1}\psi(p)
\label{Dirac}
\end{equation}
where we have made use of the freedom we have to consider the particle
moving along the $2n-1$-axis. By definition, the helicity operator in
$2n$ dimensions is
\begin{equation}
h=\Sigma^{12}\Sigma^{34}\ldots\Sigma^{2n-3,2n-2}=
\left(\frac{i}{2}\right)^{n-1}\Gamma^{1}\Gamma^{2}\ldots\Gamma^{2n-2}
\end{equation}
By applying $h$ to $\psi(p)$ we arrive at
\begin{equation}
h\psi(p)=
\left(\frac{i}{2}\right)^{n-1}\Gamma^{1}\Gamma^{2}\ldots\Gamma^{2n-2}
\psi(p)=\left(\frac{i}{2}\right)^{n-1}\bar{\Gamma}\psi(p)
\end{equation}
making use of the Dirac equation (\ref{Dirac}). This means
that in Minkowski space with $d=4k$, charge conjugation flips helicity,
while for $d=4k+2$ it does not.

In order to further investigate the possibility of finding $B$ such that
$C^{2}=1$, we consider $B$ as the matrix relating
$\Gamma^{m}$ to $-(\Gamma^{m})^{*}$
\begin{equation}
-(\Gamma^{m})^{*}=B\Gamma^{m} B^{-1},
\end{equation}
and we can write
\begin{equation}
\Gamma^{m}=-B^{*}(\Gamma^{m})^{*}B^{-1}=(B^{*}B)\Gamma^{m}(B^{*}B)^{-1}
\end{equation}
This implies that $B^{*}B$ commutes with all the  $\Gamma$'s. Since
they form an irreducible representation of the Clifford algebra
(\ref{Clifford}), using Schur's lemma we can choose $B$ satisfying
\begin{equation}
B^{*}B=\epsilon I
\label{B-condition}
\end{equation}
with $|\epsilon|=1$. In fact, taking the complex conjugate of the last
expression one can see (for example taking the trace) that
$\epsilon$ must be real, so $\epsilon =\pm 1$.
The standard charge conjugation matrix is the one
relating $\Gamma^{m}$ and $-(\Gamma^{m})^{T}$
\begin{equation}
(\Gamma^{m})^{T}=-C \Gamma^{m} C^{-1}
\end{equation}
Since in Minkowski space
$(\Gamma^{m})^{\dagger}=\Gamma^{0}\Gamma^{m}\Gamma^{0}$ we can write, after
normalizing properly
\begin{equation}
B^{\dagger}B=1
\end{equation}
But, using of (\ref{B-condition}) we can finally arrive at
\begin{equation}
B^{T}=\epsilon B
\end{equation}
and
\begin{equation}
C=B\Gamma^{0}
\end{equation}

To study the values of $\epsilon $ as a function of the dimension we
will work in an explicit basis for $d=2n$
\begin{eqnarray}
\Gamma^{0}&=&\sigma_{x}\otimes 1\otimes\ldots \otimes 1
\nonumber \\
\Gamma^{1}&=&i\sigma_{y}\otimes 1\otimes\ldots\otimes 1
\nonumber \\
\Gamma^{2}&=&i\sigma_{3}\otimes \sigma_{x}\otimes\ldots\otimes 1
\nonumber \\
\Gamma^{3}&=&i\sigma_{3}\otimes \sigma_{y}\otimes\ldots\otimes 1
\nonumber \\
&\vdots &
\nonumber
\\
\Gamma^{2n-2}&=&
i\overbrace{\sigma_{3}\otimes\ldots\otimes\sigma_{3}}^{n-1}
\otimes\sigma_{x}\otimes 1\otimes\ldots \otimes 1
\\
\Gamma^{2n-1}&=&i\sigma_{3}\otimes\ldots\otimes\sigma_{3}\otimes
\sigma_{y}\otimes 1\otimes\ldots\otimes 1
\end{eqnarray}
and
\begin{equation}
\bar{\Gamma}=\sigma_{3}\otimes\sigma_{3}\otimes\ldots\otimes\sigma_{3}
\end{equation}
where $\sigma_{x}$, $\sigma_{y}$ and $\sigma_{3}$ are the
$2$-dimensional Pauli matrices. The charge
conjugation operator becomes:
\begin{equation}
C=\sigma_{y}\otimes\sigma_{x}\otimes\sigma_{y}\otimes\ldots
\end{equation}
the number of $\sigma_{y}$'s being $k$ if $d=4k$ or $k+1$ if $d=4k+2$.
Thus we have the following values of $\epsilon $
\begin{eqnarray}
\epsilon =+1 \hspace{2cm} d=2,4 \hspace{5mm} mod \hspace{2mm} 8
\\
\epsilon =-1 \hspace{2cm} d=0,6 \hspace{5mm} mod \hspace{2mm} 8
\end{eqnarray}
The final result is that the Majorana condition can only be imposed in
$d=2,4 \hspace{2mm} mod \hspace{2mm} 8$, since only in this case
$C^{2}=(BB^{*})^{-1}=1$. Moreover, we have found above that
$[\bar{\Gamma},C]=0$ only if $d=4k+2$, so Weyl and Majorana conditions
can be simultaneously defined only in $d=2 \hspace{2mm} mod
\hspace{2mm}8$.

\subsection{When can we expect to find anomalies?}

Up to now we have met two kinds of anomalies. The first type is called
singlet anomaly and consists of an anomaly in a global current
(exemplified by the ABJ anomaly). As we said before the anomalies of
this kind do not jeopardize the consistency of the theory and their
appearance is crucial in order to explain some physical
processes
(for example the $\pi^{0} \rightarrow 2\gamma$ decay). The second
class of anomalies we mentioned
are gauge anomalies which lead to a breakdown of gauge
invariance in the quantum theory. Contrary to the first case these
anomalies are very dangerous and its presence can render the theory
inconsistent.

Besides these two kinds of anomalies there is a third type, the
gravitational anomalies \cite{G-A,LAG-Witten}. They appear in theories
coupled to gravity and
they imply a failure of diffeomorphism invariance in the quantum theory.
Unless the total anomaly cancels after summing over all fermion species,
theories with this kind of anomalies cannot be consistently coupled to
gravity.

Both, gauge and gravitational anomalies, can be further classified into
two groups. The first one are the local gauge (gravitational) anomalies,
in which the failure of gauge (diffeomorphism) invariance of the theory
is restricted to transformations that belong to the connected component
of the identity. They can appear in perturbation theory. This is the
case
for example of the potential gauge anomaly appearing in the standard
model.
Secondly we have global gauge (gravitational) anomalies. In this case,
although the theory is invariant under infinitesimal gauge
transformations (diffeomorphisms) it may not be so under transformations
that cannot be continously joined to the identity. The most celebrated
example of this kind is Witten's $SU(2)$ anomaly (see below).

We begin by analyzing how the properties of the quantum effective action
action for fermions may affect the conservation of a current
\cite{LAG-Witten}. Define the one-loop fermion effective action by
\begin{equation}
e^{-\Gamma_{eff}(A_{\mu})}=\int {\cal D}\psi {\cal D}\bar{\psi}
\exp{\left[-\int d^{d}x \,\bar{\psi}i \Dsl
\left(\frac{1-\bar{\Gamma}}{2}\right)\psi\right]},
\label{effective}
\end{equation}
where $\Dsl=\Gamma^{\mu}(\partial_{\mu}+iA_{\mu}^{a}\lambda^{a})$ is
the covariant Dirac operator. Under an infinitesimal gauge tranformation
of the gauge field
\begin{equation}
\delta A_{\mu}^{a}=D_{\mu}\epsilon ^{a}
\end{equation}
the effective action changes by
\begin{equation}
\delta \Gamma_{eff}=tr\int d^{d}x\, (D_{\mu}\epsilon^{a}) \frac{\delta
\Gamma_{eff}}{\delta A_{\mu}^{a}}=-tr\int d^{d}x\, \epsilon^{a} D_{\mu}
\left(\frac{\delta \Gamma_{eff}}{\delta A_{\mu}^{a}} \right)\;.
\end{equation}
Now, by taking the functional derivative in both sides of
(\ref{effective}) we find
\begin{eqnarray}
\frac{\delta \Gamma_{eff}}{\delta A_{\mu}^{a}}&=&
-e^{\Gamma_{eff}(A_{\mu}^{a})}\int {\cal D}\psi {\cal D}\bar{\psi}
\left[\bar{\psi}\Gamma^{\mu}\lambda^{a}\left(\frac{1-\bar{\Gamma}}{2}
\right) \psi\right] e^{-S(\psi,\bar{\psi},A)}
\nonumber
\\
&=&-\left \langle \bar{\psi}
\Gamma^{\mu}\lambda^{a}\left(\frac{1-\bar{\Gamma}}{2}\right) \psi
\right \rangle,
\end{eqnarray}
finally, we can rewrite the variation of the effective action as
\begin{eqnarray}
\delta \Gamma_{eff}&=&tr\int d^{d}x\, \epsilon^{a} D_{\mu}
\left \langle \bar{\psi}
\Gamma^{\mu}\lambda^{a}\left(\frac{1-\bar{\Gamma}}{2}\right) \psi
\right \rangle \nonumber \\
&=&tr\int d^{d}x\, \epsilon^{a} D_{\mu}\langle j_{a}^{\mu} \rangle .
\end{eqnarray}
Thus, the breakdown in the conservation of the expectation value of the
current $\langle j_{a}^{\mu} \rangle$ implies that the fermion
effective action is not invariant under infinitesimal gauge
tranformations.

A similar calculation can be done for gravitational anomalies. Now, we
perform an infinitesimal coordinate tranformation $x^{\mu}\rightarrow
x^{\mu}+\xi^{\mu}(x)$. The metric tensor transforms according to
\begin{equation}
\delta g_{\mu \nu} =D_{\mu}\xi_{\nu}+D_{\nu}\xi_{\mu}.
\end{equation}
The variation of the fermion effective action is
\begin{equation}
\delta \Gamma_{eff}=\int d^{d}x \sqrt{g}\, \xi_{\mu}D_{\nu} \langle
T^{\mu \nu} \rangle\;,
\end{equation}
where $\langle T^{\mu \nu}\rangle$ is the expectation value of the
energy-momentum tensor of the matter field and $D_{\mu}$ is the
covariant derivative. Again we see that the failure in the
conservation of the energy-momentum tensor makes the quantum theory
non invariant under diffeomorphisms.

Anomalies are expected to occur only in parity
violating theories; otherwise we can always construct a gauge invariant
mass term and regularize the theory using the Pauli-Villars method.
The anomalies are  associated
only with the imaginary part of the effective action. Take the
fermions in a complex representation $R(G)$ of the gauge group. Since
we cannot construct a gauge invariant mass term there is no evident
way to regularize the theory preserving the symmetries. Thus the
fermion effective action $\Gamma_{R}(A)$ is complex and not necessarily
gauge invariant. Now, we can consider fermions in the complex conjugate
representation ${\bar R}(G)$. In this case the discussion along the
similar lines implies that the effective action
$\Gamma_{\bar R}(A)$ (which is
the complex conjugate of $\Gamma_{R}(A)$) is again potentially
anomalous.
If we consider fermions in the real representation
$R(G)\oplus{\bar R}(G)$ it is possible to construct a gauge
invariant mass term and the theory is anomaly free. Since the
path integral is gaussian in the fermion fields, the
effective action for this theory is simply  $\Gamma_{R}(A)+\Gamma_{\bar
R}(A)=2\,Re\,\Gamma_{R}(A)$. So we conclude that the real part of
$\Gamma_{R}(A)$ is gauge invariant or can be made gauge invariant by
suitable local counterterms, and anomalies can
only be present in the imaginary part of the fermion effective action.

The usual way in which we look for anomalies is by considering
one loop diagrams
coupled to external gauge fields or gravitons by means of chiral
currents or energy-momentum tensors
\begin{eqnarray}
\bar{\psi}T^{a}\Gamma_{\mu}P_{+}\psi &\leftrightarrow & A_{\mu}^{a}\;,
\\
\bar{\psi}(D_{\mu}\Gamma_{\nu}+D_{\nu}\Gamma_{\mu})\psi &\leftrightarrow
& h_{\mu \nu}\;.
\end{eqnarray}
These amplitudes, which appear in the perturbative analysis of the Ward
identities, have  parity preserving and a parity violating parts; the
anomaly can only come from the last one, since the parity preserving
part is just like the effective action for Dirac fermions and there is
no problem with its regularization.

Since it is impossible to define a parity violating amplitude satisfying
all physical principles there are two standard definitions of the
amplitude \cite{Bardeen-Zumino}. The first way, which is the hardest one
from a computational
point of view, is to define the parity violating part of the amplitude
preserving Bose symmetry on the external lines. The anomaly so obtained
is called the consistent anomaly because it satisfy the Wess-Zumino
consistency condition \cite{Wess-Zumino}.

The second alternative (the Adler-Rosenberg method
\cite{ABJ,Rosenberg}) consists of
considering the same diagram with a single axial vector current in one
vertex and vector currents in the others. We then calculate the anomaly
by imposing vector current conservation and Bose symmetry on the
external vector lines. This form of the anomaly, called the covariant
anomaly,
does not satisfy the consistency conditions, but it is much easier to
calculate. There is a standard formalism interpolating
between consistent and covariant form of the anomaly
\cite{LAG-Ginsparg1,LAG1}, so we
can calculate the covariant anomaly (which is easier) and from it obtain
the consistent form.

We now determine what diagrams are potentially anomalous. Let us
consider a $k+1$ poligon with $k$ external vector currents and one axial
vector current. Since the amplitude is parity violating, it contains a
Levi-Civitta tensor $\epsilon_{i_{1}\ldots i_{2n}}$ (we are in $2n$
dimensions). We want to check conservation of the axial vector current,
so the polarization vector in the axial channel is proportional to the
incoming momentum $P_{\mu}$. Besides, we have the momenta in the other
vertices $p_{1},\ldots,p_{k}$ and the corresponding polarization vectors
$\epsilon_{1}^{\mu},\ldots,\epsilon_{k}^{\mu}$. We have to saturate the
indices
of the $\epsilon$ tensor and we have the momentum conservation
law $P+\sum
p_{i}=0$. The only alternative is to take $k=n$ so the simplest
diagram which is potentially anomalous is a $n+1$ polygon. In this case
we can write the amplitude associated to this diagram as
\begin{equation}
A(p,\epsilon)\left[\sum_{T_{L}}\,str(T_{L}^{a_{1}}\ldots
T_{L}^{a_{n+1}})-\sum_{T_{R}}\,str(T_{R}^{a_{1}}\ldots
T_{R}^{a_{n+1}})\right]\;.
\end{equation}
Instead of calculating $ str(T^{a_{1}}\ldots T^{a_{n+1}})$ it is easier,
and completely equivalent, to calculate $tr\,H^{n+1}$ with $H$ any
element of the Lie algebra of the gauge group; for $d=4k$ we would have
$tr\,H^{2k+1}$ and $tr\,H^{2k+2}$ for $d=4k+2$. Let us consider the case
in which fermions are in a real or pseudoreal representation of the
gauge group. In this case $H$ satisfy
\begin{equation}
H^{T}=-S^{-1}HS\;,
\end{equation}
with $S^{T}=\pm S$ ($+1$ for real representations and $-1$ for
pseudoreal). Then for $d=4k$ we have
\begin{equation}
tr\,H^{2k+1}=tr\,(H^{2k+1})^{T}=tr\,(-S^{-1}HS)^{2k+1}=-tr\,H^{2k+1}=0\;.
\end{equation}
So in $d=4k$ there is no gauge anomaly if the gauge group has only real
or pseudoreal representations. Thus we are sure that chiral theories
with gauge groups $SO(2n+1)$, $G_{2}$, $F_{4}$, $E_{7}$ or $E_{8}$ are
always anomaly free. Then the only dangerous possibilities are the
gauge group to be $SO(2n)$, $SU(N)$ or $E_{6}$.

In the last paragraphs we studied gauge anomalies. For
gravitational
anomalies we have again that the first diagram from which we can
expect an anomalous contribution is
a polygon with $n+1$ vertices ($d=2n$) \cite{LAG-Witten}. The same
reasoning as in the case of
gauge anomalies applies, since the polarization vector of the graviton
is a second rank symmetric tensor. We may also wonder when we can expect
pure gravitational anomalies; for a given fermion $\psi_{+}$ in four
dimensions, we have both particles and antiparticles inside the loop
of the diagram carrying opposite helicity since $\{\bar{\Gamma},C\}=0$.
But for the gravitons in the external
lines, particles and antiparticles are not distinguishable and the
coupling ``looks'' vector like. This is also applicable to $d=4k$ in
which the anticommutation of $\bar{\Gamma}$ and $C$ also holds. The
result is that in $4k$ dimensions we have no purely gravitational
anomalies.

This is not the case in $d=4k+2$. Now $[\bar{\Gamma},C]=0$ and particles
have the same helicity as their antiparticles.
The gravitational interaction is genuinely chiral and we can expect
anomalies in the conservation of the energy-momentum tensor.

Pure gravitational anomalies can only
be present in $4k+2$ dimensions. As an example let us compute the
gravitational anomaly for a spin-{1/2} fermion in two dimensions
\cite{LAG-Witten}. In the presence of a weak gravitational field
$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$, at linearized level the
perturbation $h_{\mu\nu}$ couple to the fermion field through the
fermion energy-momentum tensor by the lagrangian $\Delta {\cal L}=
-\frac{1}{2}h^{\mu\nu}T_{\mu\nu}$. If we consider a chiral fermion
obeying $\bar{\Gamma} \psi=-\psi$, the only non-vanishing component of
$T_{\mu\nu}$ is $T_{++}$, and we are interested in the two-point
function
\begin{equation}
U(p)=\int d^{2}x \langle \Omega |T\left(T_{++}(x)T_{++}(0)\right)\;.
|\Omega\rangle e^{ipx}
\end{equation}
{}From naive conservation of $T_{\mu\nu}$, $\partial_{-}T_{++}(x)=0$, we
get the Ward identity
\begin{equation}
p_{-}U(p)=0\;,
\end{equation}
implying $U(p)=0$ for all $p_{-}\neq 0$. But by analyticity we will
have $U(p)=0$ for all $p_{-}$, and this is impossible for the two-point
function of a hermitian operator, so we must have an anomaly in the
Ward identity. This anomaly can be explicitly calculated
from fig. \ref{fig1} \cite{LAG-Witten}, and the result is
\begin{equation}
p_{-}U(p)=\frac{i}{24\pi}p_{+}^{3}\;.
\end{equation}
\begin{figure}
\vspace{3cm}
\caption{The gravitational anomaly in two dimensions}
\label{fig1}
\end{figure}
With the value of $U(p)$ we can construct the effective action to second
order in the perturbation $h_{\mu\nu}$ simply by coupling each vertex in
the figure to $-\frac{1}{2}h_{--}$
\begin{equation}
S_{eff}(h_{\mu\nu})=-\frac{1}{192\pi}\int d^{2}p \frac{p_{+}^{3}}{p_{-}}
h_{--}(p)h_{--}(-p)\;,
\label{effective-grav}
\end{equation}
where $h_{--}(p)$ is the Fourier transform of $h_{--}(x)$. Now we can
test the invariance of this effective action under coordinate
transformations and verify that it is not;
we still have the freedom of adding local terms to the effective
action in order to make it diffemorphism invariant. Considering the most
general possible counterterm with the proper dimension
\begin{eqnarray}
\Delta S= \int
d^{2}p\left[Ap_{+}^{2}h_{--}(p)h_{+-}(-p)+Bp_{+}p_{-}h_{+-}(p)h_{+-}(-p)
\right. \nonumber
\\
\left.+Cp_{+}p_{-}h_{++}(p)h_{--}(-p)+Dp_{-}^{2}h_{++}(p)
h_{+-}(-p)\right]\;,
\end{eqnarray}
it can be checked that there is no way of choosing the constants
in order to make $S_{eff}+\Delta S$ invariant under coordinate
transformations.

The situation is quite different if we consider Dirac fermions in $1+1$
dimensions. In this case, to get the fermion effective action, we have
only to add to (\ref{effective-grav}) its parity transformed
\begin{equation}
S_{eff}=-\frac{1}{192\pi}\int
d^{2}p\left[\frac{p_{+}^{3}}{p_{-}}h_{--}(p)h_{--}(-p)+
\frac{p_{-}^{3}}{p_{+}}h_{++}(p)h_{++}(-p)\right]\;.
\label{effective2}
\end{equation}
This action is non-invariant under general coordinate
transformations, but now we can add local couterterms to obtain an
invariant action
\begin{equation}
\bar{S}=-\frac{1}{192\pi}\int d^{2}p \frac{R(p)R(-p)}{p_{+}p_{-}}\;,
\end{equation}
where $R(p)=p_{+}^{2}h_{--}+p_{-}^{2}h_{++}-2p_{+}p_{-}h_{+-}$ is the
linearized form of the scalar curvature.

{}From this result we can obtain the trace anomaly of the energy
momentum
tensor. As we indicated $T_{+-}$ vanishes classically. The result of
this is that
$h_{+-}$ does not appear in the effective action (\ref{effective2}).
However in order to preserve diffeomorphism invariance we have to add
counterterms that contain $h_{+-}$, so we get a non-zero expectation
value for $T_{+-}(p)$ given by
\begin{equation}
\langle 2T_{+-}(p) \rangle=-2\frac{\delta \bar{S}}{\delta h_{+-}(-p)}=
-\frac{1}{24\pi}R(p)\;,
\end{equation}
which means that classical conformal invariance is broken by quantum
corrections.

\subsection{The Atiyah-Singer Index Theorem and the computation of
anomalies}

In the last two decades, sophisticated technology borrowed from topology
has been applied to the computation of anomalies. A specially useful
tool has been the Atiyah-Singer index theorem
\cite{Atiyah-Singer,Atiyah-Pattodi-Singer} and its generalization,
the Atiyah-Singer index theorem \cite{Atiyah-Singer-2} for families of
elliptic operators.
By applying the first one it is possible to compute singlet anomalies,
while the use of the second one allows us to calculate both local and
global anomalies.

In what follows we will not review the whole technology for computing
anomalies, which is available in a number of reviews
\cite{LAG-Ginsparg1,LAG1}. Instead
we will present the basic results and how to use them. Since, as we will
see, the problem of computing anomalies is intimately related to the
index theorem for the Dirac operator (as well as that for
Rarita-Schwinger operator), we start by reviewing the main features
of the index theorem for the Dirac operator.
Given an operator ${\cal O}$ we define the index as the
difference between the dimension of the kernel of ${\cal O}$ and that of
its adjoint
\begin{equation}
ind\,{\cal O}= dim\,ker\,{\cal O}-dim\,ker\,{\cal O}^{\dagger}\;.
\end{equation}
The Atiyah-Singer index theorem relates the index of an elliptic
operator with the topological properties of the manifold in which it is
defined. We will use this theorem in the particular case in which ${\cal
O}$ is the Weyl operator $D_{+}=i\Dsl P_{+}$.

Let $M$ be a manifold and let $S_{+}\otimes E$ and
$S_{-}\otimes
E$ be two vector bundles over $M$. $S_{\pm}$ is the space of spinors of
positive and negative chirality respectively and $E$ is the space
carrying the representation of the gauge group. Let $A=A_{\mu}dx^{\mu}$
be the connection $1$-form over $E$, which transforms under a gauge
tranformation $g$ as
\begin{equation}
A \rightarrow g^{-1}(a+d)g\;,
\end{equation}
and associated with it the curvature $F$
\begin{equation}
F=dA+A^{2}\;,
\end{equation}
where we assume wedge product between forms (i.e. $A\wedge A=A^{2}$). In
the presence of gravity we have, besides the
gauge bundle, the vector bundle of orthonormal or coordinates frames. In
the first case we will work with the spin connection
$\omega=\omega_{\mu}dx^{\mu}$ and in the second one with the Christoffel
connection $\Gamma=\Gamma_{\mu}dx^{\mu}$; the curvature associated
$R$ has an expression similar to that of the gauge curvature $R=d\omega+
\omega^{2}$. The Dirac operator $\Dsl$ is given explicitly by
\begin{equation}
\Dsl =
\Gamma^{\mu}(\partial_{\mu}+\frac{1}{2}\omega_{\mu\,ab}\Sigma^{ab}
+A_{\mu}^{a}T^{a})\;,
\label{dir-op}
\end{equation}
In terms of (\ref{dir-op}) we define the Weyl operators $D_{\pm}=i \Dsl
P_{\pm}$ mapping $S_{\pm}\otimes E$ into $S_{\mp}\otimes E$
\begin{equation}
D_{\pm}:S_{\pm}\otimes E \longrightarrow S_{\mp}\otimes E\;,
\end{equation}
and verify $D_{+}^{\dagger}=D_{-}$. In a base in which $\bar{\Gamma}$ is
diagonal the Dirac operator can be written as
\begin{equation}
\Dsl = \left(
\begin{array}{cc}
0     &     D_{-}  \\
D_{+} &     0
\end{array}
\right)\;.
\end{equation}
Now, we can define the index of $D_{+}$ as
\begin{equation}
ind\,D_{+}=dim\,ker\,D_{+}-dim\,ker\,D_{-}\;.
\end{equation}
$ind\,D_{+}$ is a topological invariant; using the Atiyah-Singer index
theorem we can express it in terms of the topological invariants of the
bundle. Before doing so we have to introduce some elementary notions
from the theory of characteristic classes
\cite{LAG-Ginsparg1,LAG1,Nash}. Let $\Omega$ be some
matrix-valued
two-form taking values in the Lie algebra of some group $G$ and let
$P(\Omega)$ be an invariant polynomial, i.e., a polynomial in $\Omega$
satisfying
\begin{equation}
P(g^{-1}\Omega g)=P(g)
\end{equation}
for any element $g$ of $G$. Let us consider the particular case in which
$\Omega$ is either the gauge field strength tensor $F$ or the curvature
$R$. In this case any invariant polynomial $P(\Omega)$ satisfies the
following properties:
\begin{itemize}
\item{$P(\Omega)$ is a closed form, i.e., $dP(\Omega)=0$.}
\item{The integrals of $P(\Omega)$ are topological invariants.}
\end{itemize}
To prove these two properties it is enough to consider monomials of the
form $P_{m}(\Omega)=tr(\Omega^{m})$ since these are the building blocks
of any invariant polynomial. It is straighforward to show
that $dP(\Omega)=0$.
\begin{equation}
dP_{m}=m\,tr(d\Omega\,\Omega^{m-1})=m\,tr(D\Omega \, \Omega^{m-1})=0\;,
\end{equation}
where $D$ is the covariant derivative and we have used the
Bianchi identity
\begin{equation}
D\Omega=d\Omega+\Omega \omega-\omega \Omega=0
\end{equation}
with $\omega$ the connection $1$-form.

The second statement is not as evident as the first one. We have to
prove that
the integral of the polynomial $P_{m}$ over $M$ is independent of the
connection $\omega$. Consider a one-parameter family of
connections interpolating between two given connections $\omega_{0}$ and
$\omega_{1}$
\begin{equation}
\omega_{t}=\omega_{0}+t(\omega_{1}-\omega_{0})\;,
\end{equation}
with $0\leq t \leq 1$, and the family of curvatures associated to them
\begin{equation}
\Omega_{t}=d\omega_{t}+\omega^{2}_{t}\;.
\end{equation}
Now we evaluate the partial derivative of $\Omega_{t}$ with respect to
the parameter $t$ (we denote this derivative by a dot)
\begin{eqnarray}
\frac{\partial P_{m}(\Omega_{t})}{\partial
t}&=&m\,tr\left(\dot{\Omega}_{t}
\,\Omega^{m-1}_{t}\right)=-m\,tr\left(D_{t}
\dot{\omega}_{t}
\Omega^{m-1}_{t}\right)= \nonumber \\
& & m\,d\,tr\left(
\dot{\omega}_{t}\Omega^{m-1}_{t}\right)\;,
\label{top-invariant}
\end{eqnarray}
since we have that
\begin{equation}
\dot{\Omega}_{t}=d\dot{\omega}_{t}+\dot{\omega}_{t}\omega_{t}+
\omega_{t}\dot{\omega}_{t}\;.
\end{equation}
Now, by integrating (\ref{top-invariant}) with respect to $t$ from $0$
to $1$ we obtain
\begin{equation}
P_{m}(\Omega_{1})-P_{m}(\Omega_{0})=m\,d\int
dt\,tr(\dot{\omega}_{t}\Omega_{t}^{m-1})=
dQ_{2n-1}^{0}(\omega_{t},\Omega_{t})\;.
\label{Chern-Simons}
\end{equation}
where $Q_{2n-1}^{0}$ is called the Chern-Simons form.
If we now integrate over any closed $2m$-dimensional submanifold
$M_{2m} \subset M$ we have
\begin{equation}
\int_{M_{2m}} P_{m}(\Omega_{1}) = \int_{M_{2m}} P_{m}(\Omega_{0})
\end{equation}
Then the integral of $P_{m}(\Omega)$ over $M_{2m}$ is independent of the
connection, so it is a topological invariant and the
second statement is proven . Moreover, this integral is also invariant
under
deformations of $M_{2m}$. Let us take a deformation $M_{2m}^{'}$ of
$M_{2m}$
and let $B_{2m+1}$ be a manifold such that $\partial B_{2m+1}=
M_{2m}-M_{2m}^{'}$. In this case, since $dP_{m}(\Omega)=0$ we have,
using Stokes theorem, that
\begin{equation}
0=\int_{B_{2m+1}} dP_{m}({\Omega}) = \int_{M_{2m}} P_{m}(\Omega)-
\int_{M_{2m}^{'}} P_{m}(\Omega).
\end{equation}

We now introduce some well
known classes, in term of which the index of the Weyl operator will be
expressed. First we take a bundle with group $U(n)$, so the
curvature $\Omega$ is a hermitian matrix of two forms. The
the total Chern class is defined as
\begin{equation}
c(\Omega)=det\,\left(1+\frac{i}{2\pi}\Omega\right)\;.
\end{equation}
This total class can be expanded as a sum of the form
\begin{equation}
c(\Omega)=1+c_{1}(\Omega)+c_{2}(\Omega)+\ldots\; ,
\end{equation}
where $c_{i}(\Omega)$ is a $2i$-form called the $i$-th Chern class. To
determine the precise form of the Chern classes we use the
following
trick: since the matrix $\Omega$ is hermitian we can formally
diagonalize it and write
\begin{equation}
\frac{i}{2\pi}\Omega=\left(
\begin{array}{ccc}
x_{1}	&	 &	\\
	& \ddots &	\\
	&	 &  x_{n}
\end{array}
\right)\; ,
\end{equation}
where the eigenvalues $x_{i}$ are $2$-forms. Then the
total Chern class becomes
\begin{equation}
c(\Omega)=\prod_{i=1}^{n} (1+x_{i})=1+\sum_{i=1}^{n} x_{i} +
\sum_{i<j} x_{i}x_{j} +\ldots +\prod_{i=1}^{n} x_{i}\; .
\end{equation}
Now, we can identify the terms in the expansion as
\begin{eqnarray}
c_{1}(\Omega)&=&\sum_{i=1}^{n} x_{i}=\frac{i}{2\pi}\, tr\,\Omega
\nonumber
\\
c_{2}(\Omega)&=&\sum_{i<j}
x_{i}x_{j} = \left(\frac{i}{2\pi}\right)^{2}\left[(tr\,\Omega)^{2}-
tr\,(\Omega^{2})\right]
\nonumber
\\
&\vdots &
\nonumber
\\
c_{n}&=&\prod_{i=1}^{n} x_{i} =
det\,\left(\frac{i}{2\pi}\Omega\right)\;,
\end{eqnarray}
and we express the Chern classes in terms of $\Omega$. The total
Chern class has good properties with respect to the direct sum (Whitney
sum) of bundles. If $E$ and $F$ are two $U(n)$ bundles; the total Chern
character of the direct sum $E\oplus F$ is
\begin{equation}
c(E\oplus F)=c(E)c(F).
\end{equation}

This behavior of the total Chern character with respect to
direct sums is not maintained for tensor
products. We can define another useful polynomial, the Chern
character, associated with a $U(n)$ bundle by
\begin{equation}
ch(\Omega)=tr\,e^{\frac{i}{2\pi}\Omega}=
ch_{0}(\Omega)+ch_{1}(\Omega)+\ldots\; ,
\label{Ch.1}
\end{equation}
where $ch_{i}(\Omega)$ is $i$-th Chern character. The identifications of
these Chern characters follows from expanding the exponential:
\begin{equation}
tr\,e^{\frac{i}{2\pi}\Omega}=\sum_{k=0}^{\infty}
\frac{1}{n!}tr\,\left(\frac{i}{2\pi}\Omega\right)^{k}.
\end{equation}
This sum contains always a finite number of terms since $\Omega^{k}$ is
a
$2k$-form and then
$2k\leq dim\,M$. We obtain the $j$-th Chern character as
\begin{eqnarray}
ch_{0}(\Omega)&=&2n \\
ch_{j}(\Omega)&=&\frac{1}{j!}\left(\frac{i}{2\pi}\right)^{j}\,tr\,
\Omega^{j},
\hspace{1cm} 2\leq 2j \leq \,dim\, M.
\label{Ch.2}
\end{eqnarray}
The total Chern character enjoys simple properties with respect to both
direct sums and tensor products of bundles
\begin{eqnarray}
ch(E\oplus F)&=&ch(E)+ch(F)
\nonumber
\\
ch(E\otimes F)&=&ch(E)ch(F).
\end{eqnarray}

Up to now we have defined invariant polynomials for $U(n)$ bundles in
which the curvature is a hermitian matrix of $2$-forms.
Let us now consider the case in which the group bundle is $SO(n)$,  as
in Riemannian geometry. We define the total Potrjagin class as:
\begin{equation}
p(\Omega)=det\left(1+\frac{1}{2\pi}\Omega\right)=
1+p_{1}(\Omega)+p_{2}(\Omega)+\ldots\; ,
\end{equation}
with $p_{i}(\Omega)$ the $i$-th Potrjagin class.
The curvature $\Omega$ is an antisymmetric matrix of $2$-forms and it
cannot be diagonalized by a similarity transformation but it can
be always brought to the skew-diagonal form
\begin{equation}
\frac{1}{2\pi}\Omega=\left(
\begin{array}{ccccc}
0      &   x_{1} &  &  &   \\
-x_{1} &    0 &  &  &	\\
       & &    0   & x_{2} &   \\
       & & -x_{2} &   0   &   \\
       & &  &  &  \ddots
\end{array}
\right).
\end{equation}
The antisymmetry of $\Omega$ implies that the only non-zero polynomials
in $\Omega$ are of even degree; thus $p_{j}(\Omega)$ is a $4j$-form.
The total Potrjagin class can be written in terms of the formal
eigenvalues as
\begin{equation}
p(\Omega)=\prod_{i}(1+x_{i}^{2})=1+\sum_{i} x_{i}^{2}+\sum_{i<j}
x_{i}^{2}x_{j}^{2}+\ldots
\end{equation}

We are now ready to formulate the Atiyah-Singer index theorem for the
Weyl operator
\begin{equation}
D_{+}=i\Gamma^{\mu}(\partial_{\mu}+
\frac{1}{2}\omega_{\mu\,ab}\Sigma^{ab}+A_{\mu})P_{+}\;.
\end{equation}
The theorem states that the index of $D_{+}$ is given by
\begin{equation}
ind\,D_{+}=\int_{M}\left[\hat{A}(M)ch(F)\right]_{vol}
\end{equation}
where $\hat{A}(M)$, called the Dirac genus of M (or the $A$-roof genus),
is a polynomial in
the $2$-forms $x_{i}$,
and the subindex {\it vol} indicates that we only retain
from the product the volume form. The Dirac genus is given by
\begin{equation}
\hat{A}(M)=\prod_{a} \frac{x_{a}/2}{\sinh{(x_{a}/2)}}\;.
\end{equation}
It can be rewritten in terms of the invariant polynomials
$tr(R^{m})$ with $R$ the curvature form of $M$
\begin{equation}
\hat{A}(M)=1+\frac{1}{(4\pi)^{2}}\frac{1}{12}\,tr\,R^{2}+
\frac{1}{(4\pi)^{4}}\left[\frac{1}{288}(tr\,R^{2})^{2}+
\frac{1}{360}\,tr\,R^{4}\right]+\ldots
\end{equation}
This equation, together with the expansion of the total Chern character
(\ref{Ch.1}) and (\ref{Ch.2}), allows us to write explicitly the
index of $D_{+}$, for example, in $d=4$ dimensions
\begin{equation}
ind\,D_{+}=\frac{1}{(2\pi)^{2}}\int_{M}\left(\frac{i^{2}}{2}\,tr\,F^{2}+
\frac{r}{48}\,tr\,R^{2}\right) \hspace{2cm} d=4
\end{equation}
with $r$ the dimension of the vector bundle.

The index of the Rarita-Schwinger operator is also of interest in the
computation of anomalies. Now we have to be careful because ghost fields
are needed in order to remove unphysical degrees of freedom. The
constraint
\begin{equation}
k_{\mu}\psi^{\mu}=0,
\end{equation}
together with the invariance under
\begin{equation}
\psi^{\mu}\rightarrow \psi^{\mu}+k^{\mu}\chi,
\end{equation}
remove two spin-$\frac{1}{2}$ degrees of freedom of the same chirality,
while the constraint
\begin{equation}
\Gamma^{\mu}\psi_{\mu}=0,
\end{equation}
removes one spin-$\frac{1}{2}$ degree of freedom with the opposite
chirality. The final form of the index theorem for this operator is
obtained by substracting from the contribution of a
$\frac{1}{2}$-spinor
with a vector index the contribution of a $\frac{1}{2}$-spinor, namely
\begin{equation}
ind\,\Dsl_{3/2}=
\int_{M}\left[\hat{A}(M)(tr\,e^{iR/2\pi}-1)ch(F)\right]_{vol}.
\end{equation}
Finally, when we study anomalies in ten dimensions (for example in
the case
of the low-energy field theory corresponding to superstrings) we may
expect anomalies coming from antisymmetric tensor fields whose field
strength is self-dual. The index theorem for these fields is
\begin{equation}
ind\,\left(iD_{A}\right)=\frac{1}{4}\int_{M} [L(M)]_{vol}\; ,
\end{equation}
where $L(M)$ is the Hirzebruch polynomial defined by
\begin{equation}
L(M)\equiv 2^{n}\prod_{a} \frac{x_{a}/2}{\tanh{(x_{a}/2)}}\; .
\end{equation}

All the technology we have introduced can be
directly
applied to the computation of anomalies. We are going to begin by
showing how, by using Atiyah-Singer index theorem, we can obtain
the singlet chiral $U(1)$ anomaly of sec. 2.1. To do that
we use Fujikawa's approach \cite{Fujikawa}. The
one-loop effective action for a Dirac spinor in Euclidean space is
\begin{equation}
e^{-\Gamma_{eff}(A)}=\int {\cal D}\psi {\cal D}\bar{\psi}
\exp{(-\int\sqrt{g}\,d^{2n}x\,\bar{\psi}\,i\Dsl \psi)}
\label{5-action}
\end{equation}
with $\Dsl=\Gamma^{\mu}(\partial_{\mu}+A_{\mu})$.
In Euclidean space we have to consider $\psi$ and $\bar{\psi}$ as
independet variables. The action is classically
invariant under the global infinitesimal chiral transformation
\begin{eqnarray}
\delta \psi &=& i\alpha\bar{\Gamma}\;, \psi \\
\nonumber
\delta \bar{\psi} &=& i\alpha \bar{\psi} \bar{\Gamma}\;.
\label{chiral-trans}
\end{eqnarray}
By Noether's theorem, we have a classically conserved current
\begin{equation}
j_{5}^{\mu}=\bar{\psi}\bar{\Gamma}\Gamma^{\mu} \psi\;.
\end{equation}
In the quantum theory if we look at the expression for the fermion
effective action (\ref{5-action}) we see that the path integral
is invariant under (\ref{chiral-trans}) if the action and the
integration measure ${\cal D}\psi{\cal D}\bar{\psi}$ are both invariant.
If we make a chiral
transformations (\ref{chiral-trans}) taking the parameter $\alpha$
as a function of $x$ the action varies according to
\begin{equation}
\int (dx)\,\bar{\psi}\,\Dsl\,\psi \longrightarrow
\int (dx)\,\bar{\psi}\,\Dsl\,\psi+ \int (dx)\,\alpha(x)
\nabla_{\mu}j_{5}^{\mu}\; ,
\end{equation}
$(dx)$ we denotes the whole volume element and
$\nabla_{\mu}=\partial_{\mu}+\omega_{\mu}$ is the covariant derivative.
We need to compute as well the jacobian ${\cal J}$ induced in
the integration measure by the transformation $\psi \rightarrow
\psi+\delta\psi$ (and the corresponding one for $\bar{\psi}$)
\begin{eqnarray}
& &\int {\cal D}\psi {\cal D}\bar{\psi} \exp{\left(\int
(dx)\,\bar{\psi}\,i\Dsl\,\psi\right)} \nonumber \\
&=&\int {\cal D}\psi {\cal D}\bar{\psi}\,{\cal J}\,\exp{\left(\int
(dx)\,\bar{\psi}\,i\Dsl\,\psi+
\int (dx)\,\alpha(x)\nabla_{\mu}j_{5}^{\mu}\right)}\;.
\label{eff.act.j5}
\end{eqnarray}
Were the jacobian be equal to unity, by simply expanding the exponential
to first order in $\alpha(x)$, we would obtain the conservation
of the axial current $\langle \nabla_{\mu}j_{5}^{\mu}\rangle=0$.
However, this is not necessarily so. To compute ${\cal J}$,
we expand $\psi$ and $\bar{\psi}$ in terms of the eigenfunctions of
the Dirac operator
\begin{equation}
i\Dsl \psi_{n} = \lambda_{n} \psi_{n}
\end{equation}
as
\begin{eqnarray}
\psi&=&\sum_{n}a_{n}\psi_{n}\;,
\\
\bar{\psi}&=&\sum_{n}\bar{b}_{n}\psi^{\dagger}_{n}\;,
\end{eqnarray}
where $a_{n}$ and $b_{n}$ are Grassmann parameters.
Now the integration measure is simply $\prod_{n}d\bar{b}_{n}da_{n}$ and
the action can be rewritten as
\begin{equation}
\int(dx)\,\bar{\psi}\,i\Dsl\,\psi=\sum_{n} \lambda_{n}
\bar{b}_{n}a_{n}\; .
\end{equation}
The number of positive eigenvalues of $\Dsl$
equals the number of negative ones; since $\{ \Dsl ,\bar{\Gamma}\}=0$
for any eigenfunction $\psi$ with positive eigenvalue $\lambda$,
$\bar{\Gamma}\psi$ is such that
$\Dsl \bar{\Gamma}\psi=-\lambda\bar{\Gamma}\psi$. This implies that
the
only asymmetry in the spectrum of the Dirac operator
is restricted
to the zero modes. There is no difficulty in computing the jacobian in
the basis chosen. The result is
\begin{equation}
{\cal J}=\exp{(-2i\sum_{n}\langle
\psi_{n}|\alpha(x)\bar{\Gamma}|\psi_{n}\rangle)}\simeq
1-2i\sum_{n}\langle \psi_{n}|\alpha(x)\bar{\Gamma}|\psi_{n}\rangle\; ,
\label{jacobian}
\end{equation}
where the inner product is defined in the usual way
\begin{equation}
\langle \psi|\alpha(x)\bar{\Gamma}|\psi \rangle=
\int (dx) \alpha(x)\psi^{\dagger}(x)\bar{\Gamma}\psi(x).
\label{inner}
\end{equation}
This jacobian is unfortunately infinite and it is neccesary to
regularize it. This can be accomplished by introducing a gaussian
cut-off
\begin{eqnarray}
& &-2i\sum_{n}\int (dx)
\alpha(x)\psi^{\dagger}_{n}(x)\bar{\Gamma}\psi_{n}(x)
e^{-\frac{\lambda^{2}_{n}}{M^{2}}} \nonumber \\
&=&-2i\sum_{n}\langle
\psi_{n}|\alpha(x)\bar{\Gamma}
e^{-\frac{{\not D}^{2}}{M^{2}}}|\psi_{n}\rangle\;.
\label{preanomaly}
\end{eqnarray}
Since the non-zero eigenvalues of the Dirac operator
are paired, $\psi_{n}$ and
$\bar{\Gamma}\psi_{n}$ have opposite eigenvalues and therefore
\begin{equation}
\langle \psi_{n}|\bar{\Gamma}|\psi_{n}\rangle=0
\end{equation}
provided $i\Dsl\psi_{n}\neq 0$. In the limit
$\partial_{\mu}\alpha(x)\rightarrow 0$, (\ref{preanomaly}) receives
contributions only from the zero mode sector,
\begin{equation}
\alpha\lim_{M\rightarrow \infty}\langle\psi_{n}|\bar{\Gamma}
e^{-\frac{{\not D}^{2}}{M^{2}}}|\psi_{n}\rangle=\alpha
\sum_{zero\,\,modes} \langle\psi_{n}|\bar{\Gamma}|\psi_{n}\rangle
\end{equation}
but this last sum equals the number of zero modes of $D_{+}$ minus
the number of zero modes of $D_{-}$, that is, the index of $D_{+}$.
So the regularized jacobian is simply the index of the Weyl operator,
and we can write the final expression of the integrated axial anomaly
as
\begin{equation}
\int (dx) \langle \nabla_{\mu}j_{5}^{\mu}\rangle = 2\int_{M}
[\hat{A}(M)ch(F)]_{vol}
\end{equation}
where we have used the Atiyah-Singer index theorem stated
previously. This expression gives the form of the axial anomaly
in $2n$ dimensions in the presence of both gauge and gravitational
fields. In particular, in four dimensions, using the expressions for
the Dirac genus and the Chern character we obtain
\begin{equation}
\int_{M} \langle \nabla_{\mu} j_{5}^{\mu} \rangle=
\frac{1}{(2\pi)^{2}}
\int_{M}\left(-tr\,F^{2}+\frac{r}{24}\,tr\,R^{2}\right)
\end{equation}
which agrees with (\ref{anomaly1}) in the flat space-time limit ($R=0$).

We are now ready to face the computation of local gauge and
gravitational anomalies using the Atiyah-Singer index theorem for
families of operators. Once again the fermion effective action is
\begin{equation}
e^{-\Gamma_{eff}(A)}=\int {\cal D}\psi {\cal D}\bar{\psi}
\exp{(-\int d^{2n}x\,\bar{\psi}\,iD_{+}\,\psi)}\;,
\end{equation}
$D_{+}=\Dsl P_{+}$. There is a problem in defining this
path integral, since as we saw $D_{+}$ does not map a space into itself,
but
\begin{equation}
D_{+}:S_{+}\otimes E \longrightarrow S_{-}\otimes E
\end{equation}
This means that we do not have a well posed eigenvalue problem
$D_{+}\psi=\lambda \psi$ and then there is no natural definition of the
path integral in terms of the determinant of $D_{+}$. We can
define a new operator
$i\hat{D}$ with a well defined eigenvalue problem and whose
determinant can be identified with the path integral
\begin{equation}
\int {\cal D}\psi {\cal D}\bar{\psi}
\exp{\left(-\int d^{2n}x\,\bar{\psi}\,iD_{+}\,\psi\right)}=det\,\hat{D}
\end{equation}
where some suitable regularization of the determinant is assumed.
Moreover, since we know that the anomaly
comes from the imaginary part of the effective action (see
above), $|det\,\hat{D}|$ has to be gauge invariant. We also have to make
sure that the perturbative expasion for $\hat{D}$ is the same as that
for $D_{+}$.

We will work with the operator $\hat{D}$ \cite{LAG-Ginsparg2} defined by
\begin{equation}
\hat{D}=\left(
\begin{array}{cc}
  0	    &	D_{+} \\
 \dsl_{-} &   0
\end{array}
\right)
\end{equation}
which satisfies the properties listed. For example, the modulus
of $det\,\hat{D}$ is given by
\begin{equation}
(det\,\hat{D})^{2}=det\,(\hat{D}^{\dagger}D)=
det(\dsl_{+}\dsl_{-})\,det(D_{+}D_{-})
\end{equation}
where $det\,(\dsl_{+}\dsl_{-})$ does not depend of the gauge fields
and therefore is gauge invariant. Since $det\,(D_{+}D_{-})$
equals $det\,i\Dsl$ (see above), it is also gauge
invariant.

The computation of anomalies is carried out by relating them to the
Atiyah-Singer index theorem for a Dirac operator in $2n+2$ dimensions
\cite{LAG-Ginsparg2}. We compactify effectively the space-time to
$S^{4}$ by choosing appropiate boundary conditions.
Let us consider the one parameter family of gauge
transformations
$g(\theta,x):S^{1}\times S^{2n} \rightarrow G$ subject to the boundary
conditions $g(0,x)=g(2\pi,x)=1$. Associated with them we can define the
family of connections
\begin{equation}
A^{\theta}=g^{-1}(\theta,x)(A+d)g(\theta,x)\;,
\end{equation}
where $A$ is a reference connection chosen in such a way that
the associated Dirac operator has no zero modes.
Defining the corresponding set of operators $\hat{D}(A^{\theta})$ and
using the fact that $|det\,\hat{D}(A^{\theta})|$ is invariant, we may
write
\begin{equation}
det\,\hat{D}(A^{\theta})=\left[det\,i\Dsl(A)\right]^{\frac{1}{2}}
e^{iw(A,\theta)},
\end{equation}
i.e., the anomaly is in the phase of the determinant.
But we have identified the determinant of $\hat{D}$ with the exponential
of minus the effective action, so we have that, under an infinitesimal
variation of the parameter $\theta$, the effective action
behaves according to
\begin{equation}
\delta\Gamma_{eff}(A)=-i\frac{\partial w(A,\theta)}{\partial \theta}
\delta\theta\;,
\end{equation}
Thus we have related the gauge anomaly with the derivative of
$w(\theta,A)$. The exponential of this function defines a map
$\exp{[iw(\theta,A)]}:S^{1}\rightarrow S^{1}$, whose winding
number is given by
\begin{equation}
\frac{1}{2\pi}\int_{0}^{2\pi}d\theta\,\frac{\partial
w(\theta,A)}{\partial \theta}=m\;,
\end{equation}
and we can identify the anomaly with the winding number density.

This winding number density is obtained from the index theorem.
Define the two-parameter family of connections:
\begin{equation}
A^{t,\theta}=tA^{\theta}
\end{equation}
with $t\in [0,1]$. This family defines in the space
of all connections a disc ${\cal D}$ whose boundary is the circle
$\{A^{\theta}\}$.
Now, the determinant $det\,D(A^{t,\theta})$ may
vanish since $A^{t,\theta}$ and $A$ are no longer related by a gauge
transformation for $0\leq t <1$. By deforming the boundary
$S^{1}$ of the disc towards the interior the total winding number is
equal to the signed sum of the winding numbers associated with each of
the internal zeroes of the determinant. Moreover, it is possible to
show that these winding numbers can only be $\pm 1$ and coincide with
the chirality
of the eigenfunction of the Dirac operator whose eigenvalue vanishes at
that point \cite{LAG-Ginsparg2} (fig. \ref{fig-4}). These arguments
allow us to identify the winding
number with the index of the Dirac operator in $S^{2n}\times {\cal D}$
\begin{equation}
ind\,\Dsl_{2n+2}=\frac{1}{2\pi}\int_{0}^{2\pi}d\theta\,\frac{\partial
w(\theta,A)}{\partial \theta}\;.
\label{localdensity}
\end{equation}

\begin{figure}
\vspace{3cm}
\caption{The disc ${\cal D}$ which parametrizes the two-parameter family
of gauge fields $A^{t,\theta}$ with polar coordinates $t,\theta$}
\label{fig-4}
\end{figure}

Since we are interested in relating the anomaly with the topology of our
bundle we have to make use of the index theorem. Now our manifold
has a boundary and we cannot use the Atiyah-Singer index theorem
but its generalization to manifolds with boundary, the
Atiyah-Pattodi-Singer index theorem \cite{Atiyah-Pattodi-Singer}.
However
we can proceed in a different way. We construct a $G$-bundle over
$S^{2}\times S^{2n}$ in the following way. Divide $S^{2}$ into
two parts corresponding to the two hemispheres $S^{2}_{+}$ and
$S^{2}_{-}$. In the upper hemisphere, with coordinates $(t,\theta)$, we
take the gauge field to be
\begin{equation}
{\cal A}(x,t,\theta)=A^{t,\theta}+tg^{-1}d_{\theta}g
\end{equation}
while in the lower hemisphere, now with coordinates $(s,\theta)$, the
gauge field will be simply ${\cal A}(x,s,\theta)=A(x)$. By $d_{\theta}$
we denote the exterior differentiation with respect to $\theta$, i.e.
$d_{\theta}=d\theta \partial/\partial \theta$. At the equator
$s=t=1$ both connections are related by a gauge transformation as
required. The Atiyah-Singer index theorem is thus applied to
$S^{2}\times S^{2n}$
\begin{eqnarray}
ind[i\Dsl_{2n+1}({\cal A})]&=&
\int_{S^{2}\times S^{2n}} [ch({\cal
F})]_{vol}  \nonumber \\
&=&\frac{1}{(n+1)!}\left(\frac{i}{2\pi}\right)^{n+1}
\int_{S^{2}\times S^{2n}}tr\,({\cal F}^{n+1})\;,
\end{eqnarray}
where ${\cal F}$ is the curvature associated with ${\cal A}$
\begin{equation}
{\cal F}=(d+d_{\theta}+d_{t}){\cal A}+{\cal A}^{2}\;,
\end{equation}
and the definition of $d_{t}$ is analogous to that of $d_{\theta}$.
Next we split this integral into two integrals over
$S_{\pm}^{2}\times S^{2n}$. In each one, from
(\ref{Chern-Simons}) setting $\omega_{0}=0$, we see that $tr\,{\cal
F}^{n+1}$
can be written as $dQ_{2n+1}({\cal A})$ with $Q_{2n+1}({\cal A})$ the
$2n+1$-Chern-Simons form. Then by applying Stokes theorem we can write
the index of $\Dsl_{2n+2}$ as the difference of two integrals over
$S^{1}\times S^{2n}$
\begin{eqnarray}
ind[i\Dsl_{2n+1}({\cal A})]&=&
\frac{1}{(n+1)!}\left(\frac{i}{2\pi}\right)^{n+1}
\left[\int_{S^{1}\times S^{2n}} \left.Q_{2n+1}({\cal A})\right|_{t=1}
\right. \nonumber \\
&-&\left. \int_{S^{1}\times S^{2n}} \left.Q_{2n+1}({\cal
A})\right|_{s=1}\right]\;.
\end{eqnarray}
To connect this expression with the anomaly we need the local
winding density $d_{\theta}w(\theta,A)$. First of all let us notice that
the second integral
does not contribute to the anomaly, because we need a
$d\theta$ component to saturate the integral over $S^{1}$. Then we can
write
\begin{equation}
ind[i\Dsl_{2n+1}({\cal A})]=
\frac{1}{(n+1)!}\left(\frac{i}{2\pi}\right)^{n+1}
\int_{S^{1}\times S^{2n}} Q_{2n+1}(A^{\theta}+\hat{v},F^{\theta})
\end{equation}
where $\hat{v}=g^{-1}d_{\theta}g$ and $F^{\theta}$ is the gauge
curvature associated with $A^{\theta}$. From here it is now
straightforward to obtain the local winding density as
\begin{equation}
id_{\theta}w(A,\theta)=\frac{i^{n+2}}{(2\pi)^{n}(n+1)!}\int_{S^{n}}
Q^{1}_{2n}(\hat{v},A^{\theta},F^{\theta})
\end{equation}
where $Q^{1}_{2n}(\hat{v},A^{\theta},F^{\theta})$ is the term of
$Q_{2n+1}(\hat{v},A^{\theta},F^{\theta})$ linear in $\hat{v}$.
This form of the anomaly verifies the Wess-Zumino consistency
conditions. We can also derive the descent equations of Stora and
Zumino \cite{Zumino83}. We consider the case where we have a
family of gauge transformations $g(\theta^{\alpha},x)$ parametrized by
a collection of angles $\theta^{\alpha}$. Defining
\begin{eqnarray}
\bar{A}&=&g^{-1}(A+d)g \nonumber \\
\hat{v}&=&g^{-1}\delta g\;,
\end{eqnarray}
with $\delta$ the differential with respect to the parameters
$\theta^{\alpha}$, it is possible to show that (see
for example \cite{LAG-Ginsparg1})
\begin{equation}
(d+\delta)Q_{2n+1}(\bar{A}+\hat{v},\bar{F})=dQ_{2n+1}(\bar{A},\bar{F})
\;,
\label{N-parameter}
\end{equation}
where $\bar{F}$ is the curvature associated with $\bar{A}$. Now we
expand the Chern-Simons form $Q_{2n+1}(\bar{A}+\hat{v},\bar{F})$ in
powers of $\hat{v}$
\begin{equation}
Q_{2n+1}(\bar{A}+\hat{v},\bar{F})=
Q_{2n+1}^{0}(\bar{A},\bar{F})+Q_{2n}^{1}(\hat{v},\bar{A},\bar{F})+\ldots
+Q_{0}^{2n+1}
\end{equation}
where the superscript of the $Q$'s indicates the number of powers of
$\hat{v}$. Then, by substitutying this expansion into
(\ref{N-parameter}) we get the Stora-Zumino descent equations
\begin{eqnarray}
\delta Q_{2n+1}^{0}+dQ_{2n}^{1}&=&0 \nonumber \\
\delta Q_{2n}^{1} + dQ_{2n-1}^{2} &=&0 \nonumber \\
&\vdots& \nonumber \\
\delta Q_{1}^{2n}+dQ_{0}^{2n+1} &=& 0 \nonumber \\
\delta Q_{0}^{2n+1}=0\;,
\end{eqnarray}
the second equality being equivalent to the Wess-Zumino consistency
condition.

We have applied index theorem to the computation of gauge anomalies.
There, instead of applying the Atiyah-Patodi-Singer
\cite{Atiyah-Pattodi-Singer} index theorem for manifold with boundary
we constructed a $G$-bundle over a closed
manifold and from it obtained the form of the anomaly. We would like to
find a similar procedure to compute gravitational and mixed anomalies.
To do that, we will introduce the Atiyah-Singer index theorem for
families of elliptic operators \cite{Atiyah-Singer-2}.

Let us consider a family of elliptic operators $D_{p}$ with $p$ lying in
some manifold $P$. The index of $D_{p}$ over the manifold $P$ is
constant, since it is invariant under deformations, as long as the
operator varies smoothly over $P$. Nonetheless this is not true for both
$dim\,ker\,D_{p}$ and $dim\,ker\,D_{p}^{\dagger}$ separately and then
$ker\,D_{p}$ and $ker\,D_{p}^{\dagger}$ are not necessarily well defined
vector bundles over the parameter space $P$. However, in the context
of the $K$-theory for $P$ \cite{Atiyah} we can have a well defined
vector bundle over $P$ by defining the vector bundle $Ind\,D_{p}$ as
\begin{equation}
Ind\,D_{p}=ker\,D_{p} \ominus ker\,D_{p}^{\dagger}
\end{equation}
where the ordinary index for $D_{p}$ is given by $ch_{0}(Ind\,D_{p})$.
Using the Atiyah-Singer index theorem for families of operators we
may write \cite{Alvarez-Singer-Zumino}
\begin{equation}
ch(Ind\,D)=\int_{M_{2n}}\hat{A}(Z)ch(V)
\label{A-Sfamily}
\end{equation}
where $Z$ is a fiber space with base $P$ and fiber $M_{2n}$, and $V$ is
the vector bundle. In the case in which we are dealing with pure
gravity, $V$ is of purely geometrical origin; for example for a
spin-$\frac{1}{2}$ fermion $V=0$ and for a Rarita-Schwinger field
$V=TM_{2n}$, the tangent bundle of $M_{2n}$. In the case of mixed
anomalies $V$ also contains the gauge bundle. In fact,
when we made the computation of gauge anomalies, we used a version of
the Atiyah-Singer index theorem for a family of operators with
$M_{2n}=S^{2n}$, $P=S^{2}$ and the vector bundle was constructed using
the gauge transformation $g(\theta,x)$.

When the theory we are interested in is in presence of a gravitational
field, we have to be sure that the quantum theory is invariant under
the symmetries of the classical theory, i.e., coordinate diffeomorphisms
and local Lorentz transformations. In principle, we then may expect two
kind of anomalies, Einstein anomalies and Lorentz anomalies. In the
first case the breakdown of diffeomorphism invariance is translated into
a non-conservation of the energy-momentum tensor
\begin{equation}
\langle\nabla_{\mu\nu}T^{\mu}\rangle\neq 0
\end{equation}
while in the second case, since the variation of the effective action
under a local Lorentz transformation $\delta
e^{a}_{\,\, \mu}=\alpha^{a}_{b}e^{b}_{\,\, \mu}$ is
\begin{equation}
\delta_{\alpha}\Gamma_{eff}=-\int dx\,\alpha^{a}_{b}e^{b}_{\,\mu}
\frac{\delta \Gamma_{eff}}{\delta e^{a}_{\,\,\mu}}=-\int
dx\,e\,\alpha^{ab}\langle T_{ab} \rangle
\end{equation}
we have that, being $\alpha_{(ab)}=0$, $\delta_{\alpha}\Gamma_{eff}\neq
0$ implies $\langle T_{[ab]} \rangle \neq 0$. However it can be shown
that both types of anomalies (Eintein and Lorentz)
\cite{Bardeen-Zumino,LAG-Ginsparg1} are related. In fact, by adding a
local term to the action, it is always possible to  switch from one
type of anomaly to the other; we usually prefer to think in terms of
Einstein anomalies.

After this remark we give the prescription for the computation of
gravitational anomalies. Following the same argument that
gave us the gauge anomaly, the local winding number density is obtained
from the characteristic polynomial (\ref{A-Sfamily}) for a suitable
$2n+2$-dimensional Dirac operator. The characteristic polynomial for the
fields that contribute to the anomaly are the following. For
spin-$\frac{1}{2}$ fermions
\begin{equation}
I_{1/2}=\prod_{i} \frac{x_{i}/2}{\sinh{(x_{i}/2)}}\;.
\end{equation}
For Rarita-Schwinger fermions
\begin{equation}
I_{3/2}=\prod_{i}\frac{x_{i}/2}{\sinh{(x_{i}/2)}}
\left[tr\left(e^{R/2\pi}-1\right)+4k-3\right]\;.
\end{equation}
And finally for the self-dual antisymmetric tensor
\begin{equation}
I_{s.d.}=-\frac{1}{8}\prod_{i}\frac{x_{i}}{\tanh{x_{i}}}\;.
\end{equation}
The minus sign in the last case indicates the bosonic nature of the
antisymmetric field. Here $x_{i}$ are the skew-eigenvalues of $R/2\pi$,
where now the indices $a$, $b$ in $R^{a}_{\,\,b}$ are those appropiated
for $2n$ dimensions but the form part of $R$ may contain components in
the extra dimensions. If we are interested in computing mixed anomalies
we have to multiply these polynomials by $ch({\cal F})$ where ${\cal F}$
is the 2n+2-dimensional gauge curvature.

\subsection{Examples: Green-Schwarz cancellation mechanism and Witten's
$SU(2)$ global anomaly}

As an application of all this machinery we present the
non-trivial cancellation of gravitational anomalies for some
supergravity theories \cite{LAG-Witten}. In $d=2$ dimensions the
relevant characteristic polynomials of order $4$ are
\begin{eqnarray}
I_{1/2}&=&-\frac{1}{24}p_{1} \nonumber \\
I_{3/2}&=&\,\,\frac{23}{24}p_{1} \nonumber \\
I_{s.d.}&=&-\frac{1}{24}p_{1}\;,
\end{eqnarray}
and there are many ways of cancelling the anomaly.
Let us note that $I_{1/2}=I_{s.d.}$ indicates the fact that a positive
chirality fermion is equivalent to a right-moving scalar because of
two-dimensional bosonization \cite{Coleman}. In $d=6$,
\begin{eqnarray}
I_{1/2}&=&\frac{1}{5760}(7p_{1}^{2}-4p_{2}) \\
I_{3/2}&=&\frac{1}{5760}(275p_{1}^{2}-980p_{2}) \\
I_{s.d.}&=&\frac{1}{5760}(16p_{1}^{2}-112p_{2})\;.
\end{eqnarray}
The minimal cancellation ocurs for
\begin{equation}
21I_{1/2}-I_{3/2}+8I_{s.d.}=0
\end{equation}
so we must have $21$ Weyl spin-$\frac{1}{2}$ fermions, $1$ gravitino
with the opposite helicity and $8$ self-dual antisymmetric tensors as a
multiplet free from gravitational anomalies. Finally in $d=10$ we have
\begin{eqnarray}
I_{1/2}&=&\frac{1}{967680}(-31p_{1}^{3}+44p_{1}p_{2}-16p_{3}) \\
I_{3/2}&=&\frac{1}{967680}(225p_{1}^{3}-1620p_{1}p_{2}+7920p_{3}) \\
I_{s.d.}&=&\frac{1}{967680}(-256p_{1}^{3}+1664p_{1}p_{2}-7936p_{3})
\end{eqnarray}
and the minimal solution is found to be \cite{LAG-Witten}
\begin{equation}
-I_{1/2}+I_{3/2}+I_{s.d.}=0
\end{equation}
Thus the gravitational anomaly cancels for a spin-$\frac{1}{2}$ Weyl
fermion, a gravitino field of opposite chirality, and a self-dual tensor
field. These fields appear in the $N=2$ chiral supergravity multiplet in
$d=10$ \cite{LAG-Witten}.

To conclude with the subject of local anomalies we present the
cancelation of anomalies for type-$I$ superstring found by Green and
Schwarz \cite{Green-Schwarz} (we follow the presentation in
\cite{Green-Schwarz-West}). The low energy field theory for type-$I$
superstrings is $N=1$ super Yang-Mills coupled to $N=1$ SUGRA in ten
dimensions. The $N=1$ SUGRA multipled contains the graviton
$e^{\;\;a}_{\mu}$, the left-handed Weyl-Majorana gravitino
$\psi^{\mu}$,
the right-handed Weyl-Majorana fermion $\lambda$, the antisymmetric
tensor $B_{\mu\nu}$ and the dilaton $\Phi$. The super Yang-Mills
multiplet has the ten-dimensional gluon $A_{\mu}^{a}$ and the
Weyl-Majorana gluino $\lambda^{a}$ in the adjoint representation of the
gauge group $G$. Shifting all gravitational anomalies to the
local-Lorentz transformations we get the $12$-form characterizing the
anomaly is:
\begin{eqnarray}
I_{12}&=&-\frac{1}{15}tr\,F^{6}+\frac{1}{24}tr\,F^{4}\,tr\,R^{2}
\nonumber \\
&-&\frac{1}{960}tr\,F^{2}\left[5(tr\,R^{2})^{2}+4tr\,R^{4}\right]+
\frac{N-496}{7560}tr\,R^{6} \\
&+&\left(\frac{N-496}{5760}+\frac{1}{8}\right)
tr\,R^{4}\,tr\,R^{2}+\left(\frac{N-496}{13824}+\frac{1}{32}\right)
(tr\,R^{2})^{3}\;, \nonumber
\end{eqnarray}
where the trace of the gauge curvature is in the adjoint representation
and $N$ is the dimension of the gauge group. From this polynomial we
obtain the complete set of anomalies for the low-energy field theory of
type-$I$ superstrings.

To describe the cancellation of anomalies discovered by Green and
Schwarz, we note, first of all, that if we consider the leading
terms with $tr\,F^{6}$
and $tr\,R^{6}$, the anomalies they originate cannot
be cancelled by the addition of local counterterms to the
effective action.
In order to cancel the anomaly coming from the leading terms the gauge
group is restricted to have dimension $N=496$ and such that in the
adjoint representation $tr\,F^{6}$ is not an independent Casimir. This
could be accomplished for example if
\begin{equation}
tr\,F^{6}=\alpha\,tr\,F^{2}\,tr\,F^{4}+\beta(tr\,F^{2})^{3}
\end{equation}
The crucial point is that all anomalies can be cancelled
provided $I_{12}$ factorizes according to
\begin{equation}
I_{12}=(tr\,R^{2}+k\,tr\,F^{2})I_{8}
\end{equation}
where $I_{8}$ is an
invariant $8$-form constructed in terms of $F$ and $R$.  It can be shown
that this factorization is possible if $k=-1/30$ and
\begin{equation}
tr\,F^{6}=\frac{1}{48}tr\,F^{2}\,tr\,F^{4}-
\frac{1}{14400}(tr\,F^{2})^{3} \end{equation} so $I_{8}$ is equal to
\begin{eqnarray}
I_{8}&=&\frac{1}{24}tr\,F^{4}-\frac{1}{7200}(tr\,F^{2})^{2}-\frac{1}{240}
tr\,F^{2}\,tr\,R^{2}\nonumber \\
&+&\frac{1}{8}tr\,R^{4}+\frac{1}{32}(tr\,R^{2})^{2}\;.
\end{eqnarray}
All these conditions (factorization and $N=496$) can be fulfilled if and
only if the gauge group is $SO(32)$, $E_{8}\times E_{8}$ or $E_{8}\times
U(1)^{248}$. The most important feature of Green and Schwarz result is
that by simply imposing the cancellation of both gauge and gravitational
anomalies we have narrowed down the possible gauge groups to just three
possible choices. Soon after the Green-Schwarz anomaly cancellation
appeared, the heterotic string \cite{Pricenton-quartet} was formulated.
Although we have heterotic strings for both $SO(32)$ and $E_{8}\times
E_{8}$, it is the latter which seems more promising in making contact
with the SM at low energies.

After checking that all local anomalies cancel we still have to make
sure
that the theory is not afflicted by global (gauge or gravitational)
anomalies. Global anomalies lead to a breakdown of
gauge (diffeomorphism) invariance under transformations that do not lie
in the connected component of the identity of the symmetry group.
The best known example was discovered by Witten \cite{WittenSU(2)} for
gauge theories with an odd number of left-handed $SU(2)$ doublets. We
briefly review this case.

The relevant mathematical requirement for the existence of Witten's
anomaly is
the fact that the fourth homotopy group of the gauge group is non
trivial. This only happens for $Sp(n)$, $\pi_{4}[Sp(n)]={\bf Z}_{2}$. In
particular $Sp(1)=SU(2)$, and we have
\begin{equation}
\pi_{4}[SU(2)]={\bf Z}_{2}
\end{equation}
Impose boundary conditions to  effectively compactify space-time to
$S^{4}$. The gauge trasformations $U(x)$ are mappings
$U:S^{4}\rightarrow
SU(2)$ with are classified by $\pi_{4}[SU(2)]$. Since this homotopy
group is non-trivial we have transformations that cannot be continously
deformed to the identity. To make the discussion concrete imagine that
we have one $SU(2)$ left-handed doublet. Since for a theory with a
doublet of Dirac fermions the path integral
is given by
\begin{equation}
\int {\cal D}\psi {\cal D}\bar{\psi} e^{-\int d^{4}x\,\bar{\psi}i{\not
D} \psi}=det\,(i\Dsl)\;.
\end{equation}
In our case the path integral will be equal to the square root
of the determinant of $\Dsl$
\begin{equation}
\int ({\cal D}\psi {\cal D}\bar{\psi})_{Weyl}  e^{-\int
d^{4}x\,\bar{\psi}i{\not D}\psi}=[det\,(i\Dsl)]^{\frac{1}{2}}\;.
\end{equation}
The problem is then to assign a definite sign to the square root.
Since we are only interested in relative signs we may, for
example, choose the positive branch of the square root. However, for
this choice to be consistent we must have that the sign is preserved by
gauge transformations; this is evidently true for infinitesimal gauge
transformations, but for the topologically non-trivial ones,
\begin{equation}
[det\,i\Dsl(A_{\mu})]^{\frac{1}{2}}=-
[det\,i\Dsl(A_{\mu}^{U})]^{\frac{1}{2}}\;.
\label{Sign-change}
\end{equation}
This spells trouble, because $A_{\mu}$ and $A_{\mu}^{U}$
can be continously connected, and we have no consistent way of excluding
$A_{\mu}^{U}$ from the path integral. It follows that
the partition function is zero because the contribution
coming from $A_{\mu}$ is exactly cancelled by that from $A_{\mu}^{U}$.

We may understand (\ref{Sign-change}) in terms of eigenvalue flows.
We know that the number of positive and negative non-zero modes of
$i\Dsl$ are equal. Then we may define $[det\,i\Dsl]^{\frac{1}{2}}$
as the regularized product of the positives eigenvalues of the Dirac
operator
\begin{equation}
[det\,i\Dsl]^{\frac{1}{2}}=\prod_{\lambda_{n}>0}\lambda_{n}\;.
\label{det}
\end{equation}
Since we can continuously interpolate between $A_{\mu}$ and
$A_{\mu}^{U}$, consider the family of
connections $A_{\mu}^{t}=tA_{\mu}^{U}+(1-t)A_{\mu}$ with $t\in [0,1]$.
We can study the flow of the eigenvalues of $i\Dsl(A_{\mu}^{t})$ as a
function of $t$. The condition implying that the determinant changes
sign in passing from $A_{\mu}$ to $A_{\mu}^{U}$ is that the number of
eigenvalues $\lambda_{n}(t)$ passing from positive to negative
values is odd as we go from $t=0$ to $t=1$. This result
follows from a slightly exotic version
of the Atiyah-Singer index theorem, called the mod two index theorem, to
a certain five-dimensional Dirac operator. The five-dimensional Dirac
operator of interest is
\begin{equation}
{\cal D}=i\bar{\Gamma}\frac{\partial}{\partial t}+\Dsl[A(t)]\;.
\end{equation}
The interpolating parameter between $A_{\mu}(x)$ and $A_{\mu}(x)^{U}$
plays the r\^{o}le of the fifth coordinate. The operator ${\cal D}$ is
real
and antisymmetric, hence its non-zero eigenvalues are purely imaginary
and come in complex conjugate pairs, and the number of zero modes is
well defined modulo two. In our case the five-dimensional space is
$S^{4}\times{\bf R}$, $A_{\tau}=0$, and
$A_{\mu}(x,\tau=-\infty)=A_{\mu}(x)$,
$A_{\mu}(x,\tau=+\infty)=A_{\mu}^{U}(x)$. An application of the mod $2$
index theorem led Witten to conclude that for this configuration the
index is one, this means that as we interpolate between  $A_{\mu}$ and
$A_{\mu}^{U}$ there is a level crossing of the four-dimensional Dirac
operator. From (\ref{det}) it is the clear that we cannot assign a
definite sign to the square root and we cannot implement gauge
invariance. Thus $SU(2)$ gauge theory with an odd number of fermion
doublets is afflicted by a global anomaly and the physical Hilbert space
is empty.




\section{String Theory I. Bosonic String}

The lesson to be extracted from our journey in chapter one through the
various proposals of how to quantize gravity is that all of them are
afflicted by problems. The situation is somewhat remminiscent of Fermi's
theory of the weak interactions. It was able to explain a variety of low
energy phenomena, but it failed to be renormalizable and it also
contained unitarity violations at high energy. The cure of its problems
came with the SM, based on completely different dynamical principles. It
seems unreasonable to treat Einstein's Theory of Relativity to be a
fundamental theory all the way from cosmological scales down to the
Planck length. There are more than sixty orders of magnitude in between.
It seems more appropiate (specially in view of the difficulties
described) to think of GR as an effective theory, and we should strive
to identify the appropiate framework capable of encompassing a quantum
theory of space-time which in the long distance limit agrees with
Einstein's theory. It is clear from previous discussions that such a
theory will provide profound insights into many of the riddles of the
SM, like the origin of mass, symmetry breaking and the chiral nature of
quarks and leptons. We have seen in our brief study of anomalies that as
soon as the space-time dimension goes beyond four, the constraints
imposed by the existence of a rich chiral structure together with
reasonable gauge interactions are extremely restricted, and we were led
for instance in $d=10$ to the low energy spectrum of the heterotic
string \cite{Pricenton-quartet}.

So far the only candidate which incorporates gravity with other
interactions at high energies is String Theory. There are still many
puzzles and difficulties to be resolved in String Theory, but one should
also mention that many of the problems encountered in the treatment of
Quantum Gravity in terms of local field theories disappear in this
context. There are strong arguments (and explicit computations) showing
the ultraviolet finiteness of superstring theories. There are quite
interesting results (duality, finite temperature properties; see below)
which indicate that String Theory defined on spaces of different sizes
(an even different topologies) are completely equivalent. For instance a
string propagating on a circle of radius $R$ is equivalent to one
propagating on a circle of radius $1/R$ (in string units, with the Regge
slope set equal to one).

String Theory contain a fundamental length, and still have
local interactions. This is something which cannot be achieved in terms
of local field theories. We also know (see below) from the heterotic
string that one can in principle accomodate in this framework the
quantum numbers and interactions of the SM. Our knowledge of string
theory is unfortunately too rudimentary to have a glimpse on its
physical basis. However, the results obtained so far make us optimistic.
It is likely that many of the new ingredients brought in by String
Theory will appear in the correct theory of Quantum Gravity.

We begin in this chapter our exploration of the formulation and
properties of String Theory. We will focus on general properties and on
the formalism which has been developed in order to compute string
amplitudes to any order of perturbation theory. Along the way we will
encounter how the Einstein equations appear in String Theory, we will
study the simplest form of duality, and we will also describe the
geometrical interpretation of string infinities and physical states. Our
study will center on the bosonic string for the time being. This makes
the discussion simpler. In the next chapter we will briefly review the
properties of fermionic strings.

\subsection{Bosonic String}
We review the basic features of the bosonic string
\cite{Green-Schwarz-Witten,Kaku,Lust-Theisen}. We start with the
case of a free relativistic
point particle in Minkowski space. The action for such a system is
constructed by taking the simplest Lorentz invariant we can
associate
with the particle's trajectory, that is, the length of its world-line.
Then if the particle moves in Minkowski space along the world-line
$x^{\mu}(s)$ with $s$ being the proper time, the action functional is
given by \begin{equation}
S[x^{\mu}(s)]=-m\int_{a}^{b}
ds\,\sqrt{\eta_{\mu\nu}\dot{x}^{\mu}(s)\dot{x}^{\nu}(s)}\;.
\label{particle}
\end{equation}
When dealing with strings we have a one-dimensional
object, whose points we parametrize by a coordinate $\sigma$
running from $0$ to $\pi$, in a $d$-dimensional Minkowski space
(the target space). Its time evolution is
represented by a two-dimensional surface, the world-sheet of the
string. This two-dimensional surface is parametrized
by two coordinates $(\sigma,\tau)$, the first one being that introduced
to label the points on the string and $\tau$ playing the role of
the proper time in the case of the point particle. The embedding of the
world-sheet in Minkowski space is described by $d$ functions
$X^{\mu}(\sigma,\tau)$ which form a vector under Lorentz
transformations in the target space.

Now we construct the action functional for the bosonic string
by similarity with the case of the
relativistic particle: we will the action to be proportional to the area
of the world-sheet swept out by the string. The metric induced on the
world-sheet by that of the target space is
\begin{equation}
\Gamma_{ab}=\eta_{\mu\nu}\frac{\partial X^{\mu}}{\partial \sigma^{a}}
\frac{\partial X^{\nu}}{\partial \sigma^{b}}
\end{equation}
where $a=0,1$ and $(\sigma^{0},\sigma^{1})=(\tau,\sigma)$, and
the induced area equals the integral of the
square root of minus the determinant of $\Gamma_{ab}$. The action
functional for the strings is taken to be
\begin{equation}
S[X^{\mu}]=T\int d^{2}\sigma
\sqrt{-det\left[\eta_{\mu\nu}\frac{\partial
X^{\mu}}{\partial \sigma^{a}}\frac{\partial X^{\nu}}{\partial
\sigma^{b}}\right]}
\end{equation}
where $T$ is a constant, the string tension. It is the
mass or energy per unit length.
This form of the action was first proposed by Y. Nambu and T. Goto
\cite{Nambu-Goto} in the early seventies and it is known as the
Nambu-Goto action. By using the expression of the Minkowski metric
and the definitions
\begin{equation}
\dot{X}^{\mu}=\frac{\partial X^{\mu}}{\partial \tau} \hspace{2cm}
X^{'\mu}=\frac{\partial X^{\mu}}{\partial \sigma}
\end{equation}
we can rewrite Nambu-Goto action as
\begin{equation}
S[X^{\mu}(\tau,\sigma)]=\int d\tau d\sigma \sqrt{\dot{X}^{2} X^{'2}-
(\dot{X}\cdot X^{'})^{2}}\;.
\label{Nambu-Goto}
\end{equation}

Although (\ref{Nambu-Goto}) has a very simple
physical interpretation its form is not very
pleasant specially because of the presence of the square root. We would
like to find another action classically equivalent to
(\ref{Nambu-Goto}) but without its unpleasant features. The idea of
using auxiliary fields to simplify the action was introduced by Brink,
DiVechia and Howe \cite{Brink} and by Deser and Zumino
\cite{Deser-Zumino}. Now, besides the embeddings $X^{\mu}(\tau,\sigma)$
we introduce a new independent field
$g_{ab}(\tau,\sigma)$ which is the intrinsic metric on the world-sheet.
The simplified action is
\begin{equation}
S_{P}[X^{\mu},g_{ab}]=-\frac{T}{2}\int
d^{2}\sigma\sqrt{-g}\,g^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}
\eta_{\mu\nu}
\label{Polyakov}
\end{equation}
This form of the action was used by Polyakov \cite{Polyakov} to
formulate String Theory in terms of path integrals.

It is worth noticing that there is a counterpart of the
action (\ref{Polyakov}) in the case of the relativistic particle. We can
introduce an
auxiliary field $e(s)$, which plays the role of the intrinsic metric on
the world-line, and write the action
\begin{equation}
S=\frac{1}{2}\int ds\left(\frac{\dot{x}^{2}(s)}{e(s)}-m^{2}e(s)\right)
\end{equation}
which is classically equivalent to (\ref{particle}). This action has the
supplementary advantage that it can be applied to the massless case in
which (\ref{particle}) is not defined.

The string tension $T$ can be rewritten in a more useful way by
defining a new constant $\alpha^{'}$
(which was called the Regge
slope in the times of the dual models of hadronic resonances
\cite{GVeneziano}) defined by
\begin{equation}
T=\frac{1}{2\pi\alpha^{'}}
\end{equation}
Neither $T$ nor $\alpha^{'}$ affect in any way the classical string
solutions, since they are overall factor in the action.
In the quantum
theory, in a path integral approach, each configuration is weighted by
\begin{equation}
\exp{[i\frac{T}{\hbar}S^{'}]}\;,
\end{equation}
where $S^{'}$ is the action divided by $T$.
Then from $\alpha^{'}$ and $\hbar$ we can construct a constant with
dimensions of length as (in units in which $c=1$)
\begin{equation}
\lambda_{S}=\sqrt{\alpha^{'}\hbar}\;,
\end{equation}
which is the fundamental length in String Theory. This, together
with the vacuum expectation value of the dilaton (which give us the
coupling constant of the string) determines all couplings in String
Theory.

The action (\ref{Polyakov}) has a number of symmetries.
First of all it is invariant under world-sheet
diffeomorphism $\sigma^{a}\rightarrow
\sigma^{a}+\xi^{a}$
\begin{eqnarray}
\delta X^{\mu}&=&\xi^{a}\partial_{a}X^{\mu} \nonumber \\
\delta g_{ab}&=&\nabla_{a}\xi_{b}+\nabla_{b}\xi_{a}\;.
\end{eqnarray}
Secondly we have invariance under Weyl rescalings of the metric
\begin{eqnarray}
\delta g_{ab}&=&\delta\Omega(\sigma)g_{ab} \nonumber \\
\delta X^{\mu}&=&0 \;.
\end{eqnarray}
Finally the action is invariant under Poincar\'e transformations
of the target space (we are taking the target space to be Minkowski
space)
\begin{eqnarray}
\delta X^{\mu}&=&a^{\mu}_{\,\nu}X^{\nu}+b^{\mu} \hspace{1cm}
a_{(\mu\nu)}=0 \nonumber \\
\delta g_{ab}&=&0 \;.
\end{eqnarray}
Invariance under Weyl transformations
implies that the two-dimensional classical field theory described by
(\ref{Polyakov}) is a conformal field theory. As we will see the
preservation of this property after quantization lead to severe
constraints on the theory.

The classical equations of motion for (\ref{Polyakov}) give the free
field equations for $X^{\mu}$
\begin{equation}
\nabla_{\sigma}\nabla^{\sigma} X^{\mu}=0\;,
\end{equation}
together with the equations of motion for $g_{ab}$. The two-dimensional
energy-momentum tensor $T_{ab}$ is given by:
\begin{equation}
T_{ab}=-\frac{2}{T}\frac{1}{\sqrt{g}}\frac{\delta S}{\delta g^{ab}}
\end{equation}
and the variation of (\ref{Polyakov}) with respect to $g_{ab}$ yields:
\begin{equation}
T_{ab}=\partial_{a}X\cdot\partial_{b}X-
\frac{1}{2}g_{ab}g^{cd}\partial_{c}X\cdot\partial_{d}X=0\;.
\label{constrainT}
\end{equation}
The energy-momentum tensor of the two-dimensional field theory
described by the action (\ref{Polyakov}) vanishes identically.
As a subproduct of these equation we have that the energy-momentum
tensor is traceless; this is a consequence of the invariance of the
action under Weyl rescalings of the metric.
{}From these two equations it is easy to see that, whenever the
constraint on the energy-momentum tensor (\ref{constrainT})
is satisfied, (\ref{Polyakov}) and the Nambu-Goto action are classically
equivalent; this does not necessarily imply the equivalence
of both theories at the quantum level.

The invariances of the action implies that we
have a lot of freedom in choosing a gauge for the
intrinsic metric $g_{ab}$. For example, by making a suitable choice of
local coordinates, we can write (locally) the metric in the form
\begin{equation}
g_{ab}=e^{\phi(\sigma)}\eta_{ab}\;,
\end{equation}
with $\eta_{ab}$ the flat Minkowski metric on the world-sheet. This is
the conformal gauge.

We now analyze a free propagating closed string.
First we impose the periodicity condition of the
bosonic coordinates $X^{\mu}(\sigma,\tau)$
\begin{equation}
X^{\mu}(\tau,\sigma+\pi)=X^{\mu}(\tau,\sigma)\;.
\end{equation}
The world-sheet is a cylinder $S^{1}\times {\bf R}$ parametrized by
the coordinates $0\leq\sigma< \pi$ and $-\infty<\tau<\infty$.
By choosing the conformal gauge and using light-cone coordinates in the
world-sheet $\sigma^{\pm}=\tau\pm\sigma$ we can write the action as
\begin{equation}
S=-\frac{T}{2}\int d^{2}\sigma \partial_{+}X^{\mu}\partial_{-}X^{\nu}
\eta_{\mu\nu}\;.
\end{equation}
If we make the Wick rotation $\tau \rightarrow -i\tau$, we can define
the complex coordinates
\begin{eqnarray}
w&=&2(\tau-i\sigma)\;, \nonumber \\
\bar{w}&=&2(\tau+i\sigma)\;,
\end{eqnarray}
It is possible now
to transform the cylinder into the punctured complex
plane ${\bf C}^{*}={{\bf C}}-\{ 0\}$ by the conformal transformation
\begin{equation}
z=e^{w}
\end{equation}
The field equations for $X^{\mu}$ are
\begin{equation}
\partial_{z}\partial_{\bar{z}}X^{\mu}(z,\bar{z})=0
\label{embedding}
\end{equation}
and
\begin{eqnarray}
\partial_{z}X\cdot \partial_{z}X=0 \nonumber \\
\partial_{\bar{z}}X\cdot \partial_{\bar{z}}X=0
\end{eqnarray}
for the constraints.
Equation (\ref{embedding}) can be solved by noting that the most
general solution is a sum of a holomorphic and an antiholomorphic
part
\begin{equation}
X^{\mu}(z,\bar{z})=X^{\mu}(z)+X^{\mu}(\bar{z})\;,
\end{equation}
Therefore the solution to (\ref{embedding}) can be expressed as a
Laurent series:
\begin{equation}
X^{\mu}(z,\bar{z})=q^{\mu}-\frac{i}{4}p^{\mu}\log{|z|^{2}}+\frac{i}{2}
\sum_{n\neq 0}
\frac{\alpha_{n}^{\mu}}{n}z^{-n}+\frac{i}{2}\sum_{n\neq
0}\frac{\bar{\alpha}_{n}^{\mu}}{n}\bar{z}^{-n}\;.
\label{expansion}
\end{equation}

In general there are three possible approaches to the quantization of
the string.

a) Light-cone gauge. We take in the target space a light-cone frame,
i.e., coordinates $(X^{\pm},X^{i})$ with $X^{\pm}=X^{0}\pm
X^{d-1}$ and $i=1,\ldots,d-2$. Next
we use the freedom we still have in the
conformal gauge to set (see \cite{Green-Schwarz-Witten} for details)
\begin{equation}
X^{+}=q^{+}+p^{+}\tau
\end{equation}
In this gauge it is possible to show that the constrains can be
explicitly solved and that by substituying them into the solution of
(\ref{embedding}) we eliminate all the unphysical degrees of
freedom so the theory depends only on the transverse modes.
However, it is necessary to test that the quantum theory preserves the
Lorentz invariance of the classical action. To verify this we check
whether the conserved charges $\{J^{\mu\nu}\}$ associated with the
Lorentz invariance satisfy the Lorentz algebra. For example
classically we
would have $[J^{i-},J^{j-}]=0$. In the quantum theory on the other
hand, the result is \cite{Green-Schwarz-Witten}
\begin{equation}
[J^{i-},J^{j-}]=-\frac{1}{(p^{+})^{2}}\sum_{m=1}^{\infty}\Delta_{m}
(\alpha^{i}_{-m}\alpha^{j}_{m}-\alpha^{j}_{-m}\alpha^{i}_{m})
\end{equation}
where the coefficient $\Delta_{m}$ is given by
\begin{equation}
\Delta_{m}=m\left(\frac{26-d}{12}\right)+\frac{1}{m}\left(\frac{26-d}{12}
+2(1-b)\right)
\end{equation}
and $b$ is a normal ordering constant that appears because of the
ambiguity in the order of the operators (see also below). We see that
Lorentz invariance is only recovered if we take $d=26$ and $b=1$.
Hence this procedure is only consistent in the critical dimension
$d=26$.

b)Old covariant approach. Consider the components of the
energy-momentun tensor in the complex coordinate basis $(z,\bar{z})$.
Energy-momentum conservation and tracelessness imply,
\begin{equation}
\partial_{z}T_{\bar{z}\bar{z}}=\partial_{\bar{z}}T_{zz}=0\;.
\end{equation}
We can Laurent expand around $z=\bar{z}=0$
\begin{eqnarray}
T_{zz}(z)=\sum_{n\in {\bf Z}}L_{n}z^{-n-2}\;, \nonumber \\
T_{\bar{z}\bar{z}}(\bar{z})=\sum_{n\in {\bf Z}}\bar{L}_{n}z^{-n-2}\;,
\end{eqnarray}
where $L_{n},\bar{L}_{n}$ are called the Virasoro generators and
they satisfy $L_{n}^{*}=L_{-n}$ (and the same for barred
quantities). Using the expansion
(\ref{expansion}) and the canonical
Poisson brackets of the classical theory,
we obtain the algebra of the Virasoro generators
\begin{eqnarray}
\{L_{n},L_{m}\}&=&-i(n-m)L_{n+m}\;, \nonumber \\
\{\bar{L}_{n},\bar{L}_{m}\}&=&-i(n-m)\bar{L}_{n+m}\;, \nonumber \\
\{L_{n},\bar{L}_{m}\}&=&0\;.
\end{eqnarray}
After quantization, we obtain that the previous algebra is not
exactly recovered as the algebra of the operators
$L_{n},\bar{L}_{m}$. Instead we get a central term in the
commutation relations
\begin{eqnarray}
[L_{m},L_{n}]&=&(m-n)L_{m+n}+\frac{d}{12}m(m^{2}-1)
\nonumber \\
{[}\bar{L}_{m},\bar{L}_{n}]&=&(m-n)\bar{L}_{m+n}+
\frac{d}{12}m(m^{2}-1)
\nonumber \\
{[}L_{m},\bar{L}_{n}]&=&0
\end{eqnarray}

In general, whenever
we have a classical system with a set of first class constrains
$\{\phi_{i}\}$, physical states are defined as those that are
annihilated by the operators $\{\hat{\phi}_{i}\}$.
When we try to quantize the bosonic
string this procedure is not available, since the gravitational
anomaly in $2$ dimensions (see above) makes its impossible to impose
$\hat{T}_{ab}|phys\rangle=0$ and the best we can do is
to use Gupta-Bleuler method \cite{Itzykson-Zuber}. It is possible to
show
that the presence of the gravitational anomaly in two dimensions and the
existence of a central term in the Virasoro algebra are equivalent.

Since we cannot impose that physical states are annihilated by all the
Virasoro generators, we proceed in a way similar to the
Gupta-Bleuler quantization of the electromagnetic field.
The physical states are annihilated by the positive
frequency part of the energy-momentum tensor, namely
\begin{eqnarray}
L_{n}|phys\rangle&=&\bar{L}_{n}|phys\rangle=0\;, \hspace{1cm} n>0
\nonumber \\
(L_{0}-b)|phys\rangle&=&(\bar{L}_{0}-b)|phys\rangle=0\;,
\end{eqnarray}
where $b$ is a normal ordering constant that we have to include because
of the ambiguity in the order of the operators $a_{n}$, $\bar{a}_{n}$ in
the definition of $L_{0}$ and $\bar{L}_{0}$. As in the Gupta-Bleuler
method, we have to ensure that the spectrum of the theory
has no ghost states, no physical states $|\chi\rangle$
with negative norm $\langle \chi|\chi \rangle<0$. Brower, Goddard
and Thorn \cite{Brower-Goddard-Thorn} proved a no-ghost theorem in which
they showed that there are ghost states as long as $d>26$. However, if
$d=26$ and $b=1$ or $d<26$ and $b<1$ the spectrum is always ghost-free.
This result implies an upper bound on the dimensionality of the
target space.

c)Modern covariant quantization. In this approach we begin with
the path integral \cite{Polyakov,KPZ}
\begin{equation}
Z=\int{\cal D}g{\cal D}X \exp{\left[\frac{1}{4\pi\alpha^{'}}\int
d^{2}\sigma\sqrt{-g}\,g^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}
\eta_{\mu\nu}\right]}\;.
\end{equation}
In order to perform the path integration without overcounting it is
necessary to make a gauge fixing. For example we may use the conformal
gauge
\begin{equation}
g_{ab}=e^{\phi(\sigma)}\hat{g}_{ab}
\label{conformal-factor}
\end{equation}
where $\hat{g}_{ab}$ is a fiducial metric. Since the action is also
invariant under diffeomorphisms we have to integrate not over all
$\hat{g}_{ab}$ but only over those that are not related by
a world-sheet diffeomorphism. In fact
it can be shown that the gauge slice we have to choose is a
finite-dimensional space \cite{D'Hoker-Phong}, the moduli space of
Riemann surfaces.

The gauge fixing with respect to the
group of diffeomorphisms is made using Fadeev-Popov ghosts.
After factoring out the volume of the group of diffeomorphisms we are
left with:
\begin{equation}
Z=\int d\mu\,{\cal D}_{e^{\phi}\hat{g}}X {\cal D}_{e^{\phi}\hat{g}}\phi
{\cal D}_{e^{\phi}\hat{g}} b {\cal D}_{e^{\phi}\hat{g}}c\,
\exp{\left\{-S_{P}[X,\hat{g}]-S_{gh}[b,c,\hat{g}]\right\}}
\end{equation}
where ${\cal D}_{e^{\phi}\hat{g}}$ indicates that the integration
measure is evaluated for the original metric $g_{ab}$, $d\mu$ is the
measure in
the space of fiducial metrics (moduli space), and $S_{P}$ and $S_{gh}$
are respectively the string action and the action for the ghost fields
\begin{equation}
S_{gh}[b,c,\hat{g}]=\frac{i}{\pi\alpha^{'}}\int
d^{2}\sigma\sqrt{\hat{g}} \hat{g}^{ab} b_{bc}\nabla_{a}c^{c}\;.
\end{equation}
We would like to write the integration measures with respect to the
fiducial metric, in order to make explicit the dependence in the field
$\phi(\sigma)$. This can be done for the $X^{\mu}$ fields
\cite{Alvarez-Gaume-91}
\begin{equation}
{\cal D}_{e^{\phi} \hat{g}}X=e^{(d/48\pi)S_{L}[\phi,\hat{g}]}
{\cal D}_{\hat{g}}X\;,
\end{equation}
and for the ghost fields
\begin{equation}
{\cal D}_{e^{\phi} \hat{g}}b{\cal D}_{e^{\phi}
\hat{g}}c=e^{(-26/48\pi)S_{L}[\phi,\hat{g}]}
{\cal D}_{\hat{g}}b {\cal D}_{\hat{g}}c\;,
\end{equation}
where $S_{L}[\phi,\hat{g}]$ is the Liouville action for the conformal
factor
\begin{equation}
S_{L}[\phi,\hat{g}]=\int d^{2}\sigma
\sqrt{\hat{g}}\left(\frac{1}{2}\hat{g}^{ab}\partial_{a}\phi
\partial_{b}\phi+R\phi+\mu e^{\phi}\right)\;.
\end{equation}
We finally arrive at
\begin{eqnarray}
Z &=&
\int d\mu\,{\cal D}_{e^{\phi}\hat{g}}\phi {\cal D}_{\hat{g}}X  {\cal
D}_{\hat{g}} b {\cal D}_{\hat{g}}c  \nonumber \\
&\times &\exp{\left\{-S_{P}[X,\hat{g}]-
S_{gh}[b,c,\hat{g}]-
\frac{26-d}{48\pi}S_{L}[\phi,\hat{g}]\right\}}
\nonumber
\end{eqnarray}
When $d\neq 26$ the conformal mode of the metric (the Liouville field)
becomes a dynamical field. Hence the realization of the conformal
symmetry is very different depending on whether $d=26$ or $d\neq 26$.
Non-critical strings are those with $d\neq 26$, and the Liouville field
is necessary in order to satisfy the requirement of conformal
invariance. When $d=26$, the Liouville field decouples from the action
and we can absorbe the integration over $\phi$ in the normalization of
the path integral. This is the case for critical ($d=26$) strings. The
only remainder of the integration over metrics in this case is the
integration over moduli $d\mu$:
\begin{equation}
Z=\int d\mu\,{\cal D}_{\hat{g}}X  {\cal
D}_{\hat{g}} b {\cal D}_{\hat{g}}c
\,\exp{\left\{-S_{P}[X,\hat{g}]-S_{gh}[b,c,\hat{g}]\right\}}
\end{equation}
Under some plausible assumptions we can extract the dependence of the
measure on the conformal factor:
\begin{eqnarray}
Z&=& \int {\cal D}_{\hat{g}}\phi {\cal D}_{\hat{g}}X  {\cal
D}_{\hat{g}} b {\cal D}_{\hat{g}}c   \\
&\times &\exp{\left\{-S_{P}[X,\hat{g}]-
S_{gh}[b,c,\hat{g}]-
\frac{25-d}{48\pi}S_{L}^{'}[\phi,\hat{g}]\right\}}
\nonumber
\end{eqnarray}
where $S_{L}^{'}[\phi,\hat{g}]$ is given by
\begin{equation}
S_{L}^{'}[\phi,\hat{g}]=\int d^{2}\sigma \sqrt{\hat{g}}\left(
\frac{1}{2}\hat{g}^{ab}\partial_{a}\phi\partial_{b}\phi+ R\phi+
\mu e^{\gamma \phi}\right)
\end{equation}
with
\begin{equation}
\gamma=\frac{25-d-\sqrt{(25-d)(1-d)}}{12}
\end{equation}
This expression was obtained first by David and Distler and Kawai
\cite{DDK} by imposing a general ansatz for the jacobian and requiring
the resulting theory to be well defined. It is clear that the results
obtained  look reasonable only for $d<1$. For $d>1$ the na\"{\i}ve
arguments
in \cite{DDK} break down. For $d\geq 26$ the formulae may still hold as
long as we think of the Liouville field as a time variable.

In the critical dimension we are left with a two-dimensional
quantum field theory of $d$ scalar fields and the reparametrization
ghosts. The system of {\it X-fields + ghosts}
preserves conformal invariance at the quantum level. This can be seen by
constructing the Virasoro generators for the ghost fields
which satisfy the algebra
\begin{equation}
[L_{m}^{gh},L_{n}^{gh}]=(m-n)L_{m+n}^{gh}+\frac{1}{6}m(1-13m^{2})
\delta_{m+n,0}
\end{equation}
and correspondingly for $\bar{L}_{m}^{gh}$. Considering now the total
Virasoro generator (matter+ghosts)
\begin{eqnarray}
L_{m}^{tot}&=&L_{m}+L_{m}^{gh}-b\delta_{m,0} \nonumber \\
\bar{L}_{m}^{tot}&=&\bar{L}_{m}+\bar{L}_{m}^{gh}-b\delta_{m,0}
\end{eqnarray}
we find that they satisfy the classical Virasoro algebra without central
term
\begin{eqnarray}
[L_{m}^{tot},L_{n}^{tot}]&=&(m-n)L_{m+n}^{tot} \nonumber \\
{[}\bar{L}_{m}^{tot},\bar{L}_{n}^{tot}]&=&(m-n)\bar{L}_{m+n}^{tot}
\end{eqnarray}
so our quantum theory is invariant under the two-dimensional conformal
group.

The handling of ghosts fields makes BRST formalism
\cite{Green-Schwarz-Witten}
specially well suited. We define the BRST operator from the matter and
ghost fields satisfying $Q^{2}=0$.
Then physical states are defined as those in the cohomolgy of
the $Q$ operator with minimum ghost number.

The quantum field theory on the
world-sheet describing the embedding in the target space has the special
property of having a traceless
energy-momentum tensor; in other words, it is a two-dimensional
conformal field theory. Up to now we have worked with a flat target
space metric. We could instead start
with a general target space with metric $G_{\mu\nu}$. Since to
construct a string theory we need a two-dimensional conformal field
theory, we demand that the $\beta$-function
$\beta_{\mu\nu}(G)$ associated with the quantum field
theory on the world-sheet vanishes (as it is required for conformal
invariance). Then we would obtain constrains on the background
metric in the form $\beta_{\mu\nu}(G)=0$. These equations, in the
lowest order in powers of $\alpha^{'}$, will give us the Einstein
equations for $G_{\mu\nu}$ (see below).

\subsection{Conformal Field Theory}

As we have seen, conformal field theories (CFT) are the starting point
for the construction of classical string models. We now review
the
basic properties of these theories. Two-dimensional CFT's were
studied in a seminal paper by Belavin, Polyakov and Zamolodchikov
\cite{Belavin-Polyakov-Zamolodchikov} which has become the basic
reference on the subject (for a general review see also
\cite{Ginsparg}).

Consider a field theory in $d$ dimensions and its
conserved energy-momentum
tensor $T_{\mu\nu}$. For each vector field $\xi^{\mu}(x)$ we can
construct the current $j_{\nu}=\xi^{\mu}(x)T_{\mu\nu}$ which will be
conserved ($\partial_{\mu}j^{\mu}=0$) provided $\partial_{\mu}\xi_{\nu}
+\partial_{\nu}\xi_{\mu}=0$ since
\begin{equation}
\partial_{\mu}j^{\mu}=\partial_{\mu}\xi_{\nu}T^{\mu\nu}=
\frac{1}{2}(\partial_{\mu}\xi_{\nu}+\partial_{\nu}\xi_{\mu})T^{\mu\nu}\;.
\label{current-conserv}
\end{equation}
If we take the generator of dilatations
$\xi^{\mu}(x)=x^{\mu}$ we would have that, by appliying
(\ref{current-conserv}), the corresponding
current is conserved if the energy-momentum tensor is traceless.
This conservation implies that the theory is invariant under scale
transformations.

A conformal trasformation is a change of coordinates $x^{\mu}\rightarrow
x^{'\mu}$ characterized by the relation
\begin{equation}
g_{\mu\nu}(x^{'})dx^{'\mu}dx^{'\nu}=
\Omega(x)g_{\mu\nu}(x)dx^{\mu}dx^{\nu}\;.
\label{ct}
\end{equation}
By considering an infinitesimal transformation $\Omega(x)=1+\lambda(x)$
and $x^{'\mu}=x^{\mu}+\xi^{\mu}(x)$, (\ref{ct}) implies the
relation
\begin{equation}
\partial_{\mu}\xi_{\nu}+\partial_{\nu}\xi_{\mu}-
\frac{2}{d}\eta_{\mu\nu}\partial_{\sigma}\xi^{\sigma}=0
\label{killing}
\end{equation}
Vector fields $\xi_{\mu}(x)$ that generate conformal transformations are
called conformal Killing vectors. In the case in which $d>2$ the
general solution to this equation is \begin{equation}
\xi_{\mu}=b_{\mu}+\omega_{\mu\nu}x^{\nu}+(c\cdot x)x_{\nu}-
\frac{1}{2}c_{\mu}x^{2}
\end{equation}
with $\omega_{(\mu\nu)}=0$. Hence the conformal group in $d>2$
dimensions
is finite-dimensional, since conformal transformations are labeled by a
finite number of parameters $(b_{\mu},\omega_{\mu\nu},c_{\mu})$. For a
space of signature $(t,s)$, the conformal group is $SO(t+1,s+1)$.

The situation is quite different in the two-dimensional case. There we
can use light-cone coordinates and (\ref{killing}) becomes
\begin{equation}
\partial_{+}\xi_{-}=\partial_{-}\xi_{+}=0
\end{equation}
the general solution being $\xi_{+}=\xi(\sigma^{+})$ and
$\xi_{-}=\xi(\sigma^{-})$. With the metric in light-cone coordinates
$ds^{2}=d\sigma^{+}d\sigma^{-}$ the conformal transformations take the
form
\begin{equation}
\sigma^{+}\rightarrow f(\sigma^{+}) \hspace{1cm}
\sigma^{-}\rightarrow g(\sigma^{-})
\end{equation}
If, after Wick rotating, we use complex coordinates $z$, $\bar{z}$,
the conformal transformations correspond to holomorphic and
antiholomorphic trasformations for $z$ and $\bar{z}$ respectively
\begin{equation}
z^{'}=f(z) \hspace{1cm} \bar{z}^{'}=\bar{f}(\bar{z})
\label{conformal-trans}
\end{equation}
In these coordinates the energy-momentum tensor $T_{\mu\nu}(\sigma)$
has only two non-vanishing components, namely, $T_{zz}(z)=T(z)$ and
$T_{\bar{z}\bar{z}}(\bar{z})=\bar{T}(\bar{z})$ while the component
$T_{z\bar{z}}=T_{\bar{z}z}$ vanishes because of the tracelessness
of $T_{\mu\nu}$. Now it is easy to see that the conformal group in two
dimensions is infinite-dimensional, being generated by
\begin{equation}
l_{n}=-z^{n+1}\frac{d}{dz}\;,
\end{equation}
and satisfying the Virasoro algebra $[l_{m},l_{n}]=(m-n)l_{m+n}$.

We define primary fields $\Phi_{h\bar{h}}(z,\bar{z})$ with conformal
weights $(h,\bar{h})$ as those fields such that
$\Phi_{h\bar{h}}(z,\bar{z})(dz)^{h}(d\bar{z})^{\bar{h}}$ is invariant
under conformal transformations
\begin{equation}
\Phi_{h\bar{h}}(z,\bar{z})(dz)^{h}(d\bar{z})^{\bar{h}}=
\Phi_{h\bar{h}}[f(z),\bar{f}(\bar{z})](dz^{'})^{h}
(d\bar{z}^{'})^{\bar{h}}\;.
\end{equation}
This implies the following transformation property for the fields under
(\ref{conformal-trans})
\begin{equation}
\Phi_{h,\bar{h}}(z,\bar{z})=\Phi_{h,\bar{h}}[f(z),\bar{f}(\bar{z})]
|f^{'}(z)|^{2}\;,
\end{equation}
or, if we consider infinitesimal conformal transformations
$z^{'}=z+\epsilon(z)$,
\begin{equation}
\delta_{\epsilon}\Phi_{h,\bar{h}}(z,\bar{z})=
\left(\epsilon\partial_{z}+h\partial_{z}
\epsilon\right)\Phi_{h\bar{h}}(z,\bar{z})\;,
\end{equation}
and similarly for the antiholomorphic part. Since
\begin{equation}
T(\epsilon)=\oint_{|z|=1}\frac{dz}{2\pi i}\epsilon(z)T(z)
\end{equation}
is the generator of conformal transformations we could equally write
\begin{equation}
\delta_{\epsilon}\Phi_{h\bar{h}}=[T(\epsilon),\Phi_{h\bar{h}}(z,\bar{z})]
\label{commutator}
\end{equation}
For simplicity, we are going to consider from now on only the
holomorphic part. Any holomorphic function can be
written as a power series in $z$, so the whole conformal group can
be constructed from the generators associated with
$\epsilon_{n}=z^{n+1}$
\begin{equation}
L_{n}=\oint_{|z|=1}\frac{dz}{2\pi i} z^{n+1}T(z)
\end{equation}
then we can rewrite $T(z)$ as
\begin{equation}
T(z)=\sum_{n\in{\bf Z}}L_{n}z^{-n-2}
\end{equation}
where $L_{n}$ are the Virasoro generators introduced in the previous
section in the context of String Theory.
In the quantum theory they will satisfy the central extension of the
Virasoro algebra
\begin{equation}
[L_{m},L_{n}]=(m-n)L_{m+n}+\frac{c}{12}m(m^{2}-1)\delta_{m+n,0}
\end{equation}
where $c$ is called the central charge. It is easy to see that $L_{\pm
1}$ and $L_{0}$ form an $SL(2,{\bf C})$ subalgebra. From
(\ref{commutator}) we can determine the commutator of a Virasoro
generator with a primary field
\begin{equation}
[L_{n},\Phi_{h}(z)]=h(n+1)z^{n}\Phi_{h}(z)+
z^{n+1}\partial_{z}\Phi_{h}(z)\;.
\label{comm-L-Phi}
\end{equation}

We learned in the previous section that by means of a conformal
transformation
we can map the cylinder into the punctured complex plane ${\bf C}^{*}$.
Time ordering in the original description is converted into
radial ordering in ${\bf C}^{*}$, $\tau=-\infty$ corresponds
to $z=0$ and $\tau=\infty$ to $z=\infty$. Associated with
every primary field $\Phi_{h}(z)$ we have a state in the Fock
space defined as the action of this field on the vacuum at $z=0$
($\tau=-\infty$)
\begin{equation}
|h\rangle =\Phi_{h}(0)|0\rangle\;.
\end{equation}
In the case of the energy-momentum tensor for its action on the vacuum
state to be well defined it is neccesary that
\begin{equation}
L_{n}|0\rangle =0\;; \hspace{1cm} n\geq-1\;,
\end{equation}
so the vacuum state $|0\rangle$ is $SL(2,{\bf C})$ invariant.
By using (\ref{comm-L-Phi}) the state
$|h\rangle$ associated with the primary field $\Phi_{h}(z)$ satisfies
\begin{eqnarray}
L_{0}|h\rangle &=&h|h\rangle \\
L_{n}|h\rangle &=& 0\;, \hspace{1cm} n>0
\end{eqnarray}
due to the second property, states created by primary fields are
called highest weight states. On the other hand,
by applying $L_{-n}$ with $n>0$, to a highest weight state we obtain the
so-called descendent states which are labeled by the eigenvalue of
$L_{0}$ (we have from the Virasoro algebra that
$[L_{0},L_{-n}]=nL_{-n}$)
\begin{eqnarray}
|h\rangle &\hspace{1cm}& L_{0}=h \nonumber \\
L_{-1}|h\rangle &\hspace{1cm}& L_{0}=h+1 \nonumber \\
L_{-1}^{2}|h\rangle , L_{-2}|h\rangle &\hspace{1cm}& L_{0}=h+2
\\
&\vdots& \nonumber
\label{family}
\end{eqnarray}
The representation of the Virasoro algebra spanned by these states is
called a Verma module
$V(h,c)$ and depends only on the conformal weight of the primary field
and on the central charge of the Virasoro algebra.

Given a highest weight state $|h\rangle$ we associate with it a
primary field $\Phi_{h}(z)$ such that $|h\rangle=\Phi_{h}(0)|0\rangle$;
in the same way, given a descendent state $L_{-n_{1}}\ldots
L_{-n_{k}}|0\rangle$ we have the corresponding descendant field defined
by
\begin{equation}
\Phi^{n_{1}+\ldots +n_{k}}(z)=L_{-n_{1}}(z)\ldots
L_{-n_{k}}(z)\Phi_{h}(z)
\end{equation}
where the operators $L_{n}(z)$ are defined by
\begin{equation}
L_{n}(z)=\oint_{C_{n}}\frac{dw}{2\pi i}(w-z)^{-n+1}T(w)
\end{equation}
and the contours are taken in such a way that $C_{n_{k}}\subset \ldots
\subset C_{n_{1}}$. This allows us to make a one-to-one correspondence
between fields and states.

Equal time (radius) commutator can be related to the operator product
expansion (OPE) of the operators involved in the commutator. Let us
consider a complete set of operators $\{\Phi_{k}(z)\}$. In general, the
product of any two operators in this set $\Phi_{i}(z)\Phi_{j}(w)$ will
be divergent in the limit $z\rightarrow w$. Since the
set of operators we are considering is complete, we can write
\begin{equation}
\Phi_{i}(z)\Phi_{j}(w)=\sum_{k}C_{ij}^{k}(z,w) \Phi_{k}(w)
\label{OPE}
\end{equation}
which is called the operator product expansion (OPE) of
$\Phi_{i}(z)\Phi_{j}(w)$.
Thus, the divergence of the product in the limit in which $z$ and $w$
coincides is given by the behavior of $C_{ij}^{k}(z,w)$ in that limit.
Take now two analytic fields $A(z)$ and $B(z)$ and two
arbitary functions $f(z)$ and $g(z)$, and construct the operators
\begin{eqnarray}
A(f)&=&\oint_{|z|=1}\frac{dz}{2\pi i}f(z)A(z) \nonumber \\
B(g)&=&\oint_{|w|=1}\frac{dw}{2\pi i}g(w)B(w)
\end{eqnarray}
We write the equal-time commutator as
\begin{eqnarray}
[A(f),B(g)]&=&\oint_{C_{1}}\frac{dz}{2\pi i}f(z)A(z)
\oint_{C_{2}}\frac{dw}{2\pi i}g(w)B(w) \nonumber \\
&-&\oint_{C_{2}}\frac{dw}{2\pi
i}g(w)B(w)\oint_{C_{1}}\frac{dz}{2\pi i}f(z)A(z)
\label{comm1}
\end{eqnarray}
which is schematically represented in fig. \ref{contours}a. By deforming
the contour as in fig. \ref{contours}b we obtain the result
\begin{equation}
[A(f),B(g)]=\oint_{|w|=1}\frac{dw}{2\pi i}g(w)\oint_{w}\frac{dz}{2\pi i}
A(z)B(w)f(z)
\label{OPE-commutator}
\end{equation}
where the second integral is performed along a little circunference
around $w$.
\begin{figure}
\vspace{3cm}
\caption{Integration contours to compute the equal-radius commutator}
\label{contours}
\end{figure}
This expression tell us that the integral in $z$ is
determined by the singularities in the OPE of $A(z)B(w)$. From
(\ref{comm-L-Phi}) we get the OPE
\begin{equation}
T(z)\Phi_{h}(w)=\frac{h}{(z-w)^{2}}\Phi_{h}(w)+
\frac{1}{z-w}\partial_{w}\Phi_{h}(w)+ regular \; terms
\end{equation}
where the regular terms are analytic in the limit $z\rightarrow w$.
Using (\ref{OPE-commutator}) we can also represent the Virasoro algebra
in terms of an OPE
\begin{eqnarray}
T(z)T(w)&=&\frac{c/2}{(z-w)^{4}}+\frac{2}{(z-w)^{2}}T(w) \nonumber \\
&+&\frac{1}{z-w}\partial_{w}T(w)+regular \; terms\;,
\end{eqnarray}
where $c$ is the central charge. Now we can easily get the
vacuum expectation value for the product $T(z)T(w)$:
\begin{equation}
\langle T(z)T(w) \rangle = \frac{c/2}{(z-w)^{4}}\;,
\end{equation}
and we recover the gravitational anomaly calculated in the previous
chapter
\begin{equation}
\int d^{2}x \langle T_{++}(x)T_{++}(0)\rangle e^{ipx} \sim
\frac{p_{+}^{3}}{p_{-}}\;.
\end{equation}

The presence of a central charge in the OPE for $T(z)T(w)$ makes
the
energy-momentum tensor to transform not as a primary field. In fact,
$T(z)$ can be written as
\begin{equation}
T(z)=L_{-2}(z){\bf I}(z)\;,
\end{equation}
where ${\bf I}(z)$ is the identity operator (which is trivially a
primary
field of conformal weight $h=0$). This means that the energy momentum
tensor is a descendant field of the identity operator. If instead of
considering infinitesimal conformal transformations we deal with a
finite one $z \rightarrow f(z)$, $T(z)$ transforms according to
\begin{equation}
T(z)\rightarrow (\partial_{z}f(z))^{2}T(z)+\frac{c}{12}\{f,z\}\;,
\label{T-transform}
\end{equation}
where $\{f,z\}$ is the Schwarzian derivative defined by
\begin{equation}
\{f,z\}=\frac{\partial_{z}f(z)\partial_{z}^{3}f(z)-
\frac{3}{2}(\partial_{z}^{2}f(z))^{2}}{(\partial_{z}f(z))^{2}}
\end{equation}

Recall that we started with a conformal field theory defined
over the cylinder with complex coordinate $w$,	and that we have
passed from the cylinder
to the punctured complex plane by applying the
conformal transformation $z=e^{w}$. If we use the transformation
rule for the energy-momentum tensor (\ref{T-transform}) we get the
following relation between the energy-momentum tensor in the cylinder
and in the complex plane
\begin{equation}
T_{cyl}(w)=z^{2}T(z)-\frac{c}{24}\;,
\end{equation}
which in terms of the Virasoro generators means that the only generator
that changes is $L_{0}$, namely
\begin{equation}
(L_{0})_{cyl}=(L_{0})_{plane}-\frac{c}{24}\;.
\label{cyl-plane}
\end{equation}
We can give an interpretation to the trasformations induced by
$L_{0}$ and $\bar{L}_{0}$. From their definition in terms of the
energy-momentum tensor,
\begin{eqnarray}
L_{0}=\oint_{|z|=1}\frac{dz}{2\pi i}zT(z)\;, \nonumber \\
\bar{L}_{0}=\oint_{|\bar{z}|=1}\frac{d\bar{z}}{2\pi
i}\bar{z}\bar{T}(\bar{z})\;,
\end{eqnarray}
we see that $L_{0}+\bar{L}_{0}$ is the generator of dilatations
$z\rightarrow e^{\lambda} z$ in ${\bf C}^{*}$ with $\lambda$ real. But
this dilatations are
time traslations $\tau \rightarrow \tau+\lambda$ in the cylinder and
then $L_{0}+\bar{L}_{0}$ plays the role of the hamiltonian in our radial
quantization formalism.
The interpretation of $L_{0}-\bar{L}_{0}$ is also straighforward; this
is the generator of the trasformations $z\rightarrow e^{i\lambda} z$
for real $\lambda$, which are the rotations with center $z=0$. Going
back to the cylinder we see that this correspond to traslations in the
coordinate $\sigma$ (i.e., a change in the origin of the coordinate
$\sigma$) so it can be viewed as a momentum operator.

We are interested now in finding the partition function of our CFT,
which counts the number of states per energy level. It is
also interesting to formulate the CFT not on the cylinder
(or the punctured complex plane) but
on a Riemman surface of arbitrary genus, in order to apply our results
to String Theory. It is possible to construct a CFT on the torus
starting from our formulation in the cylinder (see \cite{Ginsparg} for
the details); to get a torus we identify the two ends of our cylinder.
Then we have two periods, one associated to time traslations
and the other associated to the spatial ones which, as we have seen,
are generated by $(L_{0})_{cyl}\pm(\bar{L}_{0})_{cyl}$ respectively.
First we redefine the complex coordinate $w$ as $iw$ so the
spatial period is simply $w \equiv w+2\pi$. The remaining period
can be defined by a complex number $\tau=\tau_{1}+i\tau_{2}$ as $w
\equiv w+2\pi \tau$, where $\tau$ is called the modular parameter of the
torus and parametrizes different tori (see fig. \ref{torus}).
\begin{figure}
\vspace{3cm}
\caption{Torus with modular parameter $\tau$}
\label{torus}
\end{figure}
Now we calculate the partition fuction; making use of the relation
(\ref{cyl-plane}) derived earlier, the result is \cite{Ginsparg}
\begin{equation}
Z=Tr\,q^{L_{0}-c/24}\bar{q}^{\bar{L}_{0}-c/24}
\end{equation}
where $q$ is defined as $q=e^{2\pi i\tau}$.

The partition function counts the number
of states associated with a given conformal family. Let $V(h,c)$ be the
Verma module associated with a primary field $\Phi_{h}(z)$
\begin{equation}
V(h,c)=\{|h\rangle, L_{-1}|h\rangle, L_{-2}|h\rangle,
L_{-1}^{2}|h\rangle,\ldots \}
\end{equation}
We are interested in the number of states per level in $V(h,c)$.
Define the character
\begin{equation}
\chi_{h}(q)=Tr_{V(h,c)}q^{L_{0}-c/24}
\end{equation}
which is the holomorphic part of the partition function defined earlier
and restricted to states in $V(h,c)$. By evaluating the trace we have
\begin{equation}
\chi_{h}=\sum_{m} d_{m} q^{m}
\end{equation}
where $d_{m}$ is the number of states in $V(h,c)$ with $L_{0}=m+c/24$.
For example if $V(h,c)$ contains a single highest weight vector the
character can be explicitly evaluated, with the result:
\begin{equation}
\chi_{h}=q^{h-c/24}\prod_{n=1}^{\infty}\frac{1}{1-q^{n}}
\end{equation}

As an example to illustrate our discussion on CFT, we
analize a massless free scalar field in two dimensions.
Using complex coordinates $(z,\bar{z})$ we can write the action
(we set from now on $\partial=\partial_{z}$ and
$\bar{\partial}=\partial_{\bar{z}})$
\begin{equation}
S=\frac{1}{4\pi}\int d^{2}z
\partial\phi(z,\bar{z})\bar{\partial}\phi(z,\bar{z})
\label{action-scalar}
\end{equation}
and the field equations are
\begin{equation}
\partial\bar{\partial}\phi(z,\bar{z})=0
\end{equation}
The general solution to this equation can be written as
$\phi(z,\bar{z})=\phi(z)+\phi(\bar{z})$. Since
$\partial\phi$ is a holomorphic field, we can make the Laurent
expansion
\begin{equation}
\partial\phi(z)=\sum_{n\in {\bf Z}}\alpha_{n}z^{-n-1}\;,
\end{equation}
and equivalently for $\bar{\partial}\phi$
\begin{equation}
\bar{\partial}\phi(\bar{z})=\sum_{n\in {\bf
Z}}\bar{\alpha}_{n}\bar{z}^{-n-1}\;.
\end{equation}
{}From (\ref{action-scalar}) we can evaluate the propagator:
\begin{equation}
\langle \phi(z,\bar{z})\phi(w,\bar{w})\rangle=-\log|z-w|^{2}\;,
\end{equation}
and the energy-momentum tensor
\begin{equation}
T(z)=-\frac{1}{2}:\partial\phi(z)\partial\phi(z):
\end{equation}
and similarly for the antiholomorphic component.
In order to calculate the central charge we compute the OPE for
two energy-momentum tensors. Using Wick's theorem for radial
quantization, we obtain
\begin{equation}
T(z)T(w)=\frac{1/2}{(z-w)^{4}}+ other \; terms
\end{equation}
and the central charge associated to a massless scalar field in two
dimensions is $c=1$. From the OPE
\begin{equation}
T(z)\partial\phi(w) =
\frac{1}{(z-w)^{2}}\partial\phi(w)+\frac{1}{z-w}\partial^{2}\phi(w)
\end{equation}
we see that $\partial\phi(z)$ is a primary field with conformal weight
$(h,\bar{h})=(1,0)$. We would like to construct
other primary fields with differents weights. This can be achieved
by using the vertex operators
\begin{equation}
V_{k}(z,\bar{z})= :e^{ik\phi(z,\bar{z})}:
\end{equation}
which, by a simple OPE computation have conformal dimensions
\begin{equation}
h(V_{k})=\bar{h}(V_{k})=\frac{k^{2}}{2}
\end{equation}

\subsection{Quantization of the Bosonic String}

The example of the two-dimensional massless scalar field provides a
natural connection
with the quantization of the closed bosonic string. Recall that we
started with the action
\begin{equation}
S_{P}[X,g]=-\frac{1}{4\pi \alpha^{'}}\int d^{2}\sigma \sqrt{g}g^{ab}
\partial_{a}X^{\mu}\partial_{b}X^{\nu} \eta_{\mu\nu}\;.
\end{equation}
After fixing the conformal gauge and performing the mapping of the
cylindrical world-sheet into ${\bf C}^{*}$ we obtain a gauge-fixed
action \begin{equation}
S_{P}=-\frac{1}{4\pi \alpha^{'}}\int dzd\bar{z} \,\partial X^{\mu}
\bar{\partial} X^{\nu} \eta_{\mu\nu}\;.
\end{equation}
This action describes a set of $d$ massless scalar fields. Working in
the critical dimension $d=26$ where the Liouville field decouples makes
the analysis of the free bosonic string particulary easy. From now on we
restrict our considerations to this case. The field $X^{\mu}$ is
expanded as
\begin{equation}
X^{\mu}(z,\bar{z})=q^{\mu}-\frac{i}{4}p^{\mu}\log{z\bar{z}} +
\frac{i}{2}\sum_{n\in{\bf Z}}
\frac{\alpha_{n}^{\mu}}{n}z^{-n}+\frac{i}{2}\sum_{n\in{\bf Z}}
\frac{\bar{\alpha}_{n}^{\mu}} {n}\bar{z}^{-n}\;.
\label{general-solution}
\end{equation}
The vertex operators \cite{Polyakov92} can be interpreted geometrically
as the Fourier transforms of the operators
\begin{equation}
V(X)\sim \int d^{2}\sigma \sqrt{g} \delta(X^{\mu}-X^{\mu}(\sigma))\;,
\end{equation}
which pin a string at the point $X$. Since only $(1,1)$ fields
can be integrated in a conformal invariant way we have that
physical states satisfy
\begin{equation}
L_{0}|phys\rangle=|phys\rangle\;,  \hspace{1cm}
\bar{L}_{0}|phys\rangle=|phys\rangle\;.
\end{equation}
These conditions fix the value of the normal ordering constant $b$
introduced in sec. 3.1.

Now we proceed to obtain the spectrum of the closed bosonic string
(we will work in a system of units in which $\alpha^{'}=\frac{1}{2}$).
Canonical quantization produces the following commutation relations
\begin{eqnarray}
[q^{\mu},p_{\nu}]&=&i\eta^{\mu\nu} \nonumber \\
{[}\alpha_{m}^{\mu},\alpha_{n}^{\nu}]&=&m\eta^{\mu\nu}\delta_{m+n,0}
\nonumber \\
{[}\bar{\alpha}_{m}^{\mu},\bar{\alpha}_{n}^{\nu}]&=&
m\eta^{\mu\nu}\delta_{m+n,0}
\label{commutation-rel}
\end{eqnarray}
and all other commutators vanish. We split the general
solution (\ref{general-solution}) into the right and left-moving parts
(holomorphic and antiholomorphic respectively)
\begin{eqnarray}
X_{R}^{\mu}(z)=
x_{R}^{\mu}-\frac{i}{4}p_{R}^{\mu}\log{z}+ \frac{i}{2}\sum_{n\in{\bf Z}}
\frac{\alpha_{n}^{\mu}}{n}z^{-n} \nonumber \\
X_{L}^{\mu}(\bar{z})=
x_{L}^{\mu}-\frac{i}{4}p_{L}^{\mu}\log{\bar{z}}+\frac{i}{2}
\sum_{n\in{\bf Z}} \frac{\bar{\alpha}_{n}^{\mu}}{n}\bar{z}^{-n}\;,
\end{eqnarray}
with $x_{R}^{\mu}=x_{L}^{\mu}=q^{\mu}/2$ and $p_{R}^{\mu}=p_{L}^{\mu}=
p^{\mu}$.
Using these expressions we can write $L_{0}$ in terms of operators
$\alpha_{m}^{\mu}$ as
\begin{equation}
L_{0}=\frac{1}{4}p^{2}+\sum_{n\geq 1}\alpha_{-n}\cdot
\alpha_{n}
\end{equation}
and the similarly for $\bar{L}_{0}$ changing $\alpha_{n}^{\mu}$
for $\bar{\alpha}_{n}^{\mu}$. Setting $L_{0}=\bar{L}_{0}=1$ and taking
into account that the mass of a physical state is defined by
$m^{2}=-p_{\mu}p^{\mu}=m_{R}^{2}+m_{L}^{2}$ we obtain the mass formula
\begin{equation}
\frac{1}{4}m^{2}=\sum_{n\geq 1} \alpha_{-n}\cdot\alpha_{n}+
\sum_{n\geq 1} \bar{\alpha}_{-n}\cdot\bar{\alpha}_{n}-2\;.
\end{equation}
The condition $L_{0}-\bar{L}_{0}$, which now states that the theory is
invariant under redefinitions of the $\sigma$-origin,
gives the level matching condition
\begin{equation}
\sum_{n\geq 1}\alpha_{-n}\cdot\alpha_{n}=
\sum_{n\geq 1}\bar{\alpha}_{-n}\cdot\bar{\alpha}_{n}\;.
\end{equation}
Looking back to the commutation relations (\ref{commutation-rel}) we see
that $\sum_{n\geq 1}\alpha_{-n}\cdot\alpha_{n}$ can be interpreted as
the ocupation number
associated to the right-moving modes (the same for the left-movers).
This allows us to rewrite the mass formula and the level matching
condition in a simpler form
\begin{eqnarray}
\frac{1}{4}m^{2}&=&N_{R}+N_{L}-2\;, \nonumber \\
N_{R}&=&N_{L}\;.
\end{eqnarray}

We are now ready to construct the spectrum of the closed bosonic string.
We start with the vacuum state $|0,p\rangle$ defined by
\begin{eqnarray}
\alpha_{n}^{\mu}|0,p\rangle&=&\bar{\alpha}_{n}^{\mu}|0,p\rangle=0\;,
\hspace{1cm} n\geq 1 \nonumber \\
\hat{p}^{\mu}|0,p\rangle&=&p^{\mu}|0,p\rangle
\end{eqnarray}
where we have written $\hat{p}^{\mu}$ to distinguish the operator from
its eigenvalue. This vacuum state is clearly a tachyon, since
$N_{R}=N_{L}=0$ and $m^{2}<0$. This tachyon is created by the
action of the tachyon vertex operator defined by
\begin{equation}
V_{tachyon}(p)=:e^{ip\cdot X}:
\end{equation}

In the first excited level we have only one posibility surviving the
level matching condition $N_{L}=N_{R}=1$, so a general state will be of
the form
$\zeta_{\mu\nu}(p)\alpha_{-1}^{\mu}\bar{\alpha}_{-1}^{\nu}|0,p\rangle$
with $\zeta_{\mu\nu}(p)$ the polarization vector with an
associated vertex operator given by
\begin{equation}
V(p)=\zeta_{\mu\nu}(p):\partial X^{\mu}
\bar{\partial}X^{\nu}e^{ip\cdot X}:
\end{equation}
The states in the first level are massless as one can see either from
the mass formula or from the condition that the previous
vertex operator has conformal
weights $(1,1)$. This also gives the
relation $p^{\mu}\zeta_{\mu\nu}(p)=0$ which is the on-shell condition
for a massles tensor particle. We can decompose a general state in the
first excited level into three states that transform irreducibly under
$SO(1,25)$; the first one, with $\xi_{\mu\nu}=\xi_{\nu\mu}$,
$\eta^{\mu\nu}\zeta_{\mu\nu}(p)=0$, is a spin $2$ particle that we
will identify with the graviton. Secondly we have the antisymmetric
rank-$2$ tensor field for which $\zeta_{(\mu\nu)}=0$. Finally if we set
$\zeta_{\mu\nu}=\eta_{\mu\nu}$ we have a scalar particle that is known
as the dilaton. Thus at the massless level of the bosonic string we have
three states, the graviton $g_{\mu\nu}$, the antisymmetric tensor
$B_{\mu\nu}$ and the dilaton $\Phi$.

Without much effort we have been led to the presence of spin $2$
particles. In principle we cannot still identify these states with
gravitons since we are dealing with free bosonic strings and we have no
interaction between them. However, when we introduce
interactions in String Theory following the Polyakov prescription (see
below) it is possible to show that the scattering amplitudes for these
states in the low energy limit are described by the linearized
Einstein-Hilbert action, so they can be consistently identified with the
graviton. This is one of the main features of String Theory: every
closed string theory (and all of them in the interacting case) contains
a massless state of spin $2$.

We should remark that the existence of a graviton state is also a
consequence of working in the critical dimension $d=26$. Gravitons seem
to be excluded from the spectrum in non-critical dimensions. Thus, if we
want to use String Theory to unify gravity with other interactions we
should work in the critical dimension. More generally, in String Theory
it is now clear that the notion of dimension is encoded in the central
extension of the matter Virasoro algebra. We want to have $c=26$,
although the contribution from ``geometrical'' dimensions may be smaller
than $26$. By adding up different CFT's without a geometrical
interpretation we can reach the critical value for $c$.

\subsection{Interaction in String Theory and the characterization
of the moduli space}

After studying the theory of free strings it is time to
introduce interacting strings. This is done by introducing a
perturbative expansion in which the basic vertex describes
the splitting of one initial string into two final ones (fig.
\ref{vertex})
\begin{figure}
\vspace{3cm}
\caption{Fundamental vertex in string perturbation theory}
\label{vertex}
\end{figure}
If we use a particle analog, this vertex would
correspond to the vertex of a $\phi^{3}$ scalar field theory.
However the situation in String Theory is
quite different from that in QFT: in the case of a $\phi^{3}$ theory in
order to compute a given amplitude we have to sum over all
Feynman graphs with a given number of
external states. For example, for the four point amplitude we have to
sum the contributions of all graphs with four external legs (fig.
\ref{phi3})
\begin{figure}
\vspace{3cm}
\caption{Feynman diagrams contributing to the four-point function in
$\phi^{3}$ scalar theory}
\label{phi3}
\end{figure}
In String Theory we have to compute amplitudes with a given number of
external string states. To do so we have to sum over all world-sheets
that connect {\it in} with {\it out} strings (fig.
\ref{strings-graphs}).
\begin{figure}
\vspace{4cm}
\caption{String graphs contributing to the scattering of four strings}
\label{strings-graphs}
\end{figure}
All the surfaces we have to sum over
can be constructed from the fundamental vertex of fig. \ref{vertex}.
It is easy to check that the number of vertices we need to construct a
given surface depends only on the genus $g$ of the surface and the
number $n$ of external strings and it is equal to $2g+n-2$. If $g_{st}$
is the string coupling constant associated with the fundamental vertex,
the amplitudes can be schematically written as
\begin{eqnarray}
\lefteqn{A(1,\ldots,n)=}  \\
& &\sum_{topologies}g_{st}^{2g+n-2}\sum_{metrics}
\frac{1}{Vol(\mbox{Diff})\times Vol(\mbox{Weyl})}\sum_{embeddings}
e^{-S_{P}[X,g]} \nonumber
\label{amplitude}
\end{eqnarray}
where the first sum indicates that we sum over two-dimensional
surfaces of genus $g$ and with $n$ external strings. We also divide out
by the volume of the group of diffeomorphisms and of the group of Weyl
transformations because these are (in the critical dimension) the gauge
groups of the quantum theory. Nevertheless it seems quite difficult to
perform the sum over surfaces that have external ``legs'' extending to
infinity. This can be solved if we make use of the Weyl invariance of
the theory: we may cut external strings from the surfaces in
the perturbative expansion and then we are left with two-dimensional
surfaces with $n$ boundaries. Then, since by a conformal mapping we can
transform a cylinder into the punctured complex plane, we will attach to
every boundary the punctured unit circle ${\cal C}^{*}
=\{z\in {\bf C}|\,0<|z|\leq 1\}$. External string states are mapped to
the punctures by
means of local operators (vertex operators) that carry the quantum
numbers of the external strings. When all this information is put
together we can write the amplitude (\ref{amplitude}) as
\begin{eqnarray}
\lefteqn{A(1,\ldots,n)=}\nonumber \\
& &\sum_{g=1}^{\infty} g_{st}^{-\chi} \int_{\Sigma_{g,n}}
\frac{{\cal D}X{\cal D}g}{Vol(\mbox{Diff})\times Vol(\mbox{Weyl})}
\prod_{i=1}^{n} V_{\Lambda_{i}}(k_{i}) e^{-S_{P}[X,g]}
\label{amplitude2}
\end{eqnarray}
where $V_{\Lambda_{i}}(k_{i})$ is the vertex operator representing a
external
string with momentum $k_{i}$ and quantum numbers $\Lambda_{i}$, and
$\chi$ is the Euler characteristic for a genus $g$ manifold with $n$
boundaries $\chi=2-2g-n$.

This is the Polyakov prescription for the computation of amplitudes in
String Theory. We sum over all two-dimensional surfaces with genus $g$
and $n$ punctures non-equivalent under the joined action of the group of
diffeomorphisms and Weyl transformations. We can, for every surface
$\Sigma_{g,n}$, write the metric in the form (up to diffeomorphisms)
\begin{equation}
g_{ab}=e^{\phi(x)}\hat{g}_{ab}
\end{equation}
where $\hat{g}_{ab}$ is a fiducial metric. Since we are in the critical
dimension the dependence over the conformal factor $\phi(x)$ drops out
from (\ref{amplitude2}) and the integration over $\phi(x)$ is cancelled
by the $Vol(\mbox{Weyl})$ in the denominator. Then we are faced with the
problem of clasifying fiducial metrics $\hat{g}_{ab}$. We will
see that the set $\{\hat{g}_{ab}\}/\mbox{Diff}(\Sigma_{g,n})$ is
finite-dimensional.

We work with compact orientable two-dimensional surfaces of
genus $g$ with punctures. If we take complex coordinates in such
surfaces the problem of classifying them into conformal classes is
equivalent to the classification of complex structures on the manifold:
then the sum over conformal classes is transformed into a sum over
Riemann surfaces of genus $g$ and $n$ punctures $\Sigma_{g,n}$.

However after performing the sum over conformal classes, we are still
left with the integration over Riemann surfaces that are
not equivalent under diffeomorphisms. We have to classify
Riemann surfaces according to their complex structure; the space of
parameters
that label differents classes constitutes the moduli space for the
surface ${\cal M}_{g,n}$. We begin with the case $n=0$.
In order to characterize the moduli space of a Riemann surface of
genus $g$ we use the uniformization theorems
due to Klein, Poincar\'e and Koebe. Any given Riemann surface $\Sigma$
can be constructed by starting with one of the simply connected Riemann
surfaces (their universal cover $\hat{\Sigma}$)
\begin{itemize}
\item The sphere $S^{2}$ with the round metric
\begin{equation}
ds^{2}=\frac{dzd\bar{z}}{(1+z\bar{z})^{2}}
\end{equation}

\item The complex plane ${\bf C}$ with the flat metric
$ds^{2}=dzd\bar{z}$.

\item The upper half plane ${\bf H}$ with the metric of constant
negative curvature
\begin{equation}
ds^{2}=2\frac{dzd\bar{z}}{(z-\bar{z})^{2}}
\end{equation}

\end{itemize}
We recover the Riemann surface from its universal cover by dividing it
by its fundamental group $\pi_{1}(\Sigma)$ which acts in the cover
without fixed points. We also have the Gauss-Bonnet theorem
\begin{equation}
\frac{1}{2\pi}\int_{\hat{\Sigma}} d^{2}z R =2(1-g)
\end{equation}
The sphere is the covering space for genus zero Riemann surfaces, the
complex plane for $g=1$ and the upper half plane for any surface with
$g>1$. The conclusion is that we will classify all the Riemann surfaces
into diffeomorphism classes by finding out the action of the fundamental
group $\pi_{1}(\Sigma)$ on the universal covering space $\hat{\Sigma}$.

Let us begin with a genus zero Riemann surface. There is a theorem that
states that any Riemann surface homeomorphic to the sphere is also
isomorphic to it. This means that the moduli space for a genus zero
Riemann surface ${\cal M}_{0}$ has only one point. In the case of
$g=1$ the situation is not so simple. Since in the complex plane the
only group that acts without fixed points is the group of
translations, to get a torus we
have to divide ${\bf C}$ by the action of a subgroup of
discretes translations isomorphic to ${\bf Z}+{\bf Z}$ which
is the fundamental group of the torus. Thus we take two
complex numbers $\omega_{1}$ and $\omega_{2}$ and construct the
lattice generated by them. It is easy to see that without loss of
generality we can take one of the complex numbers equal to one
($\omega_{1}=1$) and we are left only with one complex modular
parameter $\omega_{2}=\tau$; moreover we can take the imaginary part
of $\tau$ positive since the lattices
generated by $\tau$ and $-\tau$ are isomorphic.

The tori parametrized by $\tau$ are equivalent under
transformations that lie in the connected component of the identity.
However we have to restrict the values of $\tau$ in order to avoid
overcounting because different values of $\tau$ may correspond to
tori equivalent under diffeomorphisms that are not connected with the
identity. It can be seen that this is the case for the tori
characterized by $\tau$
and
\begin{equation}
\tau^{'}=\frac{a\tau+b}{c\tau+d} \hspace{1cm} ad-bc=1, \hspace{1cm}
a,b,c,d \in {\bf Z}
\end{equation}
because the lattices defined by $\tau$ and $\tau^{'}$ are equivalent,
they differ only in the choice of the fundamental cell, hence
we have to restrict the values of $\tau$ to the fundamental region of
the group $SL(2,{\bf Z})$, which we take to be
\begin{equation}
{\cal F}=\{\tau\in {\bf C} | Im\tau>0,
-\frac{1}{2}<Re\tau\leq\frac{1}{2},|\tau|>1\}
\end{equation}

We have found that the modular space for a genus zero Riemann surface
${\cal M}_{0}$ consists of a single point and for a genus
one surface ${\cal M}_{1}={\cal F}$. The case with $g>1$ is
more complicated. To construct the moduli space
${\cal M}_{g}$ with $g>1$ we are going to use the sewing technique.
Let us consider a genus-$g$ Riemann surface with two punctures
(fig. \ref{sewing}) located at $P_{1}$ and $P_{2}$ and let us also
consider
two annuli around these points with coordinates $z_{1}$ and $z_{2}$
such that $z_{1}(P_{1})=z_{2}(P_{2})=0$. We can sew the Riemann surface
by identifying the points on the annuli via the transformation
\begin{equation}
z_{1}z_{2}=q\;,
\end{equation}
with $q\in {\bf C}-\{0\}$ the sewing parameter. After this is done we
get a Riemann surface with genus $g+1$.
\begin{figure}
\vspace{3cm}
\caption{Sewing of two punctures in a genus $g$ Riemann surface to
obtain a genus $g+1$ Riemann surface}
\label{sewing}
\end{figure}

This construction allow us to establish a relation between the
dimensions of the moduli spaces ${\cal M}_{g}$ and ${\cal M}_{g+1}$. In
fact, the parameters characterizing the Riemann surface after
sewing two of its points are the moduli parameters characterizing  the
original Riemann surface of genus $g$ plus the three complex parameters
($z_{1},z_{2},q$) that label the position of the points which we
have sewed and the sewing parameter. Thus we arrive at the
relation
\begin{equation}
dim{\cal M}_{g+1}=dim{\cal M}_{g}+3\;.
\end{equation}
If we start with a genus one surface we have that since we can
always locate one of the points at $z=0$, due to the isometries of the
flat metric on the torus, the number of moduli parameters for
a genus two surface is three. Applying the sewing technique again and
again we obtain that the dimension of the moduli space for a
genus-$g$ surface is given by
\begin{eqnarray}
dim{\cal M}_{0}&=&0 \nonumber \\
dim{\cal M}_{1}&=&1 \nonumber \\
dim{\cal M}_{g}&=&3g-3 \hspace{1cm} g>1
\end{eqnarray}

As we have seen above, in the computation of string amplitudes we work
with Riemann surfaces of genus $g$ and $n$ puctures. To derive
the dimension of the moduli space ${\cal M}_{g,n}$ for such surfaces we
use a slightly different method from the one used for
${\cal M}_{g}={\cal M}_{g,0}$.
We start with a sphere with $2g+n$ punctures (we consider first the
case $2g+n>3$); by conformal invariance \cite{Ginsparg} we can
take three of the punctures to be located at $z=0$, $z=1$ and $z=\infty$
respectively so we are left with $2g+n-3$ free complex parameters to
locate the remaining punctures. Now by sewing $2g$ punctures together
with $g$ sewing parameters $q_{1},\ldots,q_{g}$ we have that the final
number of independent complex parameters in the moduli space of a genus
$g$ Riemann surface with $n$ punctures is $3g+n-3$ for $g>1$ (see fig.
\ref{construction}). In the case with $2g+n\leq 2$ the situation is
different because
in that cases all the punctures in the sphere are in fixed positions.
Then the dimension of the moduli space only depends upon the number of
sewing parameters which is equal to $g$.
\begin{figure}
\vspace{4cm}
\caption{Construction of a genus $g$ Riemann surface with $n$ puctures
from a sphere with $2g+n$ punctures}
\label{construction}
\end{figure}

We have seen that the space of two-dimensional metrics modulo
differmorphisms and Weyl rescalings is finite-dimensional. This means
that the path integration over metrics has to be transformed into an
integration over the moduli space of the corresponding surface
\begin{equation}
\int_{\Sigma_{g,n}}{\cal D}g \longrightarrow \int_{{\cal M}_{g,n}}
d\mu(g,n)
\end{equation}
where $d\mu(g,n)$ is the integration measure over the moduli space. In
general the problem we face when computing amplitudes is to characterize
this measure.

\subsection{Bosonic strings with background fields. ``Stringy''
corrections to the Einstein equations}

Up to now we have considered bosonic strings propagating in Minkowski
space. We are also insterested in studying the propagation of
strings in general manifolds with metric $G_{\mu\nu}(X)$. This means
that we have to modify the string action in order to include the
effect of the background metric $G_{\mu\nu}(X)$
\begin{equation}
S_{1}=-\frac{1}{4\pi \alpha^{'}}\int d^{2}\sigma
\sqrt{g}g^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}G_{\mu\nu}(X)\;.
\label{graviton-background}
\end{equation}
We have seen that the massless sector of the bosonic string
contains, besides the graviton, the antisymmetric tensor and the
dilaton. Thus, it seems natural that if we include a background value
for the graviton field we should also consider background values
for the other massless states, $B_{\mu\nu}$ and $\Phi$. The
general form of the action is
\begin{eqnarray}
S&=&-\frac{1}{4\pi\alpha^{'}}\int d^{2}\sigma
\sqrt{g}g^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}G_{\mu\nu}(X)
\label{action-background}
\\
&-&\frac{1}{4\pi\alpha^{'}}\int d^{2}\sigma
\epsilon^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}B_{\mu\nu}(X)+
\frac{1}{4\pi}\int d^{2}\sigma \sqrt{g}R^{(2)}(g)\Phi(X) \nonumber
\end{eqnarray}
Where $G_{\mu\nu}(X)$, $B_{\mu\nu}(x)$ and $\Phi(X)$ are the background
values of the fields in the massless sector and $\epsilon^{ab}$ is the
completely antisymmetric tensor with two indices.

However for the string theory described by (\ref{action-background}) to
be consistent, we have to impose that this action define a CFT; this
will constrain the values of the background fields. The best way to
study the conformal invariance of (\ref{action-background}) is to write
the metric $g_{\mu\nu}$ as $e^{\phi}\hat{g}^{\mu\nu}$ in $2+\epsilon$
dimensions and then to see what are the conditions that have to be
fulfilled for the conformal factor to disappear from the effective
action in the limit
$\epsilon\rightarrow 0$. Doing so we obtain the following equations for
the background fields to lowest order in $\alpha^{'}$
\cite{CFMP}
\begin{eqnarray}
R_{\mu\nu}+\frac{1}{4}H_{\mu}^{\,\lambda\rho}H_{\nu\lambda\rho}-
2D_{\mu}D_{\nu}\Phi &=& 0 \nonumber \\
D_{\lambda}H^{\lambda}_{\,\mu\nu}-2D_{\lambda}\Phi\,H^{\lambda}_{\mu\nu}
&=&0 \nonumber \\
4(D_{\mu}\Phi)^{2}-4D_{\mu}D^{\mu}\Phi+R+\frac{1}{12}H_{\mu\nu\lambda}
H^{\mu\nu\lambda} &=& 0
\label{background-eqs}
\end{eqnarray}
where $H_{\mu\nu\lambda}$ is the field strength associated with the
antisymmetric tensor $B_{\mu\nu}$
\begin{equation}
H_{\mu\nu\lambda}=\partial_{\mu}B_{\nu\lambda}+\partial_{\lambda}
B_{\mu\nu}+\partial_{\nu}B_{\lambda\nu}
\end{equation}
An important remark is that (\ref{background-eqs}) can be derived
as the field equations associated to the action functional
\begin{equation}
\hspace{-.5cm} S_{d=26}=-\frac{1}{2\kappa^{2}}\int d^{26}x \sqrt{G}
e^{-2\Phi}\left( R-4D_{\mu}\Phi D^{\mu}\Phi+
\frac{1}{12}H_{\mu\nu\lambda}H^{\mu\nu\lambda}\right).
\end{equation}
When we include higher order effects in $\alpha^{'}$ we obtain
corrections
to the Einstein-Hilbert action for the background fields. For example,
if we consider the action (\ref{graviton-background}) and compute the
$\beta$-function at one loop order we obtain (for the evaluation of
$\beta$-functions in non-linear $\sigma$-models see \cite{Friedan})
\begin{equation}
\beta_{\mu\nu}(X)=-\frac{1}{2\pi}R_{\mu\nu}(X)
\end{equation}
this means that, since for conformal invariance we
must have $\beta_{\mu\nu}(X)=0$, we
obtain the vacuum Einstein equations for the background metric.
However if
we include the two-loop correction to the $\beta$-function the result is
\cite{Friedan}
\begin{equation}
\beta_{\mu\nu}(X)=-\frac{1}{4\pi}\left(R_{\mu\nu}(X)+
\frac{\alpha^{'}}{2}R_{\mu\lambda\sigma\tau}
R_{\nu}^{\;\;\lambda\sigma\tau} \right)
\end{equation}
so we obtain a correction to the Einstein equations for the background
metric of order $\alpha^{'}$ coming from String Theory
\begin{equation}
R_{\mu\nu}(X)+\frac{\alpha^{'}}{2}
R_{\mu\lambda\sigma\tau}R_{\nu}^{\;\;\lambda\sigma\tau}= 0\;.
\end{equation}
We do not know the form of the $\beta$-function to all orders in
$\alpha^{'}$.

\subsection{Toroidal compactifications. $R$-duality}

In this section we would like to discuss a genuine property of strings
which has no analogue in QFT. This is the notion of duality.
All known critical string theories are formulated in dimension higher
that four ($d=26$ for the bosonic string and $d=10$ for the super
and the heterotic string). This means that, in order to derive
phenomenological implications from String Theory we have to pass from
$26$ or $10$ to $4$ dimensions. One way in which this can be
accomplished is a Kaluza-Klein compactification.
Starting with a, say, $26$-dimensional manifold we compactify
the $22$ extra dimensions in a $22$-dimensional torus $T^{22}$. Then we
are left with four open dimensions and $22$ compactified ones;
if we assume that the typical radii of compactification is of the order
of the Planck length, the existence of extra dimensions is
completely unobservable from the energies available in
present experiments. Nevertheless, this
does not mean that they have no effect
in low energy phenomenology. For example, the isometry group of the
internal manifold appears as part of the gauge group of the low energy
effective field theory. In last few years a great effort has been
devoted
to the search of a four-dimensional string theory with acceptable
phenomenological properties (by using toroidal compactifications or
more sophisticated mechanisms as orbifolds, Calabi-Yau manifolds etc.).

We illustrate the main features of toroidal
compactifications in the context of the closed bosonic string
\cite{Lust-Theisen}. Assume that we want to compactify $D$
bosonic coordinates in a $D$-dimensional torus $T^{D}$. We will denote
by $X^{i}$ the $26-D$ open dimensions and by $X^{I}$ the compactified
ones. We define the $D$-dimensional torus by simply identifying the
points in the target space according to the rule
\begin{equation}
X^{I}\sim X^{I}+\pi\sqrt{2}\sum_{i=1}^{D}n_{i}R_{i}e_{i}^{I}=
X^{I}+2\pi L^{I}  \hspace{2cm} n_{i}\in {\bf Z}
\label{identification}
\end{equation}
where ${\bf e}_{i}$ are $D$ linearly independent vectors normalized to
${\bf e}_{i}^{2}=2$ and $R_{i}$ are the radii of the compact dimensions.
The vectors $L^{I}$ define a $D$-dimensional lattice $\Lambda$ and the
torus $T^{D}$ is represented as
\begin{equation}
T^{D}={\bf R}^{D}/(2\pi \Lambda)
\end{equation}
The momenta in the compact dimensions $p^{I}$ are the generators of
translations of the center of mass $x^{I}$. From the
single valuedness of the wave function $\exp{(ix\cdot p)}$ we have that
momenta $p^{I}$ have to lie in the lattice dual to $\Lambda^{D}$,
$(\Lambda^{D})^{*}$, and they can be expanded as
\begin{equation}
p^{I}=\sqrt{2}\sum_{i=1}^{D} \frac{m_{i}}{R_{i}}e^{*I}_{i}\;,
\end{equation}
with ${\bf e}_{i}\cdot {\bf e}_{j}^{*}=\delta_{ij}$ and
${\bf e}_{i}^{* 2}=1/2$.

Let us now consider a closed bosonic string on the torus $T^{D}$.
We have the periodicity condition $X^{\mu}(\sigma+\pi,\tau)=
X^{\mu}(\sigma,\tau)$. When we have compactified
dimensions, we have to take into account the
twisted sector in which the periodicity
condition is fulfilled modulo the identification (\ref{identification})
\begin{equation}
X^{I}(\tau,\sigma+\pi)=X^{I}(\tau,\sigma)+2\pi L^{I}=
X^{I}(\tau,\sigma)+\pi\sqrt{2}\sum_{i=1}^{D}n_{i}R_{i}e_{i}^{I}\;,
\end{equation}
which means that the string wrap around the compactified
dimensions. The integer $n_{i}$ is the winding number and
measures the number of times the string wraps around the
$i$th compactified dimension. Because of this
new boundary conditions we have to modify the mode expansions for
$X^{I}$ \begin{eqnarray}
X^{I}_{L}(\tau+\sigma)=\frac{1}{2}x^{I}+
(\frac{1}{2}p^{I}+L^{I})(\tau+\sigma)+ \frac{i}{2}\sum_{n\neq
0}\frac{\bar{\alpha}^{I}_{n}}{n} e^{-2in(\tau+\sigma)} \nonumber \\
X^{I}_{R}(\tau-\sigma)=\frac{1}{2}x^{I}+
(\frac{1}{2}p^{I}-L^{I})(\tau-\sigma)+ \frac{i}{2}\sum_{n\neq
0}\frac{\alpha^{I}_{n}}{n} e^{-2in(\tau-\sigma)}
\end{eqnarray}
Defining $p_{R}^{I}=\frac{1}{2}p^{I}+L^{I}$ and
$p_{L}^{I}=\frac{1}{2}p^{I}-L^{I}$ we rewrite the mass formulae as
\begin{equation}
\frac{1}{4}m_{L}^{2}=\frac{1}{2}p_{L}^{2}+N_{L}-1 \hspace{2cm}
\frac{1}{4}m_{R}^{2}=\frac{1}{2}p_{R}^{2}+N_{R}-1\;.
\end{equation}
Then the total mass formula and the level matching condition
become
\begin{eqnarray}
\frac{1}{4}m^{2}=\frac{1}{4}m^{2}_{R}+\frac{1}{4}m^{2}_{L}
&=&N_{R}+N_{L}-2+\sum_{I=1}^{D}(\frac{1}{4}p^{I}p^{I}+
L^{I}L^{I}) \nonumber \\
N_{L}-N_{R}&=&\sum_{I=1}^{D}p^{I}L^{I}
\end{eqnarray}
It is worth noticing that the spectrum of the compactified bosonic
string contains in the untwisted sector all the states of the
uncompactified theory. This means that, in general, tachyons cannot be
removed
by na\"{\i}ve toroidal compactifications (they can be removed if the
compactification is made in the presence of Wilson lines
\cite{Narain-Sarmadi}).
With respect to the twisted sector, it constains
$2D$ massless vectors $\alpha^{\mu}\bar{\alpha}^{I}|0\rangle$ and
$\alpha^{I}\bar{\alpha}^{\mu}|0\rangle$ as well as $D^{2}$ massless
scalars $\alpha^{I}\bar{\alpha}^{J}|0\rangle$. The massless vector
bosons are associated with the isometry group of the torus $U(1)^{D}$
and play the role of gauge bosons.

Let us consider the simpler case in which we have only one compactified
dimension with radius $R$. This means that we have the identification
\begin{equation}
X^{25}\sim X^{25}+2\pi RL
\end{equation}
with $L$ the winding number. The mass formula is
\begin{equation}
\frac{1}{4}m^{2}=\frac{M^{2}}{4R^{2}}+L^{2}R^{2}+N_{L}+N_{R}-2
\end{equation}
with $N_{L}-N_{R}=ML$, $M,L\in{\bf Z}$. We see one important
property of
the spectrum of the bosonic string with one compactified dimension:
the mass formula is invariant under the replacement \cite{duality}
\begin{equation}
M\rightarrow L \hspace{2cm} L\rightarrow -M \hspace{2cm}
R \rightarrow \frac{\alpha^{'}\hbar}{R}
\label{replace}
\end{equation}
where we have restored the constants $\alpha^{'}$ and $\hbar$.
However, the invariance of the mass formula under (\ref{replace}) does
not ensure that the whole (interacting) theory is invariant under such
transformations. We have to check that the amplitudes are also
invariant.

It was shown in \cite{Ginsparg-Vafa} using the low energy field theory
that if the theory was invariant under (\ref{replace}) then the string
coupling constant (i.e., the vacuum expectation value of the dilaton
field) has to transform according to
\begin{equation}
g_{st} \longrightarrow	g_{st}\frac{\sqrt{\alpha^{'}\hbar}}{R}
\end{equation}
where $\Phi$ is the vacuum expectation value of the dilaton. This
transformation for $\Phi$ can also be obtained in a $\sigma$-model
approach by demanding the transformed $\sigma$-model
to preserve conformal invariance at zero order in $\alpha^{'}$
\cite{Buscher}. In \cite{Alvarez-Osorio-3} it was finally proved that
the duality transformation
\begin{equation}
R\rightarrow \frac{\alpha^{'}\hbar}{R}\, \hspace{1cm}
g_{st}\rightarrow g_{st}\frac{\sqrt{\alpha^{'}\hbar}}{R}
\label{g}
\end{equation}
is a symmetry of the whole string perturbation theory either for the
bosonic and the heterotic string. This symmetry can also be shown to
hold in the $c=1$ non-critical string for the whole perturbative
expansion when the matrix model is truncated to the sector without
vortices \cite{Gross-Klebanov}.

The invariance under (\ref{g}) is called $R$-duality and means that
the theory at radius
$R$ is equivalent to the one at radius $\alpha^{'}\hbar/R$ with an
appropiate transformation of the couplings (for a
general review see \cite{J-Schwarz}).
Duality is a typical ``stringy'' property without any counterpart in
field theory, since it depends crucially on the extended nature of the
string.

$R$-duality is not an exclusive property of the bosonic string. It
appears
also in theories with fermions like the heterotic and closed
superstrings. In recent years this property has been considered as
lending support to the idea that string theory has a fundamental minimal
length. This and the study of the high energy string scattering has led
to the formulation
of a generalized uncertainity principle \cite{Veneziano} which reads
\begin{equation}
\Delta x \sim \frac{\hbar}{\Delta E}+\alpha^{'}\Delta E
\label{Veneziano}
\end{equation}
It is easy to see that there is a lower bound for the right hand side of
(\ref{Veneziano}) corresponding to $\sqrt{\alpha^{'}\hbar}$. This
would
impose a lower bound to any measurable distance that would be of the
order of the self-dual radius since, by duality, we are unable to
distinguish between spaces of size $R$ and $(\alpha^{'}\hbar)/R$. In
some sense this means that strings cannot probe distances shorter than
its proper length (which is of the order of the Planck length).

\subsection{Operator formalism}

Now we are going to study the operator formalism for the bosonic string
in a genus $g$ Riemann surface \cite{Alvarez-Gaume-Gomez-Moore-Vafa}.
This formalism provides a general framework where we can analyze many of
the properties of String Theory to arbitrary orders of perturbation
theory. In particular we can easily understand the origin of potential
infinities in string amplitudes, and also the geometrical interpretation
of the physical state conditions. Before doing so, it would be useful
to review some features of Riemann surfaces that we will use later
\cite{Alvarez-Gaume-Moore-Vafa}. It is well known that every
orientable two-dimensional surface is topologically characterized by its
genus $g$ (or equivalently by the Euler characteristic $\chi=2-2g$).
The homology groups of a genus $g$ surface have dimensions
\begin{equation}
dim\,H_{0}(\Sigma_{g})=1 \hspace{1cm} dim\,H_{1}(\Sigma_{g})=2g
\hspace{1cm} dim\,H_{2}(\Sigma_{g})=1\;.
\end{equation}
We choose a basis for the first homology group $H_{1}(\Sigma_{g})$ by
considering the $2g$ cycles $a_{i}$, $b_{i}$ ($1\leq i \leq g$) as shown
in fig. \ref{cycles}.
\begin{figure}
\vspace{3cm}
\caption{Basis of the first homology group $H_{1}(\Sigma_{g})$}
\label{cycles}
\end{figure}
Any closed curve on the surface $\Sigma$ can be uniquely decomposed
in terms of curves belonging to the homology classes of $(a_{i},b_{i})$.
Moreover, once we have chosen a homology basis we can represent the
genus $g$ surface by a $4g$-sided polygon with proper identifications
on its boundary. In fig. \ref{g=2} we can see how this works for a
genus $2$ surface. We have an octagon, and by gluing
$a_{i}a_{i}^{-1}$ and $b_{i}b_{i}^{-1}$ we obtain the original surface.
\begin{figure}
\vspace{3cm}
\caption{Octagon obtained by cutting a genus $2$ Riemann surface along
the elements of the basis of $H_{1}(\Sigma_{2})$}
\label{g=2}
\end{figure}
With the basis $\{a_{i},b_{i}\}$ we can associate a set of $1$-forms
$\{\alpha_{i},\beta_{i}\}$ ($1\leq i \leq g$) satisfying
\begin{eqnarray}
\int_{a_{i}}\alpha_{j}&=&\delta_{ij}\;, \hspace{2cm}
\int_{a_{i}}\beta_{j}=0 \nonumber \\
\int_{b_{i}}\alpha_{j}&=&0\;, \hspace{2.3cm}
\int_{b_{i}}\beta_{j}=\delta_{ij}\;.
\end{eqnarray}
When we introduce a complex structure in the two-dimensional surface
we have a Riemann surface, and we can decompose the cotangent space
$T^{*}\Sigma$ into two sets
\begin{equation}
T^{*}\Sigma=T^{*(1,0)}\Sigma\oplus T^{*(0,1)}\Sigma
\end{equation}
depending of whether the $1$-forms are locally of the form
$f(z,\bar{z})dz$ or $f(z,\bar{z})d\bar{z}$. In particular there exists
a set of $g$ holomorphic $(1,0)$ differentials $\omega_{i}$
(locally of the form $\omega_{i}=f_{i}(z)dz$) and which can be
normalized according to
\begin{equation}
\int_{a_{i}}\omega_{j}=\delta_{ij}\;.
\end{equation}
This normalization determines completely the differentials $\omega_{i}$
(called Abelian differentials of the first kind). The integration of
$\omega_{i}$ along the cycles $b_{j}$ define the period matrix of the
Riemann surface
\begin{equation}
\int_{b_{i}} \omega_{j}=\Omega_{ij}\;.
\end{equation}
It can be shown that the period matrix verifies
$\Omega_{ij}=\Omega_{ji}$ and $Im\,\Omega_{ij}>0$. The set of matrices
with these properties define the Siegel upper half plane ${\bf H}_{g}$.
In fact, ${\bf H}_{g}$ contains the Teichm\"{u}ller space for a genus
$g$ Riemann surface, i.e., the space
\begin{equation}
{\cal
T}_{g}=\frac{\mbox{Metrics}}{\mbox{Weyl}\times
\mbox{Diff}_{0}^{+}(\Sigma)}
\end{equation}
with $\mbox{Diff}_{0}^{+}(\Sigma)$ the set of orientation preserving
diffeomorphisms
on $\Sigma$ connected with the identity. The moduli space ${\cal M}_{g}$
\footnote{The space of inequivalent complex structures on the surface
$\Sigma_{g}$.} introduced above is related to the Teichm\"{u}ller
space by \begin{equation}
{\cal M}_{g}=\frac{{\cal T}_{g}}{\Omega(\Sigma)}
\end{equation}
with the mapping class group defined by
\begin{equation}
\Omega(\Sigma)
=\frac{\mbox{Diff}^{+}(\Sigma)}{\mbox{Diff}_{0}^{+}(\Sigma)}
\end{equation}

Another important subject in the theory of Riemann surfaces is the
definition of
spin structures. In a genus $g$ Riemann surface there are $2^{2g}$
spin structures corresponding, roughly speaking, to the assignment of
periodic or
antiperiodic boundary conditions for the fermions along each generator
of the homology group $H_{1}(\Sigma)$. These spin structures are
classified into two groups; a spin structure is even if the number of
zero modes of the chiral Dirac operator is even and it is odd otherwise.
In the case of the torus, for example, we have four spin structures
corresponding to $(P,P)$, $(P,A)$, $(A,P)$ and $(A,A)$ with $P$ ($A$)
indicating periodic (antiperiodic) boundary conditions along the
homology basis $(a,b)$. With the flat metric on the torus we can see
that $(P,P)$ is the only odd spin structure. The other three are even.

After this brief review about some topics on Riemann surfaces, we
analize
the origin of the divergences appearing in the partition function of the
bosonic string \cite{Nelson}. The analysis in the case of fermionic
strings is similar. To do so we have to convince
ourselves that, in the sum over world-sheets, the divergences come
from those surfaces lying on the boundary of moduli space  ${\cal
M}_{g}$. The presence of a tachyon in the spectrum of the bosonic
string makes the contribution coming from surfaces with long narrow
tubes divergent (fig. \ref{tubes})
\begin{figure}
\vspace{3cm}
\caption{Riemann surface containing a long narrow tube. Divergences in
the bosonic string amplitudes appear when summing the contributions from
this type of surfaces in the limit in which the length of the tube
tends to infinity}
\label{tubes}
\end{figure}
This can be easily seen from the partition function on the cylinder
constructed in sec. 3.2. The tachyon in the spectrum makes the
integration over the length of the cylinder divergent since the
integrand grows exponentially with the length.

In order to study the divergences of String Theory it is convenient to
add to the moduli space ${\cal M}_{g}$ all the points on its boundary to
construct the compactified moduli space $\bar{\cal M}_{g}$.
Points in $\bar{\cal M}_{g}-{\cal M}_{g}$ represent Riemann
surfaces with infinitely long narrow tubes. Equivalently we can
consider the conformally related surfaces in which we have nodes, i.e.,
in which some curve is pinched off (fig \ref{pinched}).
\begin{figure}
\vspace{3cm}
\caption{Riemann surface with a cycle pinched off}
\label{pinched}
\end{figure}
It is easy to see that a neigbourhood of that nodes is not topologically
equivalent to a piece of ${\bf C}$ but to two disks joined by their
centers (fig. \ref{node}).
\begin{figure}
\vspace{3cm}
\caption{Local aspect of a node}
\label{node}
\end{figure}
We will say that one of this surfaces in the boundary of the moduli
space is of type
$\Delta_{i}$ if the degenerate surface splits a genus $g$ surface into
two joined surfaces of genus $i$ and $g-i$ (fig. \ref{Delta}). By
$\Delta_{0}$ we denote
those surfaces in which a homologically non-trivial cycle
is pinched
off and we obtain a genus $g-1$ surface with two points
identified. This surfaces can be obtained by sewing Riemann surfaces
with sewing parameter $q=0$.
\begin{figure}
\vspace{3cm}
\caption{Splitting of a genus $g$ Riemann surface into two Riemann
surface by pinching off a homologically trivial cycle}
\label{Delta}
\end{figure}

We now present the construction of the operator formalism for higher
genus Riemann surfaces. In sec. 3.3 we studied the operator formalism
in the cylinder by passing to the punctured complex plane where any
correlation function can be represented as
\begin{equation}
\langle 0|\Phi_{1}(z_{1})\ldots \Phi_{N}(z_{N})|0\rangle\;.
\end{equation}
For higher genus surfaces we would like to construct a
state $|W\rangle$ characterizing the details about the topology and
complex structure of the surface, and such that
\begin{equation}
\frac{\langle 0|\Phi_{1}(z_{1})\ldots
\Phi_{N}(z_{N})|W\rangle}{\langle 0|W\rangle}
\end{equation}
coincides with the path integral computation of the corresponding
correlator. To achieve this goal we begin by defining the augmented
moduli space ${\cal P}(g,n)$ which is
the moduli space of genus $g$ surfaces with $n$ parametrized boundaries,
i.e., with $n$ distincted points $P_{i}$ and $n$ local parameters
$z_{i}(P_{i})$. This set has the property that the mapping class group
of a genus $g$ Riemann surface with $n$ distinguished points
$\Sigma_{g,n}$
\begin{equation}
\Omega(\Sigma_{g,n})
=\mbox{Diff}(\Sigma_{g,n})/\mbox{Diff}_{0}(\Sigma_{g,n})
\end{equation}
equals the fundamental group $\pi_{1}({\cal P}(g,n))$. Moreover, this
set is by far more convenient than ${\cal M}_{g,n}$ because in order
to define oscillators, creation and anihilation operators, etc., in the
neighborhood of $P_{i}$ we need local parameters $z_{i}$ to define
the Laurent expansion of the various fields. Besides, ${\cal P}(g,n)$
resolves the orbifold singularities of ${\cal M}_{g,n}$.

The construction of the operator formalism for a general Riemann surface
amounts to the construction of a map between the space ${\cal P}(g,n)$
and
${\cal H}\otimes\ldots\otimes{\cal H}$ where ${\cal H}$ is the Hilbert
space of the theory. This means that to every CFT we associate
a ray in ${\cal H}^{\otimes n}$ for every point of ${\cal P}(g,n)$. By
making use of the sewing technique
we can define a kind of semigroup structure in ${\cal P}(g,n)$ by
defining a ``composition''
\begin{eqnarray}
{\cal P}(g_{1},n_{1})\times {\cal P}(g_{2},n_{2}) &\rightarrow &
{\cal P}(g_{1}+g_{2},n_{1}+n_{2}-2) \\
P\in {\cal P}(g_{1},n_{1}), Q\in {\cal P}(g_{2},n_{2}) &\rightarrow
& P_{i}\infty_{j} Q \in {\cal P}(g_{1}+g_{2},n_{1}+n_{2}-2) \nonumber
\end{eqnarray}
which correspond to sewing together the $i$th puncture of the
first surface with the $j$th puncture of the second one (fig.
\ref{operations}). We can also define the operation
\begin{eqnarray}
{\cal P}(g,n) &\longrightarrow & {\cal P}(g+1,n-2) \nonumber \\
P \in {\cal P}(g,n) &\longrightarrow & P8^{i}_{j} \in {\cal P}(g+1,n-2)
\end{eqnarray}
in which we sew the $i$th and $j$th punctures of $P$ (see fig.
\ref{sewing}).
\begin{figure}
\vspace{3cm}
\caption{Construction of a genus $g_{1}+g_{2}$ Riemann surface by sewing
two Riemann surfaces with genera $g_{1}$ and $g_{2}$}
\label{operations}
\end{figure}
This two operations allows us to construct ${\cal P}(g,n)$ starting with
${\cal P}(0,3)$ and ${\cal P}(0,2)$ (fig. \ref{buildings}).
\begin{figure}
\vspace{3cm}
\caption{Buildings blocks for the construction of any element of ${\cal
P}(g,n)$}
\label{buildings}
\end{figure}
These operations can be represented in the space of states as follows:
let $\{|n\rangle\}$ be an orthonormal basis in ${\cal H}$.
We will associate with the Riemann surface in fig \ref{buildings}-b the
state
\begin{equation}
|S_{ij}\rangle=\sum_{n}|n\rangle_{i}|n\rangle_{j}
\end{equation}
If we have two surfaces $P\in {\cal P}(g_{1},n_{1})$, $Q \in{\cal
P}(g_{2},n_{2})$ with associated states $|P\rangle \in {\cal H}^{\otimes
n_{1}}$, $|Q\rangle \in {\cal H}^{\otimes n_{2}}$, the state associated
with $P_{i}\infty_{j}Q$ is given by
\begin{equation}
|R\rangle=\langle S_{ij}|P\rangle\otimes |Q\rangle\;.
\end{equation}
We see that starting with
${\cal P}(0,3)$ and using $|S_{ij}\rangle$ we arrive in principle to
any surface, so our procedure is quite similar to a Feynman diagram
technique \footnote{This procedure has been made explicite in the
construction of a closed String Field Theory by B. Zwiebach (see lecture
by B. Zwiebach and \cite{SFT}).}.

Once we have a correspondence between elements in ${\cal P}(g,n)$ and
the states in ${\cal H}^{\otimes n}$ we can reformulate Polyakov's
prescription in an operator language. For example if we want to evaluate
the amplitude for $n$ external strings in the on-shell states
$|\chi_{i}\rangle \in {\cal H}$ we have that the contribution coming
from a Riemann surface $W\in {\cal P}_{g,n}$ to the amplitude will be
given by
\begin{equation}
\langle\chi_{1}|\ldots\langle\chi_{n}|W\rangle\;,
\end{equation}
so the total amplitude $A(1,\ldots,n)$ can be schematically written as
\begin{equation}
A(1,\ldots,n)=\sum_{g=0}^{\infty}g_{st}^{-\chi}\sum_{W}
\langle\chi_{1}|\ldots\langle\chi_{n}|W\rangle\;,
\label{amp.}
\end{equation}
where the sum over $W$ is given by an integral over the
moduli parameters.

It is worth noticing that only for CFT's we obtain a ray associated with
a point of ${\cal P}(g,n)$. Furthermore, for $g>1$ the first homology
group
$H_{1}[{\cal P}(g,n)]$ vanishes (Harer theorem \cite{Harer}); this is a
very useful
result in order to characterize the Polyakov measure since it implies
that any flat line bundle is necessarily trivial. From (\ref{amp.}) it
is clear now that the aim of the operator formalism is to obtain an
operator representation of ``scattering'' measures on ${\cal M}_{g,n}$.

The problem that arises is to determine the action of the Virasoro
generators on the space of states.
Let be $R\in {\cal P}(g,n)$ and let $P_{i}$ be a point in $R$ with
local parameter $z_{i}$ (see fig. \ref{disk}). We now
cut an annulus off
the disk around the point $P_{i}$ and transform it by the action of a
meromorphic vector field $v(z_{i})$
\begin{equation}
z_{i}\rightarrow z_{i}+\epsilon v(z_{i})\;,
\label{vector}
\end{equation}
($v(z_{i})$ may have poles at $P_{i}$).
When this is done, we fill the inside of the annulus to get a disk and
glue it back on the surface. This is a deformation of the
original Riemann surface and we may wonder what is the relation of the
new surface with the old one. There are three posibilities
\begin{figure}
\vspace{3cm}
\caption{Deformation of the Riemann surface by cutting out a disk and
gluing it back after deforming it}
\label{disk}
\end{figure}

\begin{itemize}
\item The vector field $v(z_{i})$ is holomorphic on the disk and
vanishes at $P_{i}$.
In this case the transformation (\ref{vector}) is equivalent to choosing
different coordinates at $P_{i}$. In the case in which the vector field
is holomorphic on the disk but does not vanish at $P_{i}$
we have an infinitesimal translation of the point $P_{i}$.

\item The vector field can be extended to a holomorphic vector field on
the rest of the surface (i.e., $\Sigma-P_{i}$). In this case the
transformed surface is
identical to the original one, since any transformation (\ref{vector})
can be undone by a transformation generated by $v$ in the rest of the
surface.

\item We have none of the preceeding posibilities. $v(z)$ is holomorphic
in the annulus but does not extend holomorphically either to the disk or
to $\Sigma-P_{i}$. Then (\ref{vector})
is an infinitesimal moduli deformation, i.e., an infinitesimal motion in
the space ${\cal M}_{g,n}$.

\end{itemize}

The next problem is to represent moduli deformations, or any of the
changes induced by $v(z)$ in ${\cal
H}^{\otimes n}$. This is acomplished by an operator $O(v)$ acting
on the	states $|W\rangle$. Because of the fundamental r\^{o}le played
by the
Virasoro algebra in any CFT we will consider the energy-momentum tensor
in a neighborhood of the point $P_{i}$
\begin{equation}
T(z)=\sum_{n\in{\bf Z}} L_{n}z^{-n-2}
\end{equation}
and construct the operator $T(v)$ associated with the deformation
induced by $v(z)$ as
\begin{equation}
T(v)=\frac{1}{2\pi i}\oint_{P_{i}}dz\,T(z)v(z)
\end{equation}
The change in $|W\rangle$ by the deformation generated by $v(z)$
is given by
\begin{equation}
\delta_{v}|W\rangle=[T(v)+\bar{T}(\bar{v})]|W\rangle
\label{delt-v}
\end{equation}
This should be taken as the defining property of the action of the
Virasoro algebra on the states of the theory. Since for matter fields
the Virasoro algebra has a central extension, equation (\ref{delt-v})
makes sense only in terms of rays.

A final consistency condition in the construction of $|W\rangle$ for
$P\in {\cal P}(g,n)$ requires that $L_{0}-\bar{L}_{0} \in {\bf Z}$. This
condition has to be imposed since we can always make a Dehn twist
around the point $P$, which in terms of the local
holomorphic coordinate consists of
a trasformation $z \rightarrow \exp{(2\pi i\theta)}z$ with $0\leq \theta
\leq 1$. This is done by acting with the
operator $\exp{[2\pi i\theta(L_{0}-\bar{L}_{0})]}$ on the state
$|W\rangle$. In the case $\theta=1$ for the state to be invariant we
must have $L_{0}-\bar{L}_{0}\in {\bf Z}$, condition which is necessary
for modular invariance.

We now leave the general set-up and present a few useful examples which
should clarify the machinery explained so far.
The first example we will study of the operator formalism is
a pair of spin $\frac{1}{2}$ holomorphic spinors
\begin{eqnarray}
b(z)dz^{\frac{1}{2}}=\sum_{n\in{\bf
Z}+\frac{1}{2}}b_{n}z^{n-\frac{1}{2}}dz^{\frac{1}{2}}\;, \nonumber \\
c(z)dz^{\frac{1}{2}}=\sum_{n\in{\bf
Z}+\frac{1}{2}}c_{n}z^{n-\frac{1}{2}}dz^{\frac{1}{2}}\;,
\end{eqnarray}
with the anticommutation relations
\begin{equation}
\{b_{m},c_{n}\}=\delta_{m+n,0}\;, \hspace{1cm}
\{b_{m},b_{n}\}=\{c_{m},c_{n}\}=0\;,
\end{equation}
which means that $b(z)$ can be represented as the ``translation''
operator for $c(z)$
\begin{equation}
b(z)\sim \frac{\delta}{\delta c(z)}\;,
\end{equation}
The action for these fields is
\begin{equation}
S=\frac{1}{\pi}\int d^{2}z\,b\bar{\partial}c\;.
\end{equation}

\begin{figure}
\vspace{3cm}
\caption{Riemann surface divided into two parts by the circle $S^{1}$}
\label{surface}
\end{figure}

We will try to associate a state $\Psi$ with each side of the surface in
fig.
\ref{surface}, $\Sigma_{1}$ and $\Sigma_{2}$, in such a way that on the
Fock space of the circle $S^{1}$ the partition function becomes
\begin{equation}
Z=\int {\cal D}b{\cal D}c \exp{\left(-\int d^{2}z\,
b\bar{\partial}c\right)}=\langle\Psi_{2}|\Psi_{1}\rangle\;.
\end{equation}
We represent the state $\Psi$ as a functional given by
\begin{equation}
\Psi_{1}[f]=\int_{c|_{S^{1}}=f}{\cal D}b{\cal D}c\exp{\left(\,
\int_{\Sigma_{1}}d^{2}z\,c\bar{\partial}b + \oint_{S^{1}} dz\,cb
\right)}\;,
\label{psi-1}
\end{equation}
where the boundary term in the exponential accounts for the flux of the
fermionic current through $S^{1}$ and we impose no boundary condition
on $b(z)$\footnote{For spinor $b$ and $c$ are canonically conjugate
variables, and (\ref{psi-1}) is a hamiltonian functional integral
representation of the wave function.}. If $w_{n}$ is the boundary
value at $S^{1}$ of a holomorphic spinor on $\Sigma_{1}$ and we consider
$\Psi_{1}[f+w_{n}]$, after making a shift in the $c$-field
$c\rightarrow c-w_{n}$ it is easy to verify that
\begin{equation}
\Psi_{1}[f+w_{n}]=\Psi_{1}[f]
\label{Qinvariance}
\end{equation}
Since, as we indicated above, $b(z)$ plays the role of the translation
operator for $c(z)$, we can define the conserved charge
\begin{equation}
Q(w_{n})=\frac{1}{2\pi i}\oint_{S^{1}}dz b(z)w_{n}(z)\;.
\end{equation}
and the invariance (\ref{Qinvariance}) becomes
\begin{equation}
Q(w_{n})\Psi_{1}=0
\label{c-1}
\end{equation}
Furthermore, $Q(w_{n})$ depends only on the homology class of
the contour.

Notice also that the path integral is invariant under the translation
of $b(z)$ by a spinor $\tilde{w}_{n}$ holomorphic off $P$ (see fig.
\ref{P}). This invariance can be implemented by defining the operator
\begin{equation}
\tilde{Q}_{m}=Q(\tilde{w}_{n})=\frac{1}{2\pi i}\oint_{S^{1}}dz
\,c(z)\tilde{w}_{n}\;.
\end{equation}
thus $\Psi_{1}[f]$ satisfies
\begin{equation}
\tilde{Q}_{n}\Psi_{1}=0\;.
\label{c-2}
\end{equation}
The two charges $Q_{n}=Q(w_{n})$ and $\tilde{Q}_{m}=Q(\tilde{w}_{m})$
verifies the anticommutation relation
\begin{equation}
\{Q_{n},\tilde{Q}_{m}\}=\frac{1}{2\pi i}\oint_{S^{1}} dz\,w_{n}(z)
\tilde{w}_{m}(z)
\end{equation}
which vanishes by deforming the contour of integration, and by the fact
that $w_{n}$, $\tilde{w}_{n}$ are holomorphic off $P$
\begin{equation}
\{Q_{n},\tilde{Q}_{m}\}=0\;.
\end{equation}

\begin{figure}
\vspace{3cm}
\caption{Riemann surface with a single puncture at $P$}
\label{P}
\end{figure}

$Q_{n}$ and $\tilde{Q}_{m}$ give a maximal set of conditions on the
state $\Psi_{1}$. To associate a state with a given Riemann surface
$W\in{\cal P}(g,n)$ we look for all the meromorphic sections $w_{n}$,
$\tilde{w}_{m}$ of the spinor bundle which are allowed to have poles
only at the punctures, we then construct the conserved charges
$Q(w_{n})$, $Q(\tilde{w}_{n})$ and define the state by
(\ref{c-1},\ref{c-2}). As a first example let us consider the case of
a sphere with a single puncture at $z=0$. In this case the meromorphic
sections with poles only at $z=0$ are of the form
\begin{equation}
w_{n}(z)=z^{-n}dz^{\frac{1}{2}}\;, \hspace{1cm} n>0\;.
\end{equation}
Then, taking into account the mode expansion for $b(z)$ and $c(z)$ we
find that the operators $Q_{n}$, $\tilde{Q}_{m}$ are
\begin{eqnarray}
Q_{n}=\int_{P}\frac{dz}{2\pi i}b(z)z^{-n}=b_{n-\frac{1}{2}}\;, \nonumber
\\ \tilde{Q}_{m}=\int_{P}\frac{dz}{2\pi
i}c(z)z^{-n}=c_{{n-\frac{1}{2}}}\;,
\end{eqnarray}
with $n>0$. Then the state $|W\rangle$ associated with the sphere with
one puncture is defined as
\begin{equation}
b_{n-\frac{1}{2}}|W\rangle=c_{{n-\frac{1}{2}}}|W\rangle=0\;,
\hspace{1cm} n>0\;,
\label{0-rangle}
\end{equation}
so $|W\rangle$ is the $SL(2,{\bf C})$-invariant vacuum state.

We can explicitly construct the state associated with a genus-$g$
Riemann surface with a single puncture at $P$. To do so we first have to
find meromorphic sections that have poles only at $P$. The Riemann-Roch
theorem implies that these sections exist with poles of arbitrary order
at the point $P$. The starting point is the Szeg\"{o} kernel for spin
$\frac{1}{2}$. Let us consider a odd spin structure over $\Sigma$
labeled by $(\alpha,\beta)$; then the Szeg\"{o} kernel, which is the
two-point function for fermions $\langle c(z)b(w)\rangle$ on $\Sigma$,
is given by (for a constructive derivation see
\cite{Alvarez-Gaume-Gomez-Reina})
\begin{equation}
S(z,w)=\frac{\vartheta\left[
\begin{array}{c}
\alpha \\
\beta
\end{array}
\right](z-w|\Omega)}{\vartheta\left[
\begin{array}{c}
\alpha \\
\beta
\end{array}
\right](0|\Omega)E(z,w)}
\end{equation}
where $\Omega$ is the period
matrix and $E(z,w)$ is the prime
form associated to $\Sigma$. The prime form is defined as the unique
$(-\frac{1}{2},-\frac{1}{2})$ differential such that
$E(z,w)=-E(w,z)$ and vanishes only when $z=w$. In order to construct it
we begin with the function defined for an odd non-singular spin
structure (with just one zero mode)
\begin{equation}
f(z,w)=\vartheta\left[
\begin{array}{c}
\alpha \\
\beta
\end{array}
\right](\int_{w}^{z}\omega|\Omega)
\end{equation}
By using the Riemann Vanishing Theorem \cite{Alvarez-Gaume-Moore-Vafa}
we find that this function vanishes in $g-1$ points
$P_{1},\ldots,P_{g-1}$ as a function of $z$ with $w$ fixed. Moreover,
since it also vanish when $z\rightarrow w$, this function, when
considered as a function of $w$ with $z$ fixed also vanish at the same
points. It can be seen that when $z\sim w\sim P_{i}$, $f(z,w)\sim
(z-w)(z-P_{i})(w-P_{i})$ so after differentiation with respect to $z$ we
can define the holomorphic form
\begin{equation}
h^{2}(z)=\sum_{j=1}^{g}\omega_{i}(z)\frac{\partial}{\partial u_{i}}
\left.\vartheta \left[
\begin{array}{c}
\alpha \\
\beta
\end{array}
\right](\int_{z}^{w}\omega|\Omega)\right|_{z=w}\;,
\end{equation}
where the $u_{i}$ stand for the arguments of the theta function.
$h^{2}(z)$
has second order zeros at the points $P_{i}$, and therefore we can take
the square root without introducing cuts. The prime form is defined as
\begin{equation}
E(z,w)=\frac{\vartheta\left[
\begin{array}{c}
\alpha \\
\beta
\end{array}
\right](z-w|\Omega)}{h(z)h(w)}\;.
\end{equation}
The sections we are looking for are given by differentiation of the
Szeg\"{o} kernel with respect to $z$ and setting $w=0$. For $z$, $w$
in a neigborhood of $P\in \Sigma$ we have
\begin{equation}
w_{n}(z)=\left.\frac{1}{(n-1)!}\frac{\partial^{n-1}}{\partial w^{n-1}}
S(z,w)\right|_{w=0}=
\frac{1}{z^{n}}+\sum_{m=1}^{\infty}B_{nm}z^{m-1}\;,
\end{equation}
where $B_{nm}$ is given by
\begin{equation}
B_{nm}=\frac{1}{(n-1)!(m-1)!}\frac{\partial^{m-1}}{\partial z^{m-1}}
\frac{\partial^{n-1}}{\partial
w^{n-1}}\left(S(z,w)-\frac{1}{z-w}\right)\;.
\end{equation}
Looking at the anticommutation relations it is easy to see that we can
find a representation of the operators $c_{n}$, $b_{n}$ ($n\in {\bf
Z}+\frac{1}{2}$) in terms of differential operators in which $b_{-n}$
and $c_{-n}$ for $n>0$ acts as multiplication operators and
\begin{equation}
b_{n}=\frac{\partial}{\partial c_{-n}}\;, \hspace{1cm}
c_{n}=\frac{\partial}{\partial b_{-n}}\;, \hspace{1cm} n>0\;.
\end{equation}
Then, the charges $Q_{n}$, $\tilde{Q}_{m}$ become:
\begin{eqnarray}
Q_{n}=b_{n-\frac{1}{2}}+\sum_{m=1}^{\infty}B_{nm}b_{-m+\frac{1}{2}}
\nonumber \\
\tilde{Q}_{n}=
c_{n-\frac{1}{2}}+\sum_{m=1}^{\infty}B_{nm}c_{-m+\frac{1}{2}}
\label{Bogoliubov}
\end{eqnarray}
and we obtain a system of differential equations for
$\Psi(c_{-n},b_{-n})$. The resulting state is
\begin{equation}
|\Psi\rangle=C\,\exp{\left(-\sum_{n,m=1}^{\infty}B_{nm}c_{-n+\frac{1}{2}}
b_{-m+\frac{1}{2}}\right)}|0\rangle\;.
\end{equation}
This is just a Bogoliubov transformation on
the standard vacuum state $|0\rangle$ defined by (\ref{0-rangle}). The
charge operators $Q_{n}$, $\tilde{Q}_{m}$, which are linear combinations
of both creation and annihilation operators with respect to
$|0\rangle$, are the annihilation operators with respect to the new
vacuum state $|\Psi\rangle$. All the geometric properties are neatly
summarized by the Bogoliubov transformation.

In the case in which $\Sigma \in {\cal P}(g,n)$ we have to find
meromorphic sections of the spinor bundle with poles at
$P_{1},\ldots,P_{n}$. The simplest example is the sphere with two
punctures that we can arbitrarily locate at $0$ and $\infty$. With this
surface we have associated the ``sewing'' state $|S_{12}\rangle$.
The meromorphic sections on the sphere with poles at
$z=0$ and $z=\infty$ are generated by
\begin{equation}
w_{n}=z^{n}dz^{\frac{1}{2}}\;, \hspace{1cm} n \in {\bf Z}\;.
\end{equation}
It is easy to get the conserved charges. Integrating in patches around
the two points we have
\begin{eqnarray}
Q_{n}=\oint_{P_{1}}\frac{dz_{1}}{2\pi i}b^{(1)}(z_{1})w_{n}(z_{1})+
\oint_{P_{2}}\frac{dz_{2}}{2\pi i}b^{(2)}(z_{2})w_{n}(z_{2})\;,
\nonumber \\
\tilde{Q}_{n}=\oint_{P_{1}}\frac{dz_{1}}{2\pi
i}c^{(1)}(z_{1})w_{n}(z_{1})+
\oint_{P_{2}}\frac{dz_{2}}{2\pi i}c^{(2)}(z_{2})w_{n}(z_{2})\;,
\end{eqnarray}
and solving the corresponding set of differential equations, we obtain
\begin{equation}
|S_{12}\rangle=\prod_{m=1}^{\infty}
\exp{\left(c^{(1)}_{-m+\frac{1}{2}}b^{(2)}_{-m+\frac{1}{2}}+
c^{(2)}_{-m+\frac{1}{2}}b^{(1)}_{-m+\frac{1}{2}}\right)}
|0\rangle_{1}\otimes|0\rangle_{2}\;,
\end{equation}
which satisfies
\begin{equation}
(Q_{1}+Q_{2})|S_{12}\rangle=(\tilde{Q}_{1}+\tilde{Q}_{2})|S_{12}\rangle
=0\;.
\end{equation}

Previously we saw that by sewing two Riemann surfaces
$P\in {\cal P}(g_{1},n_{1})$, $Q\in {\cal P}(g_{2},n_{2})$ with sewing
parameter $q=1$ we obtained a new Riemann surface $R=P_{i}\infty_{j}Q
\in{\cal P}(g_{1}+g_{2},n_{1}+n_{2}-2)$. As an example of the use of
the operator formalism we are going to verify that the state associated
with $R$
\begin{equation}
|R\rangle=\langle S_{ij}|P\rangle\otimes |Q\rangle
\end{equation}
verifies the condition
\begin{equation}
(Q_{1}+\ldots+Q_{i-1}+Q_{j+1}+\ldots+Q_{n_{1}+n_{2}})|R\rangle=0
\end{equation}
and correspondingly for the $\tilde{Q}_{j}$ operators.
Since (fig. \ref{remnants}) in $R$ we will have some remnants of the two
discs around the $i$th and $j$th punctures with coordinates $t_{i}$ and
$t_{j}$ respectively, if we expand holomorphic spinors in $R$ away from
the points $P_{1},\ldots,P_{i-1},P_{j+1},\ldots,P_{n_{1}+n_{2}}$ we
could express them in terms of the coordinates $t_{i}$, $t_{j}$.
These spinors will have at most poles at $t_{i}=t_{j}=0$, therefore
\begin{equation}
\langle S_{ij}|(Q_{i}+Q_{j})=0
\end{equation}
and we can write
\begin{eqnarray}
& &(Q_{1}+\ldots+Q_{i-1}+Q_{j+1}+\ldots+Q_{n_{1}+n_{2}})\langle
S_{ij}|P\rangle\otimes|Q\rangle \\
&=&\langle
S_{ij}|(Q_{1}+\ldots+Q_{i})+(Q_{j}+Q_{j+1}+\ldots+Q_{n_{1}+n_{2}})
|P\rangle\otimes |Q\rangle=0   \nonumber
\end{eqnarray}
as was to be shown.
\begin{figure}
\vspace{3cm}
\caption{Remnants of the two disks located around $P_{i}$ and $P_{j}$
with local parameters $t_{i}$, $t_{j}$ after sewing them together}
\label{remnants}
\end{figure}

Another case of interest is a single-valued scalar field
$\phi$. We proceed along the lines of the spin-$\frac{1}{2}$ case
and construct the functional associated with the Riemann surface
$\Sigma_{1}$ with boundary $S^{1}$ (fig. \ref{scalar-field})
as the path integral
\begin{equation}
\Psi[f]=\int_{\phi|_{S^{1}}=f}{\cal D}\phi
\exp{\left(-\frac{1}{2}\int_{\Sigma_{1}}d\phi\wedge *d\phi\right)}
\end{equation}
with $f$ a real function defined on $S^{1}$. It is possible to show that
there exists a harmonic function defined on $\Sigma_{1}$ and
which coincides with $f$ along the boundary $S^{1}$. We also denote this
function by $f$. Then we can make a shift in the scalar field
$\phi$ by $f$
\begin{equation}
\phi=\phi^{'}+f
\label{change-phi}
\end{equation}
where the new field $\phi^{'}$ satisfies the boundary condition
$\phi^{'}|_{S^{1}}=0$. Substituying (\ref{change-phi}) into the
definition of $\Psi[f]$ yields
\begin{equation}
\Psi[f]=\int_{\phi|_{S^{1}}=f}{\cal D}\phi\,e^{-S[\phi]}=
\int_{\phi^{'}|_{S^{1}}=0}{\cal D}\phi^{'}\,e^{-S[\phi^{'}+f]}
\end{equation}
\begin{figure}
\vspace{3cm}
\caption{Riemann surface with boundary $S^{1}$}
\label{scalar-field}
\end{figure}
We now evaluate $S[\phi^{'}+f]$
\begin{eqnarray}
S[\phi^{'}+f]&=&\frac{1}{2}\int_{\Sigma_{1}} d(\phi^{'}+f)\wedge
*d(\phi^{'}+f)= S[\phi^{'}]+\frac{1}{2}\int_{\Sigma_{1}} df\wedge *df
\nonumber \\
&+&\int_{\Sigma_{1}} d\phi^{'}\wedge *df=
S[\phi^{'}]+\frac{1}{2}\oint_{S^{1}} f*df\;,
\end{eqnarray}
where we have used the Stokes theorem, the boundary condition on
$\phi^{'}$ and the fact that $f$ is a harmonic fuction $d*df=0$ on
$\Sigma_{1}$. Finally the state functional becomes:
\begin{equation}
\Psi[f]=(det^{'}\,\Delta)_{\phi^{'}|_{S^{1}}=0}^{-\frac{1}{2}}
\exp{\left(-\frac{1}{2}\oint_{S^{1}} f*df\right)}\;.
\end{equation}
The conserved charges in this case can be written using the Green's
theorem
\begin{equation}
Q(h_{n})=\oint_{S^{1}}(h_{n}*d\phi-\phi*dh_{n})
\label{charge-scalar}
\end{equation}
where $h_{n}$ is a harmonic function analytic in $\Sigma_{1}$.
It is easy to see that since field equations for $\phi$ gives
$\partial\bar{\partial}\phi=0$ the operator $Q(h_{n})$ does not change
under deformations of $S^{1}$ and only depends on the homology class of
the boundary. As in the fermionic case, the state $\Psi$ is constructed
by requiring that it is annihilated by the conserved charges. By
splitting the function $h_{n}$
into holomorphic and antiholomorphic parts $h_{n}^{H}+h_{n}^{A}$, we
find that the charge (\ref{charge-scalar}) can be derived from the
conserved current
\begin{equation}
j(h_{n})=\partial\phi h_{n}^{H}-\bar{\partial}\phi h_{n}^{A}\;.
\end{equation}

The problem is to find an explicit expression for the functions
$h_{n}$. For example consider a genus $g$ Riemann surface with one
puncture at $P$. The simplest solutions are the real and imaginary parts
of a holomorphic function on $\Sigma-P$ with poles at $P$. Unfortunately
the Weierstrass gap theorem states that the order of the pole of a
holomorphic function at $P$ must be bigger than $g$. The problem arises
because we need $g$ extra conditions in order to determine completely
the state $\Psi$. This means that we cannot restrict to holomorphic
objects and we have to consider functions with holomorphic and
antiholomorphic parts. We start by considering a holomorphic
function with poles at $P$ of order less than $g$ but which, in order to
bypass Weierstrass' theorem, is allowed to be multivalued
around the cycles. Then we add some antiholomorphic piece
to obtain a single-valued harmonic function.
We begin with the multivalued meromorphic differentials:
\begin{equation}
\eta_{n}(t)=\frac{1}{(n-1)!}\left.\frac{\partial}{\partial t}
\frac{\partial^{n}}{\partial y^{n}} \log{E(t,y)}\right|_{y=0}\;.
\end{equation}
where $E(t,y)$ is the prime form defined earlier. With these
differentials we define the functions
\begin{equation}
h_{n}(t)=
\int^{t}[\eta_{n}(t)-A_{n}(Im\,\Omega)^{-1}(\omega-\bar{\omega})]\;,
\label{hn-func}
\end{equation}
where $\omega_{i}$ are the Abelian differentials and the coefficients
$A_{ni}$ are defined in terms of the local coordinate at $P$ through the
expansion
\begin{equation}
\omega_{i}(t)=\sum_{n=1}^{\infty}A_{ni}t^{n-1}dt\;.
\end{equation}
The $\eta_{n}(t)$ differentials satisfy
\begin{equation}
\oint_{b_{i}}\eta_{n}(t)=A_{ni}
\end{equation}
Using the properties of the prime form $E(t,y)$ \cite{Mumford} it is
possible
to check that the functions $h_{n}$ defined by (\ref{hn-func}) are
single-valued.

Now we expand the field $\phi(t)$ in oscillators using the familiar
expression
\begin{equation}
\phi(t)=q+ip\log{t}+ip\log{\bar{t}}+\sum_{n\neq
0}\left(\frac{a_{n}}{n}t^{n}+\mbox{c.c.}\right)\;,
\end{equation}
with the non-vanishing commutation relations
\begin{equation}
[q,p]=i\;, \hspace{1cm} [a_{n},a_{m}]=n\delta_{n+m,0}\;, \hspace{1cm}
[\bar{a}_{n},\bar{a}_{m}]=n\delta_{n+m,0}\;.
\end{equation}
By using these commutation relations and Green's theorem for harmonic
functions it is possible to show that the charges $Q(h_{n})$ commutes
\begin{equation}
[Q(h_{n}),Q(h_{m})]=0\;.
\end{equation}
We can construct a representation for operators $(a_{n},\bar{a}_{n})$ by
introducing variables $(x_{n},\bar{x}_{n})$ and defining
\begin{equation}
a_{n}=\frac{\partial}{\partial x_{n}}\;, \hspace{1cm}
a_{-n}=nx_{n}\;, \hspace{1cm} n\geq0\;,
\end{equation}
and the corresponding expressions for barred quantities. Then we impose
the conditions on the state $|\Psi\rangle$
\begin{equation}
Q(h_{n})|\Psi\rangle=\bar{Q}(\bar{h}_{n})|\Psi\rangle=0\;.
\end{equation}
Substituying the representation of the operators $(a_{n},\bar{a}_{n})$
in terms of $(x_{n},\bar{x}_{n})$ we arrive at a set of differential
equations on $\langle x,\bar{x}|\Psi\rangle$ that upon integration
yield:
\begin{equation}
\langle x,\bar{x}|\Psi\rangle=\exp{\left[(x,\bar{x})M\left(
\begin{array}{c}
x \\
\bar{x}
\end{array}
\right)\right]}\;,
\end{equation}
where the matrix $M$ is given by
\begin{equation}
M=\left(
\begin{array}{cc}
Q_{nm}+\frac{\pi}{2}A_{n}(Im\,\Omega)^{-1}A_{m} &
-\frac{\pi}{2}A_{n}(Im\,\Omega)^{-1}\bar{A}_{m} \\
-\frac{\pi}{2}\bar{A}_{n}(Im\,\Omega)^{-1}A_{m} &
\bar{Q}_{nm}+\frac{\pi}{2}\bar{A}_{n}(Im\,\Omega)^{-1}\bar{A}_{m}
\end{array}
\right)\;,
\end{equation}
and
\begin{equation}
Q_{nm}=\frac{1}{2(n-1)!(m-1)!}\frac{\partial^{n}}{\partial t^{n}}
\frac{\partial^{m}}{\partial y^{m}}
\log{\left(\frac{E(t,y)}{t-y}\right)}\;.
\end{equation}

As a particular case we can compute the state associated with a sphere
with two puctures (the sewing state $|S_{12}\rangle$) the result being
\begin{equation}
|S_{12}\rangle=\int dp \prod_{n=1}^{\infty}
\exp{\left(a_{n}^{(1)+}a_{n}^{(2)+}+\bar{a}_{n}^{(1)+}\bar{a}_{n}^{(2)+}
\right)}|p\rangle_{1}\otimes|p\rangle_{2}\;,
\end{equation}
where the state $|p\rangle$ carries momentum $p$. For a genus $g$
surface with $n$ points $\Sigma_{g,n}$ we introduce the
function
\begin{equation}
G(z,w)=-2\pi\left(Im\,\int_{z}^{w}\omega\right)(Im\,\Omega)^{-1}
\left(Im\,\int_{z}^{w}\omega\right)-\log{|E(z,w)|^{2}}\;,
\end{equation}
and define
\begin{eqnarray}
g(z_{i},w_{j})=\left\{
\begin{array}{ll}
G(z_{i},w_{j})+\log{|z_{i}-w_{j}|^{2}} & \mbox{if
$z_{i},w_{j}$ lie in the same patch} \\
G(z_{i},w_{j}) & \mbox{otherwise}\;.
\end{array}
\right.
\nonumber \\ \mbox{  }
\end{eqnarray}
We can split $g(z_{i},w_{j})$ into four parts corresponding to the
holomorphic or antiholomorphic contribution in each of the two variable.
After constructing with these four pieces the matrix $M(z_{i},w_{j})$ we
can write the unique ray associated with $\Sigma_{g,n}$ as
\begin{eqnarray}
& &|\Psi_{g}^{n}\rangle=\int dp_{1}\ldots dp_{n} \delta(\sum_{i}p_{i})
\nonumber \\& &\times \exp{
\left\{-\frac{1}{(2\pi i)^{2}}\sum_{i,j}\oint_{P_{i}}\oint_{P_{j}}
(\partial\phi(z_{i}),\bar{\partial}\phi(z_{i}))M(z_{i},z_{j})\left(
\begin{array}{c}
\partial\phi(z_{i}) \\
\bar{\partial}\phi(z_{j})
\end{array}
\right)\right\}} \nonumber \\
& &\times|p_{1}\rangle\otimes\ldots\otimes |p_{n}\rangle\;.
\end{eqnarray}

The last case we shall consider is the ghost system containing two
anticommutating fields: $b(z)$, with spin $2$, and
$c(z)$ with spin $-1$. In a sphere with a single puncture at $P$ with
$z(P)=0$ we can expand the fields as
\begin{equation}
b(z)=\sum_{n\in {\bf Z}}b_{n}z^{n-2}\;, \hspace{1cm} c(z)=\sum_{n\in
{\bf Z}}c_{n}z^{n+1}
\end{equation}
with the anticommutation relations
\begin{equation}
\{b_{n},c_{m}\}=\delta_{m+n,0}
\end{equation}
We have to look for vector fields holomorphic off $P$ and quadratic
differentials with the same property.
Using the Riemann-Roch theorem \cite{Griffiths-Harris} on the sphere it
is
possible to show that the only vector fields with the desired properties
are
\begin{equation}
\{z^{2}, z, 1, z^{-1},\ldots\} \frac{\partial}{\partial z}
\end{equation}
and for the quadratic differentials
\begin{equation}
\{z^{-4}, z^{-5}, z^{-6},\ldots\}dz^{2}
\end{equation}
The conserved charges associated with these sets are, in the case of
$b(z)$
\begin{equation}
\oint_{P}\frac{dz}{2\pi i}b(z)z^{k}=b_{-k+1}\;, \hspace{1cm}
k=2,1,0,-1,\ldots\;,
\end{equation}
and for $c(z)$
\begin{equation}
\oint_{P}\frac{dz}{2\pi i}c(z)z^{-k}=c_{k-2}\;, \hspace{1cm}
k=4,5,\ldots\;.
\end{equation}
Thus the state $|\phi_{0}\rangle$ associate with the sphere with one
puncture and the ghost system is defined by
\begin{eqnarray}
b_{n}|\phi_{0}\rangle&=&0 \hspace{1cm} n>-2 \nonumber \\
c_{n}|\phi_{0}\rangle&=&0 \hspace{1cm} n>1
\end{eqnarray}
which is an $SL(2,{\bf C})$-invariant state,
because it is annihilated by $L_{0},L_{\pm 1}$, where $L_{n}$ are the
Virasoro generators associated with the ghost system. We define the
ghost current by
\begin{equation}
j_{gh}(z)=:c(z)b(z):
\end{equation}
and the associated charge
\begin{equation}
Q_{gh}=\oint_{P}\frac{1}{2\pi i}j(z)
\end{equation}
Our first task will be to determine the ghost charge of the vacuum
$|\phi_{0}\rangle$. If we evaluate the commutator of $L_{0}$ with the
ghost fields we get
\begin{eqnarray}
[L_{0},b_{n}]=-nb_{n} \nonumber \\
{[}L_{0},c_{n}]=-nc_{n}
\end{eqnarray}
which shows that the state $|\phi_{0}\rangle$ is not a highest weight
state of the oscillators algebra, since it is not annihilated by
negative energy modes.
Moreover, the zero modes $b_{0}$, $c_{0}$ form a Clifford algebra since
$\{b_{0},c_{0}\}=1$. Then we can define the highest-weight states
$\{|+\rangle, |-\rangle\}$ which provide a representation of the
Clifford algebra:
\begin{eqnarray}
c_{n}|+\rangle&=&b_{n}|-\rangle=0\;, \hspace{1cm} n\geq 0 \nonumber \\
c_{0}|-\rangle&=&|+\rangle\;, \hspace{1cm} b_{0}|+\rangle=|-\rangle\;.
\end{eqnarray}
It can be argued \cite{Green-Schwarz-Witten} that the natural assignment
of ghost number to these states is
$Q|\pm\rangle=\pm\frac{1}{2}|\pm\rangle$. We can construct these states
from the $SL(2,{\bf C})$-invariant vacuum as
\begin{equation}
|-\rangle=c_{1}|\phi_{0}\rangle \hspace{1cm}
|+\rangle=c_{0}c_{1}|\phi_{0}\rangle
\end{equation}
and it is now straighforward to verify that
$L_{0}|\pm\rangle=-|\pm\rangle$. This is the origin of the tachyon.

We next construct the sewing state $|S_{12}\rangle$. For
a sphere with two punctures (one at $0$ and the other one at
$\infty$) the quadratic differentials and vector fields
holomorphic off those points are respectively
\begin{equation}
\mu=z^{n}dz^{2}\;, \hspace{1cm} v=z^{n+1}\frac{d}{dz}\;, \hspace{1cm}
n\in{\bf Z}\;.
\end{equation}
We proceed as in the previous cases. We impose
the conditions
\begin{eqnarray}
(b^{(1)}_{n}-b^{(2)}_{-n})|S_{12}\rangle&=&0 \nonumber \\
(c^{(1)}_{n}-c^{(2)}_{-n})|S_{12}\rangle&=&0
\end{eqnarray}
and obtain the solution
\begin{equation}
|S_{12}\rangle=\prod_{m=1}^{\infty}\exp{\left(-c_{-m}^{(1)}b_{-m}^{(2)}-
c_{-m}^{(2)}b_{-m}^{(1)}\right)}(b_{0}^{(1)}-b_{0}^{(2)})|+\rangle_{1}
\otimes |-\rangle_{2}
\end{equation}
If we evaluate the total ghost charge associated with the sewing state
we obtain
\begin{equation}
Q_{gh}|S_{12}\rangle=(Q^{(1)}_{gh}+Q^{(2)}_{gh})|S_{12}\rangle=0
\end{equation}

To generalize the construction to genus $g$ Riemann surfaces with $n$
punctures we have to take into account that for $g>1$ the Riemann-Roch
theorem states that there are $3g-3$ holomorphic quadratic
differentials. In the neighborhood of a point $P$ we can
write them using the local parameter $z$ as
\begin{equation}
\psi_{n+1}=z^{n}+\sum_{m\geq q}C_{nm}^{(2)}z^{m} \hspace{1cm}
q=3g-3
\end{equation}
with $n=0,\ldots,3g-4$. In a similar way we can write quadratic
differentials with poles at $P$ as
\begin{equation}
s_{n}=z^{-n}+\sum_{m\geq q}B_{nm}^{(2)}z^{m}
\end{equation}
and the corresponding expressions for the vectors. The coefficients
$B^{(2)}_{nm}$ could in principle be written in terms of prime forms
and $\vartheta$-functions. It is worth noticing that since
\begin{equation}
\oint_{P_{i}}\frac{dz}{2\pi i}c(z)\psi_{n+1}(z)=c_{-n-2}+\sum_{m\geq q}
C^{(2)}_{nm}c_{-m-2}
\end{equation}
and
\begin{equation}
\oint_{P_{i}}\frac{dz}{2\pi i}c(z)s_{1}(z)=c_{-1}+\sum_{m\geq q}
B^{(2)}_{1m}c_{-m-2}
\end{equation}
contains only creation operators, the only way in which the associated
state $|\phi^{n}_{g}\rangle$ can be annihilated by the these charges is
if they appear
explicitly in the state. In the case of quadratic differentials with
poles of higher order we also get annihilation operators. As we did
before we use a representation for the operators $c_{n}$, $b_{n}$
($n>0$)
\begin{equation}
c_{n}=\frac{\partial}{\partial b_{-n}} \hspace{1cm}
b_{n}=\frac{\partial}{\partial c_{-n}} \hspace{1cm} n>0
\end{equation}
and $c_{-n}$, $b_{-n}$ act as multiplication operators.
The conserved charges conditions lead to:
\begin{eqnarray}
|\phi_{g}^{n}\rangle&=&C_{1}\ldots C_{3g-3}A^{(1)}_{1}\ldots A^{(n)}_{1}
\nonumber \\
&\times &\exp{\left(-\sum_{m\geq q, n\geq 2} B^{(2)ij}_{mn}
c^{(i)}_{-m-2}b^{(j)}_{-n+2}\right)}
|+\rangle_{1}\otimes\ldots\otimes|+\rangle_{n}\;,
\end{eqnarray}
where
\begin{equation}
A_{1}^{(i)}=\sum_{j}\oint_{P_{j}}s_{-1}^{(i)}(z_{j})c^{(j)}(z_{j})\;,
\end{equation}
$s_{-1}^{(i)}(z)$ being the quadratic differential with a single
pole at $P_{i}$, and
\begin{equation}
C_{i+1}=\sum_{j}\oint_{P_{j}}\psi_{i}(z_{j})c^{(j)}(z_{j})\;.
\end{equation}
The ghost charge associated with the state $|\phi_{g}^{n}\rangle$ can be
evaluated if we take into account that $Q_{gh}(b)=-Q_{gh}(c)=1$.
We obtain
\begin{equation}
Q_{gh}|\phi_{g}^{n}\rangle=(3g-3+\frac{3n}{2})|\phi_{g}^{n}\rangle
\end{equation}
We finally check that this ghost charge associated with a genus $g$
Riemann surface and $n$ punctures is consistent with the sewing rules.
In fact, if we take two states $P\in{\cal P}(g_{1},n_{1})$ and
$Q\in{\cal P}(g_{2},n_{2})$ taking into account that the sewing state
does not carry ghost charge at all, the ghost charge associated with
$P_{i}\infty_{j}Q$ will be equal to
\begin{equation}
(3g_{1}-3+\frac{3n_{1}}{2})+(3g_{2}-3+\frac{3n_{2}}{2})=
3(g_{1}+g_{2})-3+\frac{3(n_{1}+n_{2}-2)}{2},
\end{equation}
consistent with the sewing prescription. It is also possible to
apply the previous formalism to the commuting ghosts appearing in the
quantization of superstrings (see \cite{LAG-Gomez-Nelson-Sierra-Vafa}
and references therein).

Now we apply the operator formalism to
the bosonic string. In this case we have to consider $26$
bosons together with the ghost system, in such a way that the total
central charge (matter+ghost) is equal to zero. This condition
guarantees that energy-momentum tensor acts without a central extension.
If $v_{1}$,
$v_{2}$ are two vector fields in the neighborhood of a point $P$ we have
\begin{equation}
[T(v_{1}),T(v_{2})]=T([v_{1},v_{2}])
\end{equation}
where $[v_{1},v_{2}]$ is the commutator of the two vector fields.
In the free bosonic string we work with $26$th tensor products of the
one boson Hilbert space described above, times the space of the ghost
system. In the combined matter+ghosts Hilbert space we are given a ray
by the operator formalism for any point in ${\cal P}(g,n)$. Next we have
to worry about the relative normalization of the ray for different
points in ${\cal P}(g,n)$. Since infinitesimal motions in ${\cal
P}(g,n)$ are generated by $T(v)$, and for the total system $c=0$, there
is no normalization ambiguity between the normalization at different
points as long as we can make the comparision along contractible paths.
We may encounter problems only along non-trivial elements of
$H_{1}[{\cal P}(g,n)]$. However, Harer's theorem \cite{Harer}
guarrantees that
this group vanishes for $g>2$. In this case, the state $|\phi\rangle$
associated to the bosonic string is globally well defined and free of
normalization ambiguities. For $g\leq 2$ one can reach the same
conclusion using sewing.

Since the state $|\phi\rangle$ is globally defined, we want to obtain
the string measure in moduli space ${\cal
M}_{g}$ using the operator formalism. This measure is useful in
computing the partition function for the bosonic string.
Starting for simplicity with ${\cal P}(g,1)$, we can obtain a projection
map
\begin{eqnarray}
\begin{array}{cc}
{\cal P}(g,1) & \\
\downarrow & \pi \\
{\cal M}_{g} &
\end{array}
\nonumber
\end{eqnarray}
by simply forgetting the puncture and the local parameter.
To construct a measure in ${\cal M}_{g}$ we have to associate with a
basis of holomorphic tangent vectors at a point $P\in{\cal M}_{g}$
$\{V_{1},\ldots,V_{3g-3}\}$ a volume element that we will denote as
\begin{equation}
\mu(P)(V_{1},\ldots,V_{3g-3},\bar{V}_{1},\ldots,\bar{V}_{3g-3})
\end{equation}
Given a point $P\in{\cal M}_{g}$ and a basis of tangent vectors at that
point $\{V_{i}\}$ we can find at some point in $\pi^{-1}(P)\subset {\cal
P}(g,1)$ a set of vector fields $\{v_{i}\}$ in the neigborhood of the
puncture which get mapped into $\{V_{i}\}$ under the projection map
$\pi$. For any such set of vectors $\{v_{i}\}$ we define the measure
\begin{eqnarray}
& &\mu(P)(V_{1},\ldots,V_{3g-3},\bar{V}_{1},\ldots,\bar{V}_{3g-3})=
\nonumber \\
& &\langle 0|\prod_{i=1}^{3g-3}b(v_{i})
\prod_{i=1}^{3g-3}\bar{b}(\bar{v}_{i})|\phi\rangle_{P}
\label{measure}
\end{eqnarray}
where the operator $b(v)$ is defined as
\begin{equation}
b(v)=\oint_{P}\frac{dz}{2\pi i}b(z)v(z)
\end{equation}
$|0\rangle$ is the $SL(2,{\bf C})$ invariant vacuum and
$|\phi\rangle_{P}$ is the state associated with the point in ${\cal
P}(g,1)$ (these states belongs to ${\cal F}_{matter}\otimes{\cal F}_{bc}
\otimes{\cal F}_{\bar{b}\bar{c}})$. Notice first of all
that the ghost numbers match in (\ref{measure}).

It can be shown that (\ref{measure}) defined above is a measure on
moduli space. First if we change the representative $\{v_{i}\}$ in
${\cal P}(g,1)$ of $\{V_{i}\}$ by $v_{i}\rightarrow v_{i}+\epsilon_{i}$,
if $\epsilon_{i}$ extends holomorphically away from $P$ there is no
change in the measure since $b(\epsilon_{i})|\phi\rangle=0$.
Furthermore, if we change the local parameter, or move $P$
infinitesimally, since $L_{n}|0\rangle=0$ for $n\geq -1$ and the total
central charge vanishes, there is no change in the measure.
There is a last condition that the measure has to verify in order to be
identified with the bosonic string measure. $\mu(P)$ has to verify the
Belavin-Knizhnik theorem \cite{Belavin-Knizhnik}
\begin{equation}
\partial\bar{\partial}\log{\mu}=-13\,\partial\bar{\partial}
\log{(Im\,\Omega)}\;,
\label{B-K}
\end{equation}
therefore
\begin{equation}
\mu=\frac{\rho\wedge\bar{\rho}}{(det\,Im\,\Omega)^{13}}
\end{equation}
where $\rho$ is a holomorphic $3g-3$ form in the moduli space. To prove
(\ref{B-K}) we make use of (\ref{delt-v}):
\begin{eqnarray}
\partial\bar{\partial}\log{\mu (\ldots V\ldots)}&=&
\frac{\langle0|\ldots T(\,)\bar{T}(\,)
\ldots|\phi\rangle}{\langle0|\phi\rangle} \nonumber \\
&- &\frac{\langle0|\ldots T(\,)|\phi\rangle\langle0|\ldots
\bar{T}(\,)|\phi\rangle}{\langle0|\phi\rangle^{2}}
\end{eqnarray}
Since the only obstruction to holomorphic factorization comes from the
matter
sector, we can restrict the calculation to the matter states. Using
Rohrlich's formula \cite{Fay}
\begin{equation}
\delta\Omega_{ij}=-\oint_{P}\omega_{i}\omega_{j}\nu
\end{equation}
we obtain
\begin{equation}
\partial\bar{\partial}\log{\mu(\ldots
V\ldots)}=-13\,\partial\bar{\partial} \log{det\,Im\,\Omega}
\end{equation}
So $\mu(P)$ is a well defined measure over the moduli space.

As an application we are going to show that the measure has second order
poles at the boundary of moduli space. Let us consider a surface
which is pinched off along a non-contractible loop (see sec. 3.4). This
can be obtained by sewing two surfaces with sewing parameter $q=0$. Near
the $q=0$ region the measure looks like
\begin{equation}
\left(\frac{dq}{q}\frac{d\bar{q}}{\bar{q}}\right)
\langle
S_{ij}|b_{0}\bar{b}_{0}q^{L_{0}}\bar{q}^{\bar{L}_{0}}|\phi\rangle
\end{equation}
since the lowest eigenvalue of $L_{0}$, $\bar{L}_{0}$ is $-1$ (which
correspond to the tachyon) as $q\rightarrow 0$ we obtain
\begin{equation}
\left|\frac{dqd\bar{q}}{q^{2}\bar{q}^{2}}\right|
\end{equation}
This means that $\rho$ has a second order pole at the boundary of
moduli space associated with the equivalence class of Riemann surfaces
with a node. This structure completes the identification of
(\ref{measure}) with the Polyakov measure. It also exhibits in a
rigorous way the intuitive expectation that string infinities originate
in the exchange of tachyons along thin and long tubes between different
elements of the surface.

As a final application, we study string scattering amplitudes to obtain
a geometrical interpretation of the physical state conditions. If we
have a scattering amplitude with $n$ external states, we want to
associate with it a measure on ${\cal M}_{g,n}$ if we wish to compute
the amplitude to order $g$. The operator formalism provides us with a
well-defined state $|\phi\rangle_{P}$ for any $P\in{\cal P}(g,n)$. Once
again we have a projection map
\begin{eqnarray}
\begin{array}{cc}
{\cal P}(g,n) & \\
\downarrow & \pi \\
{\cal M}_{g,n} &
\end{array}
\nonumber
\end{eqnarray}
by simply forgetting the punctures and the local parameters. Now we
construct a measure similar to (\ref{measure})
\begin{equation}
\langle\chi_{1}|\ldots\langle\chi_{n}|b(\,)\ldots\bar{b}(\,)
|\phi\rangle_{P}
\label{measure-n}
\end{equation}
In this case, however, for (\ref{measure-n}) to be a well defined
measure we need
a $(3g-3+n,3g-3+n)$ form so we must have $3g-3+n$ $b$'s. This means that
since $|\phi\rangle_{P}$ has ghost number $3g-3+3n/2$ the ghost number
of the external states has to be $-\frac{n}{2}$. Distributing the ghost
number equally among the external states, we learn that
$\langle\chi_{i}|$ must have ghost number $-\frac{1}{2}$.
Apply the other requirements neccesary to have a well defined measure
over ${\cal M}_{g,n}$ we arrive to the conditions
\begin{eqnarray}
L_{n}|\chi_{i}\rangle=\bar{L}_{n}|\chi_{i}\rangle&=&0\;, \nonumber \\
b_{n}|\chi_{i}\rangle=\bar{b}_{n}|\chi_{i}\rangle&=&0\;, \hspace{1cm}
n\geq 0\;,
\end{eqnarray}
and we have no constraint coming from $L_{-1}$ because the measure
depends on the position of the punctures. These conditions, together
with
the condition over the ghost number of $|\chi_{i}\rangle$ are solved by
\begin{equation}
|\chi\rangle=\sum |\psi\rangle_{matter}\otimes|-\rangle\;,
\end{equation}
where $|\psi\rangle_{matter}$ satisfy
\begin{eqnarray}
L_{n}|\psi\rangle&=&\bar{L}_{n}|\psi\rangle=0 \hspace{1cm}\;, n>0\;,
\nonumber \\
L_{0}|\psi\rangle&=&\bar{L}_{0}|\psi\rangle=|\psi\rangle\;,
\end{eqnarray}
so we recover the standard physical state conditions.

We see that the operator formalism provide a natural framework to
analyze many of the qualitative and quantitative questions in String
Theory. From a geometrical point of view, the computation of string
amplitudes is related to the construction of measure on ${\cal M}_{g,n}$
satisfying the Belavin-Knizhnik condition. In the operator formalism we
can compute these measures using the same techniques as in CFT on the
plane. All the geometrical information is coded into the Bogoliubov
transformation relating the standard vacuum $|0\rangle$ to the state
$|\phi\rangle_{P}$ associted to $P\in{\cal P}(g,n)$. This formalism can
also be extended to the supersymmetric case with similar results. We
will not discuss this extension in detail. The interested reader can
find the details in \cite{LAG-Gomez-Nelson-Sierra-Vafa}.



\section{String Theory II. Fermionic Strings}

We now begin a brief study of fermionic strings. The aim is to show how
in a supersymmetric theory, many of the undesirable features of the
bosonic string are eliminated. We will also emphasize how some simple
consistency conditions (absence of global anomalies on the world-sheet)
provide strong constraints on the spectrum of the theory. It is quite
remarkable that once the modular anomalies (global diffeomorphism
anomalies) on the world-sheet are cancelled, and after we make the
theory invariant under the mapping class group (see previous chapter),
all space-time anomalies also cancel \cite{Schellekens-Warner}. At the
end of this chapter we will discuss some special properties of
superstring theory: finite temperature behavior, and a brief status
report on finiteness of string perturbation theory.
A brief introduction to string black holes (covered in more detail in
Polyakov's lectures) will appear in the next chapter.

\subsection{Fermionic String}

In spite of all the beautiful features found in the study of the bosonic
string, this is not a satisfactory model. The first problem we
meet is the presence of a tachyon in the spectrum which is the origin
of many difficulties. Furthermore, the resulting theory in the target
space only contains bosonic degrees of
freedom so there is no hope of phenomenological implications for this
model. We can get rid of these two problems at the same time by
adding fermions in the two-dimensional field theory on the world-sheet.
This lead to the construction of the fermionic, or more appropiately,
supersymmetric string. Now, together with the bosonic coordinates
$X^{\mu}(\tau,\sigma)$ ($\mu=1,\ldots,d$), we introduce $d$
Majorana-Weyl
spinors on the world-sheet $\psi^{\mu}(\tau,\sigma)$ transforming as
a $d$-dimensional vector under Lorentz transformations in the target
space. In the bosonic string the negative-norm states were
eliminated from the spectrum by using reparametrization invariance,
which gave rise to the Virasoro constraints. Now we also have
negative-norm states coming from the time component of the spinor field,
$\psi^{0}(\tau,\sigma)$, so we need a new local invariance
to get rid of these ghost states. This new local symmetry is
two-dimensional supergravity. To implement it we
introduce two fields, the {\it zweibein}
$e_{\alpha}^{\;\;a}(\tau,\sigma)$
\begin{equation}
h_{\alpha\beta}=e_{\alpha}^{\;\;a}e_{\beta}^{\;\;b}\eta_{ab}
\end{equation}
and the Majorana gravitino $\chi_{\alpha}$ ($a,b=1,\ldots,d$;
$\alpha,\beta=0,1$). The complete action is \cite{Brink,Deser-Zumino}
\begin{eqnarray}
S&=&-\frac{1}{4\pi\alpha^{'}} \int d\tau d\sigma
(det\;e)\eta_{\mu\nu}\left\{ h^{\alpha\beta} \partial_{\alpha}
X^{\mu}\partial_{\beta}X^{\nu}-ie^{\alpha}_{\;\;a}\bar{\psi}^{\mu}
\rho^{a}\partial_{\alpha}\psi^{\nu}\right.  \nonumber \\
&+&\left. 2e^{\alpha}_{\;\;a}e^{\beta}_{\;\;b}\bar{\chi}_{\alpha}
\rho^{b}\rho^{a}
\psi^{\mu}\partial_{\beta}X^{\nu}-\frac{1}{2}\bar{\psi}^{\mu}\psi^{\nu}
e^{\alpha}_{\;\;a}e^{\beta}_{\;\;b}
\bar{\chi}_{\alpha}\rho^{b}\rho^{a}\chi_{\beta}\right\}
\end{eqnarray}
where the $\rho^{\alpha}$'s are the Dirac matrices satisfying the
two-dimensional Clifford algebra
\begin{equation}
\{\rho^{\alpha},\rho^{\beta}\}=-2\eta^{\alpha\beta}\;.
\end{equation}
We can fix the superconformal gauge:
\begin{equation}
e_{\alpha}^{\;\;a}=e^{\phi}\delta_{\alpha}^{\;\;a}\;,
\hspace{1cm}
\chi_{\alpha}=\rho_{a}\lambda\;.
\end{equation}
In the critical dimension the classical Weyl and super-Weyl
transformations
are also symmetries of the quantum theory so $\phi$ and $\lambda$
can be gauged away. At the classical level this is always true and we
obtain the gauge fixed action
action \begin{equation}
S=-\frac{1}{4\pi\alpha^{'}}\int d\tau d\sigma \left[
\partial_{\alpha}X^{\mu}\partial^{\alpha}X^{\nu}-i\bar{\psi}^{\mu}
\rho^{\alpha}\partial_{\alpha}\psi^{\nu}\right]\eta_{\mu\nu}\;,
\label{SST-conf-gauge}
\end{equation}
which is still invariant under the supersymmetry transformations
\begin{eqnarray}
\delta_{\epsilon} X^{\mu}&=&\bar{\epsilon}\psi^{\mu} \nonumber \\
\delta_{\epsilon}\psi^{\mu}&=&-i\rho^{\alpha} \partial_{\alpha}
X^{\mu}\epsilon\;,
\end{eqnarray}
with $\epsilon$ a constant Majorana spinor. Associated with the
supersymmetry transformation we have the supercurrent
\begin{equation}
J_{\alpha}=\frac{1}{2}\rho^{\beta}\rho_{\alpha}\psi^{\mu}
\partial_{\beta} X_{\mu}\;,
\end{equation}
such that $\partial_{\alpha}J^{\alpha}=0$.
We also have the energy momentum tensor
\begin{equation}
T_{\alpha\beta}=\partial_{\alpha}X^{\mu}\partial_{b}X_{\mu}+
\frac{i}{2}\bar{\psi}^{\mu}\rho_{(\alpha}\partial_{\beta)}\psi_{\mu}-
(\mbox{Trace})\;,
\end{equation}
which is also conserved $\partial_{\alpha}T^{\alpha\beta}=0$. Classical
equation of motion are
\begin{eqnarray}
\partial_{\alpha}\partial^{\alpha}X^{\mu}&=&0 \nonumber \\
\rho^{\alpha}\partial_{\alpha}\psi^{\mu}&=&0
\end{eqnarray}
together with the constraints
\begin{eqnarray}
T_{\alpha\beta}&=&0 \nonumber \\
J_{\alpha}&=&0
\end{eqnarray}
which follow from the equations of motion for the {\it zweibein}
$e_{\alpha}^{\;\;a}$ and the gravitino $\chi_{\alpha}$ respectively.
Invariance under Weyl and super-Weyl transformations imply that
\begin{eqnarray}
T^{\alpha}_{\;\alpha}&=&0\;, \nonumber \\
\rho^{\alpha}J_{\alpha}&=&0\;.
\end{eqnarray}

We are going to study the free closed superstring. As for the
bosonic string, the world-sheet is a cylinder parametrized by the
coordinates $-\infty<\tau<\infty$ and $0\leq\sigma< \pi$. After Wick
rotating
these coordinates and conformally mapping the
cylinder into the punctured complex plane ${\bf C}^{*}$, the action
becomes
\begin{equation}
S=-\frac{1}{4\pi\alpha^{'}}\int d^{2}z (\bar{\partial}X^{\mu}
\partial X_{\mu}-\psi^{\mu}\bar{\partial}\psi_{\mu}-
\bar{\psi}^{\mu}\partial \bar{\psi}_{\mu})
\end{equation}
where the components of the two-dimensional spinor $\psi^{\mu}$ are
\begin{equation}
\psi^{\mu}= \left(
\begin{array}{c}
\psi^{\mu} \\
\bar{\psi}^{\mu}
\end{array}
\right)\;,
\end{equation}
and we have used the following representation for the two-dimensional
Dirac algebra
\begin{equation}
\rho^{0}=\left(
\begin{array}{cc}
0  &  -i \\
i  &   0
\end{array}
\right)\;, \hspace{2cm} \rho^{1}= \left(
\begin{array}{cc}
0  &   i \\
i  &   0
\end{array}
\right)\;.
\end{equation}
The equations of motion are
\begin{eqnarray}
\partial\bar{\partial}X^{\mu}(z,\bar{z})&=&0\;, \nonumber \\
\partial\bar{\psi}(z,\bar{z})=\bar{\partial}\psi(z,\bar{z})&=&0\;,
\end{eqnarray}
so $\psi(z)$, $\bar{\psi}(\bar{z})$ are respectively holomorphic and
antiholomorphic fields. In the same way we can rewrite the energy
momentum tensor and the supercurrent as
\begin{eqnarray}
T(z)&=:&T_{zz}(z)=-\frac{1}{2}\partial X^{\mu}\partial X_{\mu}-
\frac{1}{2}\partial\psi^{\mu}\psi_{\mu}\;, \nonumber \\
T_{F}(z)&=:&J_{z}(z)=-\frac{1}{2}\psi^{\mu}\partial X_{\mu}\;,
\end{eqnarray}
and similarly for barred quantities
$\bar{T}(\bar{z})=T_{\bar{z}\bar{z}}(\bar{z})$,
$\bar{T}_{F}(\bar{z})=J_{\bar{z}}(\bar{z})$.

The superfield language is specially suited to study the
supersymmetric string \cite{Green-Schwarz-Witten,Kaku}.
We introduce together with the commuting coordinates $z,
\bar{z}\in{\bf C}^{*}$ a
pair of anticommuting variables $\theta,\bar{\theta}$, and the
supercoordinates:
\begin{equation}
{\bf z}=(z,\theta) \hspace{1cm} {\bf \bar{z}}=(\bar{z},\bar{\theta})
\end{equation}
Taking into account the anticommuting character of $\theta$ and
$\bar{\theta}$ any function $f(z,\bar{z},\theta,\bar{\theta})$ can be
expanded as
\begin{equation}
f(z,\bar{z},\theta,\bar{\theta})=f_{0}(z,\bar{z})+
f_{1}(z,\bar{z})\theta+f_{2}(z,\bar{z})\bar{\theta}+
f_{3}(z,\bar{z})\theta\bar{\theta}
\end{equation}
The supersymmetric derivative is defined by
\begin{equation}
D=\frac{\partial}{\partial\theta}+\theta\frac{\partial}{\partial z}\;,
\end{equation}
and the integration over the anticommuting variables $\theta$,
$\bar{\theta}$ is made according to the Berezin rules
\cite{Berezin}
\begin{equation}
\int d\theta=0\;, \hspace{1cm} \int d\theta\, \theta=1\;.
\end{equation}

We can now reformulate the superstring action using the superspace
formalism. Defining the superfield
\begin{equation}
Y^{\mu}(z,\bar{z},\theta,\bar{\theta})=X^{\mu}(z,\bar{z})+
\theta\psi^{\mu}(z,\bar{z})+\bar{\theta}\bar{\psi}^{\mu}(z,\bar{z})
\end{equation}
we can rewrite (\ref{SST-conf-gauge}) as
\begin{equation}
S=-\frac{1}{4\pi\alpha^{'}}\int d^{2}z\,d\theta\,d\bar{\theta}
\bar{D}Y^{\mu}DY_{\mu}\;,
\end{equation}
which is invariant under superconformal transformations (for many
details and results in superconformal field theory, and the formulation
of superstrings in this language, see \cite{FMS})
We can also define the components of the super-energy-momentum tensor
${\cal T}(z,\theta)$, ${\bar{\cal T}}(\bar{z},\bar{\theta})$ as
\begin{eqnarray}
{\cal T}(z,\theta)&=&T_{F}(z)+\theta T(z) \nonumber \\
\bar{\cal T}(\bar{z},\bar{\theta})&=&\bar{T}_{F}(\bar{z})+
\bar{\theta}\bar{T}(\bar{z})
\end{eqnarray}

After quantization we obtain the following the OPE's for $T(z)$ and
$T_{F}(z)$
\begin{eqnarray}
T(z)T(w)&=&\frac{c/2}{(z-w)^{4}}+\frac{2T(w)}{(z-w)^{2}}+
\frac{1}{z-w}\partial T(w)+regular\; terms \nonumber \\
T(z)T_{F}(w)&=&\frac{3/2}{(z-w)^{2}}T_{F}(w)+\frac{1}{z-w}\partial
T_{F}(w)+\ldots \nonumber \\
T_{F}(z)T_{F}(w)&=&\frac{c/6}{(z-w)^{3}}+\frac{1}{2(z-w)}T(w)
+\ldots
\label{OPE-superconformal}
\end{eqnarray}
which imply that the field $T_{F}(z)$ is a primary field of spin
$\frac{3}{2}$.

In the path integral formalism we need to gauge
fix all local symmetries. This is done using the
Fadeev-Popov procedure. Now, together with the anticommuting
reparametrization
ghosts $b_{\alpha\beta}$, $c^{\alpha}$ we have also a pair of commuting
superconformal ghosts $\beta_{\alpha}$
and $\gamma$ with spins $\frac{3}{2}$ and $-\frac{1}{2}$
respectively. Their action is:
\begin{equation}
S_{gh}=\frac{1}{2\pi\alpha^{'}}\int d^{2}z(b_{zz}\bar{\partial}
c^{z}+\beta_{z}\bar{\partial}\gamma+\mbox{c.c.})
\end{equation}
In the superspace formalism, the ghost fields can be combined to ghosts
superfields
\begin{eqnarray}
B(z,\theta)&=&\beta(z)+\theta b(z) \nonumber \\
C(z,\theta)&=&c(z)+\theta\gamma(z)
\end{eqnarray}
and their combined action becomes
\begin{equation}
S_{gh}=\frac{1}{2\pi\alpha^{'}}\int d^{2}z d\theta d\bar{\theta}
(B\bar{D}C+\mbox{c.c.})
\end{equation}

Thus, in the free quantum superstring, we have the following fields
\begin{equation}
X^{\mu}(z), \hspace{.5cm} \psi^{\mu}(z), \hspace{.5cm} b(z),
\hspace{.5cm} c(z), \hspace{.5cm} \beta(z), \hspace{.5cm}
\gamma(z)
\end{equation}
and the corresponding antiholomorphic components.
The contribution of every field to the central charge $c$ is given by
\begin{equation}
\begin{array}{ccc}
X^{\mu}    & \longrightarrow  & d \\
\psi^{\mu} & \longrightarrow  & \frac{d}{2} \\
b,\;c	 & \longrightarrow  & -26 \\
\beta,\;\gamma& \longrightarrow  & 11
\end{array}
\end{equation}
so the total central charge is equal to $c_{tot}=\frac{3d}{2}-15$. This
means that the conformal and superconformal anomaly cancel if the
dimension of the target space is $d=10$. As in the discussion of the
bosonic string, what distinguishes critical and non-critical strings is
the decoupling of the conformal modes. In the case of fermionic strings
the critical value of $d=10$. Only in this case a graviton is found
in the spectrum. The attention is thus dedicated to critical
superstrings when one wants to understand Quantum Gravity and its
unification with other interactions.

A crucial point when dealing with fermionic string is the question of
the
boundary conditions for the fermions. In the closed bosonic string we
had the periodicity condition over the bosonic fields
$X^{\mu}(\tau,\sigma+\pi)=X^{\mu}(\tau,\sigma)$. Now we have the same
boundary condition on the $X^{\mu}$ field, but for the
fermions we can have either periodic or antiperiodic boundary
conditions. Periodic boundary conditions are called Ramond (R)
boundary condition whereas antiperiodic ones are called Neveu-Schwarz
(NS) boundary conditions
\begin{eqnarray}
\psi^{\mu}(\tau,\sigma+\pi)&=&\psi^{\mu}(\tau,\sigma) \hspace{2.25cm}
\mbox{(R)} \nonumber \\
\psi^{\mu}(\tau,\sigma+\pi)&=&-\psi^{\mu}(\tau,\sigma) \hspace{2cm}
\mbox{(NS)}
\end{eqnarray}
When mapping the cylinder onto the punctured complex plane
${\bf C}^{*}$
we have to take into account the transformation properties of spinors
under conformal transformations. Since $\psi^{\mu}(z)$ is a
$(\frac{1}{2},0)$ field, we have that if
$z=e^{w}=\exp{[2(\tau-i\sigma)]}$
\begin{equation}
\psi^{\mu}(z)dz^{\frac{1}{2}}=\psi^{\mu}(w)dw^{\frac{1}{2}}\;.
\end{equation}
This implies that under the transformation $z\rightarrow
e^{2\pi i}z$ the periodicity conditions of $\psi^{\mu}(z)$
and $\psi^{\mu}(w)$ are opposite when $w\rightarrow w+2\pi i$. Then, the
boundary conditions over the complex plane for both sectors (R and NS)
will be
\begin{eqnarray}
\psi^{\mu}(e^{2\pi i}z)&=&-\psi^{\mu}(z) \hspace{1.7cm} \mbox{(R)}
\nonumber \\
\psi^{\mu}(e^{2\pi i}z)&=&\psi^{\mu}(z) \hspace{2cm} \mbox{(NS)}
\end{eqnarray}
In the closed superstring we have four possible sectors corresponding to
the boundary conditions for left- and right-moving modes (R,R), (R,NS),
(NS,R) and (NS,NS). For one handness we
have the two boundary conditions R and NS. We can expand the fields
$\psi(z)$ in these two sectors in Fourier modes:
\begin{eqnarray}
\psi(z)&=&\sum_{n\in{\bf Z}} d^{\mu}_{n}
z^{-n-\frac{1}{2}} \hspace{1cm} \mbox{(R)}\;, \nonumber \\
\psi(z)&=&\sum_{r\in{\bf Z}+\frac{1}{2}} b^{\mu}_{r}
z^{-r-\frac{1}{2}}  \hspace{0.5cm} \mbox{(NS)}\;.
\end{eqnarray}
The field $X^{\mu}(z)$ has the same expansion as in the
bosonic string
\begin{equation}
X^{\mu}(z)=q^{\mu}-\frac{i}{4}p^{\mu}\log{z}+\frac{i}{2}
\sum_{n\in{\bf Z}}
\frac{\alpha_{n}^{\mu}}{n}z^{-n}\;.
\end{equation}

The energy-momentum tensor, being a bosonic field, can be
expanded in the usual way
\begin{equation}
T(z)=\sum_{n\in{\bf Z}}L_{n}z^{-n-2}
\end{equation}
where $L_{n}$ are the Virasoro generators. However, the boundary
conditions for the supercurrent $T_{F}(z)$ are the same as those
for the fermionic fields. Hence
\begin{eqnarray}
T_{F}(z)&=&\frac{1}{2}\sum_{n\in{\bf Z}}G_{n}z^{-n-\frac{3}{2}}
\hspace{1.5cm} \mbox{(R)}\;, \nonumber \\
T_{F}(z)&=&\frac{1}{2}\sum_{r\in{\bf Z}+\frac{1}{2}}G_{r}
z^{-r-\frac{3}{2}} \hspace{1cm} \mbox{(NS)}
\end{eqnarray}
where $G_{r}$ are the generators of the superconformal transformations.
The (anti)commutation relations follow from the OPE's
(\ref{OPE-superconformal}):
\begin{eqnarray}
[L_{m},L_{n}]&=&(m-n)L_{m+n}+\frac{\hat{c}}{8}m(m^{2}-1)\delta_{m+n,0}
\;; \nonumber \\
{[}L_{m},G_{r}]&=&\left(\frac{m}{2}-r\right)G_{m+r}\;, \nonumber \\
\{G_{r},G_{s}\}&=&2L_{r+s}+\frac{\hat{c}}{2}
\left(r^{2}-\frac{1}{4}\right)\delta_{r+s,0}\;,
\end{eqnarray}
where now $r\in{\bf Z}$ or $r\in{\bf Z}+\frac{1}{2}$ depending on
whether we are in (R) or (NS) sector and $\hat{c}=\frac{2}{3}c$.

We now proceed to the determination of the spectrum for the closed
supersymmetric string. First of all we have the following non-zero
(anti) commutation relations between oscillators,
\begin{eqnarray}
[\alpha_{m},\alpha_{n}]&=&m\eta^{\mu\nu}\delta_{m+n,0} \nonumber \\
\{d^{\mu}_{m},d^{\nu}_{n}\}&=&\eta^{\mu\nu}\delta_{m+n,0} \nonumber \\
\{b^{\mu}_{r},b^{\nu}_{r}\}&=&\eta^{\mu\nu}\delta_{r+s,0}\;.
\end{eqnarray}
As for the bosonic string we have negative norm states. In order to
eliminate
these states there are several procedures already explained in
sec. $3.1$. The constraints, as in the bosonic string, cannot be imposed
na\"{\i}vely because of the anomaly; then we have the following
conditions
over the physical states (we also have the same conditions for the
antiholomorphic parts) if we use covariant quantization:
\begin{eqnarray}
L_{n}|phys\rangle&=&0 \hspace{1cm} n>0 \nonumber \\
G_{r}|phys\rangle&=&0 \hspace{1cm} r>0\;\mbox{(NS)}\hspace{0.5cm}
r\geq 0 \;\mbox{(R)}
\end{eqnarray}
together with $(L_{0}-a)|phys\rangle=0$ and
$(L_{0}-\bar{L}_{0})|phys\rangle=0$ where $a$ is a normal ordering
constant that is $a=\frac{1}{2}$ for the NS and $a=0$ for the R sector.
To obtain the physical spectrum we will use light-cone gauge where
the constraints
can be explicitly solved. Making use of the residual invariance
left after fixing the superconformal gauge we impose the light-cone
gauge conditions
\begin{eqnarray}
X^{+}&=&q^{+}+\alpha^{'}p^{+}\tau \;, \nonumber \\
\psi^{+}&=&0\;.
\end{eqnarray}
In this gauge the theory is completely described in terms of transverse
modes.

We begin with the NS sector. The mass formula is computed following
the same steps as in the previous chapter (again we use
$\alpha^{'}=\frac{1}{2}$)
\begin{equation}
\frac{1}{2}m^{2}=\sum_{n>0}\alpha_{-n}^{i}\alpha_{n}^{i}+
\sum_{r>0}rb_{-r}^{i}b_{r}^{i}-\frac{1}{2}\;.
\label{m-super}
\end{equation}
The vacuum state $|0\rangle$ satisfies the conditions
\begin{equation}
\alpha_{n}^{i}|0\rangle=b_{r}^{i}|0\rangle=0 \hspace{1cm} n,r>0
\label{vac-super}
\end{equation}
It follows from (\ref{m-super},\ref{vac-super}) that $|0\rangle$ has
$m^{2}<0$ and it is a tachyon state. This looks like the kind
of trouble we were trying to scape from in the bosonic string. We will
see presently how the tachyon states are eliminated from the theory.
The first excited level is
constructed by acting with $b_{-\frac{1}{2}}^{i}$ on the vacuum state.
This is a massless state that transforms as a vector with respect to
$SO(8)$, the little group for massless particles in ten dimensions. In
the first massive state we have two states $\alpha_{-1}|0\rangle$ and
$b_{-\frac{1}{2}}^{i}b_{-\frac{1}{2}}^{j}b|0\rangle$ both with
$m^{2}=1$.

The mass formula in the R sector is given by
\begin{equation}
\frac{1}{2}m^{2}=\sum_{n>0}\alpha^{i}_{-n}\alpha^{i}_{n}+
\sum_{n>0}nd^{i}_{-n}d^{i}_{n}
\end{equation}
However in this sector we have a supplementary complication. The
zero modes $d_{0}^{\mu}$ form a closed subalgebra since
\begin{equation}
\{d_{0}^{\mu},d_{0}^{\nu}\}=2\eta^{\mu\nu}
\label{subalgebra}
\end{equation}
This means that, with respect to the oscillator vacuum $|0\rangle$
defined by
\begin{equation}
\alpha_{n}|0\rangle=d_{n}|0\rangle=0 \hspace{1cm} n>0
\end{equation}
all the states $d_{0}^{\mu}|0\rangle$ are also massless because
$\{d_{0}^{\mu},d_{n}^{\nu}\}=0$ ($n\neq 0$). These states provide us
with a representation of the Clifford algebra (\ref{subalgebra}),
therefore the ground state for the R sector is a ten-dimensional
spinor $|a\rangle$ with $a$ a spinor index. We know that in ten
dimensions we can impose both Weyl and
Majorana conditions; then we can choose our ground state to have a
definite chirality. We denote two possible chiral ground states by
$|a\rangle$, $|\bar{a}\rangle$, $a,\bar{a}=1,\ldots,8$.
Starting with this massless spinor state we construct the spectrum of
the R sector of the fermionic string. The first excited level has
(apart from the two possible chiralities) two states, namely,
$d^{i}_{-1}|a\rangle$ and $\alpha^{i}_{-1}|a\rangle$ both with masses
$m^{2}=2$. We obtain higher excited levels in a similar fashion.

To construct the spectrum of the closed fermionic string we take
the tensor product of the right- and left-moving states, taking into
acount the level matching condition which can be written as
$m_{L}^{2}=m_{R}^{2}$ for each of the four sectors (R,R), (R,NS), (NS,R)
and (NS,NS). Note that we have not taken care of the problems posed by
the presence of tachyons in the sector with NS boundary conditions.
Historically, a projection was made onto a set of definite $G$-parity
states \cite{GSO} thus eliminating the tachyon and also rendering the
spectrum space-time supersymmetric. We begin by presenting first how the
projection is achieved, and we will later see how this projection is
naturally understood in terms of modular invariance, the absence of
global diffeomorphism anomalies on the world-sheet. The $G$-parity
operator in the NS sector is defined to be:
\begin{equation}
G=(-1)^{F+1}=(-1)^{\sum_{r\in{\bf Z}+\frac{1}{2}}
b^{i}_{-n}b^{i}_{n}+1}\;.
\end{equation}
Hence the vacuum state in the NS sector has $G|0\rangle=-|0\rangle$. In
the R sector we construct the $G$-parity operator using a string
generalization of $\gamma_{5}$:
\begin{equation}
\Gamma=\Gamma^{0}\ldots\Gamma^{8}(-1)^{\sum_{n>0}
d^{i}_{-n}d^{i}_{n}}
\end{equation}

By using these operators we can at the same time
project out the unwanted tachyon and make the string
spectrum space-time supersymmetric. This can be accomplished by using
the GSO projection \cite{GSO} introduced by Gliozzi, Scherk and Olive.
We project the spectrum of the string onto the states with positive
$G$-parity in each handness separately. Since the fundamental state in
the NS sector has $G=-1$ this projection eliminate the tachyon from the
theory. With respect to the $\Gamma$ operator in the R sector we
have different choices: $\Gamma=\pm 1$. Since the action of $\Gamma$
on the R ground state $|a\rangle$ gives us the chirality of this state,
different choices will lead to chiral and non chiral spectra. By
taking $\Gamma=\bar{\Gamma}=1$ we have the ground states in the
left- and right-R sector will be two spinors with the same chirality
$|a\rangle$, $|b\rangle$. This kind of string is called type IIB
superstring \cite{type-IIB} and its massless sector contains the
states \footnote{Apart from the GSO projection we have also required the
level matching condition. The same remark applies in the derivation of
\ref{IIA}.}
\begin{eqnarray}
|a\rangle_{L}&\otimes& |b\rangle_{R} \nonumber \\
\bar{b}_{-\frac{1}{2}}^{i}|0\rangle_{L}&\otimes &
b_{-\frac{1}{2}}^{j}|0\rangle_{R} \nonumber \\
|a\rangle_{L} & \otimes & b_{-\frac{1}{2}}^{i}|0\rangle_{R} \nonumber \\
\bar{b}_{-\frac{1}{2}}^{i}|0\rangle_{L} & \otimes & |b\rangle_{R}
\label{IIB}
\end{eqnarray}
Decomposing these states into irreducible representations of the little
group $SO(8)$ we have the graviton,
two real scalars, two antisymmetric tensors and a rank four
antisymmetric tensor. In addition we have two spin $\frac{3}{2}$
and two spin $\frac{1}{2}$ states all with the same helicity. The states
in the massless sector are those of the chiral $N=2$ SUGRA in $d=10$
\cite{Nahm,SUGRA-d=10}.

By performing the choice $\Gamma=-\bar{\Gamma}=1$ we obtain the type IIA
superstring \cite{Green-Schwarz-Witten}. The states in the massless
sector are
\begin{eqnarray}
|\bar{a}\rangle_{L} &\otimes &|b\rangle_{R} \nonumber \\
\bar{b}_{-\frac{1}{2}}^{i}|0\rangle_{L}& \otimes &
b_{-\frac{1}{2}}^{j}|0\rangle_{R} \nonumber \\
|\bar{a}\rangle_{L} &\otimes & b_{-\frac{1}{2}}^{i}|0\rangle_{R}
\nonumber \\
\bar{b}_{-\frac{1}{2}}^{i}|0\rangle_{L} &\otimes & |b\rangle_{R}
\label{IIA}
\end{eqnarray}
Now the theory has the same states than the type IIB superstring but
with the difference that fermion states come in both chiralities.
Then the particle content for type IIA superstring is that of
the non-chiral $N=2$ SUGRA in $d=10$ \cite{Nahm} which can
also be obtained by dimensional reduction from $N=1$ SUGRA in $d=11$.

After the GSO projection, the spectrum of the fermionic string becomes
supersymmetric. This is rather surprising because having started with
a action that was supersymmetric only on the world-sheet and we
obtain in the end supersymmetry in
the target space. There is a second formalism developed by
Green and Schwarz \cite{Green-Schwarz-2,Schwarz} which starts with a
space-time supersymmetric action.

In order to compute amplitudes we will need also to consider
world-sheets with non-trivial topology. We have to be specially
careful with
fermions. Since they are represented by world-sheet spinors, we have to
decide how to treat their spin structures. On a genus $g$ surface the
first homology group has dimension $2g$. Once a spin structure is fixed,
all others differ from the fixed one by some choice of boundary
conditions along a basis of the homology cycles. Thus the number of spin
structures is $2^{2g}$. They can be divided into even and odd depending
on whether the number of holomorphic sections of the corresponding line
bundle is even or odd. There are $2^{g-1}(2^{g}+1)$ even spin structures
and $2^{g-1}(2^{g}-1)$ odd ones (for a detailed discussion of spin
structures on Riemann surfaces see \cite{Fay}). The physical
interpretation of the sum over spin structures is the implementation of
the GSO projection. Reciprocally, the geometrical interpretation of the
GSO projection lies on the sum over spin structures, the cancellation of
modular anomalies. This is best illustrated by studying in some detail
the case of genus one, the torus. The spin line bundles on the torus are
flat line bundles and we can represent them simply in terms of boundary
conditions on the two homology cycles. There are four possibilities
schematically drawn below

\unitlength=1mm
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\linethickness{0.4pt}
\begin{picture}(110.00,30.00)(20,60)
\put(40.00,70.00){\framebox(10.00,10.00)[cc]{}}
\put(60.00,70.00){\framebox(10.00,10.00)[cc]{}}
\put(80.00,70.00){\framebox(10.00,10.00)[cc]{}}
\put(100.00,70.00){\framebox(10.00,10.00)[cc]{}}
\put(38.00,75.00){\makebox(0,0)[cc]{+}}
\put(45.00,68.00){\makebox(0,0)[cc]{+}}
\put(58.00,75.00){\makebox(0,0)[cc]{+}}
\put(65.00,68.00){\makebox(0,0)[cc]{--}}
\put(78.00,75.00){\makebox(0,0)[cc]{--}}
\put(85.00,68.00){\makebox(0,0)[cc]{+}}
\put(98.00,75.00){\makebox(0,0)[cc]{--}}
\put(105.00,68.00){\makebox(0,0)[cc]{--}}
\end{picture}

The horizontal line represents the $\sigma$-variable on the world-sheet
and the vertical line the $\tau$-variable.

The mapping class group, or modular group on the torus is the group
$PSL(2,{\bf Z})$ which acts on the modular parameter $\tau$ as
\begin{equation}
\tau\longrightarrow \frac{a\tau+b}{c\tau+d} \hspace{1cm}
a,b,c,d \in {\bf Z}\;, \hspace{1cm} ab-cd=\pm 1\;.
\end{equation}
This group is generated by two elements
\begin{eqnarray}
T:\tau\longrightarrow \tau+1 \;, \nonumber \\
S:\tau\longrightarrow -\frac{1}{\tau}\;,
\end{eqnarray}
which satisfy two relations:
\begin{equation}
S^{2}=(ST)^{3}=1
\end{equation}

The action of this group changes the fermionic boundary conditions. For
instance, acting with $T$ on $(-,-)$ the fundamental cell of the torus
changes as shown in the figure

\unitlength=1mm
\special{em:linewidth 0.4pt}
\linethickness{0.4pt}
\begin{picture}(110.00,40.00)(30.00,110.00)
\put(40.00,120.00){\framebox(20.00,15.11)[cc]{}}
\put(100.00,120.00){\line(4,3){20.00}}
\put(120.00,135.11){\line(1,0){20.00}}
\put(140.00,135.11){\line(-4,-3){20.00}}
\put(120.00,120.00){\line(-1,0){20.00}}
\put(75.00,127.56){\vector(1,0){16.67}}
\put(50.33,118.22){\makebox(0,0)[cc]{$-$}}
\put(50.33,121.78){\makebox(0,0)[cc]{$a$}}
\put(38.33,127.56){\makebox(0,0)[cc]{$-$}}
\put(41.67,127.56){\makebox(0,0)[cc]{$b$}}
\put(40.00,136.89){\makebox(0,0)[cc]{$\tau$}}
\put(61.00,118.22){\makebox(0,0)[cc]{$1$}}
\put(111.00,118.22){\makebox(0,0)[cc]{$-$}}
\put(108.33,128.89){\makebox(0,0)[cc]{$+$}}
\put(118.33,137.33){\makebox(0,0)[cc]{$\tau+1$}}
\put(121.33,118.22){\makebox(0,0)[cc]{$1$}}
\put(83.67,129.33){\makebox(0,0)[cc]{$T$}}
\end{picture}

Hence, if originally we had the $(-,-)$ spin structure, we end up with
$(+,-)$. If we now act with $S$ (which effectively exchange the $a$ and
$b$ cycles) $(+,-)$ is transformed into $(-,+)$. Thus, on the torus we
have two orbits of $PSL(2,{\bf Z})$ on the spin structures: the even
orbit $(-,-)$, $(-,+)$, $(+,-)$ and the odd orbit $(+,+)$. The reader
can easily check that the operator $\bar{\partial}$ (the Dirac operator
in this case) has no zero modes in the even orbit, and one zero mode for
$(+,+)$ boundary conditions. If we let $\Gamma_{2}$
\footnote{$\Gamma_{2}$ is the subgroup of elements
\begin{eqnarray}
\left(
\begin{array}{cc}
a & b \\
c & d
\end{array}
\right)\in SL(2,{\bf Z}) \nonumber
\end{eqnarray}
such that
\begin{eqnarray}
\left(
\begin{array}{cc}
a & b \\
c & d
\end{array}
\right) \equiv \left(
\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}
\right)
\;\;\mbox{mod}\;2
\nonumber
\end{eqnarray}}
be the subgroup of $SL(2,{\bf Z})$ leaving the spin structure fixed, the
quotient $SL(2,{\bf Z})/\Gamma_{2}$ is a finite group of order six. The
issue of global gravitational anomalies on the world-sheet was analized
by Witten (see first reference in \cite{Witten-anom}). After this
brief mathematical interlude
we can pose the problem of global anomalies on the world-sheet. When we
defined scattering measures in the previous chapter, in the measure we
had to deal with the integration over metrics modulo diffeomorphisms.
Among the latter we can distinguish the normal subgroup of orientation
preserving diffeomorphisms in the identity component
$\mbox{Diff}_{0}^{+}(\Sigma)$. The mapping class group was defined as
$\Omega(\Sigma)=\mbox{Diff}^{+}(\Sigma)/\mbox{Diff}_{0}^{+}(\Sigma)$.
Since all the quantities appearing in the bosonic string were from the
beginning generally covariant, we were rather cavalier in the treatment
of $\Omega(\Sigma)$. We need to be more careful now. Since the fields
appearing in the fermionic string are acted on non-trivially by
$\Omega(\Sigma)$, we have to ensure that the quantum theory is invariant
under $\Omega(\Sigma)$. If we fix the spin structures of the fermions
the absence of global anomalies is the requirement that the theory does
not change under the action of the elements of $\Gamma_{2}$ (the
subgroup of $\Omega(\Sigma)$ preserving the spin structures). Once this
is done we can classify all possible ways of summing over spin
structures which are invariant under $\Omega(\Sigma)/\Gamma_{2}$. This
should produce all modular invariant string theories with the initial
content of two-dimensional fields. The simplest example to analyze the
action of $\Gamma_{2}$ is in the case at hand with the GSO projection.
The simplest example we are considering has the $G$-parity projection
imposed on left and right movers independently. Let us compute the toral
partition functions in the four sectors. Using $q=\exp{(2\pi i\tau)}$ we
obtain:
\begin{eqnarray}
Z_{(-,-)}&=&tr\,q^{H_{NS}} \nonumber \\
Z_{(-,+)}&=&tr\,(-1)^{F}q^{H_{NS}}  \nonumber \\
Z_{(+,-)}&=&tr\,q^{H_{R}}  \nonumber \\
Z_{(+,+)}&=&tr\,(-1)^{F}q^{H_{R}}\;,
\label{Z-traces}
\end{eqnarray}
where $H_{NS}$ and $H_{R}$ are respectively the hamiltonians
corresponding to the NS and R sectors
\begin{eqnarray}
H_{NS}&=&(L_{0})_{plane}-\frac{1}{2}=
\sum_{n>0}\alpha_{-n}^{i}\alpha_{n}^{i}+\sum_{r>0}rb_{-r}^{i}b_{r}^{i}-
\frac{1}{2} \nonumber \\
H_{R}&=&(L_{0})_{plane}-\frac{1}{2}=
\sum_{n>0}\alpha_{-n}^{i}\alpha_{n}^{i}+\sum_{n>0}nd_{-n}^{i}d_{n}^{i}
\end{eqnarray}
since in the light-cone gauge $c=8\times 1+8\times \frac{1}{2}=12$.
Note that in order to implement the periodic boundary condition in the
$\tau$-direction we need to insert $(-1)^{F}$ in the traces appearing in
(\ref{Z-traces}). This gives the first hint of the geometrical
interpretation of the GSO projection. We
have dropped the contribution of the zero modes of $\alpha_{0}^{i}$;
when computing this contribution coming from both left- and right-moving
modes, we get that it is proportional to $\tau_{2}^{-4}$. We can
evaluate the trace for the bosonic modes $\alpha^{i}_{n}$ which
is a common factor in all the contributions:
\begin{equation}
tr\, q^{\sum_{n>0}\alpha_{-n}^{i}\alpha_{n}^{i}}=
q^{\frac{1}{3}}\eta^{-8}(\tau)
\end{equation}
where $\eta(\tau)$ is the Dedekind eta function
\begin{equation}
\eta(\tau)=q^{\frac{1}{24}}\prod_{n=1}^{\infty}(1-q^{n})
\end{equation}
All the contributions to the partition function can be expressed in
terms of Jacobi theta functions \cite{Mumford,Alvarez-Gaume-Moore-Vafa}
\begin{equation}
\vartheta\left[
\begin{array}{c}
\alpha \\
\beta
\end{array}
\right](z|\tau)=\sum_{n\in{\bf Z}}e^{i\pi\tau(n+\alpha)^{2}+
2\pi i(n+\alpha)(z+\beta)}
\label{vartheta-function}
\end{equation}
with $\alpha$, $\beta \in ({\bf Z}+\frac{1}{2})/{\bf Z}$.
The computation of the traces in (\ref{Z-traces}) yields:
\begin{eqnarray}
Z_{(-,-)}&=&\eta^{-8}(\tau)\left(\frac{\vartheta\left[
\begin{array}{c}
0 \\
0
\end{array}
\right](0|\tau)}{\eta(\tau)}\right)^{4} \nonumber \\
Z_{(-,+)}&=&\eta^{-8}(\tau)\left(\frac{\vartheta\left[
\begin{array}{c}
0 \\
\frac{1}{2}
\end{array}
\right](0|\tau)}{\eta(\tau)}\right)^{4} \nonumber \\
Z_{(+,-)}&=&\eta^{-8}(\tau)\left(\frac{\vartheta\left[
\begin{array}{c}
\frac{1}{2} \\
0
\end{array}
\right](0|\tau)}{\eta(\tau)}\right)^{4} \nonumber \\
Z_{(+,+)}&=&\eta^{-8}(\tau)\left(\frac{\vartheta\left[
\begin{array}{c}
\frac{1}{2} \\
\frac{1}{2}
\end{array}
\right](0|\tau)}{\eta(\tau)}\right)^{4}=0\;.
\label{theta-contrib}
\end{eqnarray}
since the factor $q^{\frac{1}{3}}$ cancel out when computing the total
trace. We see that the odd spin structure does not contribute as could
have been predicted due to the presence of a zero mode. It is not
difficult to verify that the fourth power if the $\vartheta$-functions
appearing in (\ref{theta-contrib}) is the minimal power guaranteeing
invariance under $\Gamma_{2}$. For example, using the explicit form
(\ref{vartheta-function}) with $z=0$, we see that
the $\vartheta$-functions change by
a fourth root of unity as $\tau\rightarrow \tau+2$. This modular
transformation does not change the boundary conditions. It is an useful
exercice to verify that for any element of $\Gamma_{2}$ the change is
only a fourth root of unity. The fourth power appearing in
(\ref{theta-contrib}) is a consequence of the fact that in the
light-cone gauge we have eight Weyl-Majorana fermions with the same
spin structure. They can be combined into four Weyl fermions, and the
determinant of the Weyl operator is proportional to a
$\vartheta$-function \cite{Alvarez-Gaume-Moore-Vafa}. Now that the
global anomalies have cancelled, we can determine the full modular
invariant combination of the partition functions (\ref{theta-contrib}).
Thus we have to determine the relative phases with which each element in
(\ref{theta-contrib}) enters in the total partition function
\begin{eqnarray}
Z(\tau)&=&Z_{(-,-)}(\tau)+w(-,+)Z_{(-,+)}(\tau) \nonumber \\
&+&w(+,-)Z_{(+,-)}(\tau)+w(+,+)Z_{(+,+)}(\tau)
\label{tor-partition}
\end{eqnarray}
The phases $w(\pm,\mp)$, $w(+,+)$ are well defined due to the absence of
global anomalies. Since the $(-,-)$, $(+,-)$, $(-,+)$ spin structures
are determined by the action of $T$, $S$, we easily determine the phases
to be
\begin{equation}
w(-,+)=w(+,-)=-1
\end{equation}
Then we can write the one loop partition function as
\begin{equation}
Z(\tau)=tr\left(1+(-1)^{F+1}\right)q^{H_{NS}}-tr\,q^{H_{R}}
\end{equation}
This means that, in order to maintain modular invariance we have to
project onto the sector with $(-1)^{F+1}=1$. Thus the GSO projection
is a consequence of modular invariance of the one loop partition
function. Introducing a more convenient notation
\begin{equation}
\vartheta\left[
\begin{array}{c}
0 \\
0
\end{array}
\right]=\theta_{3} \hspace{0.5cm}
\vartheta\left[
\begin{array}{c}
\frac{1}{2} \\
0
\end{array}
\right]=\theta_{2} \hspace{0.5cm}
\vartheta\left[
\begin{array}{c}
0 \\
\frac{1}{2}
\end{array}
\right]=\theta_{4} \hspace{0.5cm}
\vartheta\left[
\begin{array}{c}
\frac{1}{2} \\
\frac{1}{2}
\end{array}
\right]=\theta_{1}\;,
\end{equation}
the partition function becomes
\begin{equation}
Z(\tau)=\eta^{-12}(\tau)[\theta_{3}^{4}(0|\tau)-
\theta_{4}^{4}(0|\tau)-\theta_{2}^{4}(0|\tau)]\;.
\label{aequatio}
\end{equation}
The combination of $\theta$-functions in (\ref{aequatio}) vanishes
identically. An identity called by Jacobi {\it aequatio identica  satis
abstrusa}. Since the
total partition function for the closed superstring is proportional to
$\tau_{2}^{-4}Z(\tau)Z(\bar{\tau})$
we have that the one loop partition function for the type II superstring
vanishes. This is a consequence of supersymmetry since the contribution
to $Z(\tau,\bar{\tau})$ from the target space bosons is exactly
cancelled by the fermions.

As a final remark we notice that $w(+,+)$ was not determined by the
previous arguments. This is not unreasonable. From the space-time point
of view, the states in the NS sector transform under single-valued
representations of the little group $SO(8)$, and we should quantize them
as bosons. In the R sector on the other hand the states are obtained by
acting with tensor operators on a spinor ground state, and therefore
these states should be treated as space-time fermions. Since there is an
ambiguity in how to define space-time chirality, it is not unreasonable
that $w(+,+)$ is not determined. To obtain constraints on the relative
phases between the NS and R sectors one should consider the
factorization properties of two-loop amplitudes. The lesson to be
learned from this lengthy discussion is that the spectrum of fermionic
string theories is strongly constrained by a simple consistency
requirement: absence of global gravitational anomalies, or
equivalently, modular invariance. If we consider simultaneously the
modular properties of both left and right-movers, more possibilities
open up. Each modular invariant combination will correspond to a
different spectrum. This way of constructing string theories has been
explored in the literature, specially in the context of heterotic
strings \cite{string-construction}.

As we pointed out earlier, by using Green-Schwarz formalism
\cite{Green-Schwarz-2}, one can
work with an action in which space-time supersymmetry is explicit. The
difficulty with this formalism is that it cannot be easily
quantized in a covariant way (it requires the use of Batalin-Vilkoviski
techniques). Here we use light-cone gauge quantization.
In this approach we have, besides the $X^{i}(\tau,\sigma)$ bosonic
coordinates, a field $S^{Aa}(\tau,\sigma)$ where $A=1,2$ is a
world-sheet spinor index and $A=1,\ldots,32$ is a space-time spinor index. We
impose on the ten-dimensional spinor both Weyl and Majorana condition,
so the $64$ complex components of $S^{Aa}$ are reduced to $32$ real
components. Imposing the light-cone gauge condition on $S^{Aa}$
\begin{equation}
(\Gamma^{+})^{ab}S^{Ab}=0
\end{equation}
with $\Gamma^{\mu}$ the ten-dimensional Dirac matrices,
we are left with $16$ real components. Since the Dirac equation reduces
further the number of real components to $8$, we have that, on-shell,
the number of fermionic space-time degrees of freedom is equal to the
number of transverse bosonic coordinates, and we can have in
principle space-time supersymmetry. The light-cone gauge action is given
by
\begin{equation}
S=-\frac{1}{4\pi\alpha^{'}}\int d\tau d\sigma (\partial_{\alpha}X^{i}
\partial^{\alpha}X^{i}-i\bar{S}^{a}\rho^{\alpha}
\partial_{\alpha}S^{a})\;.
\end{equation}
The equations of motion for the spinors $S^{Aa}$ are simply
\begin{equation}
\partial_{+}S^{1a}=\partial_{-}S^{2a}=0
\end{equation}
This, together with the periodic boundary conditions
$S^{a}(\tau,\sigma+\pi)=S^{a}(\tau,\sigma)$ for the closed
superstring, allow us to write the mode expansion
\begin{eqnarray}
S^{1a}&=&\sum_{n\in{\bf Z}}S^{a}_{n}e^{2in(\tau-\sigma)} \nonumber \\
S^{2a}&=&\sum_{n\in{\bf Z}}\tilde{S}^{a}_{n}e^{2in(\tau+\sigma)}
\end{eqnarray}
with the anticommutation relations for the modes:
\begin{equation}
\{S_{m}^{a},\tilde{S}_{n}^{b}\}=\delta^{ab}\delta_{m+n,0}
\end{equation}
We will now use $\tilde{\mbox{ }}$ to denote left-moving modes in order
to avoid confusions with the components of the conjugate spinor.
Using these operators the mass formula and the level matching
condition for the closed superstring are respectively:
\begin{eqnarray}
\frac{1}{2}\alpha^{'}m^{2}=\sum_{n>0}(\alpha_{-n}^{i}\alpha_{n}^{i}+
\tilde{\alpha}_{-n}^{i}\tilde{\alpha}_{n}^{i}+nS_{-n}^{a}S_{n}^{a}+
n\tilde{S}_{-n}^{a}\tilde{S}_{n}^{a}) \nonumber \\
\sum_{n>0}( \alpha_{-n}^{i} \alpha_{n}^{i}+nS_{-n}^{a}S_{n}^{a})=
\sum_{n>0}(\tilde{\alpha}_{-n}^{i}\tilde{\alpha}_{n}^{i}+
n\tilde{S}_{-n}^{a}\tilde{S}_{n}^{a})
\end{eqnarray}
If the states $|i\rangle$ denote the eight physical degrees of freedom
of the massless vector state, we define the supersymmetric partners
$|a\rangle$
\begin{equation}
|a\rangle=\frac{i}{8}(\Gamma_{j}S_{0})^{a}|j\rangle\;,
\label{partner}
\end{equation}
which are normalized according to
\begin{equation}
\langle i|j\rangle=\delta_{ij}\;, \hspace{1cm} \langle a|b\rangle=
\frac{1}{2}(h\Gamma^{+})^{ab}\;,
\end{equation}
where $h$ is the ten-dimensional helicity operator. The massless
spectrum contain $128$ bosonic states
\begin{equation}
|i\rangle_{L}\otimes|j\rangle_{R}  \hspace{1cm}
|a\rangle_{L}\otimes|b\rangle_{R}
\end{equation}
and the same number of fermionic states
\begin{equation}
|i\rangle_{L}\otimes|a\rangle_{R} \hspace{1cm}
|a\rangle_{L}\otimes|i\rangle_{R}
\end{equation}
As indicated above, when $|a\rangle_{L}$ and $|b\rangle_{R}$ have the
same helicity we have the type IIB superstring (chiral) whereas
when their helicities are opposite we have the type IIA superstring
(non-chiral).

As a final remark let us indicate the spectrum of the type I (open)
superstring. Although now we do not have any periodicity condition
over the fields, it is possible to show that the massless spectrum is
the same as in one of the sectors of the type II superstring: we have
the eight transverse bosonic modes $|i\rangle$ and their
fermionic partners $|a\rangle$ defined by (\ref{partner}). Then the
states in the massless sector of the open superstring are those of a
chiral $N=1$ SUGRA in $d=10$. For the open  superstring it
is possible to introduce gauge symmetries using the Chan-Paton
\cite{Chan-Paton} procedure:
we attach charges to the endpoints of the string. However by using this
method we can only have $USp(N)$, $SO(N)$ and $U(N)$ as gauge groups.
By doing so the low energy field theory is a $N=1$ SUGRA coupled
to $N=1$ super-Yang-Mills in $d=10$. Green and Schwarz
\cite{Green-Schwarz} proved that this low energy field theory is
anomaly free (from gauge and gravitational anomalies) as long as
the gauge groups are $E_{8}\times E_{8}$,
$SO(32)$ or $E_{8}\times U(1)^{248}$. Given the restriction imposed over
the possible gauge groups introduced by Chan-Paton factors we have
that the only consistent open string theory has $SO(32)$ as its gauge
group.

\subsection{Heterotic String}

Up to now we have studied two kinds of string models: the closed bosonic
string in which the dimension of the
target space is $26$, and the type II superstring for which
the dimension is ten. In both cases we have to deal with two
sectors, the left- and the right-moving modes without mixing.
Besides, in these models strings are ``neutral''
objects, in the sense that they cannot have attached charges
by means of Chan-Paton factors.

The independence between the left and right moving modes in the closed
string models can be used to construct a hybrid theory, the heterotic
string \cite{Pricenton-quartet} (for a general review see also
\cite{Green-Schwarz-Witten,Lust-Theisen,Gross}). We combine the $10$
right-moving modes of a type-II superstring with the $26$ left-movers
of a bosonic string after compactifying $16$ of them into an internal
manifold. We work in the light cone gauge. The physical
degrees of freedom in the right-moving sector are the $8$ transverse
bosons $X^{i}(\tau-\sigma)$ and one Majorana-Weyl ten-dimensional
fermion $S^{a}(\tau-\sigma)$ ($a=1,\ldots,8$). In the
left-moving sector we
have $24$ transverse bosonic coordinates $X^{i}(\tau+\sigma)$ and
$X^{I}(\tau+\sigma)$ ($I=1,\ldots,16$). The last $16$ bosonic
coordinates
can be interpreted as parametrizing an internal manifold. Consistency of
the resulting theory impose that this internal manifold has to be a
$16$-dimensional torus ${\bf T}$
\begin{equation}
{\bf T}={\bf R}^{16}/\Lambda_{16}
\end{equation}
where $\Lambda_{16}$ is a $16$-dimensional lattice. The action for the
free heterotic string can be written in term of the transverse modes as
\begin{equation}
S=-\frac{1}{4\pi\alpha^{'}}\int d\tau d\sigma
[\partial_{\alpha}X^{i}\partial^{\alpha}X^{i}+
\partial_{\alpha}X^{I}\partial^{\alpha}X^{I}+
i\bar{S}\Gamma^{-}(\partial_{\tau}+\partial_{\sigma})S]
\label{het-action}
\end{equation}
together with the constraints
\begin{equation}
\Phi\equiv (\partial_{\tau}-\partial_{\sigma})X^{I}=0 \hspace{1cm}
I=1,\ldots,16
\label{Phi-constr}
\end{equation}
The action (\ref{het-action}) is invariant under the supersymmetry
transformations
\begin{eqnarray}
\delta X^{i}&=&\frac{1}{\sqrt{p^{+}}}\bar{\epsilon}\Gamma^{i}S \nonumber
\\ \delta S &=& \frac{i}{\sqrt{p^{+}}}\Gamma_{-}\Gamma_{\mu}
(\partial_{\tau}-\partial_{\sigma})X^{\mu}\epsilon
\end{eqnarray}
where $\epsilon$ is a right-moving Majorana-Weyl light-cone spinor.

It is quite easy to obtain the spectrum for the heterotic string, since
what we have to do is to put together the results for the bosonic
and supersymmetric strings. Expanding the different fields into modes,
we get the set of operators
\begin{equation}
\{\alpha_{n}^{i},S^{a}_{n},\tilde{\alpha}_{n}^{i},
\tilde{\alpha}^{I}_{n}\}
\end{equation}
with $n\in{\bf Z}$ and $a,i=1,\ldots,8$. Besides these
operators we also have the center of mass positions $x^{i}$ and
$x^{I}$ and the momenta $p^{i}$ and $p^{I}$.
The commutation relations for the modes of the fields $X^{I}$ satisfying
the constraints (\ref{Phi-constr}) are obtained using Dirac brackets
(see \cite{Pricenton-quartet}). The commutation relations for the modes
are
\begin{eqnarray}
[\alpha_{n}^{i},\alpha_{m}^{j}]&=&
[\tilde{\alpha}_{n}^{i},\tilde{\alpha}_{m}^{j}]=
n\delta^{ij}\delta_{m+n,0} \nonumber \\
\{S_{n}^{a},S_{m}^{b}\}&=&(\Gamma^{+}h)^{ab}\delta_{n+m,0} \nonumber \\
{[}\tilde{\alpha}_{n}^{I},\tilde{\alpha}_{m}^{J}]&=&
n\delta^{IJ}\delta_{m+n,0} \nonumber \\
{[}q^{i},p^{j}]&=&i\delta^{ij} \hspace{1cm}
[q^{I},p^{J}]=\frac{i}{2}\delta^{IJ}
\end{eqnarray}
We define the normal-ordered number operators
\begin{eqnarray}
N&=&\sum_{n>0}(\alpha_{-n}^{i}\alpha_{n}^{i}+\frac{n}{2}\bar{S}_{-n}
\Gamma^{-}S_{n})\;, \nonumber \\
\tilde{N}&=&\sum_{n>0}(\tilde{\alpha}_{-n}^{i}\tilde{\alpha}_{n}^{i}+
\tilde{\alpha}_{-n}^{I}\tilde{\alpha}_{n}^{I})\;,
\end{eqnarray}
in terms of which the mass formula becomes
\begin{equation}
\frac{1}{2}\alpha^{'}m^{2}=N+\tilde{N}+
\frac{1}{2}\sum_{I=1}^{16}(p^{I})^{2}
\end{equation}
As in the case of bosonic and supersymmetric strings, we have to
implement the level matching condition:
\begin{equation}
N=\tilde{N}+\frac{1}{2}\sum_{I=1}^{16}(p^{I})^{2}-1\;.
\end{equation}

We still need to determine the internal manifold in which we have
compactified the $16$ extra bosonic coordinates. The constraints on
$\Lambda_{16}$ come, as might be expected from previous discussions,
from the requirement of modular invariance. Once again this requirement
determines the spectrum. In the case at hand, it will be shown below
that $\Lambda_{16}$ must be an even integral euclidean self-dual
lattice. These lattices are
very rare; in fact there are only two such lattices, the weight lattice
of $Spin(32)/{\bf Z}_{2}$ (the modding by ${\bf Z}_{2}$ eliminates
one of the spin representation, and the root lattice of $SO(32)$ is a
sublattice) and the root lattice of $E_{8}\times E_{8}$.

Now we proceed to determine the spectrum of the heterotic string. The
ground state $|0\rangle$ for left-moving modes is defined by
\begin{equation}
\tilde{\alpha}^{i}_{n}|0\rangle=\tilde{\alpha}^{I}_{n}|0\rangle=0\;,
\hspace{1cm} n>0\;.
\end{equation}
In the right-moving sector we have the eight bosonic states $|i\rangle$
and the their fermionic partners $|a\rangle$.
The mass formula, the level matching condition and the structure of
$\Lambda_{16}$ imply that there are no tachyons in the spectrum. The
massless states are
\begin{eqnarray}
\tilde{\alpha}_{-1}^{i}|0\rangle_{L}&\otimes &|i\rangle_{R}
\label{grav-multip-1} \\
\tilde{\alpha}_{-1}^{i}|0\rangle_{L}&\otimes &|a\rangle_{R}
\label{grav-multip-2} \\
\tilde{\alpha}_{-1}^{I}|0\rangle_{L}&\otimes &|i\rangle_{R}
\label{super-YM-1}\\
\tilde{\alpha}_{-1}^{I}|0\rangle_{L}&\otimes &|a\rangle_{R}
\label{super-YM-2} \\
|p^{I}\rangle_{L}&\otimes &|i\rangle_{R} \label{super-YM-3}\\
|p^{I}\rangle_{L}&\otimes &|a\rangle_{R} \label{super-YM-4}
\end{eqnarray}
with $(p^{I})^{2}=2$. The states (\ref{grav-multip-1}) and
(\ref{grav-multip-2}) form a $N=1$, $d=10$ supergravity
multiplet containing the graviton and the gravitino together with
the antisymmetric tensor, the dilaton and its supersymmetric partners.
States (\ref{super-YM-1})-(\ref{super-YM-4}) form the $N=1$, $d=10$
super-Yang-Mills multiplet with gauge group $E_{8}\times E_{8}$ or
$Spin(32)/{\bf Z}_{2}$. Then the low energy field theory
($\alpha^{'}\rightarrow 0$) of the heterotic string is $N=1$ SUGRA
coupled to $N=1$ super-Yang-Mills in $d=10$ with gauge group
$E_{8}\times E_{8}$ or $SO(32)$.
Then the heterotic string provide us with the first
example of a string theory with gauge group $E_{8}\times E_{8}$ which
could not be obtained in the open string case.

We see, by applying Green-Schwarz cancellation mechanism, that the low
energy field theory for the heterotic string is free from both
gravitational and gauge anomalies. Moreover, it can be shown that the
theory is anomaly-free at the string level if the gauge groups are
$E_{8}\times E_{8}$ or $Spin(32)/{\bf Z}_{2}$ \cite{Gross-Mende}. So
the cancellation of anomalies with these gauge groups not only works
at the level of the low energy effective field theory but it can be
verified for the whole string theory.

Let us compute the one loop partition function for the heterotic string.
Had we formulated the heterotic string using the old superstring
formalism (which is explicitly world-sheet supersymmetric) we would
have had to include the GSO-projected NS and R sectors of the right
movers. We have also the contribution from the $8$
transverse bosonic coordinates of the left-moving sector and the $16$
internal degrees of freedom. When all the contributions are included
the one loop partition function for
the heterotic string (up to constant factors) becomes:
\begin{equation}
Z_{het}(\tau,\bar{\tau})= \tau_{2}^{-4}
\frac{\bar{\Theta}(\bar{\tau})}{\bar{\eta}^{24}(\bar{\tau})}
\eta^{-12}(\tau)[\theta_{3}^{4}(0|\tau)-
\theta_{3}^{4}(0|\tau)-\theta_{2}^{4}(0|\tau)]\;,
\label{het-aequatio}
\end{equation}
where $\bar{\Theta}(\bar{\tau})$ is the theta function associated with
the internal lattice $\Lambda_{16}$
\begin{equation}
\bar{\Theta}(\bar{\tau})=\sum_{{\bf p}\in \Lambda_{16}}
\bar{q}^{\frac{1}{2}{\bf p}^{2}}\;.
\end{equation}
In order for the partition function $Z_{het}(\tau,\bar{\tau})$ to be
modular invariant, we need $\bar{\Theta}(\bar{\tau})$ to transform
under the $T$ and $S$ according to
\begin{eqnarray}
T:\bar{\Theta}(\bar{\tau}) &\longrightarrow& \bar{\Theta}(\bar{\tau})
\nonumber \\
S:\bar{\Theta}(\bar{\tau}) &\longrightarrow&
\bar{\tau}^{8}\bar{\Theta}(\bar{\tau})
\end{eqnarray}
It is easy to check that the first condition implies that
${\bf p}^{2}$ is an even integer. This means that $\Lambda_{16}$ has
to be an even lattice. To impose the condition with respect to $S$
we use the Poisson summation formula:
\begin{equation}
\bar{\Theta}\left(-\frac{1}{\bar{\tau}}\right)=
\frac{\bar{\tau}^{8}}{\sqrt{{\mbox{det}g}}}
\sum_{{\bf p}\in \Lambda_{16}^{*}}\bar{q}^{\frac{1}{2}{\bf p}^{2}}
\end{equation}
where $\Lambda_{16}^{*}$ is the dual lattice of $\Lambda_{16}$ and
$g_{ij}$ is the metric of the lattice. Then the $S$ and $T$ conditions
imply that $\Lambda_{16}$ is even and self-dual. Let us note that, as
in the superstring case, the
partition function (\ref{het-aequatio}) vanishes because of Jacobi's
famous theta function identity. This
is again a consequence of the space-time supersymmetric character of the
heterotic string.

\subsection{Strings at finite temperature}

In the Standard Cosmological Model (see for example \cite{Weinberg}), at
very early times the Universe was
in a very hot state. Close to the Planck time this temperature could be
of the order of the Planck mass. Thus it seems reasonable to think that
if String Theory is a theory of Quantum Gravity, it should have a lot to
say about the very early Universe
\cite{EAlvarez-cosm,Tye,Brandenberger-Vafa,Tseylin-Vafa}.
A first step in this direction is
to study the thermal properties of strings. This is what we do in this
section. As we will see later, modular invariance also plays a r\^{o}le
in determining the structure of the string free energy.

When studying (critical) strings at finite temperature we meet two
properties which are typically ``stringy''. The first one is the
fact that for every string theory there is a temperature at which the
canonical partition function diverges. The presence of this temperature,
called the Hagedorn temperature \cite{Hagedorn}, is due to the fact
that the
number of states for a given mass in any critical string theory grows
exponentially with the
mass. For example, in the bosonic string the asymptotic
density of states with mass $m$ is given by
\begin{equation}
\rho(m)\sim m^{-\frac{25}{2}}e^{4\pi\sqrt{\alpha^{'}}m}\;.
\label{asymptotic-density}
\end{equation}
The canonical partition function $Z(\beta)$ is defined as the
Laplace transform of the density of states ($\beta=1/T$)
\begin{equation}
Z(\beta)=\int_{0}^{\infty}dE \rho(E)e^{-\beta E}\;.
\end{equation}
In our case we can write \cite{Alvarez}
\begin{equation}
Z(\beta)\sim \int_{\mu}^{\infty} dE
E^{-\frac{25}{2}}e^{(4\pi\sqrt{\alpha^{'}}-\beta)E}\;,
\end{equation}
where $\mu$ is a cutoff in order to make (\ref{asymptotic-density})
applicable. Now it is easy to see that the partition function $Z(\beta)$
is defined as long as
\begin{equation}
\beta>4\pi\sqrt{\alpha^{'}}\;.
\end{equation}
The right-hand side of the inequality defines the Hagedorn temperature
for the bosonic string
\begin{equation}
\beta_{H}=4\pi\sqrt{\alpha^{'}}
\end{equation}
On the other hand, the Helmholtz free energy of the bosonic string is,
because of the tachyon, undefined
(infinite) for every value of $\beta$. Nevertheless, the Hagedorn
temperature is present in every critical string theory even in
those which have a well defined low temperature phase, as the
superstring or the heterotic string (cf. for example
\cite{Kani-Vafa}). The Hagedorn temperature is a consequence of the
fact that the density of states grows exponentially with the energy.

The second ``stringy'' feature mentioned is duality
property (sometimes called $\beta$-duality). This is expressed by the
equality of
the canonical partition function of the theory evaluated at $\beta$ and
$constant/\beta$. For example, for the heterotic string
\cite{O'Brien-Tan}
\begin{equation}
Z(\beta)=Z\left(\frac{\pi^{2}}{\beta}\right)
\end{equation}
Note that although formally this duality property is quite similar to
the $R$-duality property that we have treated in sec. 3.6, they are in
fact quite different. We will see that the kind of compactification that
give raise to both ``dualities'' are of very different type.
$\beta$-duality
is present formally in the bosonic string (it is a symmetry for the
integrand of the free energy),
and in the heterotic string (see below); in the type II superstring
there
is no duality symmetry although it is possible to construct a kind of
``generalized'' duality relation for the one-loop free energy
\cite{Osorio-1}.

As in ordinary Statistical Mechanics, it is convenient to work with the
Helmholtz free energy $F(\beta)$:
\begin{equation}
F(\beta)=-\frac{1}{\beta}\log{Z(\beta)}
\end{equation}
In QFT, the canonical partition function in ${\bf R}^{1,d-1}$ is
equal to the vacuum energy
of the same theory in ${\bf R}^{d-1}\times S^{1}$ with the
length of the compactified circle equal to $\beta$. Using
this procedure
Polchinski computed the Helmholtz free-energy per unit volume for the
bosonic string at one-loop \cite{Polchinski} with the result
\begin{equation}
F(\beta)=-\pi^{-26}2^{-14}\int_{S}\frac{d^{2}\tau}{\tau^{2}_{2}}
\tau_{2}^{-12}|\eta(\tau)|^{-48}
\left[\theta_{3}\left(0\left|\frac{i\beta^{2}}{2\pi^{2}\tau_{2}}
\right.\right) -1\right]
\label{bosonic-S}
\end{equation}
where $\tau=\tau_{1}+i\tau_{2}$ is the modular parameter of the torus,
$\eta(\tau)$, $\theta_{3}(0|\tau)$ are respectively the Dedekind eta
function and the Jacobi theta function and the integral is evaluated
over the strip $S=\{\tau|\tau_{2}>0, -1/2<\tau_{1}<1/2\}$.

As we announced earlier, (\ref{bosonic-S})
does not converge because of the infrared divergence caused by
the tachyon as $\tau_{2}\rightarrow \infty$.
However this result does not correspond to the vacuum energy of a
string in ${\bf R}^{d-1}\times S^{1}$ since there are no winding modes
corresponding to the would-be space direction in the world-sheet and we
know that $\tau\rightarrow -1/\tau$ interchanges space and
time coordinates on the world-sheet. As a consequence,
the integral in (\ref{bosonic-S}) is not restricted to the fundamental
region of the modular group $PSL(2,{\bf Z})$ as we should expect from
a path integral calculation, since the integral over the modular
parameter comes from the integration over tori not equivalent under
diffeomorphisms \cite{McClain-Roth,O'Brien-Tan}.

It was observed by Polchinski \cite{Polchinski} that the path-integral
computation of the one-loop free energy (\ref{bosonic-S}) coincides with
what one would obtain by
adding the contributions to the free energy of the different states in
the string spectrum. This approach to the computation of the free energy
which
consists in considering the string as a collection of fields is called
the analog model
\cite{EAlvarez-cosm,Sundborg,Bowick-W,Tye,Alvarez,Alvarez-Osorio-1}
and gives actually the same result than that obtained by Polchinski.

Let us begin \cite{Alvarez-Osorio-1} with the expression for the free
energy per physical degree
of freedom and per unit volume of a bosonic (fermionic)
quantum field in $d$ dimensions
\begin{equation}
F_{B,F}(\beta)=\pm\frac{1}{\beta}\int \frac{d^{d-1}k}{(2\pi)^{d-1}}
\ln(1\mp e^{-\beta \omega_{k}})
\label{int-log}
\end{equation}
where $\omega_{k}=\sqrt{{\bf k}^{2}+m^{2}}$. After some elementary
manipulations (\ref{int-log}) can be rewritten as
\begin{equation}
F_{B}(\beta)=-\pi^{d/2}2^{d/2-1}
\int_{0}^{\infty}ds\,s^{-1-d/2}e^{-m^{2}s/2}
\left[\theta_{3}\left(0\left|\frac{i\beta^{2}}{2\pi
s}\right.\right)-1\right] \label{boson}
\end{equation}
for a bosonic field and as
\begin{equation}
F_{F}(\beta)=\pi^{d/2}2^{d/2-1}
\int_{0}^{\infty}ds\,s^{-1-d/2}e^{-m^{2}s/2}
\left[\theta_{4}\left(0\left|\frac{i\beta^{2}}{2\pi
s}\right.\right)-1\right] \label{fermion}
\end{equation}
for a fermionic field. The general procedure now is to add the
contributions of the states in the string given by the preceeding
formulae (taking into account the statistic of the corresponding state).
As we have seen in previous sections we have to impose the level
matching condition. This can be implemented by inserting the following
$\delta$-function:
\begin{equation}
\delta_{nm}=\int_{-\frac{1}{2}}^{\frac{1}{2}} d\tau_{1}
e^{i2\pi(n-m)\tau_{1}}
\end{equation}
One easily convince oneself that the free
energy obtained using the analog model corresponds to the free energy
of an ensemble of strings, since we sum the contributions of second
quantized fields. Using the analog model it is possible to evaluate
the one-loop free energy of any string theory. The extension to more
loops can be done by substituing the mass in (\ref{boson}),
(\ref{fermion}) by the renormalized
mass calculated to the same loop level minus one \cite{Osorio} (see
also \cite{Moore}).

We concentrate now on the heterotic string at finite temperature, since
this is the most interesting string model from many points of view. By
using the analog model (or evaluating the path integral in the
light-cone gauge for a single string and exponentiating the result) we
obtain that the one loop free energy is \cite{Alvarez-Osorio-1}
\begin{equation}
F(\beta)=-\frac{1}{2^{6}\pi^{10}}\int_{S}\frac{d^{2}\tau}{\tau_{2}^{2}}
\tau_{2}^{-4}\frac{\theta_{2}^{4}}{\eta^{12}}
\frac{\bar{\Theta}_{\Gamma}}{\bar{\eta}^{24}}
\theta_{2}\left(0\left|\frac{2i\beta^{2}}{\pi^{2}\tau_{2}}\right.\right)
\label{heterotic-T}
\end{equation}
As in the case of the bosonic string, our integral is performed over the
strip $S$ instead of the fundamental region of the modular group ${\cal
F}$. It is possible to obtain an explicitly
modular invariant form for the
one-loop free energy \cite{O'Brien-Tan,Alvarez-Osorio-2}. We have that
the integrand of (\ref{heterotic-T}) is invariant under a subgroup of
the full modular group, namely the subgroup $U$
generated by $T$ which is the subgroup that leaves invariant the
spin structure $(+,-)$. We can think of the fundamental
region of this subgroup (the strip $S$) as obtained by acting with
transformations belonging to $\Gamma=PSL(2,{\bf Z})$ on ${\cal F}$.
This implies that the translation subgroup $U$
can be written as the union of cosets of the modular group:
\begin{equation}
U=\bigcup_{(c,d)=1}\Gamma\gamma_{cd}\;,
\label{coset-exp}
\end{equation}
where $c$, $d$ are two co-prime integers and $\gamma_{cd}\in
SL(2,{\bf Z})$ is given by
\begin{equation}
\gamma_{cd}=\left(
\begin{array}{cc}
* & * \\
c & d
\end{array}
\right) \;,
\end{equation}
the two first entries in the matrix being any integers satisfying the
condition on the determinant. Then, if we have an integral over the
strip $S$ we can write
\begin{equation}
\int_{S}\frac{d^{2}\tau}{\tau_{2}^{2}}F(\tau)=
\int_{\bigcup\gamma_{cd}{\cal F}}\frac{d^{2}\tau}{\tau_{2}^{2}}F(\tau)=
\sum_{(c,d)=1}\int_{\gamma_{ab}{\cal F}}
\frac{d^{2}\tau}{\tau_{2}^{2}}F(\tau)\;.
\end{equation}
If we perform a change of variables for each term in the sum and commute
the sum with the integral we finally obtain
\begin{equation}
\int_{S}\frac{d^{2}\tau}{\tau_{2}^{2}}F(\tau)=
\int_{\cal F} \frac{d^{2}\tau}{\tau_{2}^{2}}\sum_{(c,d)=1}
F(\gamma_{ab}\tau)
\label{f-m-i}
\end{equation}
where we have used the modular invariance of the measure
$d^{2}\tau/\tau_{2}^{2}$. It can be easily seen that the
integrand in (\ref{f-m-i}) is modular invariant, since any modular
transformation acting on $\sum_{(c,d)=1}F(\gamma_{ab}\tau)$ amounts to
change of representatives in the coset expansion (\ref{coset-exp}).
This technique can be applied to any integral
whose integrand is invariant under any subgroup of $\Gamma$.
The equivalence between the original expression and its modular
invariant
extension depends upon the possibility of performing the change of
variables and the commutation of the integral and the sum.

Using the technique just described we can obtain an explicit (and
divergent) modular
invariant expression for the one-loop free energy of the bosonic string
which is also invariant under $U$. In the case of the heterotic string,
in order to obtain the modular invariant
extension, we have to work a bit more. The reason is that the extension
of the ``thermal'' $\theta_{2}$ function from $U$ to $\Gamma$ directly
is quite complicated. Instead we use the congruence subgroup
$\Gamma_{0}(2)\subset \Gamma$ \cite{Koblitz} as an intermediary
subgroup. Then we make
the extension from $U$ to $\Gamma_{0}(2)$ by restricting $\gamma_{cd}$
to this subgroup, the result being
\begin{equation}
F(\beta)=-\frac{1}{2^{6}\pi^{10}}\int_{F(\Gamma_{0}(2))}
\frac{d^{2}\tau}{\tau_{2}^{2}}\tau_{2}^{-4}
\frac{\bar{\Theta}_{\Gamma}}{\bar{\eta}^{24}}
\frac{\theta_{2}^{4}}{\eta^{12}} \theta \left[
\begin{array}{cc}
0  &  \frac{1}{2} \\
0  &  0
\end{array}
\right](0|4\Omega)
\end{equation}
where $F(\Gamma_{0}(2))$ is the fundamental region of the intermediary
subgroup and we have introduced Riemann theta functions \cite{Mumford}
with the period matrix
\begin{equation}
\Omega=\frac{i\beta^{2}}{2\pi^{2}\tau_{2}}\left(
\begin{array}{cc}
|\tau|^{2} & -\tau_{1} \\
-\tau_{1}  &  1
\end{array}
\right)\;.
\end{equation}
Performing now the extension from $\Gamma_{0}(2)$ to the full modular
group $\Gamma$ we obtain the modular invariant expression for the
Helmholtz free energy of the heterotic string
\begin{eqnarray}
F(\beta)&=&-\frac{1}{2^{6}\pi^{10}}
\int_{\cal F}\frac{d^{2}\tau}{\tau_{2}^{2}}\tau_{2}^{-4}
\frac{\bar{\Theta}_{\Gamma}}{\bar{\eta}^{24}}\left\{
\frac{\theta_{2}^{4}}{\eta^{12}} \theta \left[
\begin{array}{cc}
0  &  \frac{1}{2} \\
0  &  0
\end{array}
\right](0|4\Omega)\right. \nonumber \\
&+&\left.\frac{\theta_{4}^{4}}{\eta^{12}} \theta \left[
\begin{array}{cc}
\frac{1}{2}  & 0  \\
0  &  0
\end{array}
\right](0|4\Omega)
-\frac{\theta_{3}^{4}}{\eta^{12}} \theta \left[
\begin{array}{cc}
\frac{1}{2}  & \frac{1}{2}  \\
0  &  0
\end{array}
\right](0|4\Omega)\right\}
\end{eqnarray}
This second extension is quite simple because now $\Gamma_{0}(2)$ has
index $3$ with respect to $\Gamma$. If we take the limit
$\beta\rightarrow\infty$ we obtain that
$F(\beta)\rightarrow 0$, as it should be, because we are dealing with
a supersymmetric string theory.

Using this form of the one-loop Helmholtz free energy  and after some
calculations it is possible to see
that there is a critical value of $\beta$ at which $F(\beta)$ begins to
diverge, the Hagedorn temperature for the heterotic string,
which is $1/T_{H}=\pi(\sqrt{2}+1)$ \cite{Pricenton-quartet}. By using
the Poisson summation
formula on the Riemann theta functions it is also straightforward to
prove the duality relation for $F(\beta)$ \cite{O'Brien-Tan}
\begin{equation}
F(\beta)=\frac{\pi^{2}}{\beta^{2}}F\left(\frac{\pi^{2}}{\beta}\right)\;.
\end{equation}
The duality property, that we have explicitly seen at the one loop
level, is satisfied to all orders in string perturbation
theory \cite{Alvarez-Osorio-3}.
Let us consider the perturbation series for the Helmholtz free energy
\begin{equation}
F(g_{st},\beta)=\sum_{g=1}^{\infty}g_{st}^{2(g-1)}F_{g}(\beta)\;,
\end{equation}
where $g_{st}$ is the string coupling constant.
It is possible to show that the genus $g$ contribution to the free
energy has the property
\begin{equation}
F_{g}(\beta)=\left(\frac{\pi^{2}}{\beta^{2}}\right)^{g}
F_{g}\left(\frac{\pi^{2}}{\beta}\right)\;,
\end{equation}
so the whole perturbative expansion satisfies the duality relation:
\begin{equation}
F(g_{st},\beta)=\frac{\pi^{2}}{\beta^{2}}
F\left(\frac{\pi^{2}g_{st}}{\beta},\frac{\pi^{2}}{\beta}\right)
\end{equation}

When the existence of a Hagedorn temperature, and the duality relation
are considered together, we obtain a curious three phase structure for
the Helmholtz free energy \cite{O'Brien-Tan}. We know that
$F(\beta)$ is finite for all $\beta\geq\beta_{H}$ and that it diverges
for $\beta<\beta_{H}$. Duality implies that there is another
critical temperature
$\beta_{H}^{*}=\pi^{2}/\beta_{H}$ such that $F(\beta)$ is finite for
$\beta\leq\beta_{H}^{*}$ and diverges when we approximate to this value
of $\beta$ from the right. So the final result is that $F(\beta)$ is
finite for all $\beta\leq\beta_{H}^{*}$, $\beta\geq\beta_{H}$ and
diverges whenever $\beta_{H}^{*}<\beta<\beta_{H}$.

The high temperature phase ($\beta<\beta_{H}^{*}$) implied by duality
has singular
properties. For example the canonical entropy is negative in this phase.
One of the most intriguing
features of the high temperature phase is that if we evaluate the limit
of the free energy as $T\rightarrow\infty$ we obtain
\begin{equation}
\lim_{\beta\rightarrow 0}F(\beta)\sim \frac{\pi^{2}}{\beta^{2}}\Lambda
\label{F-L}
\end{equation}
where $\Lambda$ is the cosmological constant. Since in our case
$\Lambda=0$, this implies that in the high temperature limit there are
no degrees of freedom at all.

Some authors have interpreted these facts as showing that the high
temperature phase is not physical and that the Hagedorn temperature must
be the maximun temperature of the Universe \cite{Brandenberger-Vafa}.
Other authors, however, have interpreted the loss of gauge-invariant
degrees of freedom at high temperature as evidence
that in this regime the theory could be described by a topological phase
\cite{Atick-Witten}. In this case the presence of the Hagedorn
temperature would
indicate the transition between the topological and non-topological
phases. There are also a number authors \cite{Kogan-Sathiapalan} who
claim that there is a close resemblance between the possible phase
transition at the Hagedorn temperature and a Kosterlitz-Thouless phase
transition taking place at the same temperature
\cite{Kosterlitz-Thouless}. Nevertheless, these two
phenomena do not seem to be closely related since in the $c=1$
non-critical string
we find a Kosterlitz-Thouless phase transition but there is no Hagedorn
temperature \cite{Gross-Klebanov}. The problem of what
happens to the string ensemble at the Hagedorn temperature is still an
open problem.

In the canonical ensemble description it can be seen that when
approaching the Hagedorn temperature from below the system undergoes
violent fluctuations \cite{Alvarez}. This may be seen
as the breakdown of the canonical ensemble and we could expect that the
application of the microcanonical ensemble should be useful to study
the system beyond the Hagedorn temperature
\cite{Sundborg,Bowick-W,Alvarez,Bowick-Giddings,Deo-Jain-Tan}.
The density of states $\Omega(E)$ defined by
\begin{equation}
\Omega(E)=tr\,\delta(E-H)
\end{equation}
can be written as the inverse Laplace transform of the partition
function:
\begin{equation}
\Omega(E,V)=\int_{c-i\infty}^{c+i\infty}\frac{d\beta}{2\pi i}
Z(\beta,V)e^{\beta E}
\end{equation}
with $c\geq \beta_{H}$. It is expected that by analytically continuing
the partition function $Z(\beta)$ to the whole complex $\beta$-plane
and deforming the contour of integration one could get a density of
states valid in regimes different from the Hagedorn one (cf. for example
\cite{Deo-Jain-Tan,Deo-Jain-Tan-2}). Looking at the
distribution of the energy among the ensemble of strings when
the Hagedorn temperature is approached from the low temperature phase,
the phase transition would correspond to the fact that, when the number
of noncompact dimensions is higher than two, a single string
is able to absorb all the energy because of the exponential growth of
the number of state per energy level for one string
\cite{Deo-Jain-Tan-2}. In the microcanonical description this actually
implies that the specific heat is negative, showing an inestability in
the system that may be associated to a phase transition. Many issues of
finite temperature strings remain to be clarified.

\subsection{Is string theory finite?}

A very attractive feature of String Theory is that it has a real
possibility of being an ultraviolet finite quantum theory of gravity. In
the previous chapter we learned that in the bosonic strings the more
dangerous infinities came from the boundary of moduli space (surfaces
with nodes) and they were due to the presence of tachyons in the
spectrum. The integration measure is finite and well behaved except near
the boundary of ${\cal M}_{g}$. Since for the superstring and the
heterotic string the tachyon is projected out after GSO proyection, one
would intuitively expect that no further divergences should arise. This
is not completely correct, since the massless states may also contribute
some subleading divergences. We now however that for one-loop amplitudes
there are no divergences \cite{Green-Schwarz-Witten,Kaku}. For higher
loops we still lack a complete proof, but many exploratory computations
has been carried out. For the bosonic string we have two-loop
computations \cite{Several-bosonic} as well for the superstring
\cite{Verlinde-Verlinde-SST}. For the heterotic string
there are several computations of the two-loop cosmological constant
\cite{Several-heterotic}. It was finally shown in \cite{Ortin} that
there is an explicit modular invariant expression for the two-loop
cosmological constant of the heterotic string. The result as one might
expect from supersymmetry equals zero, after we sum over all spin
structures. There are several general arguments \cite{Martinec} based on
space-time supersymmetry which indicate the finiteness of superstring
perturbation theory. A detailed analysis of the many technical issues
involved in the analysis to all order can be found in
\cite{Atick-Moore-Sen,D'Hoker-Phong} and references therein.

Another approach to the proof of finiteness has been pursued by
Mandelstam \cite{Mandelstam}. There are several thorny obstacles
encountered in trying to work out a rigorous proof of finiteness. First
the moduli space of Riemann surfaces encountered in the previous chapter
is now replaced by supermoduli space, the moduli space of super-Riemann
surfaces. This space is unfortunately poorly understood. For genus $g$
supersurfaces it is a superalgebraic variety of graded dimension
$(3g-3|2g-2)$. If we denote in a given patch a coordinate system by
$(m_{\alpha},\tau_{\alpha})$ $\alpha=1,\ldots,3g-3$, $a=1,\ldots,2g-2$;
the transition function between two overlapping coordinate patches in
supermoduli will take the form:
\begin{eqnarray}
m^{'}_{\alpha}&=&f_{\alpha}(m_{\beta},\tau_{b})=
f^{(0)}_{\alpha}(m_{\beta})+f^{ab}_{\alpha}(m_{\beta})\tau_{a}\tau_{b}
\ldots
\label{1a} \\
\tau^{'}_{a}&=&g_{a}(m_{\beta},\tau_{b})=g^{(0)b}_{a}(m_{\beta})
\tau_{b}+g_{a}^{\;\;bcd}(m_{\beta})\tau_{b}\tau_{c}\tau_{d}+\ldots
\label{1b}
\end{eqnarray}
We say that the supervariety is projected if we can find a coordinate
cover such that in (\ref{1a}) only $f^{(0)}_{\alpha}(m_{\beta})$
appears;
and it is split if we can also make $g_{a}^{\;\;bcd}$ and higher order
terms vanish. In the split case the supermanifold would be equivalent to
a vector bundle. In genus one there is no problem. In genus two it was
shown by Deligne \cite{Deligne} that supermoduli space is split, but not
much is known for genus $g>2$. The integration measure for superstring
scattering amplitudes is a supermeasure on the supermoduli space, and we
can use for example the operator formalism extended to this case
\cite{LAG-Gomez-Nelson-Sierra-Vafa} in order to compute many properties
of the supermeasure. Supermoduli space also has a boundary made of
surfaces with nodes, but now these nodes comes in two types depending on
whether the punctures identified are NS or R punctures.

The first thing we would like to do is to integrate over the odd moduli
$\tau_{a}$ and be left with a measure on spin-moduli space; the moduli
space of Riemann surfaces with a fixed spin structure; since the modular
group acts transitively on the even and odd spin structures, this space
has two components. However if supermoduli space is not split we will
encounter ambiguities. All these ambiguities translates into total
derivatives which however may give non-trivial contributions on the
boundary of moduli space. Remember that the tachyon is eliminated
summing over spin structures, and the integration over spin moduli space
should include this sum. Related to these integration ambiguities is the
existence of certain spurious poles found in the superstring measure in
\cite{Verlinde-Verlinde-SST}. These brief remarks should give the reader
a flavor of the technical obstacles found in trying to actually proof
that Superstring Theory is finite. In the references quoted the
interested reader can find many more details.

A second issue of interest even accepting the finiteness of superstring
perturbation theory is its summability. This would be quite undesirable.
The non-summability of perturbation theory is a good indication of the
existence of important non-perturbative effects which should dominate
many dynamical questions. For critical strings, the compactification of
extended dimensions and the breaking of various symmetries is expected
to come from non-perturbative effects. It has been shown for instance
that in the bosonic string the perturbative expansion is not Borel
summable \cite{Gross-Periwal} (see also \cite{Kaku2}). The genus $g$
contribution for large $g$ grows as $(2g)!$, which is a rate of
divergence bigger than in ordinary QFT. One might argue that this
behavior is due to the tachyon. This interpretation is probably not
correct. In fact, in some recent studies of strings in dimensions $d<1$,
the same growth of $(2g)!$ is found (see \cite{Shenker}) and for these
dimensions there is no tachyon. This behavior is the more likely
evidence of important non-perturbative effects (\cite{Shenker} and
references therein). It is not known whether the same divergent behavior
is also characteristic of supersymmetric strings.



\section{Other Developments and Conclusions}

\subsection{String Phenomenology}

There are many areas of String Theory not presented in these lectures.
Most notably String Phenomenology \cite{phenomenology}. Two of the most
appealing features of models based on the Heterotic String are:
i)The gauge group $E_{8}\times E_{8}$ has plenty of room to
contain the SM group. ii)After compactifying it is possible to account
naturally for a chiral spectrum of low-energy fermions. Once a classical
background of the form $M_{4}\times K$ ($M_{4}$ is four-dimensional
Minkowski space and $K$ is a six-dimensional manifold, or a CFT with
the
appropiate properties) the low energy theory looks like a supersymmetric
extension of the SM. The advantage is that in principle the parameters
of the low energy effective action are computable once the background
$K$ is known.

As we learned in sec. 3.5 the motion of a string in some space-time
background requires a CFT on the world-sheet. In Superstring Theory,
this two-dimensional field theory should be superconformal invariant. If
we want furthermore the theory to be space-time supersymmetric, there is
a general argument \cite{Banks} showing that this is equivalent to $N=2$
world-sheet superconformal invariance with an integral lattice for the
$U(1)$ generator of the $N=2$ algebra. For backgrounds like $M_{4}\times
K$, the effective action describing the interactions between the light
states looks like any other $N=1$ SUGRA lagrangian in four dimensions
\cite{Cremmer-et-al}. Any of them can be characterized in terms of
three functions: the K\"{a}hler potential ${\cal K}$ which determines the
kinetic term of the scalars in the theory, the superpotential $W$
responsible among other things for the Yukawa couplings, and the gauge
kinetic function $f$ which determines de gauge coupling constants.

The subject of String Phenomenology is about the computation of the
three functions ${\cal K}$, $W$, $f$. In ordinary model building the
form of these functions depends very much on how the model is
constructed. In String Theory these functions are in principle
computable directly in terms of the CFT in the given string vacuum
solution. The couplings constant of the low-energy lagrangian are
computed in terms of correlation functions of the two-dimensional
theory. One of the massless scalars which is ubiquitous in String Theory
is the dilaton. Its vacuum expectation value is related to the
tree-level gauge coupling constants ($\langle\Phi\rangle=1/g^{2}$). One
of the specific features of four-dimensional strings is that coupling
constants are given in terms of expectation values of fields. They
depend on the dilaton field as well as on other scalar fields known as
moduli fields. An important problem encountered in the determination of
$\langle\Phi\rangle$ is that this expectation value is a flat direction
in pertubation theory, and therefore non-perturbative effects should
generate a non-trivial dilaton potential. As we will see, there are some
interesting recent results in this direction.

We should also mention that the breaking of space-time supersymmetry is
required to have special characteristics in order to obtain reasonably
realistic models. The basic scale in the problem is the Planck scale
$M_{P}$ ($\sim 10^{19}$GeV), and supersymmetry should be broken at
scales of the order of $10^{10}$-$10^{11}$ GeV. This value is not so
easy to obtain with current model building technology. In the resolution
of some of the problems listed, some properties of String Theory have
been applied with encouraging success. One of these properties is
duality (see sec. 3.6) which has no analoge in conventional Field
Theory.
The structure of the duality group is only known in few cases, but it
proved to be very useful in constraining the couplings of the low-energy
theory (for reviews and references, see \cite{dual-group}). The other
symmetry with purely ``stringy'' origin is the mirror
symmetry \cite{mirror}. This symmetry is based on the general properties
of
$N=2$
superconformal field theories, and again shows that apparently different
geometries correspond to just one string model. Using this symmetry it
is possible to compute explicitly some of the Yukawa couplings in the
superpotential $W$. The full physical implications of these symmetries
is now being vigorously explored.

In the determination of the gauge coupling constants, we need as
mentioned before to determine the non-perturbative form of the dilaton
effective potential. In QFT non-perturbative effects display a
dependence on the coupling constant of the form $\exp{(-1/g^{2})}$.
This dependence can be generated by gaugino condesation
(for a review see for example \cite{Nilles}). The effective potential
generated has the form
\begin{equation}
V_{eff}\sim \sum_{a} c_{a}\exp{\left(-\frac{24\pi^{2}}{b_{a}g_{a}^{2}}
\right)}\;,
\label{gaugino-cond}
\end{equation}
where the sum runs over the different factors on the gauge groups
responsible for gaugino condensation, $b_{a}$ are the one-loop
coefficient of the $\beta$ functions, and $\langle \Phi\rangle$ enters
in (\ref{gaugino-cond}) through the relation between $g_{a}$ and
$\langle\Phi\rangle$. This potential as written has a minimum at
$\langle\Phi\rangle=\infty$ corresponding to $g_{a}=0$. This changes
once loop corrections and threshold effects are included \cite{correct}
\begin{equation}
\frac{1}{g_{a}^{2}(p^{2})}=\langle\Phi\rangle+b_{a}\log{\left(
\frac{M_{P}^{2}}{p^{2}}\right)}+\Delta_{a}(T)
\label{loop-corr}
\end{equation}
The $\Delta_{a}(T)$ are the threshold corrections. The argument $T$
denotes collectively the dependence of the masses of the heavy fields
integrated out on the light scalar fields expectation values. These
threshold corrections are crucial to reconcile string model predictions
with the current value of the Weinberg angle. The inclusion of
(\ref{loop-corr}) into (\ref{gaugino-cond}) fixes the dilaton
expectation
value, and this may break space-time supersymmetry (for more details
and references see the second reference in \cite{phenomenology}).
Explicit studies
have been carried out so far only in few models, and hopefully a more
general picture will develop in the not very distant future. Many more
things could be said about Superstring Phenomenology, but this seems as
a good point to stop.

\subsection{Black Holes and Related Subjects}

We finally turn to some of the exciting recent work on String Black
Holes. It is well known that in GR, collapsing matter, under certain
conditions generates black holes. The singularties in the space-time
metric are supposed to always come with a horizon (Cosmic Censorship
Hypothesis). A property of classical black holes is to come with no
hair \cite{Hawking-Ellis}. The only parameters which characterize a
static black hole are
its mass, angular momentum and charge. Any other attribute of the matter
which collapses to form it dissapears. This obviously provides a rather
efficient mechanism to violate global symmetries like baryon number. In
the mid seventies Hawking made the momentous discovery that black holes
radiate with a thermal spectrum whose temperature is inversely
proportional to its mass \cite{Hawking1}. Thus, unlike better known
thermodynamics
systems it heats up as it radiates. The thermal nature of the radiation
found by Hawking lead him to assert that black holes produce an inherent
loss of quantum coherence \cite{Hawking-coher}. This has generated a big
controversy in the
last fifteen years. The computations carried out by Hawking and others
involve quantizing fields in the presence of a classical background
gravitational field, and the back-reaction is not taken into account.
Among other reasons because to properly understand the back reaction we
need a well defined quantum theory of the gravitational field.
The last sentence should of course be qualified by String Theory,
since it is alledged to provide a consistent quantum theory of gravity.
Two approaches has been followed on analyzing this problem. On the one
hand one can study some simple toy model of the gravity+dilaton system
in two dimensions suggested by the equation of motion of String Theory
(for details and references see \cite{Strominger}). These simple systems
provide a precise formulation of Hawking's paradoxes in a situation
where we have some control on the approximation used. The second
approach began after Witten \cite{Witten-BH} found an exact string
theory in two dimensions based on an $SL(2,{\bf R})/U(1)$ coset model
which describe a true string black hole in two dimensions. We know from
QFT that often in the translation of a problem form four to two
dimensions it loses most of its flavor. This does not seem to be the
case here. Many of the conceptual issues involved in the
four-dimensional problem continue to be non-trivial in the
two-dimensional case. Since Witten's solution is perfectly sensible as a
CFT (and more importantly as a String Theory), an important research
effort was initiated after Witten's paper to try to understand what
answers it gives to black hole paradoxes. Furthermore, recent revisions
of the no-hair theorem (see \cite{Coleman-Preskill-Wilczek} for details
and references) also find a setting in this two-dimensional scenario, in
a way possibly connected with the loss of quantum coherence. The authors
of \cite{Ellis-Mavromatos-Nanopoulos}, on the basis of the properties of
Witten's solution have proposed solutions to many of the riddles of
black holes and string physics.

The explanation of dilaton black hole solutions and of Witten's black
hole string theory is currently going at a fast pace. Details can be
found in the literature and in Polyakov's lectures (this volume). The
fact that some of the deepest questions in Quantum Gravity are beginning
to be addressed by String Theory is both exciting and encouraging. It is
however hard to guess at this moment what will be the outcome of these
investigations.

We have finally reached the end of these lectures. We hope to have
conveyed the impression that String Theory is an important, and active
area of research which may ultimately provide us with a consistent
picture of Quantum Gravity and of its relations with the other
interactions. We may only hope that some of the ideas and proposals put
forward in the last few years contain the seeds which will lead to this
Grand Synthesis.


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\end{document}

