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\begin{center}

%\vspace{2cm}

{\Large \bf Closed Bosonic String Partition Function in Time Independent Exact PP-Wave Background}\\
[4mm]

\large{Agapitos Hatzinikitas} \\ [5mm]

{\small University of Crete, \\
Department of Applied Mathematics, \\
L. Knosou-Ambelokipi, 71409 Iraklio Crete,\\
Greece, \\
Email: ahatzini@tem.uoc.gr}\\ [5mm]

\large{and} \\ [5mm]


\large{Ioannis Smyrnakis} \\ [5mm]

{\small University of Crete, \\
Department of Applied Mathematics, \\
L. Knosou-Ambelokipi, 71409 Iraklio Crete,\\
Greece, \\
Email: smyrnaki@tem.uoc.gr} \vspace{5mm}

\end{center}


\begin{abstract}
  The modular invariance of the one-loop partition function of the closed bosonic string in four dimensions in the 
presence of general time independent exact pp-wave background is studied.  In the absence of an axion field the 
partition function is found to be modular invariant.  In the presence of an axion field modular invariace 
is broken. This can be attributed to the light-cone gauge which breaks the symmetry in the $\sigma$-, $t$-directions.
\end{abstract}
\newpage

\section{Introduction}
\renewcommand{\thefootnote}{\arabic{footnote}}
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\par 
It has been known for some time \cite{peres} that there are certain solutions to the vacuum Einstein equations in 
four dimensions, 
with a covariantly constant null Killing vector, that 
can be interpreted as plane fronted gravitational waves, the so called pp-waves.  These are of the form 
\beqr
\label{1}
ds^2=-dX^+dX^-+ \sum_{i=1}^{2}(dX_i)^2+F(X^+,X_i){(dX^+)}^2
\feqr
where $\partial_T^2F(X^+,X_i)=0$.  It was shown in \cite{amati} that pp-waves are exact solutions of the spacetime 
string equations 
to all orders of perturbation theory.  So if this metric is used as a spacetime metric in string theory, 
the $\beta$-function is zero, so conformal invariance is preserved.  
This class of solutions can be extended by the introduction of an antisymmetric tensor and a dilaton field 
so that the $\beta$-function remains zero
\cite{horowitz}. The axion field strength corresponding to the antisymmetric tensor is given by
\beqr
\label{2}
H_{\mu \nu \rho}=A_{ij}(X^+)l_{[ \mu}\nabla_{\nu}X_i \nabla_{\rho ]}X_j
\feqr  
\noi where $l_{\mu}=\partial_{\mu}X^+$ is the null Killing vector. The metric, axion and dilaton have to satisfy the 
equation
\beqr
\label{3}
\partial_T^2F + \frac{1}{18}A_{ij}A^{ij}+ 2\Phi^{''}=0
\feqr
\noi for the $\beta$-function of the theory to be zero.
The action corresponding to the above exact solution of the spacetime string equations is 
\beqr
S=\frac{1}{2\pi}\int d\sigma dt \left(h^{ab} \partial_a X^+ \partial_b X^- - h^{ab} \partial_a X_i \partial_b X_i
-h^{ab}F \partial_a X^+ \partial_b X^+ -\frac{1}{3} A_{ij} X_j \partial_a X^+ \partial_b X_i \epsilon^{ab} \right)
\label{4}
\feqr
\noi where $\alpha'=1/2$, $h^{ab}=\sqrt{-g}g^{ab}$ and $\epsilon^{t \sigma}=-1$. Note that $h=deth_{ab}=-1$. This 
action as it stands is not quadratic in the string coordinates so it is difficult to manipulate. For a particular 
class of pp-waves, the exact plane waves, it turns out that by choosing the light-cone gauge it is possible to make 
this action quadratic and hence quantize it. Interestingly enough the transverse string coordinates become massive 
bosonic fields. Nevertheless the conformal invariance is manifested in the partition function. The modular invariance 
of the partition function for the Nappi-Witten model \cite{nappi} has been investigated in \cite{tadashi}.  
\par This paper is organised as follows. In section one the canonical momenta are used to write the phase-space 
action before fixing the light-cone gauge. From this we read off the phase-space Hamiltonian in the light-cone gauge. 
Next we specialize to time-independent exact plane waves. The transverse string coordinates and the canonical 
momenta are decomposed into oscillator modes. Finally each oscillator Hamiltonian is diagonalized through a canonical 
transformation. In section two the partition function for $A_{ij}=0$ is computed in the path integral formalism following 
a deformed zeta function regularization scheme. This partition function is explicitly modular invariant, although it does 
not split into a finite sum of holomorphic times antiholomorphic blocks in any obvious way. In section three the axion 
field is interpreted as an $O(2)$ worldsheet gauge field in the $\sigma$-direction. This is shown to break the modular 
invariance due to inequivalence of the $\sigma$ and $t$ directions in the worldsheet, 
for which the light-cone gauge is responsible. The modular invariance is only 
recovered after an integration over the twists.         
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Quantization}

\par The canonical momenta corresponding to the action \rf{4} are
\beqr
\mathcal{P}_+ &=& \frac{\partial L}{\partial \dot{X}^-}= \frac{1}{2\pi} h^{a0}\partial_a X^+ \non \\
\mathcal{P}_- &=& \frac{\partial L}{\partial \dot{X}^+}= \frac{1}{2\pi} 
\left(h^{a0} \partial_a X^- -2F\mathcal{P}^+ + \frac{1}{3} A_{ij}X_j X_i^{'}\right) \non \\
\mathcal{P}_i &=& \frac{\partial L}{\partial X_i}= - \frac{1}{2\pi} 
\left(2 h^{a0}\partial_a X_i + \frac{1}{3} A_{ij} X_j X^{+ '} \right). 
\label{5}
\feqr
\noi Substituting in the action we get
\beqr
S &=& \frac{1}{2\pi}\int d\sigma dt \Biggl[ 2\pi \mathcal{P}_+ \dot{X}^- 
+ 2\pi \mathcal{P}_- \dot{X}^+ + 2\pi \mathcal{P}_i \dot{X}_i
+ \frac{h^{01}}{h^{00}}\left(2\pi \mathcal{P}_+ X^{- '} + 2\pi \mathcal{P}_- X^{+ '} 
+ 2\pi \mathcal{P}_i X_i^{'} \right) \non \\
&-& \frac{1}{h^{00}}\left(4\pi^2 \mathcal{P}_+ \mathcal{P}_- + 4\pi^2 F (\mathcal{P}_+)^2 - \pi^2 (\mathcal{P}_i)^2 + 
X^{+ '}X^{- '} - F (X^{+ '})^2 - (X_i^{'})^2 \right) \non \\
&+& \frac{1}{h^{00}}\left(\frac{2\pi}{3}A_{ij}\mathcal{P}_+ X_j X_i^{'} + \frac{2\pi}{6} A_{ij} \mathcal{P}_i X_j X^{+ '}
+ \left(\frac{1}{6}A_{ij}X_j X^{+ '} \right)^2  \right) \Biggl]. 
\label{6}
\feqr
In the conformal gauge $h^{00}=-1$ and $h^{10}=0$ so we are left with the constraint 
\beqr
\mathcal{P}_+ X^{- '} + \mathcal{P}_- X^{+ '} + \mathcal{P}_i X_i^{'} =0
\label{7}
\feqr  
\noi and the action 
\beqr
S &=& \frac{1}{2\pi}\int d\sigma dt \Biggl[ 2\pi \mathcal{P}_+ \dot{X}^- 
+ 2\pi \mathcal{P}_- \dot{X}^+ + 2\pi \mathcal{P}_i \dot{X}_i \non \\
&+& \left(4\pi^2 \mathcal{P}_+ \mathcal{P}_- + 4\pi^2 F (\mathcal{P}_+)^2 - \pi^2 (\mathcal{P}_i)^2 + 
X^{+ '}X^{- '} - F (X^{+ '})^2 - (X_i^{'})^2 \right) \non \\
&-& \left(\frac{2\pi}{3}A_{ij}\mathcal{P}_+ X_j X_i^{'} + \frac{2\pi}{6} A_{ij} \mathcal{P}_i X_j X^{+ '}
+ \left(\frac{1}{6}A_{ij}X_j X^{+ '} \right)^2  \right) \Biggl]. 
\label{8}
\feqr
\noi From this we can read off the Hamiltonian in the light-cone gauge. Setting $X^+=p_+ t$ we obtain  
\beqr
\label{9}
H=\int \frac{d\sigma}{2\pi} \left[2\pi p_+ \mathcal{P}_- + \pi^2 (\mathcal{P}_i)^2 + {X_i}' {X_i}' -F p_+^{2}
-\frac{p_+}{3}A_{ij}{X_i}' X_j \right].
\feqr
\noi Note that $\mathcal{P}_-$ is unconstrained in the light-cone gauge since it disappears from the constraint equation 
\rf{7}.
\par Next we specialize to the time independent exact plane waves. For these we have 
\beqr
\label{10}
F(X^+,X_i)=W_{1}(X^2_1 -X^2_2)-2W_2 X_1 X_2 + \tilde{\Phi}(X_1^2 +X_2^2).
\feqr 
\noi Note that $\partial^2_T F=4\tilde{\Phi}$. Setting $A_{ij}=A\epsilon_{ij}$ the spacetime field equation \rf{3} 
demands that 
\beqr
\label{11}
\tilde{\Phi}=-\frac{1}{36} A^2 - \frac{1}{2}\Phi''.
\feqr 
\noi Substituting \rf{10} into \rf{9} we obtain
\beqr
H &=& \frac{1}{2\pi}\int d\sigma \Biggl[ 2\pi p_+ \mathcal{P}_- + \pi^2 (\mathcal{P}_i)^2 + X_i^{2 '} \non \\
&-& p_+^2 \left( W_{1}(X^2_1 -X^2_2)-2W_2 X_1 X_2 - \left(\frac{1}{36} A^2 + \frac{1}{2}\Phi'' \right)
(X_1^2 +X_2^2)\right) \non \\
&-& \frac{p_+}{3}A \left({X_1}' X_2 - {X_2}' X_1  \right) \Biggl]. 
\label{12}
\feqr
\par We proceed by expanding $X^i$ and $\mathcal{P}^i$ in oscillator modes
\beqr
X^i(\sigma, t) &=& \frac{1}{\sqrt{2}}\sum_{n \in Z}X_n^i (t) e^{in\sigma} \non \\
\mathcal{P}^i (\sigma, t) &=& \frac{1}{\pi \sqrt{2}}\sum_{n \in Z}P_n^i (t) e^{in\sigma}. 
\label{13}
\feqr
\noi Reality of $X^i$, $\mathcal{P}^i$ demands that $X^i_{-n}(t)=\bar{X}^i_n (t)$, $P^{i}_{-n}(t)=\bar{P}^{i}_{n}(t)$. 
This implies in particular that $X^i_0(t)$, $P^i_0(t)$ are 
real. The commutation relations of the $\hat{X}^i_n, \hat{P}^j_m$ in the operator formalism are given by 
\beqr
[\hat{X}^i_n, \hat{P}^j_m]=i\delta^{ij}\delta_{nm}.
\label{13a}
\feqr
\noi The Hamiltonian now becomes
\beqr
H=2\pi p_+p_- + H_0 + H_{osc.}
\label{14}
\feqr
\noi where $p_-$ is the average value of the momentum density $\mathcal{P}_-$,  
\beqr
H_0=\frac{1}{2} (P^i_0)^2 - \frac{1}{2} p_+^2 W_{1}((X^1_0)^2 - (X^2_0)^2)
+p_+^2 W_2 X^1_0 X^2_0 + \frac{1}{2}p_+^2 \left(\frac{1}{36} A^2 + \frac{1}{2}\Phi''\right) (X^i_0)^2 
\label{15}
\feqr
\noi and
\beqr
H_{osc.} &=& \frac{1}{2}\sum_{n \in Z -\{0\}} \Biggl[ |P^i_n|^2 + n^2 |X^i_n|^2 
- p_+^2 W_1 (|X^1_n|^2 - |X^2_n|^2)+ p_+^2 W_2 (X^1_n X^2_{-n}+ X^1_{-n} X^2_n) \non \\
&+& p_+^2 \left(\frac{1}{36} A^2 + \frac{1}{2}\Phi''\right) 
|X^i_n|^2 - in \frac{p_+}{3}A (X^1_n X^2_{-n} - X^1_{-n}X^2_n) \Biggl]. 
\label{16}
\feqr
\noi If we define \cite{horowitz}
\beqr
\phi_1 &=& p_+^2 \left(-W_1 + \frac{A^2}{36}+ \frac{\Phi''}{2} \right) \non \\
\phi_2 &=& p_+^2 \left(W_1 + \frac{A^2}{36}+ \frac{\Phi''}{2} \right) \non \\
\rho &=& W_2 p_+^2 \non \\
\lambda &=& \frac{1}{3} p_+ A
\label{17}
\feqr
\noi we have
\beqr
H_0 &=& \frac{1}{2} (P^i_0)^2 + \frac{1}{2} 
\left( X^1_0 \,X^2_0 \right)
\left( \matrix{ \phi_1 & \rho \cr
\rho & \phi_2 } \right)
\left( \begin{array}{c} X^1_0 \\
X^2_0 \end{array} \right) \non \\
H_{osc.} &=& \frac{1}{2} \sum_{n \in Z-\{0\}} \Biggl[ |P^i_n|^2 +n^2 |X^i_n|^2 + 
\left(X^{1 \dagger}_n \, X^{2 \dagger}_n \right)
\left( \matrix{ \phi_1 & (\rho + in\lambda) \cr
(\rho -in\lambda) & \phi_2 } \right)
\left( \begin{array}{c} X^1_n \\
X^2_n \end{array} \right) \Biggl].
\label{18}
\feqr
\noi It is possible to transform canonically the phase-space variables so as to diagonalize the Hamiltonian. Let 
\beqr
\tilde{X}^i_n = M^{\dagger}_{ij}X^j_n \quad  \tilde{P}^i_n = M_{ij}P^j_n 
\label{19}
\feqr 
\noi where
\beqr
(M_{ij})=i \sqrt{\rho^2 + n^2 \lambda^2} \left( \matrix{  \alpha_- e^{i\theta}& \alpha_+ \cr
\alpha_+ & -\alpha_- e^{-i\theta}} \right),
\label{20}
\feqr
\noi  
\beqr
\alpha_{\pm} &=& \frac{1}{\sqrt{G_{\pm}^2 + \rho^2 + n^2 \lambda^2}} \non \\
G_{\pm} &=& \pm \left(\frac{\phi_1 -\phi_2}{2}\right) + 
\sqrt{\left(\frac{\phi_1 -\phi_2}{2}\right)^2 + \rho^2 + n^2 \lambda^2 }
\label{21}
\feqr
\noi and $\theta=arg(\rho -in\lambda)$. The diagonalized Hamiltonians become
\beqr
H_0 &=& \frac{1}{2} (\tilde{P}^i_0)^2 + 
\frac{1}{2} \left( S_{0,+}^2 (\tilde{X}^1_0)^2 + S_{0,-}^2 (\tilde{X}^2_0)^2 \right)  \non \\
H_{osc.} &=& \frac{1}{2} \sum_{n \in Z- \{0\}} \Biggl[ |\tilde{P}^i_n|^2 + n^2 |\tilde{X}^i_n|^2 +
 S_{n,+}^2 |\tilde{X}^1_n|^2 + S_{n,-}^2 |\tilde{X}^2_n|^2 \Biggl] \non \\
&=& \sum_{\stackrel{n > 0}{i=1,2}} H^i_n  
\label{22}
\feqr
\noi where 
\beqr
H^i_n=\tilde{P}^i_n \tilde{P}^i_{-n} + (\omega^i_n)^2 \tilde{X}^i_n \tilde{X}^i_{-n},  
\label{22a}
\feqr
\beqr
S_{n, \pm}^2= \left(\frac{\phi_1 +\phi_2}{2}\right) \pm 
\sqrt{\left(\frac{\phi_1 - \phi_2}{2}\right)^2 + \rho^2 + n^2 \lambda^2 }
\label{23}
\feqr
\noi and $(\omega^{1,2}_n )^2= n^2 +S^2_{n, \pm}$. Here we have assumed that the gravitational wave amplitudes are small 
compared to $\Phi''$ so as to 
have positivity of $S_{n, \pm}^2$. In case the gravitational waves have large amplitudes then the average number of
excitation modes of the string diverges exponentially and a string singularity appears \cite{horowitz}. 

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

\section{Partition Function when $A=0$}
In the particular case of $A=0$ we have $\phi_1 = p_+^2 \left(-W_1 + \frac{\Phi''}{2} \right)$, 
$\phi_2 = p_+^2 \left(W_1 + \frac{\Phi''}{2} \right)$, $\rho = W_2 p_+^2$ and $\lambda = 0$, so 
$S_{\pm}^2= \left(\frac{\phi_1 +\phi_2}{2}\right) \pm 
\sqrt{\left(\frac{\phi_1 - \phi_2}{2}\right)^2 + \rho^2 }$ becomes independent of $n$. Time independence of $\phi_1$
, $\phi_2$, $\rho$ demands that the dilaton is at most quadratic in $X_+$, $\Phi(X_+)=c_1 X^2_+ + c_2 X_+ + c_3$ and 
that $W_1$, $W_2$ are independent of $X_+$. So we have 
\beqr
S_{\pm}^2= p^2_+ \left(c_1  \pm  \sqrt{W_1^2 + W_2^2 } \right)= p^2_+ \tilde{S}^2_{\pm}.
\label{23a}
\feqr
\par The partition function now becomes
\beqr
\mathcal{Z}_{A=0}=\int_{\mathcal{F}} \frac{d\tau d\bar{\tau}}{\tau_2}Z_{A=0}(\tau, \bar{\tau}) 
\label{23b}
\feqr
\noi where 
\beqr
Z_{A=0}(\tau, \bar{\tau})= C \int dp_+ dp_- Tr\left(e^{-2i\pi \tau_2 H}e^{2i\pi \tau_1 \Pi} \right), 
\label{24} 
\feqr
\noi $\hat{\Pi}=\sum_{i=1}^2 \sum_{n=0}^{\infty} \hat{\Pi}^i_n$ is the momentum operator of the string, 
$\hat{H}=2\pi p_+p_- + \sum_{i=1}^2 \sum_{n=0}^{\infty}\hat{H}^i_n$ is the Hamiltonian and $\mathcal{F}$ is the fundamental 
domain of the modular transformations. Substituting the above operators in \rf{24} 
and using the results of appendix A we get, in Euclidean time, that
\beqr
Z_{A=0}(\tau, \bar{\tau})&=& C \int dp_+ dp_- e^{-2\pi \tau_2 p_+ p_-} \prod_{i=1}^2 \left( Z^i_0 \prod_{n>0}Z^i_n \right)
\non \\
&=& C \int dp_+ dp_- e^{-2\pi \tau_2 p_+ p_-} \prod_{i=1}^2 \left( det^{-\frac{1}{2}}(D^i_0) \prod_{n>0} det^{-1}(D^i_n)
\right) \non \\
&=& C \int dp_+ dp_- e^{-2\pi \tau_2 p_+ p_-} \prod_{i=1}^2 \prod_{n \in Z} det^{-\frac{1}{2}}(D^i_n)
\label{25} 
\feqr
\noi where $Z^i_n$ for $n\geq 0$ are given in appendix A. Using now the determinant formulae derived in appendix B we have
\beqr
\prod_{n \in Z}det (D^i_n) = \prod_{n \in Z} e^{2\pi \omega^i_n \tau_2} (1-e^{-2\pi \omega^i_n \tau_2 + 2i\pi n \tau_1})
(1-e^{-2\pi \omega^i_n \tau_2 - 2i\pi n \tau_1}) 
= e^{4\pi \tau_2 \Delta_i} f_i^2 (\tau, \bar{\tau}) 
\label{26}
\feqr
\noi 
\beqr
\Delta_{(1,2)}=\Delta_{p_+\tilde{S}_{(+,-)}} = \frac{1}{2}\sum_{n \in Z} \omega^{i}_n = \frac{1}{2} 
\sum_{n \in Z} \sqrt{n^2 + p_+^2 \tilde{S}_{(+,-)}^2}=
-\frac{1}{2\pi^2} \sum_{n=1}^{\infty} \int_{0}^{\infty} ds e^{-n^2 s - \frac{p^2_+ \tilde{S}^2_{(+, -)} \pi^2}{s}} 
\label{27}
\feqr
\noi and 
\beqr
f_i(\tau, \bar{\tau})=\prod_{n \in Z} (1-e^{-2\pi \omega^i_n \tau_2 + 2i\pi n \tau_1}).
\label{28}
\feqr
\noi Note that in formula \rf{27} we have regularized the sum by analytically continuing the formula
\beqr
\sum_{n=1}^{\infty} \frac{1}{(n^2 + c^2)^{\nu}}= -\frac{1}{2c^{2\nu}} + \frac{\sqrt{\pi}}{2c^{2\nu -1}\Gamma(\nu)}
\left[ \Gamma(\nu -\frac{1}{2})+4\sum_{n=1}^{\infty} (\pi n c)^{\nu - \frac{1}{2}} K_{\nu - 1/2}(2\pi nc)\right]   
\label{29}
\feqr 
\noi to $\nu=-1/2$ and dropping the infinite uniform vacuum energy 
$\frac{\sqrt{\pi}}{2c^{2\nu -1}\Gamma(\nu)}\Gamma(\nu -\frac{1}{2})$ \cite{farina}. 
We also make use of the fact that \cite{ryzhik}
\beqr
K_{-1}(z)=\frac{1}{z} \int_{0}^{\infty}e^{-t-\frac{z^2}{4t}}dt.
\label{30}
\feqr
\noi This regularized $\Delta_{i}$ is the Casimir energy of the theory. 
The regularization procedure we followed corresponds to a deformed zeta function regularization. 
Using the notation of deformed modular forms as defined
in appendix C we have $\prod_{n \in Z}det (D^{(1,2)}_n) = \hat{\eta}^2_{p_+ \tilde{S}_{(+, -)}}(\tau, \bar{\tau})$ so 
\beqr
Z_{A=0}(\tau,\bar{\tau}) = 
C \int dp_+ dp_- e^{-2\pi \tau_2 p_+ p_-} \left[ \hat{\eta}_{p_+ \tilde{S}_+}(\tau, \bar{\tau})
 \hat{\eta}_{p_+ \tilde{S}_-}(\tau, \bar{\tau}) \right]^{-1}. 
\label{31}
\feqr  
\par The function $\hat{\eta}_{c}(\tau, \bar{\tau})$ has the modular properties \cite{green} 
\beqr
\label{32} 
\hat{\eta}_{c}(\tau +1, \bar{\tau}+1) &=&  \hat{\eta}_{c}(\tau, \bar{\tau})  \non \\
\hat{\eta}_{c}(-\frac{1}{\tau}, -\frac{1}{\bar{\tau}}) &=&  \hat{\eta}_{c/|\tau|}(\tau, \bar{\tau})
\feqr  
\noi and in the limit $c\rightarrow 0$ it degenerates according to
\beqr
\hat{\eta}_0^R (\tau, \bar{\tau}) = \eta (\tau) \overline{\eta(\tau)}.
\label{33}
\feqr
\noi Here the symbol $R$ means that we have regularized $\hat{\eta}_0 (\tau, \bar{\tau})$ by dropping a zero 
factor in the limit $c\rightarrow 0$. 
\par Because 
of the modular properties \rf{32} and the fact we integrate over $p_+$, $p_-$ we have that the partition function 
transforms according to
\beqr
Z_{A=0}(\tau +1, \bar{\tau} +1)=Z_{A=0}(\tau , \bar{\tau}) \qquad  
Z_{A=0}(-\frac{1}{\tau}, -\frac{1}{\bar{\tau}})= |\tau|^2 Z_{A=0}(\tau , \bar{\tau}) 
\label{34}
\feqr
\noi so $Z_{A=0}(\tau , \bar{\tau}) d\tau d\bar{\tau}/\tau_2$ is modular invariant. It should be mentioned that in order 
to avoid infinities coming from the momenta integration a UV regularization is needed.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The partition function when $A \neq 0$}

When $A \neq 0$ the above calculation does not lead to a modular invariant partition function. This is because 
$A$ assumes the role of a worldsheet $O(2)$ gauge field in the $\sigma$-direction. It is possible to write the 
light-cone gauge fixed Hamiltonian
\rf{12} in the following form
\beqr
H &=& 2\pi p_+ p_- +\frac{1}{2\pi}\int d\sigma \Biggl[ 2\pi p_+ \mathcal{P}_- 
+ \pi^2 (\mathcal{P}_i)^2 + (D_{\sigma}X_i)^2 \non \\
&-& p_+^2 \left( W_{1}(X^2_1 -X^2_2)-2W_2 X_1 X_2 - \frac{1}{2}\Phi'' (X_1^2 +X_2^2)\right) \Biggl] 
\label{35}  
\feqr
\noi where $D^R_{\sigma}X_i = \partial_{\sigma}X_i -\frac{p_+ A}{6} \epsilon_{il} X_l$. This suggests that 
$\frac{p_+ A}{6} \epsilon_{ij}$ is an $O(2)$ connection along the $\sigma$-direction. 
It is more convenient to define a complex spacetime coordinate 
to turn the $O(2)$ connection to a $U(1)$ connection.  If $Z=X_1+iX_2$ then the Hamiltonian takes the form 
\beqr
H = 2\pi p_+ p_- + \frac{1}{2\pi}\int d\sigma \Biggl[ (2\pi)^2 P_z P_{\bar{z}}
- \frac{p_+^2}{2}W Z^2 -\frac{p_+^2}{2}\bar{W}\bar{Z}^2
+ \frac{p_+^2}{2}\Phi''Z\bar{Z} + D_{\sigma}Z \overline{D_{\sigma}Z}\Biggr] 
\label{36}
\feqr
\noi where now $D_{\sigma}=\partial_{\sigma}+i\frac{p_+ A}{6}$ and $W=W_1+iW_2$. 
\par If we make the gauge transformation 
\beqr
Z=e^{-i\frac{p_+ A}{6}\sigma} \tilde{Z}
\label{37}
\feqr
\noi then the Hamiltonian becomes 
\beqr
H = 2\pi p_+ p_- + \frac{1}{2\pi} \int  d\sigma   \Biggl[ (2\pi)^2 P_{\tilde{z}} P_{\bar{\tilde{z}}}- 
\frac{p_+^2}{2}W e^{-i\frac{p_+ A}{3}\sigma} \tilde{Z}^2 
-\frac{p_+^2}{2}\bar{W} e^{i\frac{p_+ A}{3}\sigma} \overline{\tilde{Z}}^2 
+ \frac{p_+^2}{2}\Phi''\tilde{Z}\overline{\tilde{Z}} + \partial_{\sigma} \tilde{Z} 
\partial_{\sigma}\overline{\tilde{Z}}\Biggr]. 
\label{38}
\feqr
\noi The boundary condition satisfied by the gauged transformed variable $\tilde{Z}$ is
\beqr
\tilde{Z} (\sigma + 2\pi, t)= e^{i\frac{\pi}{3}p_+ A}\tilde{Z}(\sigma, t).
\label{39}
\feqr
\noi Expanding the fields 
\beqr
\tilde{Z}(\sigma, t) &=& \frac{1}{\sqrt{2}} \sum_{n \in Z} Z_n(t) e^{i(n + \frac{p_+ A}{6})\sigma} \non \\
P_{\tilde{Z}}(\sigma ,t) &=& \frac{1}{2\pi \sqrt{2}} \sum_{n \in Z} P^{Z}_n(t) e^{i(n + \frac{p_+ A}{6})\sigma}
\label{40}
\feqr
\noi we get the Hamiltonian
\beqr
H  =2\pi p_+p_-+ \frac{1}{2} \sum_{n \in Z} \Biggl[ P_n^ZP_n^{\bar{Z}}-\frac{p_+^2}{2}WZ_nZ_{-n}-
\frac{p_+^2}{2}\bar{W}\bar{Z}_n\bar{Z}_{-n}+\frac{p_+^2}{2}\Phi''Z_n\bar{Z}_n+(n+\frac{p_+A}{6})^2Z_n\bar{Z}_n\Biggr].
\label{41}
\feqr
\noi This is the same Hamiltonian as \rf{14} when $A=0$ with the only difference that the term $n^2((X_n^1)^2+(X_n^2)^2)$ 
is replaced by the term $(n+\frac{p_+ A}{6})^2((X_n^1)^2+(X_n^2)^2)$.  
Also note that the momentum operator $\hat{\Pi}^i_n$ that appears in appendix A changes to 
\beqr
\hat{\Pi}_n=i(n+\frac{p_+A}{6}) \left( \hat{Z}_n \hat{P}^Z_{n} - \bar{Z}_{-n} \hat{P}^{\bar{Z}}_n \right).
\label{42}
\feqr
This momentum operator generates $\sigma $ translations in $\tilde{Z}$, $\overline{\tilde{Z}}$. This means that it 
generates covariant $\sigma $ translations on $Z$, $\bar{Z} $.   
Again the only difference is that $n$ has been replaced by $(n+\frac{p_+A}{6})$.  
So the partition function becomes
\beqr
Z_A(\tau, \bar{\tau })=C\int dp_+dp_-e^{-2\pi \tau_2p_+p_-}\prod_{i=1}^2\prod_{n\in Z}
det^{-\frac{1}{2}}(D^i_{(n+\frac{p_+A}{6})}).
\label{43}
\feqr
\noi The product of determinants has been computed in appendix B so if we define 
\beqr
\tilde{\omega}^{(1,2)}_{n+ \frac{p_+ A}{6}}=\sqrt{(n+\frac{p_+ A}{6})^2 + p_+^2 S^2_{(+,-)}} 
\label{43.5}
\feqr
\noi we have 
\beqr
&&\prod_{n \in Z}det (D^i_{(n+\frac{p_+A}{6})}) = \prod_{n \in Z} e^{2\pi \tilde{\omega}^i_{(n+p_+A/6)} \tau_2}
(1-e^{-2\pi \tilde{\omega}^i_{(n+p_+A/6)} \tau_2 + 2i\pi (n+\frac{p_+A}{6}) \tau_1}) \non \\
&\cdot&(1-e^{-2\pi \tilde{\omega}^i_{(n+p_+A/6)} \tau_2 - 2i\pi (n+\frac{p_+A}{6}) \tau_1}) 
= e^{4\pi \tau_2 \Delta_i(\frac{p_+ A}{6})} f_i[\frac{p_+ A}{6}] (\tau, \bar{\tau}) f_i[-\frac{p_+ A}{6}] (\tau, \bar{\tau}) 
\label{44}
\feqr
\noi where
\beqr
\Delta^{(1,2)}_{\frac{p_+ A}{6}}&=& \Delta_{\frac{p_+ A}{6}}(p_+ S_{(+,-)}) = 
\frac{1}{2}\sum_{n \in Z} \sqrt{(n+\frac{p_+ A}{6})^2 + p^2_+ \tilde{S}^2_{(+,-)}} \non \\ 
&=&
-\frac{1}{2\pi^2}\sum_{n=1}^{\infty}\int_0^{\infty} ds e^{-n^2 s -\frac{p^2_+ \tilde{S}^2_{(+,-)}\pi^2}{s}}
\cos (\pi \frac{p_+ A}{3}n) \non \\
f_i[\frac{p_+ A}{6}] (\tau, \bar{\tau}) &=& \prod_{n \in Z} 
(1-e^{-2\pi \tilde{\omega}^i_{(n+p_+A/6)} \tau_2 + 2i\pi (n+\frac{p_+A}{6}) \tau_1}). 
\label{45}
\feqr
\noi In terms of the deformed theta functions of appendix C the partition function becomes
\beqr
Z_A(\tau, \bar{\tau })=C\int dp_+dp_-e^{-2\pi \tau_2p_+p_-}
\left( \frac{ \hat{\Theta}_{p_+ \tilde{S}_+}\left[ \begin{array}{c} \frac{1}{2}+\frac{p_+ A}{6} \\
\frac{1}{2} \end{array} \right] (\tau, \bar{\tau})  \hat{\Theta}_{p_+ \tilde{S}_-}
\left[ \begin{array}{c} \frac{1}{2}+\frac{p_+ A}{6} \\
\frac{1}{2} \end{array} \right] (\tau, \bar{\tau})  }{\hat{\eta}_{p_+ \tilde{S}_+}(\tau, \bar{\tau})
 \hat{\eta}_{p_+ \tilde{S}_-}(\tau, \bar{\tau}) }\right)^{-\frac{1}{2}}
\label{46}
\feqr
\par This is not modular invariant since we only have twists in the $\sigma$-direction. The reason for this is the choice 
of the light-cone gauge. Consider the action \rf{4} in the conformal gauge. This can be written in the form  
\beqr
S &=& \frac{1}{2\pi}\int d\sigma dt \Biggl[ - \partial_{t} X^+ \partial_t X^- + \partial_{\sigma} X^+ \partial_{\sigma} X^- 
+ (D_tX)_i (D_tX)_i - (D_{\sigma}X)_i (D_{\sigma}X)_i \non \\ 
&+& \tilde{F} \partial_t X^+ \partial_t X^+ - \tilde{F} \partial_{\sigma} X^+ \partial_{\sigma} X^+ \Biggr] 
\label{47}
\feqr
\noi where 
\beqr
(D_t X)_i &=& \partial_t X_i - \frac{1}{6} A_{ij}(X^+) \partial_{\sigma}X^+ X_j \non \\
(D_{\sigma} X)_i &=& \partial_{\sigma} X_i - \frac{1}{6} A_{ij}(X^+) \partial_{t}X^+ X_j \non \\
\tilde{F} &=& F + \frac{1}{36}A_{ij}(X^+)A_{il}(X^+) X_j X_l.
\label{48} 
\feqr
\noi Note that \rf{3} implies that $\tilde{F}$ satisfies the equation
\beqr
\partial^2_T \tilde{F}+ 2\Phi^{''}=0.
\label{49}
\feqr
\noi The action \rf{47} implies that we also have a covariant derivative in the $t$-direction. In the light-cone 
gauge $(D_t X)_i = \partial_t X_i$ so the symmetry in the $\sigma$- and $t$-directions is broken. This explains why 
the partition function we obtained is not modular invariant.    
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}

In the case $A=0$ the partition function is explicitly modular invariant. Nevertheless it still needs to be regularized
in a way that does not break this invariance. 
In the case $A\neq 0$ the partition function is not modular invariant since we have a twist only in the $\sigma$-direction
which breaks the equivalence of the $\sigma$- and $t$-directions. One way to recover modular invariance is to 
integrate over all the possible twists in both directions. However it seems necessary to develop a covariant 
quantization procedure to justify such an integration. Another way to 
obtain the modular invariant partition function is through equivalences that exist between the bosonic string in 
certain pp-wave backgrounds and Wess-Zumino-Witten models based on non-semisimple Lie groups of the type discussed in 
\cite{kiritsis}.  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\addcontentsline{toc}{subsection}{Appendix A }
\section*{Appendix A }
\renewcommand{\theequation}{A.\arabic{equation}}
\setcounter{equation}{0}

Let us consider the generators of $t$ and $\sigma$ translations on the n-mode part of the field operator 
$\hat{X}^i(t,\sigma)$ 
expansion, $\hat{X}^i_n (t) e^{in\sigma} + \hat{X}^i_{-n} (t) e^{-in\sigma}$ for $n>0$.  
In this appendix to simplify notation we omit 
the tilde from the $\hat{X}^i_n$ and the $\hat{P}^i_n$. The generator of $t$ translations is just 
the harmonic oscillator Hamiltonian $\hat{H}^i_n$ while the generator of $\sigma$ translations is
\beqr 
\hat{\Pi}^i_n=in \left( \hat{X}^i_n \hat{P}^i_{-n} - \hat{X}^i_{-n} \hat{P}^i_n \right). 
\label{a1}
\feqr
\noi It satisfies 
\beqr
[\hat{\Pi}^i_n,\hat{X}^j_m (t) e^{im\sigma} + \hat{X}^j_{-m} (t) e^{-im\sigma}]
=-i\delta^{ij}\delta_{nm}\frac{\partial}{\partial \sigma}
\left(\hat{X}^i_n (t) e^{in\sigma} + \hat{X}^i_{-n} (t) e^{-in\sigma} \right).
\label{a2}
\feqr
\noi We want to compute the partition function of the oscillators on the torus. To do this we propagate 
oscillator states from $\sigma$, $t-\pi \tau_2$ to $\sigma + 2\pi \tau_1$, $t+\pi\tau_2$ along a path in the 
worldsheet which we discretize. Now we have
\beqr
&& <X^{i,j+1}_{\pm n}, \sigma_{j+1}, t_{j+1} | X^{i,j}_{\pm n}, \sigma_{j}, t_{j} > =
<X^{i,j+1}_{\pm n}|e^{-i\hat{H}^i_n \delta t} e^{i\hat{\Pi}^i_n \delta \sigma} |X^{i,j}_{\pm n} > \non \\
&=& \delta(X^{i,j+1}_n - X^{i,j}_n) \delta(X^{i,j+1}_{-n} - X^{i,j}_{-n}) 
- i \delta t 
<X^{i,j+1}_{\pm n}|\hat{H}^i_n|X^{i,j}_{\pm n}> \non \\
&+& i \delta \sigma <X^{i,j+1}_{\pm n}|\hat{\Pi}^i_n|X^{i,j}_{\pm n}>+ quad. 
\label{a3}
\feqr
\noi Now observe that 
\beqr
\delta(X^{i,j+1}_n - X^{i,j}_n) \delta(X^{i,j+1}_{-n} - X^{i,j}_{-n})= \frac{1}{(2\pi)^2} \int dP^{i,j}_n dP^{i,j}_{-n}
e^{iP^{i,j}_{-n} (X^{i,j+1}_{n} - X^{i,j}_{n})} e^{iP^{i,j}_{n} (X^{i,j+1}_{-n} - X^{i,j}_{-n})}, 
\label{a4}
\feqr
\beqr
<X^{i, j+1}_{\pm n}|P^{i,j}_{\pm n}>=\frac{1}{2\pi}e^{iP^{i,j}_n X^{i,j+1}_{-n}}e^{iP^{i,j}_{-n} X^{i,j+1}_{n}}
\label{a4.5}
\feqr
\beqr
<X^{i,j+1}_{\pm n}|\hat{H}^i_n|X^{i,j}_{\pm n}> 
&=& \int  dP^{i,j}_n dP^{i,j'}_n dP^{i,j}_{-n} dP^{i,j'}_{-n} <X^{i,j+1}_{\pm n}|P^{i,j'}_{\pm n}>
<P^{i,j'}_{\pm n}|\hat{P}^i_n \hat{P}^i_{-n}|P^{i,j}_{\pm n}>
<P^{i,j}_{\pm n}|X^{i,j}_{\pm n}> \non \\
&+& (\omega^i_n)^2 <X^{i,j+1}_{\pm n}|\hat{X}^i_n \hat{X}^i_{-n}|X^{i,j}_{\pm n}> \non \\
&=& \frac{1}{(2\pi)^2} \int dP^{i,j}_n dP^{i,j}_{-n}
e^{iP^{i,j}_{-n} (X^{i,j+1}_{n} - X^{i,j}_{n})} e^{iP^{i,j}_{n} (X^{i,j+1}_{-n} - X^{i,j}_{-n})}
H^i_n (P^{i,j}_{\pm n}, X^{i,j}_{\pm n})
\label{a5}
\feqr
\noi and similarly
\beqr
&& <X^{i,j+1}_{\pm n}|\hat{\Pi}^i_n|X^{i,j}_{\pm n}> = in \int dP^{i,j}_n dP^{i,j'}_n dP^{i,j}_{-n} dP^{i,j'}_{-n}\non \\ 
&& \Biggl[ X^{i,j}_{n} <X^{i,j+1}_{\pm n}|P^{i,j'}_{\pm n}>
<P^{i,j'}_{\pm n}|\hat{P}^i_{-n}|P^{i,j}_{\pm n}>
<P^{i,j}_{\pm n}|X^{i,j}_{\pm n}> \non \\
&-& X^{i,j}_{-n} <X^{i,j+1}_{\pm n}|P^{i,j'}_{\pm n}>
<P^{i,j'}_{\pm n}|\hat{P}^i_{n}|P^{i,j}_{\pm n}>
<P^{i,j}_{\pm n}|X^{i,j}_{\pm n}> \Biggl] \non \\
&=&  \frac{1}{(2\pi)^2} \int dP^{i,j}_n dP^{i,j}_{-n}
e^{iP^{i,j}_{-n} (X^{i,j+1}_{n} - X^{i,j}_{n})} e^{iP^{i,j}_{n} (X^{i,j+1}_{-n} - X^{i,j}_{-n})}
\Pi^i_n (P^{i,j}_{\pm n}, X^{i,j}_{\pm n})
\label{a6}
\feqr
\noi Putting together \rf{a4}, \rf{a5} and \rf{a6} we get 
\beqr
&&<X^{i,j+1}_{\pm n}|e^{-i\hat{H}^i_n \delta t} e^{i\hat{\Pi}^i_n \delta \sigma} |X^{i,j}_{\pm n} > \non \\
&=& \frac{1}{(2\pi)^2} \int dP^{i,j}_n dP^{i,j}_{-n}
e^{iP^{i,j}_{-n} (X^{i,j+1}_{n} - X^{i,j}_{n})} e^{iP^{i,j}_{n} (X^{i,j+1}_{-n} - X^{i,j}_{-n})}
\left( 1- i\delta t H^i_n (P^{i,j}_{\pm n}, X^{i,j}_{\pm n}) + i\delta \sigma
\Pi^i_n (P^{i,j}_{\pm n}, X^{i,j}_{\pm n})\right) \non \\
&=& \frac{1}{(2\pi)^2} \int dP^{i,j}_n dP^{i,j}_{-n}
e^{iP^{i,j}_{-n} (X^{i,j+1}_{n} - X^{i,j}_{n})} e^{iP^{i,j}_{n} (X^{i,j+1}_{-n} - X^{i,j}_{-n})}
e^{- i\delta t H^i_n (P^{i,j}_{\pm n}, X^{i,j}_{\pm n}) }e^{i\delta \sigma 
\Pi^i_n (P^{i,j}_{\pm n}, X^{i,j}_{\pm n})} + quad.
\label{a7}
\feqr
\noi Now, since the momentum and the Hamiltonian commute, the partition function is independent of the path by which 
we join $\sigma$, $t-\pi\tau_2$ to $\sigma + 2\pi\tau_1$, $t+\pi\tau_2$ so we can take this path to be a straight line.
Hence we have that $\delta \sigma = \frac{\tau_1}{\tau_2} \delta t$ so 
\beqr
Z^i_n &\equiv& <X^{i,final}_n, \sigma + 2\pi \tau_1, t+\pi \tau_2|X^{i,initial}_n, \sigma, t -\pi \tau_2> \non \\
&=& C \int \prod_j dX^{i,j}_{n} dX^{i,j}_{-n}dP^{i,j}_{n} dP^{i,j}_{-n} 
e^{iP^{i,j}_{-n} (X^{i,j+1}_{n} - X^{i,j}_{n})} e^{iP^{i,j}_{n} (X^{i,j+1}_{-n} - X^{i,j}_{-n})} \non \\
&& e^{- i\delta t H^i_n (P^{i,j}_{\pm n}, X^{i,j}_{\pm n}) }e^{i \frac{\tau_1}{\tau_2}\delta t 
\Pi^i_n (P^{i,j}_{\pm n}, X^{i,j}_{\pm n})}. 
\label{a8}
\feqr
\noi In the continuum limit this becomes
\beqr
Z^i_n = C \int DX^{i}_{n} DX^{i}_{-n}DP^{i}_{n} DP^{i}_{-n} e^{i\int \left(P^i_{-n}\dot{X}^i_n + P^i_{n}\dot{X}^i_{-n}
-H^i_n (P^{i}_{\pm n}, X^{i}_{\pm n})+ \frac{\tau_1}{\tau_2} 
\Pi^i_n (P^{i}_{\pm n}, X^{i}_{\pm n}) \right)dt}. 
\label{a9}
\feqr
\noi Analytically continuing to imaginary time by letting $t\rightarrow -it$, $\tau_2 \rightarrow -i\tau_2$ and 
performing the $P^i_{\pm n}$ integrations we get
\beqr
Z^i_n &=& C \int DX^i_n DX^i_{-n} e^{-\int X^i_n \left(-\frac{d^2}{dt^2}+2in\frac{\tau_1}{\tau_2}\frac{d}{dt} + n^2 
\frac{\tau_1^2}{\tau_2^2} + (\omega^i_n)^2 \right) X^i_{-n} dt } \non \\
&=& C det^{-1}\left(-\frac{d^2}{dt^2}+2in\frac{\tau_1}{\tau_2}\frac{d}{dt} + n^2 
\frac{\tau_1^2}{\tau_2^2} + (\omega^i_n)^2 \right). 
\label{a10}
\feqr 
\par For the zero mode $X^i_0$ formula \rf{a10} is no longer valid because we do not have a complex pair of modes, 
but instead we have a real mode. Doing a similar calculation we get
\beqr
Z^i_0 &=& C \int DX^i_0 e^{-\frac{1}{2}\int X^i_0 \left(-\frac{d^2}{dt^2}+ (\omega^i_0)^2 \right) X^i_{0} dt } 
= C det^{-\frac{1}{2}}\left(-\frac{d^2}{dt^2}+ (\omega^i_0)^2 \right). 
\label{a11}
\feqr  


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
\addcontentsline{toc}{subsection}{Appendix B }
\section*{Appendix B }
\renewcommand{\theequation}{B.\arabic{equation}}
\setcounter{equation}{0}

Consider now the operator
\beqr
D_{(n+a)} = -\frac{d^2}{dt^2}+2i(n+a)\frac{\tau_1}{\tau_2}\frac{d}{dt} + (n+a)^2 
\frac{\tau_1^2}{\tau_2^2} + \tilde{\omega}^2_{(n+a)} 
\label{b1}
\feqr
\noi acting on periodic functions on $(-\pi \tau_2, \pi \tau_2)$. The basis of eigenfuctions is 
\beqr
f_m(t)=e^{im\frac{t}{\tau_2}}, \quad m\in Z.
\label{b2}
\feqr 
\noi The corresponding eigenvalues are
\beqr
\lambda_{m}^{n+\alpha} = \frac{\left(m-(n+a)\tau_1 \right)^2}{\tau^2_2}+\tilde{\omega}^2_{(n+a)}.
\label{b3}
\feqr
\noi In computing the determinant of $D_n$ we are going to use the zeta function regularization. This means that 
\beqr
\prod_{n=-\infty}^{\infty} \alpha =\alpha^{2\zeta (0)+1}=1, \qquad 
\prod_{n=1}^{\infty} n^{\alpha}= e^{-\alpha \zeta'(0)}= (2\pi)^{\frac{\alpha}{2}} 
\label{b4}
\feqr 
\noi and
\beqr
\prod_{n=-\infty}^{\infty}(n+\alpha)=\alpha \prod_{n=1}^{\infty} (-n^2)\left(1-\frac{\alpha^2}{n^2} \right)
=2i\sin \pi \alpha.
\label{b5}
\feqr
\noi Now 
\beqr
det(D_{(n+a)}) &=& \prod_{m \in Z} \lambda^{n+\alpha}_m = \prod_{m \in Z} \frac{\tau^2_2 \tilde{\omega}^2_{(n+a)} 
+ (m-(n+a)\tau_1)^2}{\tau^2_2} \non \\
&=& \prod_{m \in Z} (m-(n+a)\tau_1 -i\tilde{\omega}_{(n+a)}\tau_2 )(m-(n+a)\tau_1 +i\tilde{\omega}_{(n+a)}\tau_2 ) \non \\
&=& -4\sin [\pi((n+a)\tau_1 + i\tilde{\omega}_{(n+a)} \tau_2)]
\sin [\pi((n+a)\tau_1 - i\tilde{\omega}_{(n+a)} \tau_2)] \non \\
&=&-e^{2\pi \tilde{\omega}_{(n+a)}\tau_2}\left( 1-e^{-2\pi \tilde{\omega}_{(n+a)}\tau_2+2i\pi(n+a)\tau_1}\right)
\left( 1-e^{-2\pi \tilde{\omega}_{(n+a)}\tau_2- 2i\pi (n+a)\tau_1}\right).
\label{b6}
\feqr

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

\addcontentsline{toc}{subsection}{Appendix C }
\section*{Appendix C }
\renewcommand{\theequation}{C.\arabic{equation}}
\setcounter{equation}{0}

It is possible to define a deformed version of the product $\eta (\tau) \overline{\eta (\tau)}$ where $\eta (\tau)$ 
is the Jacobi $\eta$-function in such a way that it transforms simply under modular transformations. Define
\beqr
\hat{\eta}_c(\tau, \bar{\tau})= e^{2\pi \tau_2 \Delta(c)}\prod_{n \in Z}
(1- e^{-2\pi \tau_2 \sqrt{n^2 + c^2} + 2i\pi n \tau_1})
\label{c1}
\feqr
\noi where 
\beqr
\Delta(c)=-\frac{1}{2\pi^2}\sum_{n=1}^{\infty}\int_{0}^{\infty}ds e^{-n^2 s - \frac{c^2 \pi^2}{s}}.
\label{c2}
\feqr
\noi The modular properties satisfied by $\hat{\eta}_{c}(\tau , \bar{\tau}) $ are given by 
\beqr
\label{c2.5} 
\hat{\eta}_{c}(\tau +1, \bar{\tau}+1) &=&  \hat{\eta}_{c}(\tau, \bar{\tau})  \non \\
\hat{\eta}_{c}(-\frac{1}{\tau}, -\frac{1}{\bar{\tau}}) &=&  \hat{\eta}_{c/|\tau|}(\tau, \bar{\tau}).
\feqr  
\noi Note that in particular $\Delta(0)=-1/12$. When $c\rightarrow 0$, $\hat{\eta}_c(\tau, \bar{\tau})\rightarrow 0$, 
but it is possible to regularize it by dropping the factor that goes to zero. In this way we obtain 
\beqr
\hat{\eta}^R_0 (\tau, \bar{\tau})\equiv \lim_{c\rightarrow 0} 
\frac{\hat{\eta}_c(\tau, \bar{\tau})}{(1-e^{-2\pi c \tau_2})}= \eta(\tau) \overline{\eta (\tau)}.
\label{c3}
\feqr 
\noi It is also possible to define deformed $\theta$-function bilinears. Define
\beqr
\hat{\Theta}_c \left[ \begin{array}{c} a \\ b \end{array} \right] (\tau, \bar{\tau})&=& 
e^{2\pi\tau_2 \hat{\Delta}_{a+\frac{1}{2}}(c)}
\prod_{n \in Z}(1-e^{-2\pi \tau_2 \sqrt{n^2 +c^2} + 2i\pi n \tau_1}) \non \\
&& (1-e^{-2\pi \tau_2 \sqrt{(n+a+1/2)^2 +c^2} + 2i\pi (n+a+1/2) \tau_1 + 2i\pi (b+\frac{1}{2})}) \non \\
&& (1-e^{-2\pi \tau_2 \sqrt{(n+a+1/2)^2 +c^2} - 2i\pi (n+a+1/2) \tau_1 - 2i\pi (b+\frac{1}{2})})
\label{c4}   
\feqr
\noi where 
\beqr
\hat{\Delta}_a (c)= -\frac{1}{2\pi^2} \sum_{n=1}^{\infty}\int_{0}^{\infty} ds 
e^{-n^2 s - \frac{c^2 \pi^2}{s}}(1+2\cos(2\pi a n)).
\label{c5}
\feqr
\noi The modular transformations of the deformed $\Theta$-functions are given by
\beqr
\hat{\Theta }_c \left[ \begin{array}{c} a \\ b \end{array} \right] (-\frac{1}{\tau}, -\frac{1}{\bar{\tau}})&=&
\hat{\Theta }_{\frac{c}{|\tau|}} \left[ \begin{array}{c} b \\ -a \end{array} \right] (\tau, \bar{\tau}) \non \\
\hat{\Theta }_c \left[ \begin{array}{c} a \\ b \end{array} \right] (\tau +1, \bar{\tau} +1)&=&
\hat{\Theta }_{c} \left[ \begin{array}{c} a \\ a+b+\frac{1}{2} \end{array} \right] (\tau, \bar{\tau}).
\label{c6}
\feqr
\noi Again, when $c\rightarrow 0$, $\hat{\Theta}_c \left[ \begin{array}{c} a \\ b \end{array} \right] 
(\tau, \bar{\tau})\rightarrow 0$, but it is possible to regularize it by dropping the factor that goes to zero. 
In this way we obtain  
\beqr
\hat{\Theta}_0^R  \left[ \begin{array}{c} a \\ b \end{array} \right] (\tau, \bar{\tau}) =
\lim_{c\rightarrow 0} \frac{\hat{\Theta}_c  \left[ \begin{array}{c} a \\ b \end{array} \right] (\tau, \bar{\tau}) }
{1-e^{-2\pi c \tau_2}}.
\label{c7}
\feqr
\noi One can easily prove that this regularized deformed $\Theta$-function is given by
\beqr
\hat{\Theta}_0^R  \left[ \begin{array}{c} a \\ b \end{array} \right] (\tau, \bar{\tau}) =
\Theta  \left[ \begin{array}{c} a \\ b \end{array} \right] (0, \tau)
\overline{\Theta  \left[ \begin{array}{c} a \\ b \end{array} \right] (0, \tau)}  
\label{c8}
\feqr
\noi where the $\Theta$-functions that appear on the righthand side of \rf{c8} are the usual theta functions with 
characteristics. It is worth mentioning that in this context the factor $\sqrt{-i\tau}$ that appears in the modular 
transformation properties of the usual $\Theta$-functions can be attributed to the necessary regularization when 
$c\rightarrow 0$ since 
\beqr
\hat{\Theta}_0^R  \left[ \begin{array}{c} a \\ b \end{array} \right] (-\frac{1}{\tau}, -\frac{1}{\bar{\tau}}) =
\lim_{c\rightarrow 0} \frac{1-e^{-2\pi\frac{c}{|\tau|}\tau_2}}{1-e^{-2\pi c \frac{\tau_2}{|\tau|^2} }}
\hat{\Theta}_0^R  \left[ \begin{array}{c} b \\ -a \end{array} \right] (\tau, \bar{\tau}) =
|\tau|\hat{\Theta}_0^R  \left[ \begin{array}{c} b \\ -a \end{array} \right] (\tau, \bar{\tau}).
\label{c9}
\feqr 
\noi Furthermore the deformed $\Theta$-functions have the following symmetry properties
\beqr
\hat{\Theta}_c\left[ \begin{array}{c} a+1 \\ b \end{array} \right] (\tau, \bar{\tau})=
\hat{\Theta}_c\left[ \begin{array}{c} a \\ b+1 \end{array} \right] (\tau, \bar{\tau})=
\hat{\Theta}_c\left[ \begin{array}{c} a \\ b \end{array} \right] (\tau, \bar{\tau})
\label{c10}
\feqr 
\noi and 
\beqr
\hat{\Theta}_c\left[ \begin{array}{c} -a \\ -b \end{array} \right] (\tau, \bar{\tau})=
\hat{\Theta}_c\left[ \begin{array}{c} a \\ b \end{array} \right] (\tau, \bar{\tau}).
\label{c11}
\feqr 
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\bibliographystyle{plain}
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\end{document}