\documentstyle[11pt,amsfonts,amssymb,amscd]{article}
\begin{document}
\title{Holography and the Deligne Conjecture}
\author{Ioannis P. \ ZOIS\thanks{izois@ihes.fr Research supported by the EU, 
contract No: HPMF-CT-1999-00088. Current address: 
School of Mathematics, Cardiff University, PO Box 926, Cardiff CF24 4YH, UK; 
e-mail address: zoisip@cf.ac.uk}\\
\\
IHES, Le Bois-Marie, 35, route de Chartres
\\
F-91440 Bures-sur-Yvette, FRANCE
\\
and
\\
Mathematical Institute, Oxford University
\\
24-29 St. Giles', Oxford, OX1 3LB, UK}
\date{}


\maketitle

\begin{center}
\emph{``In girum imus nocte et consumimur igni''}.\\
\end{center}
\begin{abstract}

In this article we try to explain a statement due
to Maxim Kontsevich back in 1999, that the \emph{Holography Principle}
in physics is related to the (higher dimensional)
\emph{Deligne Conjecture} in mathematics. That seems to suggest that the
\emph{little $d$-discs operad} (or equivalently the notion of a
\emph{$d$-algebra}) gives a new way to understand the mathematical aspects of
\textsl{quantum gravity} using \textsl{holography}. The strategy is as
follows: we would like
to learn something about quantum gravity in $(d+1)$ dimensions: we use
holography to reduce our original problem to a CFT in $d$-dimensions. The
deep origin of this dimensional reduction lies on the fact that it is the
\textsl{area} and \emph{not} the \textsl{volume} which appears in the
formula giving the entropy of
black holes as described long ago by Hawking. Then we use $d$-algebras (i.e. 
the little $d$-discs operad) to study our $d$-dim CFT.
The possible relation between $d$-dim CFT and $d$-algebras comes from the 
lesson we have learnt from strings (namely the 2-dim CFT case): the space of 
physical states
in closed string field theory (ie the BRST cohomology) has a natural
\emph{Gerstenhaber} algebra structure and this by Cohen's theorem is related to
the little 2-discs operad. The proposal then is that the relation might hold 
in higher
than 2 dimensions. This approach is algebraic although it would have been be 
much more satisfactory if we could generalise Segal's geometric approach
to CFT in higher than 2 dimensions. Hopefully the article is mathematically
self-contained.


PACS classification: 11.10.-z; 11.15.-q; 11.30.-Ly\\

Keywords: Holography, Operads, String Theory, Quantum Gravity, Conformal
Field Theory.\\
\end{abstract}

\section{Introduction}

This work was motivated by an attempt to understand an interesting
statement
by Maxim Kontsevich \cite{kon1} back in 1999: that the \emph{Holography
Principle} originally due to G. 't Hooft \cite{thooft} in
Physics might be related to the \emph{Higher Dimensional Deligne
Conjecture} in Mathematics due to Kontsevich. In 2000 Kontsevich and
Soibelman \cite{kon2} proved the original Deligne conjecture.\\

Let us roughly explain both these statements here and elaborate more on them
later: Holography is a statement about \emph{quantum gravity}; in
simple terms it says that quantum gravity must be a \textsl{topological
quantum field theory}.\\

On the other hand the original Deligne conjecture (due to Deligne as the name 
suggests) is about the
Hochschild complex of associative algebras; as it is well-known the
Hochschild complex is very useful when one wants to study the theory of
deformations of algebras.\\

We organise this article as follows: in section 2 we try to explain the
necessity for holography in physics starting from black hole puzzles; in
section 3 we give the necessary
mathematical definitions about operads and Gerstenhaber algebras
and we also state the two basic theorems: the first is due to F. Cohen and
relates Gerstenhaber algebras with the little 2-discs operad and the second
is D. Tamarkin's theorem on the formality of the little $d$-discs operad.
In the last section we explain the relation between string theory and the
little 2-discs operad.
Then as we shall see, one will be able to interpret Kontsevich's statement as
a higher dimensional analogue of the above relation.\\

We start our discussion with possibly the most mysterious objects
in the universe: \emph{Black Holes.}







\section{Understanding Black Holes from Strings: the need for a nonlocality
mechanism}







We have gained some understanding on two
important problems in
black hole physics (abreviated to ``BH'' in the sequel) by using
some recent results from string theory dualities (for more details one
can see \cite{das} which is a nice review article):\\

{\bf 1.}  In general relativity we have the
so-called \emph{``no hair''} theorem
which refers to black holes. This is the statement that the configuration
of a black hole solution given by the Schwarzchild metric
is \emph{uniquely determined} by its mass (= total energy) (we assume
no more conserved quantities like electric charge or angular momentum for
simplicity). In other words we have only {\bf 1} configuration (for a given
mass) associated to the Schwarzchild solution.

Following the usual definition for the classical \emph{entropy} of a system
$$S=k_{B}ln\Omega $$ where $\Omega $ is the number of microstates compatible
with some given values of the macroscopic parameters (eg temperature,
pressure, volume etc) and $k_{B}$ is Boltzmann's constant, we
immediately deduce that a BH must have \emph{zero} entropy classically since
$ln1=0$.\\

But then we encounter the \textsl{qualitative argument} originally due
to Beckenstein that if this was indeed the case, then any object, e.g. some
gas, falling into a BH would \textsl{contradict the second law of
thermodynamics}. To avoid that one should associate a \emph{nonzero} entropy
(positive of course) to any BH. The precise value of the
entropy $S$ of a BH was then determined by the Hawking area formula which,
ignoring constants, reads
$$S\sim A$$
where $A$ is the \emph{area} of the event horizon. The origin of BH entropy
was understood to be \emph{quantum mechanical}.\\

We know however that statistical physics gives a more fundamental explanation
of the laws of thermodynamics and moreover a correct ``would-be'' quantum
theory of gravity should explain the origin of the quantum states associated
to a BH. So the challenge was to find a statistical explanation for the
quantum states associated to a BH which give rise to its entropy described by
the Hawking formula.\\

\emph{Superstring theory} can indeed, in some cases, provide an explanation 
for the origin of
quantum states associated to \emph{multicharged extremal black holes}. And
anyway string theory is arguably the best known
candidate for a quantum theory of gravity. The argument
 which explains the microscopic origin of BH entropy starting from string
theory was originally due to Strominger, Vafa, Horowitz and Maldacena and it 
is based
on {\bf S}-duality. The later is a statement about an isomorphism between
\emph{strong} and \emph{weak} coupling regions of superstring theory;
equivalently it \textsl{interchanges} \emph{monopoles} with \emph{charges} in
the theory (or equivalently it interchanges topology and dynamics) and gives
us the ability to identify \textsl{BPS superstring states} which will either
be
\textsl{perturbative states} if they carry \emph{NS charges} or
\textsl{D-branes} if they carry \emph{R charges} in
\emph{``weak coupling''} region with \emph{extremal black holes} carrying the
analogous type of charge in \emph{``strong coupling''} region. We restrict
our attention to BPS states (these are states whose mass does not receive any
quantum corrections) because for simplicity we assume no backcreation for the
black hole (namely its mass which is equal to its energy remains constant).
Briefly then the main
idea behind this string theoretic explanation of the quantum states
associated to a black hole is that since strings live in 10-dim and BH in
4-dim, the remaining
6 compactified dimensions essentially provide a ``phase space'' which we
quantize and thus we obtain the states of the BH. (This picture is not 
utterly correct but we think captures the spirit of the argument
and gives a clear picture conceptually).\\


{\bf 2.}  The second problem we would like to consider in BH physics is the
so-called \emph{``information paradox''}. \emph{Classically},
\textsl{nothing can escape the event horizon of a BH} (since that
would require a velocity which is grater than the velocity of light; one can
use that as a definition of the \emph{``event horizon''}). Yet
quantum mechanically, since a BH has a positive entropy as we just
argued above, \emph{assuming a thermodynamical behaviour}, it should also
have a corresponding \emph{temperature} from the well-known relation in
thermodynamics
$$(dM=)dE=TdS.$$
This is the \emph{Hawking temperature}
$$T_{H}=\frac{\hbar \kappa }{2k_{B}\pi }$$
where $\kappa $ is \textsl{surface gravity} (the acceleration felt by a
static object at the horizon as measured from the asymptotic region),
and hence BH's should also \emph{radiate.}\\

This is the \emph{Hawking radiation}. Then the problem with
radiation carrying out the information of \emph{formation}, assuming
\textsl{no quantum xeroxing,} is to maintain \textsl{unitarity} of the process
as ordinary quantum mechanics requires. More precisely, the way one computes
Hawking radiation is by assuming a \emph{codim-1 foliation} of spacetime
where the normal direction is time
and this radiation process appears to be \emph{nonunitary}.\\

Thus trying to avoid a classical contradiction with the 2nd law of
thermodynamics we assumed that BH's have positive entropy (whose origin is
quantum mechanical); yet this almost immediately created another contradiction
with quantum mechanics: loss of unitarity in BH radiation. Unitarity is
absolutely crucial in any quantum theory since it reflects the
conservation of probabilities. So it seems that we didn't actually achieve
very much: we simply \emph{``pushed''} the contradiction from the classical
to the quantum realm.\\

It appears that the most \emph{economical} (i.e. requiring the fewest
changes to things we already know in physics) \textsl{way out} is to assume
that there is some \textsl{physical principle} which does not
allow this to happen. We \emph{enforce} \textsl{unitarity} throughout by
imposing a \textsl{non-locality} mechanism. One such mechanism is the
\emph{holography principle} due to G. 't Hooft. The original statement is the
following:\\

\emph{``a quantum theory of gravity on a
$(d+1)$-manifold with boundary should be equivalent to a conformal field
theory (CFT for short) on the boundary (which is a $d$-manifold) and this
conformal field
theory on the boundary must have one degree of freedom per Planck area''}.\\

Let us elaborate more on this: since for BH we seem to lose any information
passing the event horizon, it is reasonable to assume that in order to avoid
this problem (along with its quantum mechanical incarnation of nonunitary
radiation), everything that happens
inside the black hole should be described from data on its event horizon. It
 is clear we think that the motivation for this \textsl{dimensional reduction}
of quantum gravity in holography came from the formula for the black hole
entropy: the entropy of a black hole is proportional to the \emph{area} and 
\textsl{not} the volume of the event horizon.\\

Another very useful way of thinking about the holography principle is that
it simply says that for a given 3-volume $V$
in space the \emph{state of maximal entropy} in nature is given by the
\textsl{largest BH that fits inside $V$},
(silently we are making use of the Hawking formula which says that the
entropy of a BH is
proportional to the area of its event horizon).\\

This principle has a deep consequence on \emph{perturbative quantum field
theory:} BH's
provide a \textsl{natural} \emph{cut-off limit} since the
above statement says that for fermions for example one cannot have a huge
amount of energy concentrated
in a tiny region of space because that would collapse into a BH.\\

There is also a string theoretic version
of the holography principle using string theory language,
the so-called \emph{``Maldacena
conjecture''} which states that string theory on the smooth manifold
$AdS_{5}\times S^{5}$ is dual to $N=4$ SYM $SU(N)$ gauge theory on the
\emph{boundary} of $AdS_{5}$. In this talk we shall primarily
build our understanding
of Kontsevich's statement based on this string theoretic version of
holography. However let us for the moment go back to the original 't Hooft
version of holography and ask:\\

{\bf (Key Question:)} What is a $D=d$ CFT?\\

In order to answer the above question and eventually understand Kontsevich's
statement, we should
start by trying to understand the $D=2$ case first. We know from G. Segal
what a $D=2$ CFT is, so it seems that somehow we have to generalise his work.\\

A good motivation to study $D=2$ CFT comes from string theory itself, one can
say in fact that $D=2$ CFT is intimately related to string theory: strings
are 1-dim objects which in time sweep out 2-manifolds called
\emph{worldsheets};
this is the higher dimensional analogue of the paths (1-dim geometric objects)
swept out in time by point particles. However now we are talking about
M-Theory in physics which generalises string theory and M-Theory contains
the M2 and M5 branes; these are 2 and
5-dim objects respectively whose worldsheets are 3 and 6-dim manifolds. So
apart from holography, there is additional motivation coming from M-Theory to
understand higher dimensional CFT's.\\

Now we would like to describe briefly what string theory is \emph{classically}:
basically it is a $\sigma $\textsl{-model}, namely it describes
\emph{harmonic maps} $\phi :\Sigma _{g}\rightarrow X^{10}$
where $\Sigma _{g}$ is a Riemann surface of genus $g$ representing the
worldsheets of strings and $X$ is a 26-dim
Riemannian manifold with a $B$-field. The dimensionality of $X$ is fixed from
consistency arguments (compatibility with special relativity and cancellation
of the conformal anomaly). The $B$-field is a real valued 2-form
which is used as a potential to \emph{gauge} the worldsheets of the strings
in order to get our Dirac phase factors; in fact one can think of it as the
Poincare Dual (a 2-form) of the worldsheet which is a 2-manifold (it is the
analogue of the gauge
potentials in Yang-Mills theory, connection 1-forms, although now it has to
be a 2-form instead of a 1-form since we are talking about strings whose
worlsheets are 2-manifolds whereas for point particles we needed 1-forms
because their ``worlsheets'' were 1-dim objects). This picture needs to include
fermions as well in order to be complete but we shall not elaborate more on
this. The introduction of fermions along with supersymmetry reduces the
dimensionality of $X$ from 26 down to 10.\\

The \emph{quantum} theory of strings is essentially a \emph{$D=2$ CFT}
(plus a little bit
more structure as we shall see later). In order to
describe $D=2$ CFT one may use the original \emph{geometric} approach due to
G. Segal. This approach however does not lead to the statement of
Kontsevich in a straightforward way. Moreover it is not easy to see how it
can be generalised to
higher dimensions which is what we are after. Instead we shall adopt
 an \emph{algebraic approach} using the language of \emph{operads};
in fact we shall see the \emph{little 2-discs operad} $C_{2}(n)$ arising
naturally in our discussion. This is the crucial step in
order to understand Kontsevich's statement which is the higher
dimensional version of this beautiful fact. The appearence of the operad
$C_{2}(n)$ in string theory is not at all obvious and at least for us quite
surprising. The link between string theory and the little 2-discs operad
$C_{2}(n)$ comes from a deep theorem due to Fred Cohen as we shall try to
exhibit shortly. For simplicity we shall restrict our discussion to
\emph{closed strings}. But we shall do that in the last section because we
need some mathematical definitions first. The final remark here is that we
would still like to generalise Segal's work and get a geometric definition of
$D=d$ CFT. Currently this seems out of reach since there are two reasons
which make Segal's approach particularly nice for the $D=2$ case (but at the
same time act as
barriers when trying to generalise into higher dimensions): the
conformal group in this case is infinite dimensional and hence contains a lot
more information whereas in higher dimensions the conformal group is only
finite dimensional. The second reason is that
the classification of 2-manifolds is simple whereas for $D=3$ it is not
known if 3-manifolds can be classified and in dimensions $D\geq 4$ we can
only classify simply connected manifolds (in perturbative quantum field theory
that means we can only talk about tree level).\\




\section{A Mathematical Interlude}







In this section now we shall give formal definitions.\\

{\bf Definition 1:}\\
A \emph{Gerstenhaber algebra} (or a $G$-algebra) is a
graded vector space $V=\oplus _{i\in {\bf Z}}V_{i}$ with a \emph{dot product}
$x\cdot y$ defining the structure of a \textsl{graded commutative associative
algebra} along with a \emph{bracket operation} $[x,y]$ of degree -1 defining
the structure of a \textsl{graded Lie algebra} such that the bracket is a
\emph{derivation} with respect to the dot product, i.e. it satisfies the
Leibniz rule
$$[x,y\cdot z]=[x,y]\cdot z+(-1)^{(deg(x)-1)deg(y)}y\cdot [x,z]$$

{\bf Examples:}\\
{\bf i.} Let $A$ be an associative algebra and let
$C^{*}(A,A)$ be its Hochschild complex
where $C^{i}(A,A):=Hom(A^{\otimes i},A)$ and the differential $d$ is defined
as ($x\in C^{n}$):
$$(dx)(a_{1}\otimes ...\otimes a_{n+1}):=a_{1}x(a_{2}\otimes ...\otimes a_{n+1})+\sum _{i=1}^{n}x(a_{1}\otimes ...\otimes a_{i}a_{i+1}\otimes ...\otimes a_{n+1})+...$$
$$+(-1)^{n+1}x(a_{1}\otimes ...\otimes a_{n})a_{n+1}$$
On the Hochschild complex we can define the usual cup product
$$\cup :C^{k}\otimes C^{l}\rightarrow C^{k+l}$$ as follows
($x\in C^{k}$, $y\in C^{l}$ and $a_{i}\in A$):
$$(x\cup y)(a_{1}\otimes ...\otimes a_{k+l}):=(-1)^{kl}x(a_{1}\otimes ...\otimes a_{k})y(a_{k+1}\otimes ...\otimes a_{k+l})$$
Moreover we can also define the \emph{Gerstenhaber bracket} $[,]:C^{k}\otimes C^{l}\rightarrow C^{k+l-1}$ as:
$$[x,y]:=x\circ y-(-1)^{(k-1)(l-1)}y\circ x$$
where
$$(x\circ y)(a_{1}\otimes ...\otimes a_{k+l-1}):=\sum _{i=1}^{k-1}(-1)^{i(l-1)}x(a_{1}\otimes ...\otimes a_{i}\otimes y(a_{i+1}\otimes ...\otimes a_{i+l})\otimes ...\otimes a_{k+l-1})$$
The G-bracket gives after a shift of the ${\bf Z}$-grading by -1 the structure 
of a (differentiable graded lie algebra) DGLA on the Hochschild complex. The 
cup product is not graded commutative (it is only associative) but the induced 
operation on cohomology is graded commutative. Moreover the G-bracket induces 
an operation on cohomology which satisfies the Leibniz rule with respect to 
the cup product, hence the \emph{Hochschild cohomology of any associative 
algebra is in fact a $G$-algebra}.\\

We shall briefly mention three more examples of $G$-algebras:\\

{\bf ii.} \emph{Polyvector fields} on smooth manifolds with \textsl{wedge
product} and \textsl{Schouten-Nijenhuis bracket}.\\

{\bf iii.} \emph{Exterior algebra of a Lie algebra} with \textsl{wedge
product and extension of the Lie bracket.}\\

{\bf iv.} \emph{(Rational) homology of double loop space} with
\textsl{Pontrjagin product and Samelson bracket}.\\

{\bf Definition 2:}\\
An \emph{operad} $P$ (of vector spaces) consists of the following data:\\
{\bf a.} a collection of vector spaces $P(n), n\geq 0$,\\
{\bf b.} an action of the symmetric group $S_{n}$ on $P(n)$ for every $n$,\\
{\bf c.} an identity element $id_{P} \in P(1)$,\\
{\bf d.} compositions $m_{(n_{1},...,n_{k})}$
$$P(k)\otimes (P(n_{1})\otimes  ... \otimes P(n_{k}))\rightarrow P(n_{1}+...+n_{k})$$
for every $k\geq 0$ and $n_{1},...,n_{k}\geq 0$. These compositions have to be
 \emph{associative}, \emph{equivariant} with respect to the symmetric group 
actions and the \emph{identity} element $id_{P}$ has to satisfy the following 
naturality property with respect to the composition:
$$m_{(n)}(id_{P},p)=p$$
and
$$m_{(n,1,...,1)}(p,id_{P},...,id_{P})=p$$
for all $p\in P(n)$ (one can have a look at \cite{operads} for more details).\\

{\bf Example:} The \emph{``endomorphism operad''} of a vector space $V$ is
given
by $P(n):=Hom(V^{\otimes n}, V)$ where the action of the symmetric group and
the identity element are the obvious ones and the compositions are defined by
the substitutions
$$(m_{(n_{1},...,n_{k})}(\phi \otimes (\psi _{1}\otimes ... \otimes \psi _{k}))) (v_{1} \otimes ... \otimes v_{n_{1}+...+n_{k}})$$
$$:=\phi (\psi _{1}(v_{1}\otimes ... \otimes v_{n_{1}})\otimes ... \otimes \psi _{k}(v_{n_{1}+...+n_{k-1}+1}\otimes ...$$
$$ \otimes v_{n_{1}+...+n_{k}}))$$
where $\phi \in P(k):=Hom(V^{\otimes k}, V)$, $\psi _{i}\in P(n_{i}):=Hom(V^{\otimes n_{i}}, V)$ and $i=1,2,...,k$.\\

{\bf Definition 3:}

An \emph{algebra} \textsl{over an operad} $P$ (of vector spaces), or a
\emph{P-algebra}, (or equivalently a \emph{representation} of the operad $P$), 
consists of a vector space $A$
and a collection of multilinear maps $f_{n}:P(n)\otimes A^{\otimes n}
\rightarrow A$ for all $n\geq 0$ satisfying the following axioms:\\
{\bf a.} for any $n\geq 0$ the map $f_{n}$ is $S_{n}$-equivariant,\\
{\bf b.} for any $a\in A$ we have $f_{1}(id_{P}\otimes a)=a$,\\
{\bf c.} all compositions in $P$ map to compositions of multilinear
operations in $A$.\\

In other words the structure of an algebra over $P$ on a vector space $A$ is
given by a \emph{homomorphism} from $P$ to the \emph{endomorphism operad of
$A$}.\\
One can also define \emph{modules} over algebras over operads.\\

One can construct operads denoted $Assoc(n)$, $Lie(n)$, $Poisson(n)$, 
$G(n)$,
$A_{\infty }(n)$,
$L_{\infty }(n)$, $G_{\infty }(n)$
such that algebras over these operads are \emph{associative} algebras,
\emph{Lie} algebras, \emph{Poisson} algebras, \emph{Gerstenhaber} algebras 
and their \emph{homotopic} versions
respectively but there is {\bf no} operad for \emph{Hopf algebras} (for more 
details see \cite{operads}).\\

In the definition of operads we can replace our
\textsl{vector space} $V$ by a compact \emph{topological space} $X$ and hence
define \emph{operads over topological spaces} replacing
\textsl{tensor product} with \textsl{Cartesian product}.\\

The analogue of the endomorphism operad will in this case be
$P(n):=\{Continuous Maps:X^{n}\rightarrow X\}$. Then one can define algebras
over topological operads accordingly.\\

More generally one can define operads and algebras over operads over objects
of any \emph{symmetric monoidal category} $\mathcal C$, namely a category
endowed with the functor $\otimes :\mathcal C\times \mathcal C\rightarrow
\mathcal C$, the identity element $1_{\mathcal C}\in Obj(\mathcal C)$ and the
appropriate coherence isomorphisms for associativity and commutativity of
$\otimes $-product.\\

Operads themselves can be seen as algebras over the \emph{coloured operad.}
(see \cite{kon2} p.12 Remark 1.)\\

In particular we would like to consider operads in the symmetric monoidal
category $Complexes$ whose objects are {\bf Z}-graded complexes of abelian
groups and arrows morphisms of complexes. These are called \emph{differential
graded operads} or \emph{dg-operads} for short. So each component $P(n)$ of an
operad of complexes will be a complex, namely a vector space decomposed into
a direct sum $P(n)=\oplus _{i\in {\bf Z}}P(n)^{i}$ and endowed with a
differential $d:P(n)^{i}\rightarrow P(n)^{i+1}$ of degree $+1$ such that
$d^{2}=0$. Then every dg-operad $P$ has a corresponding \emph{homology operad}
denoted $H_{*}(P)$.\\

{\bf Key idea:}

There is a natural way to construct an \emph{operad of complexes} from
a \textsl{topological operad} by using essentially the
\emph{singular chain complex} of topological spaces.

Let $d\geq 1$ be an integer. Denote by $G_{d}$ the $(d+1)$-dimensional Lie
group of \emph{affine transformations} acting on ${\bf R^{d}}$ via
 $u\mapsto \lambda u+v$ where $\lambda >0$ is a real number and
$v\in {\bf R^{d}}$ is a vector.
This group acts simply transitively on the space of closed discs in
${\bf R^{d}}$ and the disc with centre $v$ and radius $\lambda $ is obtained
from the standard disc
$$D_{0}:=\{(x_{1},...,x_{d})\in {\bf R^{d}}|x_{1}^{2}+...+x_{d}^{2}\leq 1\}$$
by a transformation from $G_{d}$ with parameters $(v,\lambda )$.


{\bf Definition 4:}

The \emph{little d-discs operad} $C_{d}(n)$ is a topological operad with the
following structure:\\
{\bf a.} $C_{d}(0):=\emptyset $,\\
{\bf b.} $C_{d}(1):=*$,\\
{\bf c.} for $n\geq 2$ the space $C_{d}(n)$ is the \emph{space of
configurations
of $n$ disjoint d-discs} $(D_{i})_{1\leq i\leq n}$ inside the standard $d$-disc
$D_{0}$.\\

The composition
$$C_{d}(k)\times C_{d}(n_{1})\times ...\times C_{d}(n_{k})\rightarrow C_{d}(n_{1}+...+n_{k})$$
is obtained by applying elements from $G_{d}$ associated with discs
$(D_{i})_{1\leq i\leq n}$ in the configuration in $C_{d}(k)$ to configurations
in all $C_{d}(n_{i})$, $i=1,2,...,k$ and putting the resulting configurations
together. The action of the symmetric group $S_{n}$ on $C_{d}(n)$ is given
by renumerations of indices of discs  $(D_{i})_{1\leq i\leq n}$.\\

The \emph{little $d$-discs operad} $C_{d}(n)$ was introduced by
Peter May and Boardmann-Vogt in the 70's in order
to describe \emph{homotopy types} of \emph{$d$-fold loop spaces}, namely spaces
of continuous maps
$$Maps(S^{d}_{+}, X_{+})$$
where ``$+$'' denotes base point, $X$ is a topological space, $S^{d}$ is the 
$d$-dim sphere. The little $d$-discs operad is the most important operad in
homotopy theory.\\

The key result relating the little $d$-discs operad $C_{d}(n)$ with $d$-fold
loop spaces is that (with field coefficients) chains of $d$-fold loop spaces
become naturally $d$-algebras i.e. algebras over the operad
$Chains C_{d}(n)$. (In fact the above statement is true even without taking
``chains'' in both sides).\\

{\bf Fact:} The space $C_{d}(n)$ is \textsl{homotopy equivalent} to the
\emph{configuration space of $n$ pairwise distinct points} in ${\bf R^{d}}$:
$${\bf F}(n,{\bf R^{d}}):=({\bf R^{d}})^{n}-Diag=\{(v_{1},...,v_{n})\in ({\bf R^{d}})^{n}$$
$$|v_{i}\neq v_{j} for i\neq j\}$$\\
\\


{\bf Definition 5:}

Let $\tilde{C_{d}(n)}:={\bf F}(n,{\bf R^{d}})/G_{d}$ which is also the
\emph{Fulton-MacPherson operad} $FM_{d}(n)$.\\

For $n=2$, $FM_{d}(2)$ is homotopy equivalent to the $(d-1)$-sphere $S^{d-1}$.
For all $n\geq 3$, $FM_{d}(n)$ is a \emph{manifold with corners} which can be
identified explicitly.\\

{\bf Definition 6:}

For $d\geq 0$, a $d$-algebra is an algebra over the operad
$Chains (C_{d})$ in the category of complexes.\\

One then has:\\

{\bf Theorem 1.} (F. Cohen)

There is a natural homotopy equivalence
$$G(n)\simeq H_{*}[Chains C_{2}(n)]$$

Recall the fact that the Hochschild cohomology of any associative algebra has
a natural $G$-algebra structure. The original {\bf Deligne conjecture} was
that the
Hochschild complex of an associative algebra (or more generally the Hochschild
complex of an $A_{\infty }$-algebra as mentioned in \cite{kon2}) itself
carries a natural 2-algebra structure, i.e. it has an action of the operad
$Chains C_{2}(n)$.
Its higher dimensional version due to Kontsevich says:\\

\emph{``for any $d-algebra$ there is a natural action of a
universal (in an appropriate sense defined up to homotopy) $(d+1)-algebra$''}.\\

Useful facts:
$$Assoc(n)\simeq H_{*}[Chains C_{1}(n)]$$
$$Lie(n)\simeq H_{n-1}[Chains C_{2}(n)]$$
Since in general homotopic versions of various algebras appear when the 
product is originally defined on the cohomology and one wants to ``lift'' the 
structure to the cochain level,  one has the following general relations 
between algebras and their
``homotopic versions'':
$$Assoc(n)\simeq H_{*}[A_{\infty }(n)]$$
$$Lie(n)\simeq H_{*}[L_{\infty }(n)]$$
$$G(n)\simeq H_{*}[G_{\infty }(n)]$$


{\bf Aside:} The above discussion was about the little discs operad and 
based loop spaces.
One also has a variation of the above, the so called \emph{framed} little 
$d$-discs operad denoted $C^{f}_{d}(n)$
which is related to \emph{free} loop spaces. The framed little 2-discs operad
$C_{2}^{f}(n)$ is \emph{homotopic} to the \textsl{cactus} operad
and the (rational) homology of the framed little 2-discs operad is homotopic 
to the BV-operad. The main result is then that the (rational) homology of the 
free loop 
space $H_{*}(LX)$ where $X$ is a compact oriented manifold (after an
apropriate shift) has a BV-algebra structure, namely it is an algebra over 
the \emph{Batalin-Vilkovisky} operad. At this point we would like to remind 
the reader of the fact that BV-algebras appear in Lagrangian formulation of 
field theories whereas Gerstenhaber algebras (to be defined below) appear in 
BRST cohomology which is 
Hamiltoniam formalism of a field theory. For more details on the cactus 
operad, free loops and BV algebras we refer to \cite{oper}.\\


{\bf Theorem 2:} (D. Tamarkin 1998)

In characteristic zero, the operad $Chains (C_{d})\otimes {\bf R}$ is
{\bf formal}, i.e. it
is homotopy equivalent to its corresponding homology operad.







\section{The appearence of the operad $C_{2}(n)$ in string theory}




After giving all these mathematical definitions we now return back to physics.
G. Segal defined a $D=2$ CFT as roughly a topological vector space $H_{S^{1}}$
and to each ($\Sigma _{g}$ is a Riemann surface of genus $g$) \emph{cobordism}
(which physically represents a Feynman
diagram for strings which are 1-dim objects with 2-dim worldsheets)
$\Sigma _{g}:S^{1}\rightsquigarrow S^{1}$ we associate an operator
$U_{\Sigma _{g}}$ satisfying certain axioms to be compatible with the
principles of quantum
mechanics and conformal invariance. The case $g=0$ corresponds to
\emph{tree-level} in physics.

Now we shall modify Segal's definition using the more convenient language
of \emph{operads}, we follow \cite{operads}. We shall explain in detail
the structures appearing in the $D=2$ case and then we shall try to see how
much can be immediately generalised to higher dimensions.\\

Before doing that we would like to make a comment: there are \emph{two}
ways to construct operads related to Riemann surfaces: the first one is by
using moduli spaces of punctured Riemann surfaces and compactify them, such
operads are roughly denoted $M(n)$; the second is by decorating the puctures
with local coordinates. One can sew two Riemann surfaces together
unambiguously (up to modular equivalence) by using suitable local
coordinates.\\

Let $R(n)$ be the moduli space of nondegenerate Riemann spheres $\Sigma $
with $n$ labelled punctures and non-overlapping holomorphic discs at each
puncture (holomorphic embeddings of the standard disc $|z|< 1$ to $\Sigma $
centered at the puncture). The spaces $R(n)$, $n\in {\bf N}^{*}$ form an operad
under sewing Riemann spheres at punctures (cutting out the discs $|z|\leq r$
and $|w|\leq r$ for some $r=1-\epsilon $ at sewn punctures and identifying the
annuli $r < |z| < 1/r$ and $r < |w| < 1/r$ via $w=1/z$). The symmetric group
interchanges punctures along with the holomorphic discs.

Consider the complexification $V$ of the Virasoro algebra of complex valued
vector fields on the circle, generated by elements $L_{m}=z^{m+1}\partial /\partial z$, $m\in {\bf Z}$, with the commutators $[L_{m},L_{n}]=(n-m)L_{m+n}$.
Then one has:\\

{\bf Definition 7:}\\
A $D=2$ \emph{conformal field theory} at tree level consists of the following
data:
{\bf 1.} A topological vector space $H$ called \textsl{state space}.\\
{\bf 2.} An action $T:V\otimes H\rightarrow H$ of the Virasoro algebra $V$
on $H$.\\
{\bf 3.} A vector $|\Sigma >$ $\in Hom(H^{\otimes n}, H)$ for each $\Sigma \in
R(n)$ depending smoothly on $\Sigma $.
These data must satisfy the following relations:\\
{\bf 4.} $T(\underline{v})|\Sigma >=|\delta (\underline{v})\Sigma >$, where
$\underline{v}=(v_{1},...,v_{n})\in V$ and $\delta $ is the natural action of
$V^{n}$ on $R(n)$ by infinitesimal reparametrisations at punctures. In
particular $T(\underline{v})|\Sigma >=0$ whenever $\underline{v}$ can be
extended to a holomorphic vector field on $\Sigma $ outside of the discs.\\
{\bf 5.} The correspondence $\Sigma \mapsto |\Sigma >$ defines the structure
of an \emph{algebra over the operad} $R(n)$ on the space of states $H$.\\

So briefly, the slogan is that a $D=2$ CFT is an algebra over the operad
$R(n)$.\\

{\bf Definition 8:}\\
A \emph{string background} (at the tree level) is a $D=2$ CFT based on the
vector space $H$ with the following additional data:\\
{\bf 1.} A {\bf Z}-grading $H=\oplus _{i\in {\bf Z}}H_{i}$ on the state
space.\\
{\bf 2.} An action of the \textsl{Clifford algebra} $C(V\oplus V^{*})$ which
is denoted $b:V\otimes H\rightarrow H$ and $c:V^{*}\otimes H\rightarrow H$
for generators of the Clifford algebra, the degree of $b$ is -1 and the degree of $c$ is +1.\\
{\bf 3.} A differential $Q:H\rightarrow H$, $Q^{2}=0$, of degree +1 called
\emph{BRST operator} satisfying $Qb+bQ=T$.\\

The differential graded complex $(H,Q)$ is called the \emph{BRST complex} and
the degree is called the \textsl{ghost number}.\\

One of the nicest implications of a string background is the construction of
a morphism of complexes $\omega _{n}:Hom(H, H^{\otimes n})\rightarrow
\Omega ^{*}(R(n))$, from the complex of linear mappings between tensor powers
of the BRST complex $H$ to the de Rham complex of the space $R(n)$.\\

Taking the \textsl{cohomology} of the BRST complex
gives the \emph{space of physical states}. In physics this amounts to mod out
gauge invariance (this is a cohomological approach to \textsl{symplectic
reduction} in the case of a symplectic manifold  carrying a Lie group action
which is more convenient in infinite dimensions, i.e. field theory).\\

Then a \emph{closed string field theory} is a string background together with
a morphism of operads $\psi :M(n)\rightarrow R(n)$. This however does not
readily generalise to higher dimensions since both operads $M(n)$ and $R(n)$
are related to moduli spaces of Riemann surfaces. The aspect of closed string
field theory which will be useful for higher dimensional generalisations is
the following:\\

{\bf Key fact:} The space of physical states in closed string field theory
(namely BRST cohomology) has the structure of a \emph{Gerstenhaber algebra}.\\

Let us explain this a little bit more: from the work of G. Segal we knew that
in general $D=2$ topological quantum
field theories (and in particular closed string field theory)
are ${\bf Z}$-graded commutative associative Frobenious algebras, hence they
are \textsl{graded
associative algebras}. Yet it was observed by Witten and Zwiebach that closed
string field theory also carries the structure of a \textsl{Differential
Graded Lie Algebra} relative to another grading which is
\emph{``the associative grading - 1''}.
In fact these 2 structures can be combined together to give a \emph{G-algebra}
structure.\\

Now we make use of Cohen's theorem saying that $G$-algebras
correspond exactly to the homology of the chains of the little 2-discs
operad and of Tamarkin's result to deduce that the \emph{space of physical
states of closed string field theory (which is a special case of $D=2$ CFT)}
has a natural (defined up to homotopy) \textsl{2-algebra structure}.\\

So this is the important relation between strings and the operad $C_{2}(n)$
that plays the fundamental role to understand Kontsevich's
statement which is then simply the \emph{higher dimensional version} of the
above fact which originally holds for strings ($D=2$ case).\\

{\bf Remarks:}\\
{\bf 1.} The original Deligne conjecture, namely the case $d=1$, was
proved by Kontsevich and Soibelman in 2000
using ideas and techniques from Dan Quillen's \emph{homotopical algebra}
which roughly is a \textsl{non-linear}
generalisation of \emph{homological algebra}.\\

{\bf 2.} Why should physics care about the Deligne conjecture?
We think for 2 reasons:\\
{\bf i.} The fact that the Hochschild complex of an associative algebra is a
2-algebra (original conjecture) is related to the action
of the \emph{Grothendieck-Teichmuller (G-T) group}. The fact that the
Hochschild
complex plays the fundamental role in the theory of \emph{deformations} of
associative algebras explains why its study is important in quantum field
theory if one adopts the \emph{deformation quantization approach}.

In other
words the goal is to understand the \emph{action} of the G-T group on the
\emph{space of all
deformation quantizations} on the associative algebra of functions (in fact
one needs a little more structure, i.e. a Poisson algebra structure) on a
given (spacetime or phase space respectively) manifold and this is believed
to be related
to gauge symmetry. Yet all these are far from being clear at the moment.
Let us briefly recall that the Grothendieck-Teichmuller group can be defined
as the automorphism group of the tower of the pro-nilpotent completions of
the pure braid groups; the pure braid group of $n$ strings is the fundamental
group of the configuration space of $n$ points in the plane
${\bf F}(n, {\bf R^{2}})$.\\

{\bf ii.} If one wants to understand $(d+1)$-dim \emph{quantum gravity}, one
approach is
to follow holography. This means that equivalently one should understand
 $D=d$  CFT. From the string theory example we have learnt that $D=2$ CFT is
related to 2-algebras. It is then reasonable to expect that $D=d$ CFT must be
related
to $d$-algebras. So the hope is that this higher dimensional Deligne
conjecture
will tell us something about the \emph{``BRST complex''} whose cohomology
would describe the $D=d$ CFT.

Perhaps here we should mention another important
result: the BRST complex of closed string field theory has also the structure
of a $G_{\infty }$-algebra (see \cite{operads}). That makes someone to
speculate on a relation between the operads $G_{\infty }(n)$ and
$Chains C_{2}(n)$. For example it is unknown if double loop spaces (which are
the primary examples of 2-algebras) also carry a homotopy Gerstenhaber
algebra structure.\\

{\bf 3.} As Kontsevich explains in his article, from Deligne's conjecture
and from Tamarkin's theorem (namely the formality of the operad
$Chains C_{d}(n)$), follows almost immediately his
earlier result on deformation quantization of symplectic (Poisson) manifolds
for the case where the associative algebra of interest is just the polynomial
algebra in $n$ variables
$$A:={\bf R}[x_{1},...,x_{n}].$$
Let us recall that Kontsevich's result was that for the associative algebra
$A$ above one has that its Hochschild complex $C^{*}(A,A)$ is \emph{homotopic}
as a Lie algebra to its Hochschild cohomology $H^{*}(A,A)$. An equivalent
statement is that for the Euclidean space $X={\bf R}^{n}$, the Hochschild
complex of the associative algebra of functions on $X$ equipped with
the Gerstenhaber bracket is \emph{homotopic as DGLA} to
the ${\bf Z}$-graded superalgebra of \emph{polyvector fields} on $X$ equipped
with the Schouten-Nijenhuis bracket.

{\bf 4.} The structure of an $A_{\infty }$-algebra has appeared recently in
\emph{open strings}. Moreover let us mention that one of the main examples of
homotopy associative algebras (or $A_{\infty }$-algebras) is singular chains
of based loops.\\

{\bf 5.} A pessimist would argue that the holography principle does not really
``resolve'' the BH paradoxe; it simply pushes it even further to the realm
 of quantum gravity which is \emph{terra incognita} for today's physics.
We however would like to adopt a more positive point of view: by accepting
the validity of holography we can actually
\emph{use} it in order to learn something about quantum gravity and
at the same time we try to understand its wider implications for physics.
Finally we would like to mention that there have been already some positive
tests for the validity of holography mainly in the framework of some
calculations related to its string theoretic version (Maldacena conjecture).\\

So the main point in this article was to argue that the Hilbert space
of $(d+1)$ dimensional quantum gravity should have the following algebraic
structure: from holography principle it should be a TQFT; all TQFT's known are
\emph{cohomological} theories, namely it should be described as the 
cohomology of a complex. Following the lesson we have learnt from strings 
this Hilbert space should have a $d$-algebra structure; then the higher 
dimensional Deligne conjecture will tell us that the complex whose cohomology 
gives the Hilbert space must carry a $(d+1)$-algebra structure. One
can only hope that an explicit construction will eventually emerge after
accumulating information about the properties of the ``would-be''
quantum gravity Hilbert space. In D=2 case one uses Topological Vertex
Operator Algebras (TVOA) as a tool to construct topological Conformal Field
Theories but again their higher dimensional generalisation is not
obvious. We believe that the way ahead, which looks more like a programme 
for future research, is to try to understand in grater 
detail what happens in \emph{lower dimensions} before atacking the case of 
physical 
interest, namely quantum gravity in dimension 4, CFT in dimension 3 and the
higher dimensional Deligne conjecture.\\

 
The author would like to thank Maxim Kontsevich and Graeme Segal for useful 
discussions.






\begin{thebibliography}{20}

\bibitem{das} S.R. Das and S.D. Mathur: ``The Quantum Physics of Black Holes:
Results from String Theory'', Annual Review of Nuclear and Particle Science
Vol 50 (2000) and gr-qc/0105063.\\

\bibitem{thooft}G. 't Hooft: ``Dimensional Reduction in Quantum Gravity'',
Essay dedicated to Abdus Salam in \emph{``Salamfestschift}: a collection of
talks'', editors: A. Ali, J. Ellis and S. Randjbar-Daemi, World Scientific
1993 and gr-qc/9310026.\\

\bibitem{operads} J-L. Loday, J.D. Stasheff and A.A. Voronov: ``Operads:
Proceedings of Renaissence Conferences'', Contemp. Maths. 202 AMS 1997.\\

\bibitem{kon1} M. Kontsevich: ``Operads and Motives in Deformation
Quantization'', Lett. Math. Phys. 48.1 (1999) and QA/9904055.\\

\bibitem{kon2} M. Kontsevich and Y. Soibelman: ``Deformations of
algebras over operads and Deligne's conjecture'', Conference Moshe Flato 
1999, Vol I, Math. Phys. Studies 21, Kluwer (2000), 225-308 and QA/0001151.\\

\bibitem{oper} A.A. Voronov: ``Notes on Universal Algebra'', preprint 
QA/0111009.\\

\end{thebibliography}



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