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\title{Thermodynamic behavior of IIA string theory on a pp-wave}

\author{Seungjoon Hyun${}^{\ast}$, Jong-Dae Park$^{\dagger\ddag}$ and Sang-Heon
Yi$^{\ast}$\\
${}^{\ast}$Institute of Physics and Applied Physics, Yonsei University, Seoul 120-749, Korea\\
${}^{\dagger}$Yonsei Visiting Research Center, Yonsei University,
Seoul 120-749, Korea\\
${}^{\ddag}$Center for Theoretical Physics, Seoul National
University, Seoul  151-742, Korea\\




E-mail~:~\email{hyun@phya.yonsei.ac.kr},
\email{jdpark@phya.snu.ac.kr},
 \email{shyi@phya.yonsei.ac.kr}}




\abstract{We consider the thermal properties of IIA string theory
on the pp-wave which comes from the circle compactification of the
maximally supersymmetric eleven dimensional pp-wave. The one loop
free energy is computed and Hagedorn temperature is found as a
function of the RR-flux $\mu$. In the $\mu \rightarrow \infty$
limit, the free energy is shown to be identical with that of IIB
string theory on maximally supersymmetric pp-wave. We use two
complementary approaches, operator and path integral methods.
Several points, including the zero point energy and the modular
properties of the free energy, are more illuminating and
transparent in the path integral method. We find the complete
agreement between those two approaches in the free energy
expression and the dependency of the Hagedorn temperature on the
RR-flux.}




\keywords{Hagedorn temperature, Free energy, IIA string on
pp-wave}



\preprint{hep-th/0304239 \\
          SNU-TP 03-009}




\begin{document}

\section{Introduction and conclusion}

Sting/M theory is often considered as the theory of everything and
as such it is hoped to explain the early Universe from the
theory~\cite{Polchinski:rq}. In this sense, it is very important
to understand the finite temperature physics of the string/M
theory. As is well-known, there are some difficulties in
investigating the thermodynamics of the theory which contains the
gravity, for example, the string theory which is our main focus in
this paper. The gravity has the Jeans' instability which is deeply
related to the formation of black holes. And also the energy of
the system with gravity may be difficult to be defined. But, as
was argued in~\cite{atickwitten}, we may consider the weakly
coupled low energy limit of such theory and get some insight on
the thermodynamics of the theory. The most notable feature of the
thermodynamics of string theory is the existence of the Hagedorn
temperature~\cite{Hagedorn:st,huang}, around the Planck scale. It
is due to the exponential growth of the level density of string
states. There have been discussions whether the Hagedorn
temperature is the limiting temperature or it signals a phase
transition. So far, there is no clear understanding about the
physics above the Hagedorn temperature.

There were some interesting
works~\cite{PandoZayas:2002hh,Greene:2002cd,Sugawara:2002rs,
Brower:2002zx,Sugawara:2003qc,semenoff} on the finite temperature
physics of type IIB superstring theory on the maximally
supersymmetric ten-dimensional pp-wave spacetime~\cite{bla242}.
They calculated the free energy and found the dependence of the
Hagedorn temperature on the RR-flux. If we assume that the
Hagedorn temperature on the pp-wave geometry depends on  $\mu$
smoothly and is connected with that of the flat space in the
$\mu\rightarrow 0$ limit, then, from the dimensional argument, the
Hagedorn temperature on the pp-wave background  may be expanded as
a series of the characteristic mass scale, $\mu$, of the pp-wave
%
\be T_H = \frac{1}{2\pi\sqrt{2\a}}\Big(1 + \sum_{n=1}^\infty
a_n(\sqrt{\a}\mu)^n \Big) \,. \label{hagedorn}\ee
%
Here $a_n$ is the numerical constant depending on the details of
pp-wave geometry. The IIB pp-wave geometry can be obtained by
taking the Penrose limit of the $AdS_5 \times S^5$ spacetime, and
gives another solvable model for Green-Schwarz
superstrings~\cite{Metsaev:2001bj}. As one incarnation of AdS/CFT
correspondence~\cite{Berenstein:2002jq}, the finite temperature
behavior of IIB string theory on this pp-wave has been studied in
connection with that of the dual ${\cal N} =4$ super Yang-Mills
theory in the corresponding limit.



In this paper, we would like to study the thermodynamics of type
IIA Green-Schwarz superstring on the ten dimensional pp-wave
geometry~\cite{hyunandshin1},
%
\bea ds^2 &=& -2 dx^{+} dx^{-} - A(x^I) dx^{+}dx^{+} +
\sum_{I=1}^{8 }dx^{I}dx^{I}\,, \nn \\ && \nn \\
     && F_{+123} = \mu\,, \qquad F_{+4}{}={}
     -{\textstyle\frac{\mu}{3}}\,,
\label{pp-wave}\eea
%
where
$$ A = ({\textstyle\frac{\mu}{3}})^2(x_1^2+x_2^2+x_3^2+x_4^2)+
  ({\textstyle\frac{\mu}{6}})^2(x_5^2+x_6^2+x_7^2+x_8^2)\,.
$$
This geometry comes from the dimensional reduction of the
maximally supersymmetric eleven dimensional pp-wave~\cite{fig308},
and has one characteristic mass scale, $\mu$.

We use the canonical ensemble to study the thermal properties of
IIA string theory on this pp-wave geometry. The canonical ensemble
may not be justified in the cases when the formation of long
string is not ignored, as its fluctuations make the thermalization
for the gas of strings difficult, and in such cases the
microcanonical ensemble should be used
instead~\cite{Frautschi:1971ij,Carlitz:1972uf,Bowick:az,
Bowick:1989us,Deo:1988jj}. It would be interesting to study the
thermodynamics of IIA string theory on pp-wave using the
microcanonical ensemble and see the implications.

There are two complementary approaches to calculate the one loop
free energy of the finite temperature string theory. One is the
operator method which can be adaptable easily once the light-cone
gauge is chosen. The other is the path integral approach which is
conceptually more transparent and is readily generalized to higher
loops. In this paper we compute the one loop free energy using
both methods and obtain the Hagedorn temperature which takes the
assumed form (\ref{hagedorn}).

The Hagedorn temperature is found to be the monotonic function of
${\sqrt \a}\mu$ and goes to infinity as $\mu\rightarrow \infty$.
In the same limit, the free energy is shown to become
%
\be F = -\frac{L\pi}{6\beta^2}\,, \ee
%
which is exactly the same as that of IIB string theory on the
pp-wave~\cite{semenoff}. In general, the free energy of $d$
dimensional field theories at high temperature behaves
as~\cite{atickwitten}
%
\be F_d \sim V \beta^{-d}\,. \ee
%
This suggests that in the large $\mu$ limit, the physics on the
pp-wave geometry essentially reduces two-dimensional one. It would
be very much interesting to pursue along this line.

Our main motivation for this study has been to understand the real
degrees of freedom of IIA strings on pp-wave, eventually revealing
those of M theory on pp-wave.  As IIB pp-wave case, the
eleven-dimensional maximally supersymmetric pp-wave can be
obtained from the Penrose limit of $AdS_4\times S^7$ or
$AdS_7\times S^4$. Since the M theory on those $AdS$ spacetimes
are dual to three dimensional and six dimensional conformal field
theories, respectively, it is natural to expect, in the spirit of
the AdS/CFT correspondence, the M theory on the pp-wave is dual to
(the common sector of ) those theories in the corresponding limit.
Our study on the finite temperature physics of IIA strings on
pp-wave may give some clues to understand those dual theories.



This paper is organized as follows. In section two, we briefly
review IIA string theory on the pp-wave geometry. In section
three, we obtain the free energy of the weakly interacting string
gas on the pp-wave using the operator method. In section four, we
consider the asymptotics of free energy and find the Hagedorn
temperature which depends on the RR-flux through the difference of
boson and fermion zero point energies. In section five, we use the
path integral approach to calculate the one loop free energy. In
this approach, we clarify various points, including the zero point
energy and the modular properties of the one loop free energy. In
section six, we find complete agreements between the results from
the operator and path integral methods. We have included some
useful formulae in the Appendix.

\section{Review: IIA superstring on the pp-wave geometry}

In this section we summarize some results for the type IIA
superstring theory on the ten dimensional pp-wave spacetime,
including the mode expansions, the normal ordered worldsheet
Hamiltonian and momentum (see for more
details~\cite{hyunandshin1,hyunandshin2}). The IIA pp-wave
geometry (\ref{pp-wave}) admits 24 Killing spinors, as the torus
compactification breaks 8 supersymmetries out of 32 in the
eleven-dimensional pp-wave geometry. In the light-cone gauge, the
worldsheet description of IIA Green-Schwarz superstrings on this
pp-wave background is given by the two-dimensional, free massive
${\cal N}=(4,4)$ supersymmetric theory. As the metric of pp-wave
geometry suggests, the bosonic fields in the string action, which
are the target space coordinates, split into two multiplets with
masses proportional to $\frac{\mu}{3}$ and $\frac{\mu}{6}$ in the
light-cone gauge. Interestingly enough, the fermionic
superpartners, after kappa gauge fixing, also split into two
multiplets with the same masses as bosonic partners, classified by
the chirality of $SO(8)$ rotations in $x^I$ and $SO(4)$ rotations
in $x^{i'}$\footnote{In what follows, the indices $I=1,2,...,8$
denote the 8 transverse coordinates, among which the indices
$i=1,...,4$ denote the direction where RR field strengths are
spanned and the indices $i^{\prime}=5,...,8$ denote the other
directions. We also take the real and symmetric representation for
$SO(8)$ gamma matrices. }. This is due to the fact that eight
Killing spinors, which correspond to the dynamical supersymmetry
on the string worldsheet, do not explicitly depend on the
coordinate $x^+$. Let us denote the chirality of $SO(8)$  and
$\gamma^{1234}$ as superscripts $1,2$ and subscripts $\pm$ for
fermions, $\psi_{\pm}^{1,\,2}$,  respectively. Then one can see
that the theory contains two supermultiplets $( X^i, \psi_-^1,
\psi_+^2 )$ and $( X^{i'}, \psi_+^1,
  \psi_-^2 )$ whose masses are given by $\frac{m}{3}$ and
$\frac{m}{6}$, respectively, with
%
\be m \equiv \mu\a p^{+} \, . \ee
%

The light-cone gauge fixed action for this IIA Green-Schwarz
superstring becomes
%
\bea S_{LC} &=& -\frac{1}{4\pi\a}\int d\sigma^{0}d\sigma^{1}
\Big[\partial_{\alpha}X^I\partial^{\alpha}X^I +
({\textstyle\frac{m}{3}})^2(X^i)^2+
  ({\textstyle\frac{m}{6}})^2(X^{i'})^2
   -i\psi^1_{-}\partial_{+}\psi^1_{-}
   -i\psi^2_{+}\partial_{-}\psi^2_{+}\nn \\ && \nn \\
&&{}~~~~~~~~~~~~~~~~~~~{}
  -i\psi^1_{+}\partial_{+}\psi^1_{+}
  -i\psi^2_{-}\partial_{-}\psi^2_{-}
   + i\frac{2m}{3}\psi^2_{+}\gamma^4\psi^1_{-}
   -i\frac{m}{3}\psi^2_{-}\gamma^4\psi^1_{+}\Big]\,,
\eea
%
where $\alpha=0,1$ denotes worldsheet index and
$\partial_{\pm}=\partial_{0}\pm\partial_{1}$.

With frequencies
%
\be
    n\ge 0 \quad ; \quad
 \omega_n =  \sqrt{ ( {\textstyle\frac{m}{3}})^2 +
 n^2}\,,\qquad
 n<0  \quad ; \quad
  \omega_n =  - \sqrt{ ( {\textstyle\frac{m}{3}})^2 + n^2
    }\,,  \ee
%
each field in the multiplet $( X^i, \psi_-^1, \psi_+^2 )$ can be
mode expanded as
%
\bea X^{i}
&=&i{\textstyle\sqrt{\frac{\a}{2}}\sqrt{\frac{1}{\omega_0}}}
 {}~{} \Big(a^{i}e^{-i\omega_0\sigma^0} -a^{i\,
 \dagger}e^{i\omega_0\sigma^0}\Big) +
i{\textstyle\sqrt{\frac{\a}{2}}}\sum_{n\neq 0}
\Big({\textstyle\frac{1}{\omega_n}}\alpha^{i}_{n}
e^{-in\sigma^{1}} +{\textstyle\frac{1}{\omega_n}}
{\tilde\alpha}^{i}_{n} e^{in\sigma^{1}} \Big)
e^{-i\omega_{n}\sigma^{0}}\,,
\nn \\ && \nn \\
\psi^1_{-}  &=& -i{\textstyle\sqrt{\frac{\a}{2}}} \gamma^4
\Big(\chi\, e^{-i\omega_0\sigma^0}- \chi^\dagger\,
e^{i\omega_0\sigma^0}\Big) + \sum_{n\neq 0} c_n\Big(
\tilde{\psi}_n{}~{} e^{in\sigma^1} -i
{\textstyle\frac{\omega_n-n}{\omega_0}}\gamma^4\psi_n{}~{}
e^{-in\sigma^1}\Big) e^{-i\omega_n\sigma^0 }\,,
\nn \\ && \nn \\
\psi^2_{+}  &=& {\textstyle\sqrt{\frac{\a}{2}}}  \Big(\chi\,
e^{-i\omega_0\sigma^0} + \chi^\dagger\, e^{i\omega_0\sigma^0}\Big)
+ \sum_{n\neq 0} c_n\Big( \psi_n{}~{} e^{-in\sigma^1} +i
{\textstyle\frac{\omega_n-n}{\omega_0}}\gamma^4\tilde{\psi}_n{}~{}
e^{in\sigma^1}\Big) e^{-i\omega_n\sigma^0 }\,, \nn\eea
%
where $c_n = \sqrt{\a}/\sqrt{1+(\omega_n-n)^2/\omega_0^2}$ is the
normalization constant taken canonically. After the canonical
quantization, unprimed modes satisfy  the following commutation
relations,
%
\[  [ a^i, a^{j \dagger} ] = \delta^{ij}\,, \qquad
\big[\alpha^{i}_{n}, \alpha^{j}_{m} \big] = \omega_n \delta^{ij}
\delta_{n+m,0}\,, \]
%

\[ \{ \chi, \chi^\dagger \} = 1 \,, \qquad  \{ \psi_n , \psi_m \} =
\delta_{n+m,0} \,,  \qquad \{ \tilde{\psi}_n, \tilde{\psi}_m \} =
\delta_{n+m,0}\,.\nn
 \]
%
 Each field in the
multiplet $( X^{i'}, \psi_+^1, \psi_-^2 ) $ can also be mode
expanded  and canonically quantized similarly with  frequencies
%
\be
 n\ge 0 \quad ; \quad \omega'_n =\sqrt{ ({\textstyle\frac{m}{6}})^2 + n^2 }\,,
  \qquad n<0  \quad ; \quad
  \omega'_n = -
      \sqrt{ ({\textstyle\frac{m}{6}})^2 + n^2 }\,.\nn
\ee
%
Introducing the number operators $N^{B}_{i\, n}, N^{F}_{n}$
defined by
%
\bea  \left. \ba{cccc} & n >0 &  \qquad n=0 & \qquad n <0
\\ & & &  \\
N^{B}_{i\, n} := & \frac{1}{\omega_n}{}~{} \alpha^{i}_{-n}
\alpha^{i}_n\,, &  \qquad a^{i \dagger} a^i\,, & \qquad
 \frac{1}{~\, \omega_{-n}}{}~{} {\tilde\alpha}^i_{n} {\tilde\alpha}^i_{-n}\,,
 \\ & & &  \\
N^{F}_{n}:= &  \psi_{-n} \psi_n\,, & \qquad \chi^\dagger \chi\,, &
\qquad {\tilde\psi}_{n}{\tilde\psi}_{-n}\,, \ea\right. \eea
%
and $N^{B}_{i'\, n}, N^{\prime F}_{n}$, defined similarly, the
worldsheet momentum and Hamiltonian can be written as
%
\be {\cal P}= N-\tilde{N} = \sum_{n=-\infty}^{\infty}n
\Big(\sum_{i=1}^4 N^B_{i\, n} +\sum_{i^{\prime}=5}^8
N^{B}_{i^{\prime}\, n}+ N^{F}_{n} + N^{\prime\, F}_{n} \Big)\,,\ee
%
\be {\cal H} = \a p^{+} p^{-} =
\sum_{n=-\infty}^{\infty}\bigg[|\omega_n|\Big(
\sum_{i=1}^4N^{B}_{i\, n} + N^{F}_{n}\Big)
+|\omega_n^{\prime}|\Big(\sum_{i^{\prime}=5}^8 N^B_{i^{\prime}\,
n} + N^{\prime\,F}_{n}\Big) \bigg]\,. \label{lcham}\ee
%







%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Operator method}
In this section we use the operator method to study the
thermodynamic properties of IIA strings on the pp-wave. We
consider the canonical ensemble composed of an ideal gas of weakly
interacting strings on the given background. We mainly focus on
the thermodynamic partition function or the free energy of string
gas, from which all the other thermodynamic quantities can be
obtained.

The free energy of the string gas can be expressed in terms of the
trace, $\Tr^{\prime}$, over the one-string physical states as
%
 \bea F&=&\frac{1}{\beta} \Tr^{\prime}\, \bigg[ (-1)^{\bf F} \ln
\Big(1- (-1)^{\bf F} e^{-\beta p^0}\Big)\bigg]~, \nn \\ && \nn \\
&=& - \sum_{l=1}^{\infty} \frac{1+(-1)^l}{2l\beta}\,
\Tr^{\prime}\,\Big[(-1)^{\bf F}\,e^{-l\beta p^0}\Big]
-\sum_{l=1}^\infty
\frac{1-(-1)^l}{2l\beta}\,\Tr^{\prime}\,e^{-l\beta p^0}
 \,, \eea
%
where ${\bf F}$ is the space-time fermion number
operator\footnote{One may use $ap^{+}+bp^{-}$ instead of $\beta
p^0$ for the Boltzman factor
\cite{Greene:2002cd,Brower:2002zx,semenoff}, which is connected
with the grand canonical ensemble. The results for this
substitution can be easily read from our results as the geometry
is invariant under the boosting along the longitudinal direction
with the rescaling of $\mu$.}.

The level matching condition $N = \tilde{N}$ can be implemented by
introducing the Lagrange multiplier, $\tau_1$, as
%
\bea F &=& -\sum_{l=1}^{\infty} \frac{1+(-1)^l}{2l\beta}
 \Tr {}~{} \bigg[ \int_{-1/2}^{1/2}d\tau_1{}~{}  e^{2\pi i
\tau_1(N-\tilde{N})}{}~{}(-1)^{\bf F} e^{-l\beta p^{0}} \bigg]
 \nn \\
&&{}~{} -\sum_{l=1}^\infty \frac{1-(-1)^l}{2l\beta}\, \Tr
{}~{}\bigg[ \int_{-1/2}^{1/2}d\tau_1{}~{} e^{2\pi i
\tau_1(N-\tilde{N})}{}~{} e^{-l\beta p^{0}} \bigg]\,,
\label{free1} \eea
%
where $\Tr$ denotes the trace over the one-string Fock space
states without the level matching condition.


In the light-cone formalism which is particularly suitable for the
given pp-wave geometry, the space-time energy $p^{0}$ is split
into the dynamical variable, $p^{-}=\frac{1}{\a p^+}{\cal H}$, and
kinematical one, $p^{+}$, as
$p^0=\frac{1}{\sqrt{2}}(p^{+}+p^{-})$. In the given pp-wave
background, $p^{+}$ is continuous parameter and $p^{-}$ has the
discrete spectrum for given $p^+$ and  the one-string states are
represented by the quantum numbers of them. Accordingly, the above
$\Tr$ can be written as
%
\be \Tr = \frac{L}{\sqrt{2}\pi}\int_0^\infty dp^{+}{}~~{}
\tr_{trans}\,, \ee
%
where $L$ represents the (infinite) length of the longitudinal
($x^{9}=\frac{1}{\sqrt{2}}(x^{+}-x^{-})$) direction and
$\tr_{trans}$ denotes the trace for the transverse oscillators
without level matching constraint.

The first term in the free energy expression in (\ref{free1}) is
simply the integral over $p^+$ of the Witten index:
\[
\tr_{trans} \Big[ \int_{-1/2}^{1/2}d\tau_1{}~{} e^{2\pi i
\tau_1(N-\tilde{N})}{}~{}(-1)^{\bf F} e^{-\frac{\beta}{\sqrt{2}}
p^{-}}\Big]~,
\]
which, in our case, turns out to be unity. This Witten index
vanishes in the case of string theory on flat spacetime, due to
fermion zero modes.

Let us define
%
\be \tau_2 := \frac{1}{\sqrt{2}} \frac{l\beta}{2\pi\alpha^\prime
p^{+}}\,, \ee
%
then the mass parameter, $m$, can be rewritten as
%
\be
 m =  \frac{l\beta\mu}{2\sqrt{2}\pi\tau_2}\,. \label{mmod}\ee
%
This change of variables leads to
%
\be F = - \frac{L\pi}{24\beta^2} - \sum_{l=1,\,{\rm odd}}^\infty
\frac{L}{4\pi^2\a}\int^\infty_0 \frac{d\tau_2}{\tau^{~2}_2}{}~{}
e^{-\frac{l^2\beta^2}{4\pi\a\tau_2}} \int_{-1/2}^{1/2} d\tau_1
{}~{} \tr_{trans}\, \Big[ e^{2\pi i\tau_1{\cal P}}{}~{}
e^{-2\pi\tau_2\, {\cal H}} \Big] \,. \ee
%
Note that ${\cal H}=\a p^{+}p^{-}$ given in Eq.~(\ref{lcham}) has
$l$-dependence through $m$.

In the computation of $\tr_{trans}\,[ e^{2\pi i\tau_1{\cal
P}}{}~{} e^{-2\pi\tau_2\, {\cal H}}]$, for each mode including the
zero mode part,  we use the following formulae
%
\bea \!\!\!\!\! \prod_{i=1}^{4}\sum_{N^{B}_{i\, n}=0}^{\infty}
e^{(-2\pi\tau_2|\omega_n|+ 2\pi i \tau_1 n )N^{B}_{i\, n}} &=&
\bigg(\frac{1}{1-e^{-2\pi\tau_2|\omega_n| + 2\pi i\tau_1
n}}\bigg)^4\,, \qquad {\rm for ~~ bosons} \,,  \\ \!\!\!\!\! && \nn \\
\!\!\!\!\! \sum_{N^{F}_{n}=0}^{4}{4\choose N^F_n}
e^{(-2\pi\tau_2|\omega_n| + 2\pi i \tau_1 n) N^{F}_{n}} &=& \Big(1
+ e^{-2\pi\tau_2|\omega_n| + 2\pi i\tau_1 n}\Big)^4 \,, \qquad
{\rm for ~~ fermions} \,, \eea
%
with  similar ones for $N^{B}_{i'\, n}, N^{\prime F}_{n}$.
Finally, the free energy becomes
%
\bea  F &= &- \frac{\pi L}{24\beta^2} {}~{} - \sum_{l=1,{\rm
odd}}^\infty \frac{L}{4\pi^2\a}\int^\infty_0
\frac{d\tau_2}{\tau^{~2}_2} \int^{1/2}_{-1/2} d\tau_1 {}~{}
e^{-\frac{l^2\beta^2}{4\pi\a\tau_2}} \nn \\ &&\nn \\ &&
{}~~~~~{}~~~~~{} \prod_{n=-\infty}^\infty \left(\frac{1+ e^{-2\pi
\tau_2|\omega_n| + 2\pi i \tau_1 n } } {1-e^{-2\pi
\tau_2|\omega_n| + 2\pi i \tau_1 n }}\right)^4 \left(\frac{1+
e^{-2\pi \tau_2|\omega^\prime_n| + 2\pi i \tau_1 n } } {1-e^{-2\pi
\tau_2|\omega^\prime_n| + 2\pi i \tau_1 n }}\right)^4\,.  \eea
%
Since we have used the normal ordered Hamiltonian, the zero point
energy contribution to the free energy cancels between bosons and
fermions and does not appear in the above expression.





\section{Free energy and Hagedorn temperature}

In this section, we obtain the dependence of the Hagedorn
temperature on the RR flux, $\mu$. We will see that the Hagedorn
temperature behaves smoothly with respect to RR-flux and is a
monotonic function of $\mu$.

In order to see the asymptotic behavior of the free energy, it is
convenient to know the modular property of the following quantity
%
\be D_{b_1,\, b_2}(\tau_1,\tau_2; m)  :=
e^{2\pi\tau_2\Delta_{b_1}(m)}\prod_{n=-\infty}^\infty \Big(1 -
e^{-2\pi\tau_2\sqrt{(n+b_1)^2+m^2} + 2\pi i \tau_1(n+b_1) - 2\pi i
b_2}\Big) \,, \label{Dbb}\ee
%
where the function $\Delta_b(m)$ is defined by
%
\be \Delta_{b}(m) :=-\frac{m}{\pi}\sum_{p=1}^\infty
\frac{\cos(2\pi b p)}{p}K_1(2\pi m p) =
-\frac{1}{2\pi^2}\sum_{p=1}^\infty \cos(2\pi b p){} \int_0^\infty
d s {}~{}  e^{-p^2 s -\frac{\pi^2 m}{s}}\,, \label{delta} \ee
%
with the modified Bessel function $K_1(x)$. Note that
%
\be  \Delta_{b} (m) {}~{} {\buildrel m\rightarrow
0\over\longrightarrow} {}~~~{}\frac{1}{24}-\frac{1}{8}(2b-1)^2\,,
\qquad  \Delta_{b} (m) {}~{}{\buildrel m\rightarrow
\infty\over\longrightarrow}{}~~~{} 0\,.\ee
%
$\Delta_b(m)$ will be identified as the physical zero point (or
Casimir) energy for two dimensional massive boson or fermion field
with the twisted boundary condition in the path integral approach
given in the next section. One can see that the above $D_{b_1,\,
b_2}(\tau_1,\tau_2; m)$ has the following modular
property~\cite{Takayanagi:2002pi,Bergman:2002hv}
%
\be D_{b_1,\, b_2}\Big(\tau_1,\tau_2 ; m \Big) = D_{b_2,\, -b_1}
\Big(-\frac{\tau_1}{|\tau|^2}, \frac{\tau_2}{|\tau|^2} ;
m|\tau|\Big)=D_{b_1,\, b_2+b_1}\Big(\tau_1+1,\tau_2 ; m \Big)\,,
\label{modular} \ee
%
with $\tau =\tau_1 +i \tau_2$.

We can exhibit the free energy using $D_{a,\, b}(\tau_1,\tau_2;
m)$ as
%
\bea
 F &=& - \frac{\pi L}{24\beta^2}
 - \sum_{l=1\atop~~{\rm odd}}^\infty \frac{L}{4\pi^2\a}
\int^\infty_0 \frac{d\tau_2}{\tau^{~2}_2}
\int^{\half}_{-\half}d\tau_1
{}~~{}e^{-\frac{l^2\beta^2}{4\pi\a\tau_2}} \nn\\ && \nn \\
&& {}~~~~~~~~~~~~~~~~~~~~~~~~~~~{} \times ~
\bigg[\frac{D_{0,\,1/2}(\tau_1,\tau_2;{\textstyle\frac{m}{3}})}
{D_{0,\,0}(\tau_1,\tau_2;{\textstyle\frac{m}{3}})}\bigg]^4
\bigg[\frac{D_{0,\,1/2}(\tau_1,\tau_2;{\textstyle\frac{m}{6}})}
{D_{0,\,0}(\tau_1,\tau_2;{\textstyle\frac{m}{6}})}\bigg]^4 \,.
\label{Free}\eea
%
Since $D_{0,\, 1/2}(\tau_1,\tau_2; m)/D_{0,\, 0}(\tau_1,\tau_2;
m)$ diverges when $\tau_1,\tau_2$ and $\mu$ vanishes, we consider
the limits $\tau_1=0$ and $\tau_2\rightarrow 0$ for the integrand
of the free energy to see the seemingly divergent behavior of
it\footnote{There are conflicting reports whether the free energy
really diverges or not at the Hagedorn
temperature~\cite{PandoZayas:2002hh,Greene:2002cd,Brower:2002zx}
and, accordingly, whether it  means physically, a limiting or
phase transition temperature.}. Note that $m$ goes to infinity
when $\tau_2$ goes to zero. This make it difficult to see the
singular behavior of $D_{b_1,\, b_2}(\tau_1,\tau_2; m)$, but
(\ref{modular}) can be utilized to avoid it. This modular
transformation helps to extract the singular behavior of the
integrand. One can see that the singular nature is incorporated in
the zero point energy, $\Delta_b(m)$,
%$(Z_{1/2,\, 0}(\tau_1,\tau_2 ; m/3)/Z_{0,\, 0}(\tau_1,\tau_2;
%m/3))^4 \times (Z_{1/2,\, 0}(\tau_1,\tau_2; m/6)/Z_{0,\,
%0}(\tau_1,\tau_2 ; m/6))^4$
from the modular property~(\ref{modular}) as
%
\bea \!\!\!\!\!\!\!\!\!\! && \!\!\!\!\! \left[\frac{D_{0,\,
1/2}(\tau_1,\tau_2 ; \frac{m}{3})}{D_{0,\, 0}(\tau_1,\tau_2;
\frac{m}{3})}\right]^4 \left[\frac{D_{0,\, 1/2}(\tau_1,\tau_2;
\frac{m}{6})}{D_{0,\, 0}(\tau_1,\tau_2 ; \frac{m}{6})}\right]^4
\nn \\ \!\!\!\!\!\!\!\!\!\! &&  \!\!\!\!\!  \nn \\
\!\!\!\!\!\!\!\!\!\! &= &  \!\!\!\!\! \left[\frac{D_{1/2,\,
0}(-\frac{\tau_1}{|\tau|^2}, \frac{\tau_2}{|\tau|^2} ;
\frac{m}{3}|\tau|)}{D_{0,\,
0}(-\frac{\tau_1}{|\tau|^2},\frac{\tau_2}{|\tau|^2} ;
\frac{m}{3}|\tau|)} \right]^4  \left[\frac{D_{1/2,\, 0}
(-\frac{\tau_1}{|\tau|^2},\frac{\tau_2}{|\tau|^2} ;
\frac{m}{6}|\tau|)}{D_{0,\, 0} (-\frac{\tau_1}{|\tau|^2},
\frac{\tau_2}{|\tau|^2}  ; \frac{m}{6}|\tau|)} \right]^4
\\ \!\!\!\!\!\!\!\!\!\! &&  \!\!\!\!\! \nn \\
\!\!\!\!\! &\stackrel{\tau_1=0,\, \tau_2\rightarrow
0}{\longrightarrow}& \exp\left[\frac{2\pi}{\tau_2}\bigg\{
4\Delta_{1/2}\Big({\textstyle\frac{l\beta\mu}{6\sqrt{2}\pi}}\Big)
-4\Delta_{0}\Big({\textstyle\frac{l\beta\mu}{6\sqrt{2}\pi}}\Big)+
4\Delta_{1/2}\Big({\textstyle\frac{l\beta\mu}{12\sqrt{2}\pi}}\Big)
-4\Delta_{0}\Big({\textstyle\frac{l\beta\mu}{12\sqrt{2}\pi}}\Big)\bigg\}
\right] \,, \nn\eea
%
%
%
% \exp\bigg[\frac{2}{\tau_2}
%\frac{r\beta\mu}{3\sqrt{2}\pi}\sum_{p=1}^{\infty}\frac{1-(-1)^p}{p}
%\bigg(2K_1 \Big(\frac{r\beta\mu}{3\sqrt{2}} p\Big) +  K_1
%\Big(\frac{n\beta\mu}{6\sqrt{2}} p\Big)\bigg)\bigg] \,,\eea
%
where  $m|\tau|$ becomes $l\beta \mu/(2\sqrt{2} \pi)$ in the limit
$\tau_1=0$, $\tau_2\rightarrow 0$. The asymptotics of the free
energy, then, reads as
%
\bea \!\!\!\!\!\!\!\!\!\!  F &\sim& -\frac{L}{4\pi^2\a} \sum_{l=1,
{\rm odd}}^\infty \int^\infty_0
\frac{d\tau_2}{\tau^{~2}_2}{}~~{}\exp
\bigg[-\frac{l^2\beta^2}{4\pi\a\tau_2}\bigg]
 \nn \\\!\!\!\!\!\!\!\!\!\!   &&\nn \\ \!\!\!\!\!\!\!\!\!\!   &&{}~{}
\times \exp\bigg[ \frac{2\pi}{\tau_2} \bigg\{
4\Delta_{1/2}\Big({\textstyle\frac{l\beta\mu}{6\sqrt{2}\pi}}\Big)
-4\Delta_{0}\Big({\textstyle\frac{l\beta\mu}{6\sqrt{2}\pi}}\Big)+
4\Delta_{1/2}\Big({\textstyle\frac{l\beta\mu}{12\sqrt{2}\pi}}\Big)
-4\Delta_{0}
\Big({\textstyle\frac{l\beta\mu}{12\sqrt{2}\pi}}\Big)\bigg\}\bigg]\,.
\label{free2}\eea


The smallest temperature for the potential divergence of the above
free energy is given by $l=1$ case, which is taken as the Hagedorn
temperature $T_H = 1/\beta_H$:
%
\bea  \frac{\beta^2_H}{8\pi^2\a} &=&
4\left[\Delta_{1/2}\Big({\textstyle\frac{\beta_H\mu}{6\sqrt{2}\pi}}\Big)
-\Delta_{0}\Big({\textstyle\frac{\beta_H\mu}{6\sqrt{2}\pi}}\Big)\right]+
4\left[\Delta_{1/2}\Big({\textstyle\frac{\beta_H\mu}{12\sqrt{2}\pi}}\Big)
-\Delta_{0}\Big({\textstyle\frac{\beta_H\mu}{12\sqrt{2}\pi}}\Big)\right]
\label{Hagedorn} \\  && \nn \\ &=&
\frac{\beta_H\mu}{3\sqrt{2}\pi^2}\sum_{p=1}^\infty
\frac{1-(-1)^p}{p}
\bigg(2K_1\Big({\textstyle\frac{\beta_H\mu}{3\sqrt{2}} p}\Big) +
K_1\Big({\textstyle\frac{\beta_H\mu}{6\sqrt{2}} p}\Big) \bigg) \,.
\nn \eea
%
Hagedorn temperature is often defined as the temperature at which
the thermal winding mode of the string becomes
massless~\cite{Sathiapalan:1986db,Kogan:jd,atickwitten}. As will
be clear in the next section, this gives the same result. It is
interesting to observe that, if the effective tension of the
thermal winding string, depending on the temperature, is defined
as follows:
\[
\a_{{}_{eff}}(\beta\mu)
=\left(4\left[\Delta_{1/2}\Big({\textstyle\frac{\beta\mu}{6\sqrt{2}\pi}}\Big)
-\Delta_{0}\Big({\textstyle\frac{\beta\mu}{6\sqrt{2}\pi}}\Big)\right]+
4\left[\Delta_{1/2}\Big({\textstyle\frac{\beta\mu}{12\sqrt{2}\pi}}\Big)
-\Delta_{0}\Big({\textstyle\frac{\beta\mu}{12\sqrt{2}\pi}}\Big)\right]
\right)\a~,
\]
then, the leading term($l=1$) in the free energy~(\ref{free2}),
after a suitable rescaling in $\tau_2$, can be written as
%
\bea F \sim -\frac{L}{4\pi^2\a_{{}_{eff}}} \int^\infty_0
\frac{d\tau_2}{\tau^{~2}_2}{}~~{}\exp
\bigg[-\frac{\beta^2}{4\pi\a_{{}_{eff}}\tau_2}+
\frac{2\pi}{\tau_2}\bigg] \,, \eea
%
and the Hagedorn temperature becomes
\[
\beta_H = 2\pi\sqrt{2\a_{{}_{eff}}(\beta_H\mu)}\,.
\]
Note that these take the same forms as those of the string theory
on the flat spacetime and reduce to the flat spacetime case
smoothly as $\mu$ goes to zero.



%In the limit, $\mu \rightarrow 0$ or $\beta \rightarrow 0$, the
%effective tension reduces to the


For the small value of $\mu$, one can obtain the explicit Hagedorn
temperature expression in a perturbative series of $\sqrt{\a}\mu$.
Since the series expansion for the zero point energy is given
by~(see the Appendix for the full expression and its derivation),
%
\be \Delta_{1/2}(x)-\Delta_{0}(x) = \frac{x}{\pi}\sum_{p=1}^\infty
\frac{1-(-1)^p}{p}K_1( 2\pi x p) = \frac{1}{8} -\frac{x}{2} +
x^2\ln 2 + {\cal O}(x^3)\,, \ee
%
the Hagedorn temperature up to $\mu^2$ order is
%
\be T_H = \frac{1}{2\pi
\sqrt{2\a}}\bigg(1+\frac{1}{2}{}\sqrt{\a}\mu +
\Big(\frac{1}{8}-\frac{5}{18}\ln 2\Big)\,
(\sqrt{\a}\mu)^2\bigg)\,. \ee
%

On the other hand, the large $\mu$ asymptotics of the Hagedorn
temperature can be obtained as
%
\be \frac{\beta^2_H}{32\pi^2\a} \sim
\frac{1}{\pi}\sqrt{\frac{\beta_H \mu}{12\sqrt{2}\pi}} {}~{}
\exp\bigg[-\frac{\beta_H \mu}{6\sqrt{2}}\bigg]
~~~\stackrel{\mu\rightarrow\infty}{\longrightarrow} 0 \quad ;
\quad T_H \stackrel{\mu\rightarrow\infty}{\longrightarrow}
\infty\,, \ee
%
through the asymptotic expansion of the zero point energy~:
%
\be \Delta_{1/2}(x)-\Delta_{0}(x)=\frac{x}{\pi}\sum_{p=1}^\infty
\frac{1-(-1)^p}{p}K_1(2\pi x p) \sim \frac{\sqrt{x}}{\pi} {}~{}
e^{-2\pi x}\,.\ee
%

The free energy in this limit becomes identical with the one in
the IIB case~\cite{semenoff}
%
\be F = -\frac{L\pi}{6\beta^2}\,, \label{2dfree}\ee
%
which shows a kind of the  universality  in the $\mu\rightarrow
\infty$ limit. Note that this is a typical free energy behavior of
two dimensional field theories at high
temperature~\cite{atickwitten}. It strongly indicates, at least in
the weakly coupled regime, that strings  on the pp-wave geometry,
in the large $\mu$ limit, effectively live on the two-dimensional
spacetime. The AdS/CFT correspondence suggests that the dual
theory of the M theory on maximally supersymmetric pp-wave would
be related to some common sector of six-dimensional (2,0) theory
or ${\cal N}=8$ three-dimensional conformal field theory. Since
our IIA pp-wave geometry comes from this eleven-dimensional
pp-wave, our IIA string theory should be deeply related to these
dual theories. It would be interesting to see the implications for
dual theories  of the behavior of free energy in the large $\mu$
limit.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Path integral approach}

To obtain the genus-one free energy expression which is manifestly
modular invariant, it will be useful to use the path integral
formalism. The path integral approach is also more satisfactory
for the introduction of the zero point energy and useful for
extending to higher genera. We will present all the involved
quantities by the path integral method. In this section, we follow
the standard formalism of thermal string theory and clarify
several points. In the next section, we find the complete
agreement with the results from the operator method in the free
energy and Hagedorn temperature.


In finite temperature field theories, the time coordinate is
Wick-rotated and compactified with the circumference $\beta$. In
string theory, it corresponds to impose the following periodicity
conditions on the Euclideanized time coordinate $X^0_{E}$ :
%
\bea & X^{0}_{E}(\sigma^1+2\pi,\sigma^2) =
X^{0}_{E}(\sigma^1,\sigma^2)+r\beta\,,\nn  \\  & \label{bbc} \\  &
X^{0}_{E}(\sigma^1+2\pi\tau_1,\sigma^2+2\pi\tau_2) =
X^{0}_{E}(\sigma^1,\sigma^2)+l\beta\,, \nn \eea
%
where $r$ and $l$ correspond to the winding and momentum modes
along the compactified thermal direction, respectively. Note that
we also euclideanized worldsheet time as $\sigma^2= i\sigma^0$.
The classical configurations which satisfy these boundary
conditions are, up to constant term,
%
\be X_{wind}(\sigma^1,\sigma^2) = \frac{r\beta}{2\pi} \sigma^1 +
\frac{(l-r\tau_1)\beta}{2\pi\tau_2}\sigma^2 \,.
\label{thermwind}\ee
%
  Though the
spacetime Wick rotation on the pp-wave geometry itself
~\cite{Russo:2002rq} is problematic, we can still perform the Wick
rotation in the path integral of GS superstring on the pp-wave.

For convenience, we perform the field redefinitions in the
fermions as
%
\be  S^{a}_1=
          -\,(\gamma^4\,\psi^{1}_{-})^{a}\,,  \quad
          S^{a}_2 = \,\psi^{2~a}_{+}~~\,;\qquad
S'^{\,a}_1=\,(\gamma^4\,\psi^{1}_{+})^{a}\,, \quad
S'^{\,a}_2=\,\psi^{2~a}_{-} \,. \ee
%
We use the conformal gauge for the worldsheet metric along with
$\kappa$-symmetry gauge fixing for fermions. Then the action
becomes~\cite{hyunandshin1}
%
\bea  S_E &=&  \frac{1}{8\pi\a}\int d z d\bar{z} \bigg[
-4(\partial X^+{\bar\partial} X^- + {\bar\partial} X^+\partial
X^-) + 4\partial X^I{\bar\partial}X^I -4 A(X^I)\partial
X^+\bar{\partial}X^+
  \nn \\ &&  \nn \\
&&{}~~~~~{} +8\,i \Big\{\partial X^+ ( S_1\bar{\partial} S_1 +
S'_1\bar{\partial}S'_1) + {\bar\partial} X^+(
      S_2\partial S_2 +  S'_2\partial S'_2)
     \nn \\ && \nn \\
&&\hskip2cm -({\textstyle\frac{\mu}{6}})\partial
X^+{\bar\partial}X^+ (2S_2 S_1 + S'_2S'_1)
      \Big\}\bigg]\,, \label{ELCA}
\eea
%
where $z = \sigma^1+i\sigma^2$ and ${\bar z}= \sigma^1-i\sigma^2$
with $dz d{\bar z} = 2d\sigma^1d\sigma^2$. For the calculation of
the one loop amplitude via the path integral approach in the
Green-Schwarz(GS) superstring, we should take into account the
boundary conditions for the GS fermions, $S^a$. They are spacetime
fermions and, thus, should be antiperiodic along the thermal
direction $X^0_{E}$ in the finite temperature formalism. Therefore
the boundary conditions for the fermions, consistent with the
periodicity conditions (\ref{bbc}), are~\cite{Carlip:1986cy}
%
\be S^a_1(z+2\pi ) = (-1)^{r}S^a_1(z)\,, \quad  S^a_1(z+2\pi\tau)
= (-1)^{l}S^a_1(z)\,, \label{FerBound}\ee
%
with the same conditions for $S'_1(z)$ and similar ones for
$S_2(\bar{z}), S'_2(\bar{z})$ under $\bar{z}$ periodicity.


The one loop free energy  can be represented as a path integral
over the gauge fixed action as
%
\bea - \beta F &=&  Z_{T^2}(\beta) = \int_{\cal F}\frac{d\tau_1
d\tau_2}{2\tau_2} \sum_{r,\, l \in\, {\bf Z}}^{} \int_{T^2/(r,\, l)} {\cal
D}X^+ {\cal D}X^- {\cal D}b {\cal D}c ~ {\cal
D}X^I {\cal D}S_{1,\,2}{\cal D}S'_{1,\,2} \nn \\ && \nn \\
&&\hskip3cm \times\, e^{-S_E-S_{gh}}\,(\det
\partial X^+)^{-4}\, (\det {\bar \partial} X^+)^{-4}\,,
\label{path-integral}\eea
%
where $(r,\, l)$ denotes the boundary conditions
Eq.~(\ref{FerBound}) for the fermions $S^a_{1,\,2},~
S'^{\,a}_{1,\,2}$ on the torus, and $S_{gh}$ is the same ghost
action as in the flat space case. ${\cal F}$ denotes the
fundamental region of the torus
%
\be {\cal F}:= \Big\{~~(\tau_1\,,\tau_2)~~; ~~~ -\frac{1}{2} <
\tau_1 \leq \frac{1}{2}\,, \qquad  \tau_2 > 0 \,, \qquad |\tau|
\geq 1 ~~~\Big\}\,. \ee
%
Note that the determinant in the above expression comes from the
$\kappa$ symmetry gauge fixing\footnote{It has been
argued~\cite{Carlip:1986cy} that it is not easy to incorporate
$\kappa$ symmetry in the path integral of GS superstring
rigorously. Nevertheless, the path integral defined as
(\ref{path-integral}) will be shown to be consistent with the
 operator formalism. }.

We perform the integration over the longitudinal directions
$X^{\pm}$ first. As usual, we separate the fields into the
classical part and the quantum fluctuation one as
%
\be X^{\pm} (z,\,\bar{z}) = X^{\pm}_{cl}+ \delta X^{\pm}\,, \qquad
X^{\pm}_{cl} = x^{\pm}_0 +
{\textstyle\frac{-i}{\sqrt{2}}}X_{wind}\,.\ee
%
The classical thermal winding configuration, $X^{\pm}_{cl}$, leads
to the action value
%
\be S_E [X^{\pm}_{cl}] = \frac{\beta^2 |r\tau-l|^2}{4\pi\a
\tau_2}=: S_{\beta}(r,\, l)\,,  \ee
%
and the path integrals over $X^{\pm}$ are given by those of
$\delta X^{\pm}$ around this configuration.


One may note that the computation of the path integral for the one
loop free energy is analogous to that of the one loop vacuum
amplitude in the GS superstrings. We first rescale the fermions to
have the canonical kinetic term as shown in~\cite{Kallosh:wv},
which can also be done for our pp-wave case as
in~\cite{Hammou:2002bf}. The Jacobian from this rescaling cancels
out the factor $\det (\bar{\partial} \delta X^+)\det (\partial
\delta X^+)$ from $\det (\bar{\partial} X^+)\det (\partial X^+)$.
Next, we integrate over $\delta X^-$ which leads to the delta
functional $\delta [\,
\partial {\bar\partial}\delta X^+\,]= \delta [\delta
X^+]/\det(\partial {\bar\partial})$. The integration over $\delta
X^+$ with this delta functional restricts $\delta X^+$ to be zero
in the integrand and, as usual, $1/\det(\partial {\bar\partial})$
term cancels out with the $bc$ ghost contribution except the zero
mode, $x^{\pm}_0$, contribution.

Hence, the path integrals over $X^{\pm}$ and Faddeev-Popov ghosts
 around the classical
configurations~(\ref{thermwind})
give~\cite{Polchinski:rq,atickwitten,Polchinski:zf}
%
\be \sum_{r,\, l \in\, {\bf Z}}^{} \frac{\beta
L}{4\pi^2\a\tau_2}{}~{} e^{-S_{\beta}(r,\, l)}~~\,, \ee
%
where $L$ is the (infinite) length of the longitudinal direction
used in the operator method given earlier.  In addition, $X^{\pm}$
fields in the residual integrand are replaced with $X^{\pm}_{cl}$.
Note that $\beta L/(4\pi^2\a\tau_2)$ comes from the path integrals
of the zero modes, $x^{\pm}_0$.


After the replacement of $X^{\pm}$ fields with their classical
values, $X^{\pm}_{cl}$, and the field independent rescaling of
fermions which gives the irrelevant numerical factor for the path
integral measure, the Euclidean action for the transverse fields,
$S'^E[X^I,\,S]$, becomes
%
\bea S'^E[X^I,\,S] &=&  \frac{1}{8\pi\a}\int d z d\bar{z} \bigg[
4\partial X^I{\bar\partial}X^I
+({\textstyle\frac{m'}{3}})^2(X^i)^2 +
({\textstyle\frac{m'}{6}})^2(X^{i'})^2
  \nn \\ &&  \nn \\
&&{}~~~{} + S_1\bar{\partial} S_1 + S'_1\bar{\partial}S'_1
-S_2\partial S_2 -  S'_2\partial S'_2
+({\textstyle\frac{m'}{3}})S_2 S_1 +
({\textstyle\frac{m'}{6}})S'_2S'_1
      \bigg]\,, \qquad
\eea
%
with
%
\be m' := \frac{\mu\beta}{2\sqrt{2}\pi\tau_2}|r\tau-l|\,.
\label{mprime}\ee
%
As a result, the one loop free energy for a closed string gas can
be written as
%
\be - \beta F =  Z_{T^2}(\beta) = \int_{\cal F}\frac{d\tau_1
d\tau_2}{2\tau_2} \sum_{r,\, l \in\, {\bf Z}}^{} \frac{\beta
L}{4\pi^2\a\tau_2}{}~{} e^{-S_{\beta}(r,\, l)}{}~{}
Z_{trans}^{(r,\, l)}(\tau_1,\tau_2) \,,\label{pathfree} \ee
%
where $Z_{trans}^{(r,\, l)}(\tau_1,\tau_2)$ denotes the path
integrals over the transverse fields:
%
\be Z_{trans}^{(r,\, l)}(\tau_1,\tau_2) := \int_{T^2/(r,\, l)}
{\cal D} X^I{\cal D}S^a_{1,\,2}{}~~{} e^{-S'^E[X^I,\,S]}\,.
\label{path}\ee
%
Note that $(r,\,l)=(0,0)$ part of the above free energy expression
corresponds to the one loop vacuum amplitude, which is zero in our
path integral formulation in parallel with type IIB 
case~\cite{Hammou:2002bf}.
\newline



To get the transverse partition function~(\ref{path}),  it will be
convenient~\cite{Alvarez:1985fw,Alvarez:1986sj} to use the new
coordinates, $\xi^1,\xi^2$, defined by $z=\xi^1 + \tau\xi^2,\,
\bar{z}=\xi^1 + {\bar\tau}\xi^2$, with $dz d\bar{z} = d^2\xi{}~{}
2\tau_2$ and
$\partial\bar{\partial}=(1/2\tau_2)^2[|\tau|^2\partial_1^2
+\partial_2^2 -2\tau_1\partial_1\partial_2]$. The various fields
$X^i,X^{i'},S^a_{1,\,2},S^{'\,a}_{1,\,2}$ on the torus are Fourier
expanded as
%
\bea  X(\xi_1,\xi_2) &=& \sum_{n_1,n_2}x_{n_1,n_2} {}~{}e^{
in_1\xi_1 +  i n_2\xi_2}\,,
 \\       &&              \nn     \\
S(\xi_1,\xi_2)&=& \sum_{n_1,n_2}S_{n_1,n_2}
\exp\bigg[ i\Big(n_1+{\textstyle\frac{1-(-1)^{r}}{4}}\Big)\xi_1 +
i \Big(n_2+{\textstyle\frac{1-(-1)^{l}}{4}}\Big)\xi_2\bigg]\,,
\eea
%
and the eigenvalues of those fields are given by
%
\bea  && (-4\partial{\bar\partial}+M^{\,2})X =
E_{n_1,n_2}X\,;\quad
E_{n_1,n_2}=\Big(\frac{1}{\tau_2}\Big)^2|n_1\tau-n_2|^2
+ M^{\,2}\,, \nn\\
&&\nn\\ &&(-4\partial{\bar\partial}+M^{\,2})S =
E^{(r,\,l)}_{n_1,n_2}S\,;\quad
E^{(r,\,l)}_{n_1,n_2}=\Big(\frac{1}{\tau_2}\Big)^2|
(n_1+{\textstyle\frac{1-(-1)^{r}}{4}})\tau
-(n_2+{\textstyle\frac{1-(-1)^{l}}{4}})|^2+M^{\,2}\,, \nn\eea
%
where $M$ denotes $\frac{m'}{3}$ or $\frac{m'}{6}$.





The above type of path integrals was calculated in
\cite{Itzykson}. We now redo that calculation in our context and
clarify several points. Since the generalization to other cases
are straightforward, it will be sufficient to consider the path
integral of a massive complex bosonic field $\phi$ of mass $M$
with the twisted boundary condition
$\phi(\xi_1+2\pi,\xi_2)=e^{2\pi ib_1}\phi(\xi_1,\xi_2),\,
\phi(\xi_1,\xi_2+2\pi)=e^{2\pi ib_2}\phi(\xi_1,\xi_2)$. The energy
eigenvalues are, then, given by
%
\be E^{(b_1,b_2)}_{n_1,n_2}=\Big(\frac{1}{\tau_2}\Big)^2|
(n_1+b_1)\tau -(n_2+b_2)|^2+M^{\,2}\,,\ee
%
and the partition function becomes
%
\be Z_{\phi}(M):=\int {\cal D}\phi{\cal D}\bar{\phi} e^{-\int
d^2\xi \tau_2 \bar{\phi}(-4\partial{\bar\partial}+M^2)\phi} =
\prod_{n_1,n_2}(\tau_2E^{(b_1,b_2)}_{n_1,n_2})^{-1}\,, \ee
%
where we have absorbed the overall constant in the action by the
field redefinition of $\phi$. As usual, this determinant can be
regulated and evaluated using the $\zeta$-function method by
considering
%
\be G(s) := \sum_{n_1,n_2\in {\bf Z}}\left(
\frac{\mu_R^2}{\tau_2E^{(b_1,b_2)}_{n_1,n_2}}\right)^s\,, \qquad
{\rm Re}{}~{} s  > 1 \,,\label{zetaG}\ee
%
where we have introduced the renormalization scale $\mu_R$ to
insure the function $G(s)$ dimensionless in the $\zeta$-function
scheme. As is well-known, this corresponds to the introduction of
the cosmological constant term in the string worldsheet action as
a counter term. Note also that we have included the factor
$\tau_2$ which comes from the Lagrangian measure of the coordinate
transformation from $(z,\bar{z})$ to $(\xi^1,\xi^2)$. As was
pointed out in \cite{Itzykson}, it is required to perform the
ultraviolet renormalization in the above determinant
expression\footnote{In \cite{Itzykson}, ``specific heat'' is
treated as the quantity for the renormalization prescription.},
and our insertion of the renormalization scale, $\mu_R$, takes
care of that. By the analytic continuation of $s$, the
renormalized determinant can be evaluated through
%
\be e^{G^{\prime}(0)}=
\prod_{n_1,n_2}\frac{\mu^2_R}{\tau_2E^{(b_1,b_2)}_{n_1,n_2}}\,,
\qquad G^{\prime}(s) := \frac{d}{d s}G(s)\,.\ee
%
It is sufficient to consider $G(s)$ up to the linear order in $s$
to get $G'(0)$. Denoting $a = (n_1 + b_1)\tau_1 - b_2$ and $c =
\tau_2\sqrt{(n_1 + b_1)^2+M^2}$, we get
%
\bea G(s) &=& \sum_{n_1,n_2\in {\bf Z}}
\frac{(\mu^2_R\tau_2)^s}{[(n_2 + a)^2 + c^2]^s} \\ && \nn \\
&=& (\mu^2_R\tau_2)^s\sum_{n_1\in {\bf Z}}
\bigg[\sqrt{\pi}c^{1-2s}\frac{\Gamma(s-\half)}{\Gamma(s)} +
\frac{4\pi^s c^{\half-s}}{\Gamma(s)}
\sum_{p=1}^{\infty}\frac{\cos(2\pi a
p)}{p^{\half-s}}K_{\half-s}(2\pi c p)\bigg]\,,\nn\eea
%
where we have used the formula in the Appendix, (\ref{BesselK}),
for $n_2$ summation. Using the formula~(\ref{BesselK}) once again,
we can see that the first term in the sum can be expanded in terms
of $s$ as
%
\bea
\frac{\Gamma(s-\frac{1}{2})}{\Gamma(s)}\sum_{n_1}\frac{\sqrt{\pi}}{c^{2s-1}}
&=& \pi \frac{\Gamma(s-1)}{\Gamma(s)} \tau_2^{1-2s} M^{2-2s} +
\frac{4\pi^s \tau_2^{1-2s} M^{1-s}}{\Gamma(s)} \sum_{p=1}^{\infty}
\frac{\cos (2\pi b_1 p)}{p^{1-s}} K_{1-s}(2\pi M p)\,, \nn \\
 && \nn \\
 &=&  \pi\tau_2M^2+ s{}~{} \bigg[\pi\tau_2 M^2 \Big( -1 +
\ln M^2\tau_2^2\Big) - 4\pi\tau_2\Delta_{b_1}(M)\bigg]+ {\cal
O}(s^2)\,,\nn \eea
%
where $\Delta_b(m)$, the function defined by~(\ref{delta}),
appears naturally in this approach. The second term inside the sum
is also expanded as
%
\be   \frac{4\pi^s c^{\half-s}}{\Gamma(s)}
\sum_{p=1}^{\infty}\frac{\cos(2\pi a
p)}{p^{\half-s}}K_{\half-s}(2\pi c p) = s\,\bigg[ 4\, c^{\half}
\sum_{p=1}^{\infty}\frac{\cos(2\pi a p)}{p^{\half}}K_{\half}(2\pi
c p)\bigg] + {\cal O}(s^2)\,. \ee
%
To proceed, note that $K_{\half}(z) = \sqrt{\frac{\pi}{2z}}\,
e^{-z}$ and
%
\be 2 \sum_{p=1}^{\infty}\frac{\cos(2\pi a p)}{p}
        e^{-2 \pi c p}
= \sum_{p=1}^{\infty} \frac{1}{p}(q^p + \bar{q}^p) = -2|\ln(1-q)|
\,, \ee
%
where $q = e^{2\pi i a-2\pi c}= e^{ 2\pi i (n_1 + b_1)\tau_1 -
2\pi\tau_2
          \sqrt{(n_1+b_1)^2+M^2} - 2\pi i b_2 }$.
Finally,
 with $(\mu^2_R\tau_2)^s = 1 + s\,
\ln(\mu^2_R\tau_2)+{\cal O}(s^2)$,  we obtain
%
\bea G'(0) &=& - \pi \tau_2M^2  + \pi
       \tau_2 M^2\ln(\frac{M^2\tau_2}{\mu^2_R}) -
       4\pi\tau_2 \Delta_{b_1}(M)      \nn \\
       & &{}~~{}- 2\ln\bigg|\prod^{\infty}_{n_1=-\infty} \Big( 1 -
       e^{ 2\pi i (n_1+b_1) \tau_1 - 2\pi \tau_2
       \sqrt{(n_1+b_1)^2 + M^2} - 2\pi i b_2 } \Big) \bigg|\,.
\eea
%

In all, the partition function for the massive complex scalar
field $\phi$ is given by
%
\be
 Z_{\phi}(M) = \left[e^{2\pi\tau_2\Delta^R_{b_1}(M)}\bigg|\prod_{n_1}
\Big(1-e^{ 2\pi i (n_1+b_1) \tau_1 - 2\pi \tau_2
       \sqrt{(n_1+b_1)^2 + M^2} - 2\pi i b_2 }\Big)\bigg|\right]^{-2}\,,
       \ee
%
where
%
\be \Delta^R_{b_1}(M) = \Delta_{b_1}(M) - \quarter
M^2\Big[\ln\frac{M^2\tau_2}{\mu^2_R} -1\Big]\,.\ee
%
The above $\Delta^R_{b_1}(M)$ is the so-called zero point energy
(or Casimir energy) on the worldsheet. But we emphasize that this
is the renormalized quantity, not the physical one in the context
of the renormalizable quantum field theory~\cite{Collins:xc}. To
get the physical quantity we need to impose the renormalization
condition. We will take it by the criterion
%
\be \Delta_b^{ph}(M)\longrightarrow 0\,, \qquad  {\rm when}\quad
M\longrightarrow\infty\,, \label{rencond}\ee
%
which is the standard choice in the Casimir energy
literature~\cite{Bordag:2001qi} (see
also~\cite{Santangelo:2000fj,Milton:1999ge}). This consideration
finally gives us the physical quantity
$\Delta_{b_1}^{ph}(M)=\Delta_{b_1}(M)$, and it will be economical
to write the partition function only with the physical zero point
energy as
%
\be Z_{\phi}(M) =
\left[e^{2\pi\tau_2\Delta_{b_1}(M)}\bigg|\prod_{n_1} \Big(1-e^{
2\pi i (n_1+b_1) \tau_1 - 2\pi \tau_2
       \sqrt{(n_1+b_1)^2 + M^2} - 2\pi i b_2 }\Big)\bigg|\right]^{-2}\,\,.
\label{bosonZ}\ee
%
\newline

As a special case, the partition function of the massive real
scalar field $X$ is given by
%
\be Z_{X}(M) = \left[e^{2\pi\tau_2\Delta_0(M)}\prod_n
\Big(1-e^{-2\pi\tau_2|\omega_n|+2\pi i \tau_1
n}\Big)\right]^{-1}\,. \ee
%
Although GS fermions are not worldsheet fermions, in the path
integral formalism they are treated as Grassmann variables only
with the different boundary conditions from those of worldsheet
fermions. Hence, with a change according to the nature as the
Grassmann variables, the direct application of the above result
for a complex boson to GS fermions, $S^a$, leads to
%
\bea Z_{S}^{(r,\,l)}(M) &=& e^{2\pi\tau_2\Delta_{b_1}(M)}\prod_n
\Big(1-e^{-2\pi\tau_2\sqrt{(n+b_1)^2+M^2}+2\pi i \tau_1 (n +
b_1)-2\pi i b_2 }\Big) \nn \\ && \nn \\ &\equiv&
D_{b_1,\,b_2}(\tau_1,\tau_2;M)~~~\,; \qquad b_1 =
\frac{1-(-1)^r}{4}\,, \quad b_2 = \frac{1-(-1)^l}{4}\,.
\label{fermionZ} \eea
%
This justifies the form of $D_{b_1,\,b_2}(\tau_1,\tau_2;M)$ with
$\Delta_{b_1}(m)$ introduced in a somewhat ``ad hoc'' way
in~(\ref{Dbb}). Moreover, resurrecting the full arguments of
$G(s)$ as $G_{b_1,\, b_2}(s;\, \tau_1,\tau_2,M)$, one can observe
the following modular property of $G(s)$ from the translational
symmetry by integer steps in the infinite double summation over
$n_1$ and $n_2$
%
\be G_{b_1,\, b_2}\Big(s;\, \tau_1,\tau_2,M\Big)= G_{b_2,\,
-b_1}\Big(s;\,
-{\textstyle\frac{\tau_1}{|\tau|^2}},{\textstyle\frac{\tau_2}{|\tau|^2}}
,M|\tau|\Big)= G_{b_1,\, b_2+b_1}\Big(s;\,
\tau_1+1,\tau_2,M\Big)\,. \ee
%
Note that we can always take $0\le b_{1,\,2} < 1$ by a suitable
translation. These properties are transferred to $G'(0)$ which
explains the modular property of $D_{b_1,\,b_2}(M)$
in~(\ref{modular}).

After all these, the transverse partition function is given by
%
\bea Z_{trans}^{(r,\,l)}(\tau_1,\tau_2) &=&
\Big[Z_{X}({\textstyle\frac{m'}{3}})\,
Z_{S}^{(r,\,l)}({\textstyle\frac{m'}{3}}) \Big]^4 \Big[
Z_{X}({\textstyle\frac{m'}{6}})\,Z_{S}^{(r,\,l)}
({\textstyle\frac{m'}{6})}
\Big]^4 \nn \\
&& \label{transR} \\
&=&\bigg[\frac{D_{b_1,\,b_2}(\tau_1,\tau_2;\frac{m'}{3})}
{D_{0,0}(\tau_1,\tau_2;\frac{m'}{3})}\bigg]^4
\bigg[\frac{D_{b_1,\,b_2}(\tau_1,\tau_2;\frac{m'}{6})}
{D_{0,0}(\tau_1,\tau_2;\frac{m'}{6})}\bigg]^4 \,. \nn\eea
%
This transverse partition function has the following modular
property, as can be seen from that of $G(s)$,
%
\be Z_{trans}^{(r,\,l)}(\tau_1,\tau_2) = Z_{trans}^{(l,\,-r)}
(-{\textstyle\frac{\tau_1}{|\tau|^2}},
{\textstyle\frac{\tau_2}{|\tau|^2}}) = Z_{trans}^{(r,\,l+r
)}(\tau_1+1,\tau_2)\,. \label{Zmod}\ee
%


\section{Equivalence between operator and path integral methods}

In this section we find the complete agreement between operator
and path integral approaches. Apparently, two approaches give the
different integration and summation regions in the free energy
expression. However, as the flat space case, these two forms of
the free energy can be shown to be equivalent by the resummation
with the change of integration region. Before going ahead, a
couple of comments are in order.

First, note that we need the renormalization prescription (like
Eq.~(\ref{rencond})) in the path integral approach while it is
already done implicitly in the operator method. The normal
ordering prescription in the operator approach is a kind of
renormalization prescription for free fields, but it is not
incorporated in the path integral.


Second, the factor $\tau_2$ in Eq.~(\ref{zetaG}) was not
explicitly included in \cite{Itzykson}, contrary to the string
theory
literatures~\cite{Alvarez:1985fw,Alvarez:1986sj,PandoZayas:2002hh}.
Nevertheless, we should have the same physical quantity because it
comes from the Lagrangian measure of the coordinate
transformation. This can be understood from the fact that the
physical quantity we will get after imposing the renormalization
condition is independent of this factor (and, of course, the
renormalization scale, $\mu_R$). This is also obvious from the
fact that we can take the new renormalization scale as
$\tilde{\mu} = \mu_R/\sqrt{\tau_2}$ in Eq.~(\ref{zetaG}).

Third, the $\zeta$-function method gives us the regularized form
of the zero point energy directly, and this gives the identical
result with the one from the standard method for the zero point
(or Casimir) energy
%
\be E^{Casimir}_b(m) = \half
\sum_{n=-\infty}^{\infty}\sqrt{(n+b)^2+m^2}- \half
\int^{\infty}_{-\infty} d k \sqrt{(k+b)^2+m^2}\,,\ee
%
which satisfy the criterion $E^{Casimir}_b(m)\rightarrow 0$ when
$m\rightarrow \infty$. Though this $E^{Casimir}_b(m)$ is identical
with $\Delta_b(m)$, we do not need this regularization as a
separate treatment in the operator method or the path integral one
because we have already fixed the renormalization prescription in
the former one by the normal ordering and in the latter one by the
$\zeta$-function with a suitable renormalization condition.

Fourth, since the Hagedorn temperature dependence on the RR flux
is entirely given by the difference of bosons and fermions zero
point energy as indicated in Eq.~(\ref{Hagedorn}), it does not
depend on whether we use the renormalized zero point energy or the
physical one\footnote{The previous numerical
mismatch~\cite{PandoZayas:2002hh,semenoff} for the Hagedorn
temperature in IIB strings on pp-wave comes from the incorrect
series expansion for the zero point
energy~\cite{PandoZayas:2002hh} and it has no relation with the
formalism or the renormalization.}.

Fifth, while $Z_{X}(M)\,Z_{S}^{(0,\,0)}(M)=1$ for non-zero $M$ as
can be seen from (\ref{bosonZ}) and (\ref{fermionZ}),
$Z_{X}(0)\,Z_{S}^{(0,\,0)}(0)=0$ by the fermion zero modes. This
shows that the $M\rightarrow 0$ limit is not smooth in the
transverse partition function in our formulation.
\newline




To get the one loop free energy expression, we can also make use
of the Coleman-Weinberg
formula~\cite{Polchinski:rq,Polchinski:zf}, which results in
%
\be - \beta F =  Z_{T^2}(\beta) = -i\int_{\cal F}\frac{d\tau_1
d\tau_2}{2\tau_2} \sum_{r,\, l \in\, {\bf Z}}^{} \frac{\beta
L}{(2\pi)^2}\int d p^{+} dp^{-}{}~{} e^{2\pi\tau_2\a
p^{+}p^{-}}{}~{} e^{-S_{\beta}(r,\, l)}{}~{} Z_{trans}^{(r,\,
l)}(\tau_1,\tau_2)\,.\label{freeCW} \ee
%
Recalling the Wick rotation $p^{0}=ip^{0}_{E}$ and $p^{\pm}
=(p^{0}\pm p^{9})/\sqrt{2}$, one can see that this is identical
with the previous one loop free energy~(\ref{pathfree}) through
%
\be \frac{\beta L}{(2\pi)^2}\int d p^{+} dp^{-}{}~{}
e^{2\pi\tau_2\a p^{+}p^{-}} = i \frac{\beta L}{4\pi^2\a\tau_2} \,.
\label{factorL}\ee
%
Note that the transverse partition function can also be
represented as an operator trace~\cite{Sugawara:2002rs}
%
\be Z_{trans}^{(r,\, l)}(\tau_1,\tau_2)  \equiv
 \tr_{trans}\Big[ (-1)^{(l+1){\bf F}} e^{-2\pi\tau_2
 {\cal H}(r)+2\pi i \tau_1 {\cal P}}\Big]\,,\ee
%
where $ {\cal H}(r)$ is defined by the same form as~(\ref{lcham})
while the integer $n$ in $\omega_n$ and $\omega'_n$ is replaced by
$n+\frac{1-(-1)^r}{4}$.


The naive application to $(r,\,l)=(0,0)$ mode of the free energy
expression~(\ref{freeCW}) with the above operator form of the
transverse partition function gives
%
\bea - \beta F &=&  Z_{T^2}(\beta) \\ && \nn \\
&=& \int_{\cal F}\frac{d\tau_1 d\tau_2}{2\tau_2} \frac{-i\beta
L}{(2\pi)^2}\int d p^{+} d p^{-}{}~{}e^{2\pi\tau_2\a
p^{+}p^{-}}{}~~{}
 \tr_{trans}\Big[ (-1)^{{\bf F}} e^{-2\pi\tau_2
 {\cal H}+2\pi i \tau_1 {\cal P}}\Big]\,.\nn \eea
%
After $\beta$ is regarded as another longitudinal length, this is
identical with the one loop vacuum amplitude
in~\cite{Takayanagi:2002pi} except for the details in our IIA
setting. By following the same step as given
in~\cite{Takayanagi:2002pi,Hammou:2002bf} or by closely examining
the construction given here, one can show the modular invariance
of the closed string one loop vacuum amplitude in our IIA strings,
 which is one of the consistency check of the
given metric as a string theory background. Apparently, it seems
to be divergent, which, we believe, is the artifact of the
light-cone gauge $X^{+}\sim \sigma^{0}$ in the operator formalism
because the same quantity without the light-cone gauge fixing in
the path integral formalism was shown to be zero. Since our
background is consistent and globally supersymmetric, the one loop
vacuum amplitude should be zero by the cancellation of the bosonic
and fermionic vacuum energies. Therefore, we can  simply ignore
$(r,\,l)=(0,0)$ term in the above expression of one loop free
energy.





We return to the one loop free energy expression~(\ref{pathfree})
with~(\ref{transR}) in the path integral approach, which becomes
%
\be  - F = \int_{\cal F}\frac{d\tau_1 d\tau_2}{2\tau_2}\!
\sum_{r,\, l\in{\bf Z}} \frac{L}{4\pi^2\a\tau_2}{}~{} e^{-S_\beta
(r,\, l)} \bigg[\frac{D_{b_1,\,b_2}(\tau_1,\tau_2;\frac{m'}{3})}
{D_{0,0}(\tau_1,\tau_2;\frac{m'}{3})}\bigg]^4
\bigg[\frac{D_{b_1,\,b_2}(\tau_1,\tau_2;\frac{m'}{6})}
{D_{0,0}(\tau_1,\tau_2;\frac{m'}{6})}\bigg]^4\,, \label{freeF}\ee
%
where  $m'$ and $b_{1,2}$ are defined in (\ref{mprime}) and
(\ref{fermionZ}), respectively.
%
The above free energy is written as the integral over the
fundamental region of the torus, but through the explicit
resummation of $r$ and $l$ it can be rewritten over the half strip
region, $E$, given in the operator method, with letting $r=0$
%
\be E := \Big\{~~(\tau_1\,,\tau_2)~~; ~~~ -\frac{1}{2} \leq \tau_1
\leq \frac{1}{2}\,, \qquad  \tau_2 \geq 0 \,, ~~~\Big\}\,. \ee
%
The resummation with the change of the integration region for flat
space case~\cite{O'Brien:pn,McClain:1986id} can be summarized as
%
\be \sum^{r,\,l\in {\bf Z}}_{(r,\,l)\neq (0,\,0)}\int_{\cal
F}\frac{d\tau_1d\tau_2}{2\tau_2}{}~{}e^{-S_\beta (r,\, l)} \quad =
~\sum^{ l\in {\bf Z}}_{ l\neq
0}\int_{E}\frac{d\tau_1d\tau_2}{2\tau_2}{}~{}e^{-S_\beta (0,\,
l)}\,. \ee
%
Note that since the transverse partition function
$Z^{(r,\,l)}_{trans}(\tau_1,\tau_2)$ has the same modular
property, given in~(\ref{Zmod}), as $e^{-S_{\beta}(r,\,l)}$, the
above resummation with the change of the integration region can be
applied to our case as well. Moreover, since $(r,\,l)=(0,0)$ term
has no contribution, it does not have any effect in the
resummation with the change of the integration region.


As $m'$ becomes $m$ for $r=0$, the free energy can be written as
%
\bea \!\!\!\!\!\!\!\!\!\!\!\!\!\!\! - F
 &=&
\int_{E}\frac{d\tau_1d\tau_2}{2\tau_2}\frac{L}{4\pi^2\a\tau_2}
\Bigg\{ \sum^{l\in 2{\bf Z}}_{l \neq 0} {}~{}e^{-\frac{\beta^2
l^2}{4\pi\a \tau_2}}{}~~{} +  \\
\!\!\!\!\!\!\!\!\!\!&& \nn \\
&& \hskip3.25cm \sum^{l \in 2{\bf Z}+1}_{l}{}~{}e^{-\frac{\beta^2
l^2}{4\pi\a \tau_2}}
\bigg[\frac{D_{0,\,1/2}(\tau_1,\tau_2;{\textstyle\frac{m}{3}})}
{D_{0,\,0}(\tau_1,\tau_2;{\textstyle\frac{m}{3}})}\bigg]^4
 \bigg[\frac{
D_{0,1/2}(\tau_1,\tau_2;{\textstyle\frac{m}{6}})}
{D_{0,0}(\tau_1,\tau_2;{\textstyle\frac{m}{6}})}\bigg]^4\Bigg\}\,.
\nn\eea
%
Note that this is identical with the result from the operator
method given in Eq.~(\ref{Free}).

After the ${\cal S}$-modular transformation,
$$e^{-S_{\beta}(r,\,l)}Z_{trans}^{(r,\,l)}(\tau_1,\tau_2) =
e^{-S_{\beta}(l,\,-r)}
Z_{trans}^{(l,\,-r)}(-\frac{\tau_1}{|\tau|^2},\frac{\tau_2}{|\tau|^2}
)\,,$$ which exchanges the winding and momentum modes, we can
perform the resummation with the change of integration region.
This form, which contains the zero point energy difference leading
to the Hagedorn temperature, gives  the one loop free energy in
terms of the winding mode summation and shows  that the winding
string becomes tachyonic at the Hagedorn
temperature~\cite{atickwitten,Sugawara:2002rs}.





Now, by taking the correct $m\rightarrow 0$ limit, we can check
the free energy expression with the one obtained in the RNS
formalism~\cite{atickwitten} without difficulty. Recalling the
correct zero mass limit $Z^{(0,\,0)}_S(0)=0,\,$ $[Z_X(0)]^8=
V_8/(4\pi^2\a \tau_2)^4|\eta(\tau)|^{16} $ and
%
\be
 D_{0,\,1/2}(\tau_1,\tau_2;0)
     = \bigg|\frac{\theta_2 (\tau)}{\eta(\tau)}\bigg|\,,\quad
     D_{1/2,\,0}(\tau_1,\tau_2;0)
     = \bigg|\frac{\theta_4 (\tau)}{\eta(\tau)}\bigg|\,,\quad
 D_{1/2,\,1/2}(\tau_1,\tau_2;0) =
 \bigg|\frac{\theta_3 (\tau)}{\eta(\tau)}\bigg|\,,
 \ee
%
one can see that the one loop free energy becomes
%
\bea -F &=& L\,V_8 \int_{\cal F}\frac{d\tau_1
d\tau_2}{2\tau_2}\bigg( \frac{1}{4\pi^2\alpha'\tau_2} \bigg)^5
\bigg|\frac{1}{\eta(\tau)}\bigg|^{24} \bigg\{ \sum_{r: even \atop
l: odd}
e^{-S_{\beta}(r,\,l)} |\theta_2(\tau)|^8\nn \\ && \nn \\
&& {}~~~{}\hskip1.5cm
 + \sum_{r: odd\atop l: even}
e^{-S_{\beta}(r,\,l)} |\theta_4(\tau)|^8  + \sum_{r: odd\atop l:
odd} e^{-S_{\beta}(r,\,l)}| \theta_3(\tau)|^8 \bigg\}\,.\eea
%
Using the Jacobi identity $\theta_3^{\,4}(\tau) -
\theta_2^{\,4}(\tau) -\theta_4^{\,4}(\tau)=0$, this can be put
into the form given in~\cite{atickwitten}
%
\bea \!\!\!\!\!\!-\frac{F}{V} &=& \frac{1}{4}\int_{\cal
F}\frac{d\tau_1 d\tau_2}{2\tau_2} \bigg(
\frac{1}{4\pi^2\alpha'\tau_2} \bigg)^5
\bigg|\frac{1}{\eta(\tau)}\bigg|^{24}
 \sum_{r,\,l\in {\bf
Z}}e^{-S_{\beta}(r,\,l)} \bigg\{(\theta_2^{\,4}\bar{\theta}_2^{\,
4}+\theta_3^{\,4}\bar{\theta}_3^{\, 4} +
 \theta_4^{\,4}\bar{\theta}_4^{\, 4}) \nn \\ \!\!\!\!\!\!&& \nn \\
\!\!\!\!\!\! && {}~{} +{}~{} e^{\pi
i(r+l)}(\theta_2^{\,4}\bar{\theta}_4^{\, 4}+
\theta_4^{\,4}\bar{\theta}_2^{\, 4}) -  e^{\pi
i\,r}(\theta_2^{\,4}\bar{\theta}_3^{\, 4}+
\theta_3^{\,4}\bar{\theta}_2^{\, 4}) -  e^{\pi i\,
l}(\theta_3^{\,4}\bar{\theta}_4^{\, 4}+
\theta_4^{\,4}\bar{\theta}_3^{\, 4}) \bigg\}\,, \eea
%
where $V = L\,V_8 $.

\section*{Acknowledgments}
One of us (J.-D. P.) would like to thank the Yonsei Visiting
Research Center (YVRC) for its hospitality, where this work has
been completed. The work of S.H. was supported in part by grant
No. R01-2000-00021 from the Basic Research Program of the Korea
Science and Engineering Foundation.


%\newpage


\appendix

\begin{center}
\large{\textbf{Appendix}}
\end{center}
\setcounter{equation}{0}
\renewcommand{\theequation}{A.\arabic{equation}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\section{Useful Formulae}

In this section, we derive, following the methods in
Ref.~\cite{Ambjorn:1981xw}, the series expansion formula of the
function, $\Delta_b(m)$, defined by Eq.~({\ref{delta}) in the more
general setting than that in Ref.~\cite{semenoff}. Since this
expansion can cover the more general twisted boundary conditions,
it may be useful for studying the small $\mu$ dependence of the
Hagedorn temperature in orbifolded or DLCQ pp-wave case.

Firstly, let us consider the following function
%
\be {\cal S}(s)_{(b,m)} :=
\sum_{n=-\infty}^{\infty}\frac{1}{[(n+b)^2+m^2]^s}\,, \qquad {\rm
Re}(s) > \frac{1}{2} \,. \label{Sfunction}\ee
%
Note that we can take $0\le b <1$ without loss of the generality
from the translational symmetry. This function can be represented
through the Schwinger's proper time parametrization, $1/z^s
=\int^{\infty}_0 dt\, t^{s-1}\, e^{-tz}/\Gamma(s)$, as
%
\bea {\cal S}(s)_{(b,m)} &=&
\frac{1}{\Gamma(s)}\int_0^{\infty}dt{}~{}t^{s-1}{}~{}
e^{-tm^2}\sum_{n=-\infty}^{\infty}e^{-t(n+b)^2} \nn \\ &&\nn \\
&=&\frac{\sqrt{\pi}}{\Gamma(s)} \left[
\int_0^{\infty}dt{}~{}t^{(s-\half)-1}{}~{} e^{-tm^2} +
2\sum_{p=1}^{\infty}\cos(2\pi b p)
\int_0^{\infty}dt{}~{}t^{(s-\half)-1}{}~{}
e^{-tm^2-\frac{\pi^2p^2}{t}}\right]
\nn \\ && \nn \\
&=&\sqrt{\pi}m^{1-2s}\frac{\Gamma(s-\half)}{\Gamma(s)} +
\frac{4\pi^sm^{\half-s}}{\Gamma(s)}
\sum_{p=1}^{\infty}\frac{\cos(2\pi b
p)}{p^{\half-s}}K_{\half-s}(2\pi m p)\,, \label{BesselK}\eea
%
where we have used the Poisson resummation formula in the second
line and the integral representation of the modified Bessel
function $K_\nu(z)$ for the second term in the third line
%
\bea && \sum_{n=-\infty}^{\infty}e^{-t(n+b)^2} =
\sqrt{\frac{\pi}{t}}
\sum_{p=-\infty}^{\infty}e^{-\frac{\pi^2}{t}p^2+2\pi i b p} =
\sqrt{\frac{\pi}{t}}\Big[1+2\sum_{p=1}^{\infty}
e^{-\frac{\pi^2}{t}p^2}\cos(2\pi
bp)\Big]\,, \nn \\ && \nn \\
 &&{}~~{} K_{\nu}(z) = \half
{\textstyle(\frac{z}{2})}^{\nu}\int_0^{\infty}dt{}~{}t^{-\nu-1}
{}~{}e^{-t-\frac{z^2}{4t}}\,. \nn \eea
%

Next, note that ${\cal S}(s)_{(b,m)}$ can also be expanded as a
power series in terms of $m$ as,
%
\bea {\cal S}(s)_{(b,m)} & = &\left\{\ba{ll} \frac{1}{m^{2s}} +
2{\displaystyle\sum_{n=1}^{\infty}}
\frac{1}{n^{2s}}[1+\frac{m^2}{n^2}]^{-s}\,, & {}~~~{}b=0\,,
\\ & \nn \\
{\displaystyle\sum_{n=0}^{\infty}}\frac{1}{(n+b)^{2s}}
\Big[1+\frac{m^2}{(n+b)^{2}}\Big]^{-s}+
{\displaystyle\sum_{n=0}^{\infty}} \frac{1}{(n+1-b)^{2s}}
\Big[1+\frac{m^2}{(n+1-b)^{2}}\Big]^{-s}\,, & {}~~~{} 0 < b < 1\,,
\ea\right.
  \\ &&\nn \\
&=& \left\{\ba{ll}\frac{1}{m^{2s}}+2\zeta(2s)-2s\zeta(2s+2)m^2 +
2{\displaystyle\sum_{k=2}^{\infty}}
\frac{(-1)^k\Gamma(k+s)}{\Gamma(s)\Gamma(k+1)}\zeta(2k+2s)m^{2k}
\,, & {}~~~{} b=0\,, \\ & \\
\zeta(2s,b)+\zeta(2s,1-b)-s\Big[\zeta(2s+2,b)+\zeta(2s+2,1-b)\Big]m^2
&  \\ {}~~~~~{}+ {\displaystyle\sum_{k=2}^{\infty}}
\frac{(-1)^k\Gamma(k+s)}{\Gamma(s)\Gamma(k+1)}
\Big[\zeta(2k+2s,a)+\zeta(2k+2s,1-a)\Big] m^{2k} \,,  & 0<b<1\,,
\ea\right. \label{Series}\eea
%
where $\zeta(s)$ and $\zeta(s,q)$ are the (Riemann) zeta function
and the generalized (or Riemann-Hurwitz) zeta function,
respectively, defined as
%
\be \zeta(s) := \sum_{n=1}^{\infty}\frac{1}{n^s}\,,\qquad
\zeta(s,q) := \sum_{n=0}^{\infty}\frac{1}{(n+q)^s}\,,\quad q\neq
0,-1,-2,\cdots~. \ee
%
Some useful formulae for them are
%
\bea & \!\!\!\!\!\! \zeta(s)=\zeta(s,1)\,, & \qquad \zeta(s,\half)
 =(2^s-1)\zeta(s)\,, \nn \\ &  &\label{zetapro} \\
&{}~~~~~{} \zeta(-1,q) = \frac{1}{24}-\frac{1}{8}(2q-1)^2\,, &
\qquad \zeta(s,q) = \frac{1}{s-1} -\psi(q) + {\cal O}(s-1)\,,
 \nn\eea
%
where $\psi(q):=\frac{d}{dq}\ln\Gamma(q)$ with
$\psi(1)=-\gamma_E=-0.5772 ...$ and $\psi(1/2)=-\gamma_E-2\ln2$.
Here, $\gamma_E$ is the Euler constant.


 Now, we identify Eq.(\ref{BesselK}) with
Eq.(\ref{Series}) for sufficiently large and positive $s$-values,
and perform the analytic continuation to the entire $s$-complex
plane. Then, we can see that the function, $\Delta_b(m)$, can be
represented as
%
\be 2\Delta_b(m)\equiv -\frac{2m}{\pi}
\sum_{p=1}^{\infty}\frac{\cos(2\pi b p)}{p}K_{1}(2\pi m p) =
\lim_{s\rightarrow -\half}^{}\left[{\cal
S}(s)_{(b,m)}-\sqrt{\pi}m^{1-2s}
\frac{\Gamma(s-\half)}{\Gamma(s)}\right]\,. \label{ancon}\ee
%
When we take $s\rightarrow -\half$ in the above, the would-be pole
part $1/(s+\half)$ in the series form of ${\cal S}(s)_{(b,m)}$,
which comes from $\zeta(2s+2)$ or $\zeta(2s+2,b)+\zeta(2s+2,1-b)$
at $m^2$ order in Eq.~(\ref{Series}) , cancels out exactly with
the one from the Laurent expansion
%
\be \sqrt{\pi} m^{1-2s}\frac{\Gamma(s-\half)}{\Gamma(s)} = \half
m^2\Big[\frac{1}{s+\half} - \ln \frac{m^2}{4}+1\Big] +{\cal
O}(s+\half)\,. \ee
%
We emphasize that the above procedure through Eq.~(\ref{ancon})
does not involve the divergent expression which needs a
regularization but just gives us a mathematical relation by the
analytic continuation\footnote{Note that Eq.~(\ref{Sfunction}) is
used just as an intermediate step rather than the aim contrary to
Ref.~\cite{Ambjorn:1981xw}. We might, of course, take this
directly as a regularized definition for the zero point energy
like in Ref.~\cite{Ambjorn:1981xw}. But this requires the physical
consideration leading the renormalization condition. Therefore,
our logic is different from Ref.~\cite{Ambjorn:1981xw}~.}.




With Eq.s~(\ref{Series}),(\ref{zetapro}), we can finally get the
power series expansion for $\Delta_b(m)$ in terms of $m$ as,
%
\bea \Delta_b(m)
 = \left\{\ba{ll}-\frac{1}{12}+\half m+\quarter m^2
 \Big[\ln\frac{~m^2}{4}+2\gamma_E-1\Big]
 +{\displaystyle\sum_{k=2}^{\infty}}
 \frac{(-1)^k\Gamma(k-\half)}{\Gamma(-\half)\Gamma(k+1)}\zeta(2k-1)
 m^{2k} \,, & {}~~{} b = 0~,
 \\ & \!\!\! \\
\frac{1}{24}-\frac{1}{8}(2b-1)^2 + \quarter m^2
 \Big[\ln\frac{~m^2}{4}-\psi(b)-\psi(1-b)-1\Big] & \\ {}~~~{}
 + {\displaystyle \sum_{k=2}^\infty}
 \frac{(-1)^k \Gamma(k-\half)} {\Gamma(-\half)\Gamma(k+1)}
 \half\Big[\zeta(2k-1,b)+\zeta(2k-1,1-b)\Big] m^{2k} \,, &
 \!\!\!\!\!\! 0 < b <1~.
 \ea\right.
\eea
%
The special cases  $b=0$ and $b=1/2$ give us
%
\be \Delta_{1/2}(m) - \Delta_{0}(m) =
{\textstyle\frac{1}{8}}-\half m + m^2\ln 2 + {\displaystyle
\sum_{k=2}^\infty}
 \frac{(-1)^k \Gamma(k-\half)} {\Gamma(-\half)\Gamma(k+1)}
 (2^{2k-1}-2)\zeta(2k-1) m^{2k}\,, \label{diff}
\ee
%
which is the identical expression with the one in
Ref.~\cite{semenoff,Itzykson}.



\setcounter{equation}{0}
\renewcommand{\theequation}{B.\arabic{equation}}




\newpage


\begin{thebibliography}{99}



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%%CITATION = JTPLA,45,709;%%






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\bibitem{Russo:2002rq}
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\bibitem{Milton:1999ge}
K.~A.~Milton,
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\end{thebibliography}
\end{document}



%
\be \tr_{trans} =
\prod_{i=1}^4\prod_{i^\prime=5}^8\sum_{oscillator}\ee
%









\bea \frac{1}{\Gamma(z)} &=&
-\frac{1}{2\sqrt{\pi}}-\frac{1}{2\sqrt{\pi}}(\gamma+2\ln2)(z+\half)
+{\cal O}\Big((z+\half)^2\Big)\,, \nn \\ &&\nn \\
m^{1-2s} &=& m^2\Big[1 - (s-\half)\ln m^2\Big]+ {\cal
O}\Big((s-\half)^2\Big)\,, \nn\eea



we can see
%
\bea &{\displaystyle\lim_{s\rightarrow -\half}^{}}\Big[
-2s\zeta(2s+2)m^2-\sqrt{\pi}m^{1-2s}
\frac{\Gamma(s-\half)}{\Gamma(s)}\Big] = \half m^2
 \Big[2\gamma-1+\ln\frac{~m^2}{4}\Big]\,, \nn \\
 &\nn \\  &{\displaystyle\lim_{s\rightarrow -\half}^{}}\Big[
-s\Big[\zeta(2s+2,b)+\zeta(2s+2,b)\Big]m^2 -\sqrt{\pi}m^{1-2s}
\frac{\Gamma(s-\half)}{\Gamma(s)}\Big] = \half m^2
 \Big[\ln\frac{~m^2}{4}-\psi(b)-\psi(1-b)-1\Big]\nn \eea
%



N^{B}_{i'\, n} := &\frac{1}{\omega'_n} \alpha^{i'}_{-n}
\alpha^{i'}_n\,, & \qquad a^{i' \dagger} a^{i'}\,, & \qquad
 -\frac{1}{\omega'_n} {\tilde\alpha}^{i'}_{n}
{\tilde\alpha}^{i'}_{-n}\,, \\ & &  & \\

\\ & & & \\
N^{\prime F}_{n}:= & \psi'_{-n} \psi'_n\,, & \qquad \chi'^\dagger
\chi'\,, &\qquad  {\tilde\psi}'_{n} {\tilde\psi}'_{-n}\,,

For a reference we present the classical expression:
 Each of the contributions is expressed as
follows:
\begin{eqnarray}
E_0 &=& \frac{\pi}{p^+}
  \left( \left( \frac{p^I}{2\pi} \right)^2
       + \left( \frac{m}{3} \right)^2 (x^i)^2
       + \left( \frac{m}{6} \right)^2 (x^{i'})^2
       - \frac{i}{\pi} \frac{m}{3} \tilde{\psi}_0 \gamma^4 \psi_0
       + \frac{i}{\pi} \frac{m}{6} \tilde{\psi}'_0 \gamma^4 \psi'_0
  \right)~,
  \nonumber \\
E &=& \frac{1}{2 p^+} \sum_{n \neq 0}
  ( \alpha^I_{-n} \alpha^I_n
   + \omega_n \psi_{-n} \psi_n
   + \omega'_n \psi'_{-n} \psi'_n
  )~,
  \nonumber \\
\tilde{E}
  &=& \frac{1}{2 p^+} \sum_{n \neq 0}
  ( \tilde{\alpha}^I_{-n} \tilde{\alpha}^I_n
   + \omega_n \tilde{\psi}_{-n} \tilde{\psi}_n
   + \omega'_n \tilde{\psi}'_{-n} \tilde{\psi}'_n
  )~.
\end{eqnarray}



\begin{eqnarray}
{\cal H} &=&  \frac{1}{2} ({\cal P}^I)^2
    + \frac{1}{2} (\partial_\sigma X^I)^2
    +\frac{1}{2} \left( \frac{m}{3} \right)^2 (X^i)^2
    +\frac{1}{2} \left( \frac{m}{6} \right)^2 (X^{i'})^2
  \nonumber \\
 & & - \frac{i}{2} \psi^1_- \partial_\sigma \psi^1_-
     + \frac{i}{2} \psi^2_+ \partial_\sigma \psi^2_+
     + i \frac{m}{3} \psi^2_+ \gamma^4 \psi^1_-
  \nonumber \\
 & & - \frac{i}{2} \psi^1_+ \partial_\sigma \psi^1_+
     + \frac{i}{2} \psi^2_- \partial_\sigma \psi^2_-
     - i \frac{m}{6} \psi^2_- \gamma^4 \psi^1_+ \,. \label{can-h}
\end{eqnarray}



  {}~{}
\exp\Bigg[2\pi\frac{\tau_2}{|\tau|^2}
 \nn \\  \!\!\!\!\!\!\!\!\!\! && \!\!\!\!\!   \nn \\
 \!\!\!\!\!\!\!\!\!\!   &&
{}~~~~~{}
\bigg\{4\Delta_{1/2}\Big({\textstyle\frac{m}{3}}|\tau|\Big)
-4\Delta_{0}\Big({\textstyle\frac{m}{3}}|\tau|\Big)
+4\Delta_{1/2}\Big({\textstyle \frac{m}{6}}|\tau|\Big) -
4\Delta_{0}\Big({\textstyle\frac{m}{6}}|\tau|\Big)\bigg\}\Bigg]
\nn



&&{}~~~~~{}\times\exp\bigg[2\pi\tau_2
\Big\{4\Delta_{b_1}({\textstyle\frac{m'}{3}})
-4\Delta_0({\textstyle\frac{m'}{3}})
+4\Delta_{b_1}({\textstyle\frac{m'}{6}})
-4\Delta({\textstyle\frac{m'}{6}})\Big\}\bigg] \,
\\&& \nn \\ &&



By the Schwinger's reparametrization, this can be written as
%
\bea G(s) &=& (\mu^2\tau_2)^s
\sum_{n_1}\bigg(\frac{\sqrt{\pi}}{\Gamma(s)}
\Big\{{\textstyle\frac{\Gamma(s-\frac{1}{2})}{b^{2s-1}}} +
{\sum_{p\neq 0}} e^{2\pi i p a} \int^{\infty}_{0}d x{}~{}
x^{s-\frac{3}{2}}{}~{} e^{-(b^2 x + \frac{\pi^2 p^2}{x})}\Big\}
\bigg)\,,
\nn \\&& \nn \\
&=&  (\mu^2\tau_2)^s \sum_{n_1} \bigg(
\frac{\sqrt{\pi}}{\Gamma(s)}
\Big\{{\textstyle\frac{\Gamma(s-\frac{1}{2})}{b^{2s-1}}} +
\sum_{p\neq 0} e^{2\pi i p a}
({\textstyle\frac{\pi|p|}{b}})^{s-\frac{1}{2}} \int^{\infty}_{0}d
u{}~{} u^{s-\frac{3}{2}}{}~{} e^{-\pi b |p|(u + \frac{1}{u})}
\Big\} \bigg)\,.   \nn\eea
%
In the above second line we have used the change of variable
$u=\frac{b}{\pi|p|}x$.
%
%
%
%\frac{r\beta\mu}{3\sqrt{2}\pi}\sum_{p=1}^{\infty}\frac{1-(-1)^p}{p}
%\bigg(2K_1 \Big(\frac{r\beta\mu}{3\sqrt{2}} p\Big) +  K_1
%\Big(\frac{r\beta\mu}{6\sqrt{2}} p\Big)\bigg)\bigg] \eea
%
%
%
%\frac{2n\beta\mu}{3\sqrt{2}\pi\tau_2}\sum^\infty_{p=1}\frac{1-(-1)^p}{p}\bigg(
%2K_1\Big(\frac{n\beta\mu }{3\sqrt{2}} p\Big) +
%K_1\Big(\frac{n\beta\mu }{6\sqrt{2}} p \Big)\bigg)
%
&=&-\sum_{l=1}^{\infty} \frac{1}{l\beta}
\Tr^{\prime}_{p^{-}=0}{}~{}e^{-\frac{l\beta}{\sqrt{2}}(p^{+}+p^{-})}
-\sum_{l=1}^{\infty}\frac{1-(-1)^l}{2l\beta}
\Tr^{\prime}_{p^{-}>0}{}~{}e^{-\frac{l\beta}{\sqrt{2}}(p^{+}+p^{-})}
 and

%
\be Z_1 (\beta) = \Tr^{\prime}\, e^{-\beta p^0} =
\Tr^{\prime}_{p^{-}\geq 0}\,
e^{-\frac{\beta}{\sqrt{2}}(p^{+}+p^{-})}\,, \ee
%



The first term in the above, which is negligibly small when $\mu$
goes to zero, is shown to exist in~\cite{Sugawara:2002rs} and be
advocated by the agreement with the result of the dual super
Yang-Mills theory for large $\mu$ in Ref.~\cite{semenoff}


 Since $m$ is a function of
$\tau_2$ as given in~(\ref{mmod}), the above identities can be
rewritten as~\cite{PandoZayas:2002hh}
\[
D_{b_1,\, b_2}\Big(\tau_1,\tau_2 ; m(\tau)\Big)= D_{b_2,\, -b_1}
\Big(\tau_1', \tau_2' ; m(\tau')|\tau'|\Big)~,\] where $\tau' =
-1/\tau$. (This is a  {\large{\bf trivial}} variation of type IIB
case : see the references
\cite{PandoZayas:2002hh,Greene:2002cd,Sugawara:2002rs,Sugawara:2003qc,Brower:2002zx}
) which is the winding mode part of Eq.~(\ref{thermwind}) written
in terms of $z$ and $\bar{z}$. \footnote{We are using the
convention of Ref.~\cite{Polchinski:zf} such that there is a
factor $i$ in the one loop vacuum amplitude $Z_{T^2}$ coming from
the Wick-rotation and for the complex modular parameter $\tau
\equiv \tau_1 + i\tau_2$ we have $d\tau^2\equiv d\tau d{\bar
\tau}=2d\tau_1 d\tau_2.$} \footnote{To see this, note that in case
of $M\propto m'=\beta\mu|r\tau-l|/2\sqrt{2}\pi\tau_2$ with
designating $(r,l)$-dependence of $M$ as $M=M(r,l)$,
%
$ G_{b_1,\, b_2}(s;~ \tau_1,\,\tau_2,\,M(r,l))= G_{b_2,\,
-b_1}(s;\, -\tau_1/|\tau|,\tau_2/|\tau| ,M(l,-r))
 =G_{b_1,\, b_2+b_1}(s;\,
\tau_1+1,\tau_2,M(r,l+r))\,, $ while we put $0 \le \pm
b_{1,\,2},~b_1+b_2 <0$, for example, such that $b_1+b_2$ can be
identified with $\frac{1-(-1)^{r+l}}{4}$.} Finally, we would like
to comment on the finite temperature field theory in the pp-wave
background. As we have shown, the free energy of finite
temperature string theory on pp-wave becomes to (\ref{2dfree}) in
the large $\mu$ limit, indicating that the degrees of freedom of
the theory reduces to those in two dimensions. We expect the
analogous behavior of the field theory on pp-wave. Here we confirm
this by considering the finite temperature gas of point particles



$ S = -\half \int d\sigma^{0} \bigg[ e^{-1}\frac{d X^{\mu}}{d
\sigma^{0}} \frac{d X^{\nu}}{d\sigma^{0}}g_{\mu\nu} + M^2e\bigg] $

\[
-\beta F = \int^{\infty}_{0} \frac{dt}{2t}\frac{\beta L}{4\pi^2
t}{}~{}e^{-\pi t M^2}\sum_{l=-\infty}^{\infty}\Big(\frac{1}{2\sinh
\pi t m}\Big)^8{}~{} e^{-\frac{l^2\beta^2}{4\pi t}}
\]

\bibitem{Blau:2002dy}
M.~Blau, J.~Figueroa-O'Farrill, C.~Hull and G.~Papadopoulos,
%``Penrose limits and maximal supersymmetry,''
Class.\ Quant.\ Grav.\  {\bf 19} (2002) L87
[arXiv:hep-th/0201081].
%%CITATION = HEP-TH 0201081;%%


Despite its apparent complexity, this gauge gives the same action
as the above light-cone gauge one~(\ref{ELCA}) after the
worldsheet rotation
%
\be
 z'= e^{i\theta_{r,\,l}}z\,, \quad  \bar{z}'=
 e^{-i\theta_{r,\,l}}\bar{z}\,, \qquad \quad e^{i\theta_{r,\,l}} =
 \frac{r\bar{\tau}-l}{|r\tau-l|}\,,\ee
%

with the rescaling of fermions.



$(r,\,l)=(0,0)$ part of the free energy expression corresponds to
the one loop vacuum amplitude. In  using the path integral, one
can treat $X^{+}$ ``effectively'' as constant after rescaling the
 In this case, the mass terms also
effectively vanish which is consistent with the gauge choice,
$X^{+}=const.$, for $(r,\,l)=(0,0)$ mode giving $m'=0$.

\be
\partial X^{\pm} = -{\bar\partial}X^{\pm} =
i\frac{\beta}{2\sqrt{2}\pi\tau_2}|r\tau-l|\,, \ee
%

\be \frac{1}{\sqrt{2}} \frac{\beta}{4\pi i \tau_2}\Big[r(\tau
\bar{z}-\bar{\tau}z)+l(z-\bar{z})\Big]\,. \ee
%
The only modification is in the mass parameter in the above
action~(\ref{ELCA}) as

At first glance, the $(r,\,l)=(0,0)$ term seems to be problematic
because it means the gauge choice $X^{+} = x^{+}_{0} =constant$
and moreover the ($r, l$) dependent worldsheet rotation seems to
be not well-defined. However, ;\qquad S_{\beta}(r,\, l) =
\frac{\beta^2 |r\tau-l|^2}{4\pi\a \tau_2}\,,
