Correlation Functions in ${\textrm{T}\bar{\textrm{T}}}$-deformed Theories on
the Torus

Netanel Barel

_Department of Particle Physics and Astrophysics, Weizmann Institute of
Science,
Rehovot 7610001, Israel_

e-mail<EMAIL_ADDRESS>
###### Abstract

We study the correlation functions of local operators in unitary
${\textrm{T}\bar{\textrm{T}}}$-deformed field theories defined on a torus,
using their formulation in terms of Jackiw-Teitelboim gravity. We focus on the
two-point correlation functions in momentum space when the undeformed theory
is a conformal field theory. The large momentum behavior of the correlation
functions is computed and compared to that of
${\textrm{T}\bar{\textrm{T}}}$-deformed field theories defined on a plane. For
the latter, the behavior found was $|q|^{-\frac{tq^{2}}{\pi}}$, where $q$ is
the momentum and $t$ is the deformation parameter. For a torus, the same
behavior is found for $|q|<<L/t$, where $L$ is the torus’ length scale.
However, for $|q|>>L/t$, a different behavior is found:
$\left(\frac{4\sqrt{t}^{5}q^{2}}{\pi
eL^{3}|T|^{2}}\right)^{\frac{tq^{2}}{\pi}}$, where $T$ is the modular
parameter of the torus. Hence, at large momentum, the correlator decays and
then grows. This behavior suggests that operators carrying momentum $q$ are
smeared on a distance scale $t|q|$. The difference from the plane’s result
illustrates the non-locality of the theory and the UV-IR mixing.

###### Contents

  1. 1 Introduction and Summary
  2. 2 Preliminaries
    1. 2.1 JT-formalism
    2. 2.2 Torus’ Notations
  3. 3 PI-Calculation
    1. 3.1 Gauge-Fixing
    2. 3.2 Target-Space Coordinate PI
    3. 3.3 Conformal-factor and Orientation PI
    4. 3.4 Vielbein solution
    5. 3.5 Undeformed Fields PI
  4. 4 Saddle-point Evaluation
    1. 4.1 Constant Conformal-factor and Orientation
    2. 4.2 Modular-parameter and Coordinate
      1. 4.2.1 Plane’s Limit
      2. 4.2.2 Torus’ Limit
    3. 4.3 Intermediate Region
  5. A Modular Transformations
  6. B Determinants
    1. B.1 P
    2. B.2 Q
  7. C Special Functions
    1. C.1 Theta-function
    2. C.2 Eta-function
    3. C.3 G-function
  8. D Weyl Anomaly
  9. E Saddle on The Complex Plane
    1. E.1 contour integration
    2. E.2 Gaussian Integral
    3. E.3 Plane
    4. E.4 Torus

##  1 Introduction and Summary

In recent years it was discovered that in 1+1 dimensional space-time, special
types of irrelevant deformations of quantum field theories (QFTs) are under
control and even solvable [1, 2]. One of them is the
${\textrm{T}\bar{\textrm{T}}}$-deformation, which is the deformation
investigated in this paper. A review of this topic with a complete list of
references can be found in [3]. The ${\textrm{T}\bar{\textrm{T}}}$-operator is
defined as the determinant of the energy-momentum tensor using the following
normalization:

$T^{\alpha\beta}\equiv\frac{2}{\sqrt{g}}\frac{\delta S}{\delta
g_{\alpha\beta}}\Big{|}_{g=\delta},$ (1)

where Euclidean signature is being used throughout this work. $\sigma^{1,2}$
are coordinates in the two dimensional space, and $\delta$ is the flat metric.
The ${\textrm{T}\bar{\textrm{T}}}$-operator is
${\textrm{T}\bar{\textrm{T}}}\equiv\det\left(T_{\alpha\beta}\right)$.111In the
action this operator is integrated over space. One can show that there is no
short-distance singularity when bringing together the different components of
$T_{\alpha\beta}$. For each value of the deformation parameter $t$, one
defines the deformed Lagrangian at $t+dt$ in terms of the deformed Lagrangian
at $t$:

$\mathcal{L}^{(t+dt)}=\mathcal{L}^{(t)}+{\textrm{T}\bar{\textrm{T}}}^{(t)}dt,$
(2)

where ${\textrm{T}\bar{\textrm{T}}}^{(t)}$ is obtained using the energy-
momentum tensor derived from $S^{(t)}$.222The coefficients in the definitions
of the ${\textrm{T}\bar{\textrm{T}}}$-operator and the flow equation are
different from paper to paper, and one has to be careful when comparing
quantitative results from different papers. Only positive $t$ will be
considered in this work, for which the deformed theory was found to be
unitary.333At least at large enough volume the energies are real.

The energy levels of the theory on a circle, and thus the torus partition
function and the S-matrix, are explicitly known for any $t$. However, not much
is known about correlators and even what are the appropriate operators in the
deformed theory. Using this definition of the deformation, which is, in a
sense, perturbative, one can use the unperturbed operators as operators in the
deformed theory and calculate their correlation functions.444Correlation
functions of a deformed conformal field theory (CFT) were calculated for small
deformation parameter by perturbative methods, see [4, 5, 6, 7, 8, 9, 10, 11,
12, 13, 14], with specific attention to the correlation functions of the
components of the energy-momentum tensor, see [15, 16, 17, 18].,555Another
approach to defining operators and correlation functions for
J$\bar{\textrm{T}}$-deformed theories was suggested in [19, 20]. Other
possibilities for defining operators for
${\textrm{T}\bar{\textrm{T}}}$-deformed theories were given in [21, 22]. In
[21], the geometry of a cylinder is also discussed in addition to that of a
plane. In particular, the two-point correlation function for large deformation
parameter was calculated at large momentum in [23] at (4.27) (written in our
notations, keeping the leading contribution only)666Similar result appeared in
[24], using the random geometry approach to the
${\textrm{T}\bar{\textrm{T}}}$-deformation:

$\left(\frac{tq^{2}}{2\pi}\right)^{\frac{tq^{2}}{2\pi}}.$ (3)

A non-perturbative definition for the
${\textrm{T}\bar{\textrm{T}}}$-deformation was suggested in [25]. The
suggestion claims that ${\textrm{T}\bar{\textrm{T}}}$-deformed theories can be
described using flat space Jackiw-Teitelboim (JT) gravity.777Using this
description, the S-matrix, the spectrum on a circle, and the partition
function on the torus were reproduced, see [25, 26]. These calculations
confirm this suggestion. The JT-description also gives a natural suggestion
for what the operators in the deformed theory should be. In the previous paper
[27], correlation functions of these operators were calculated. Like the
result in the perturbative description (3), the need for a momentum-dependent
renormalization showed up,888Momentum-dependent renormalization implies that
one should not consider the momentum-dependent operator as the Fourier
transform of local operator, but rather as fundamental operators. as well as
the independence of the IR CFT data at the UV. However, the behavior of the
two-point correlator at large momentum turned out to be very different from
the perturbative result (3). Rather than exponential growth, it presented an
exponential decay999(115) in [27], with $\mu=\frac{1}{\sqrt{t}}$, see
subsection 4.2. Similar results appeared also in [28, 20] using different
frameworks.:

$\left(\frac{tq^{2}}{\pi e}\right)^{-\frac{tq^{2}}{2\pi}}.$ (4)

Both results emphasize the non-local nature of the
${\textrm{T}\bar{\textrm{T}}}$-deformation, as the high momentum limit of
correlators is not a power law $|q|^{2\alpha}$. Power low behavior is a
characteristic of local field theories, for which their UV-limit is expected
to be a CFT.

Other features of (4), namely the momentum-dependent renormalization, the
independence of the initial IR data, and the exponential decay, resemble
results that appeared in non-commutative gauge-theories [29]. Therefore, the
same interpretation for the latter also seems adequate here, namely that the
decay of the correlator results from the smearing of operators, which is of
scale $t|q|$. As we will see, a similar conclusion will appear in this work.

The calculation in [27] was performed for a theory living in the infinite two-
dimensional plane (just “plane” from now on). In this paper, we calculate the
two-point correlator for a theory living in the torus. In a local QFT, for
which the UV limit is a CFT, the small distance limit of correlators is the
same whether the spacetime is a plane, a torus, or any other two-dimensional
Riemannian manifold. However, for non-local theories, global aspects of
spacetime may affect local properties, a phenomenon known as UV-IR mixing.
Therefore, comparing the torus’ correlator to the plane’s correlator may
reveal another non-local aspect of ${\textrm{T}\bar{\textrm{T}}}$-deformed
theories.101010In a CFT, small distances correspond to high momentum, and vice
versa. In non-local theories, the last statement is no longer true. In
particular, in ${\textrm{T}\bar{\textrm{T}}}$-deformed theory, the fundamental
operators are defined in momentum space, and so we compare high momentum
behavior rather than small distance behavior.

In the plane, the deformation scale $t$ is the only scale in the deformed
theory. In contrast, in the torus, an additional scale appears, which is the
torus’ length scale $L$.111111Generally, there are two different scales for
the two edges of the torus, $L_{1}\equiv L$ and $L_{2}\sim LT_{2}$, where
$T_{2}$ is part of the complex structure of the torus $T=T_{1}+iT_{2}$. We
consider the generic case where $T_{2}\sim 1$. The relation between the two
scales is a dimensionless quantity $l^{2}\equiv L^{2}/t$, which can be
compared to the dimensionless momentum scale $n\sim|q|L$. In this terminology,
high momentum in the plane means $tq^{2}=\frac{n^{2}}{l^{2}}>>1$, and (4)
reads:

$\left(\frac{nr}{\pi e}\right)^{-\frac{nr}{2\pi}},$ (5)

where $r\equiv\frac{n}{l^{2}}\sim\frac{t|q|}{L}$. $r$ is the ratio between the
length scale associated with the smeared operators and the torus’ length
scale. It turns out that $r$ dictates the high momentum limit (with respect to
the deformation scale, i.e. $n>>l$) in the torus. At small $r$, the operators
are smeared on a length scale which is much smaller than $L$.121212But both
the area and the momentum are large; namely, we consider the following
hierarchy $1<<l<<n<<l^{2}$. Then, the plane’s result is recovered. The
interesting result comes at large $r$:

$\left(\frac{4\sqrt{t}^{5}q^{2}}{\pi
eL^{3}|T|^{2}}\right)^{\frac{q^{2}}{\pi|\Lambda|}}=\left(\frac{4r^{2}}{e\pi
l|T|^{2}}\right)^{\frac{nr}{\pi}}.$ (6)

This result implies that at very large momentum the correlator actually
diverges. In addition, this UV-result contains IR-quantities $L^{3},|T|^{2}$,
and therefore shows a UV-IR mixing.

The results at very small and very large $r$ suggest the following
interpretation. The deformed operators are smeared over a length scale $t|q|$.
For $r<<1$, this length scale is much smaller than $L$, and the correlator
decreases, as for the plane. However, at large enough momentum, the operators
will cover the whole torus, and the correlator need not decay from that point
on. In particular, (6) shows that the correlator grows.

The transition between these two $r$ limits is hard to calculate analytically,
and we found it numerically in the special case of a rectangular torus and
momentum aligned along one of its axes. The result is shown in figure 2,
decreasing at small $r$ and increasing at large $r$, as expected from the
analytic calculation.

The main text of this paper contains the detailed calculation leading to (6).
Its structure is the following. In section 2, we review the JT-formalism and
present the general quantities and notations needed for a torus geometry. In
section 3, we calculate the path integral (PI) leading to a simple integral,
out of which the large momentum limit can be evaluated. The evaluation of this
PI follows similar lines to the evaluation done in [27]. We gauge-fix the
diffeomorphism invariance by using the orthogonal gauge for the metric. Then,
some formal manipulations will be done on the gauge-fixed expression.
Regularization and renormalization procedures will be carried out to make the
expression well-defined and finite. In section 4, we implement the saddle
point approximation to the remaining integral and derive the final result (6).

Appendix A is a reminder of modular transformations, with specific attention
to what is needed in the paper. Appendix B is devoted to determinants
appearing during the evaluation of the PI. In appendix C, we describe briefly
the properties of the $\eta(\tau)$ and $\Theta(w,\tau)$ functions, as well as
those of the propagator $G$. In appendix D, the Weyl anomaly appearing in the
calculation is discussed. Finally, in appendix E, the method of saddle point
approximation in the complex plane and its usage in the paper are discussed.

##  2 Preliminaries

###  2.1 JT-formalism

The action for the ${\textrm{T}\bar{\textrm{T}}}$-deformed theory in the JT-
description is given by:

$\begin{split}S_{\textrm{T}\bar{\textrm{T}}}(\psi,g_{\alpha\beta},\varphi)&=S_{0}(\psi,g_{\alpha\beta})+S_{JT}(g_{\alpha\beta},\varphi),\\\
S_{JT}(g_{\alpha\beta},\varphi)&=\int d^{2}\sigma\sqrt{g}\left(\varphi
R-\Lambda\right).\end{split}$ (7)

In this equation, $\psi$ represents the field content of the undeformed theory
and $S_{0}(\psi,g_{\alpha\beta})$ its action. The action is coupled to a
dynamical metric $g_{\alpha\beta}$, and not to the dilaton field $\varphi$.
For a general QFT, the coupling to the metric may be chosen by the minimal
coupling. However, in the case where the undeformed theory is a CFT, as in
this paper, a more natural coupling is chosen, such that
$S_{0}(\psi,g_{\alpha\beta})$ admits Weyl invariance.131313The path-integral,
however, has a Weyl anomaly, see subsection 3.5 and appendix D. The energy
density in the vacuum $\Lambda$ is connected to the
${\textrm{T}\bar{\textrm{T}}}$-deformation parameter $t$ via:

$t=-\frac{1}{\Lambda}.$ (8)

In this formalism, the unitary deformed theory is found for
$\Lambda=-|\Lambda|<0$, and the undeformed theory is recovered in the limit
$|\Lambda|\to\infty$.

The JT-gravity action (7) has an equivalent formulation using vielbein
variables $e^{a}_{\alpha}$ and a spin-connection $\omega_{\alpha}$, instead of
the metric components
$g_{\alpha\beta}=e^{a}_{\alpha}e^{b}_{\beta}\delta_{ab}$. One introduces
Lagrange multipliers $\lambda^{a}$ to force the spin-connection
$\omega_{\alpha}$ to be compatible with the metric. Defining
$X^{a}\equiv\Lambda^{-1}\epsilon^{a}_{b}\lambda^{b}$, simple manipulations on
(7) gives:

$\begin{split}S_{\textrm{T}\bar{\textrm{T}}}(\psi,e^{a}_{\alpha},X^{a})&=S_{0}(\psi,e^{a}_{\alpha})+S_{JT}(e^{a}_{\alpha},X^{a}),\\\
S_{JT}(e^{a}_{\alpha},X^{a})&=\frac{\Lambda}{2}\int
d^{2}\sigma\epsilon^{\alpha\beta}\epsilon_{ab}(\partial_{\alpha}X^{a}-e^{a}_{\alpha})(\partial_{\beta}X^{b}-e^{b}_{\beta}).\end{split}$
(9)

The fields $X^{a}(\sigma)$ play the role of dynamical coordinates of the
target-space (TS), which is identified with the space of the undeformed theory
[25, 26], i.e. a torus in the present paper. The world-sheet (WS), on which
the actions (7,9) are formulated, has the same topology as the TS, so it is
also a torus. The dynamical coordinates describe one winding of the WS torus
(WSt) around the TS torus (TSt). Using these dynamical coordinates, local
operators can be defined in the deformed theory:

$\mathcal{O}(X)\equiv\int
d^{2}\sigma\sqrt{g(\sigma)}O(\sigma)\delta\left(X(\sigma)-X\right),$ (10)

where $O(\sigma)$ is some operator in the undeformed theory.141414We consider
scalar operators for simplicity. If the undeformed operator has a spin,
additional vielbein should be inserted. For example, for the operator
$O^{\alpha}(\sigma)$, one defines: $\mathcal{O}^{a}(X)=\int
d^{2}\sigma\sqrt{g(\sigma)}e^{a}_{\alpha}(\sigma)O^{\alpha}(\sigma)\delta\left(X(\sigma)-X\right)$.
The vielbein changes the WS index to a TS index and keeps the diffeomorphism
invariance of the deformed operator. As it turns out, the Fourier transform of
these operators is more natural to use:

$\mathcal{O}(q)\equiv\int d^{2}Xe^{iqX}\mathcal{O}(X)=\int
d^{2}\sigma\sqrt{g(\sigma)}O(\sigma)\exp\left(iq\cdot X(\sigma)\right).$ (11)

In particular, $\mathcal{O}(q)$ will be multiplied with momentum-dependent
renormalization factor (see (68)), so they should not be viewed as a Fourier
transform of the position space operators, but rather as fundamental
operators, out of which spatial operators can be defined.

The correlation function in the JT-formalism is defined as follows:

$\begin{split}C(q_{1},q_{2})\equiv&\left<\mathcal{O}(q_{1})\mathcal{O}(q_{2})\right>=\frac{1}{Z_{{\textrm{T}\bar{\textrm{T}}}}}\int\frac{\mathcal{D}e\mathcal{D}X\mathcal{D}\psi}{V_{\textrm{diff}}}\mathcal{O}(q_{1})\mathcal{O}(q_{2})e^{-S_{\textrm{T}\bar{\textrm{T}}}},\\\
Z_{{\textrm{T}\bar{\textrm{T}}}}\equiv&\int\frac{\mathcal{D}e\mathcal{D}X\mathcal{D}\psi}{V_{\textrm{diff}}}e^{-S_{\textrm{T}\bar{\textrm{T}}}}.\end{split}$
(12)

The partition function $Z_{{\textrm{T}\bar{\textrm{T}}}}$ was already
calculated in [26]. It will be omitted from now on since it does not depend on
the momentum. Plugging the definition of the deformed operators (11) inside
(12) gives:

$C(q_{1},q_{2})=\frac{1}{Z_{{\textrm{T}\bar{\textrm{T}}}}}\int\frac{\mathcal{D}e\mathcal{D}X\mathcal{D}\psi}{V_{\textrm{diff}}}\int
d^{2}\sigma_{1}d^{2}\sigma_{2}\sqrt{g(\sigma_{1})}\sqrt{g(\sigma_{2})}O(\sigma_{1})O(\sigma_{2})e^{iq_{1}\cdot
X(\sigma_{1})+iq_{2}\cdot X(\sigma_{2})-S_{\textrm{T}\bar{\textrm{T}}}}.$ (13)

###  2.2 Torus’ Notations

For the WSt we present periodic Cartesian coordinates:

$\sigma^{1}\sim\sigma^{1}+1,\ \sigma^{2}\sim\sigma^{2}+1.$ (14)

The line element in the orthogonal gauge depends on the conformal factor
$e^{\Omega}$ and the complex structure of the WSt
$\tau\equiv\tau_{1}+i\tau_{2}$:

$ds^{2}=e^{2\Omega}\left((d\sigma^{1}+\tau_{1}d\sigma^{2})^{2}+\left(\tau_{2}d\sigma^{2}\right)^{2}\right)=e^{2\Omega}|d\sigma^{1}+\tau
d\sigma^{2}|^{2}.$ (15)

From the line element, one reads the metric:

$\begin{split}[\bar{g}_{\alpha\beta}]_{\sigma}&=e^{2\Omega}\begin{pmatrix}1&\tau_{1}\\\
\tau_{1}&|\tau|^{2}\end{pmatrix}=e^{2\Omega}[\bar{g_{0}}_{\alpha\beta}]_{\sigma},\\\
[\left(g_{0}\right)_{\alpha\beta}]_{\sigma}&=M^{T}M,\\\
M&\equiv\begin{pmatrix}1&\tau_{1}\\\ 0&\tau_{2}\end{pmatrix}.\end{split}$ (16)

$[\left(g_{0}\right)_{\alpha\beta}]_{\sigma}$ is the flat metric of the
undeformed theory. The vielbein is given by:

$\begin{split}[\bar{e}^{a}_{\alpha}]_{\sigma}=e^{\Omega+\epsilon\phi}M=e^{\Omega}\begin{pmatrix}\cos(\phi)&\tau_{1}\cos(\phi)+\tau_{2}\sin(\phi)\\\
-\sin(\phi)&-\tau_{1}\sin(\phi)+\tau_{2}\cos(\phi)\end{pmatrix},\end{split}$
(17)

where $\epsilon_{ab}$ is the anti-symmetric symbol,
$\epsilon_{12}=-\epsilon_{21}=1,\epsilon_{11}=\epsilon_{22}=0$. The WS area
element is $d\bar{A}=\sqrt{g}d^{2}\sigma=e^{2\Omega}\tau_{2}d^{2}\sigma$.
After gauge-fixing the diffeomorphism invariance, all possible
$\Omega(\sigma),\phi(\sigma)$ and $\tau$ will be integrated.

It is convenient to present the complex coordinates:

$w\equiv\sigma^{1}+\tau\sigma^{2},\
\bar{w}\equiv\sigma^{1}+\bar{}\tau\sigma^{2}.$ (18)

The line element in these coordinates is $ds^{2}=dwd\bar{w}$, and the metric
and the vielbein are given by:

$\begin{split}[\bar{g}_{\alpha\beta}]_{w}&=e^{2\Omega}\begin{pmatrix}0&\frac{1}{2}\\\
\frac{1}{2}&0\end{pmatrix},\\\
[\bar{e}^{a}_{\alpha}]_{w}&=e^{\Omega+\epsilon\phi}\begin{pmatrix}\frac{1}{2}&\frac{1}{2}\\\
-\frac{i}{2}&\frac{i}{2}\end{pmatrix}.\end{split}$ (19)

Some useful relations to the Cartesian coordinates that will be used all along
the paper and are given here:

$\begin{split}d^{2}w&=2\tau_{2}d^{2}\sigma,\\\
\delta(w)&=\frac{1}{2\tau_{2}}\delta(\sigma),\\\ \begin{pmatrix}\partial\\\
\bar{\partial}\end{pmatrix}&=\frac{i}{2\tau_{2}}\begin{pmatrix}\bar{}\tau&-1\\\
-\tau&1\end{pmatrix}\begin{pmatrix}\partial_{1}\\\
\partial_{2}\end{pmatrix},\\\
4\partial\bar{\partial}&=\frac{1}{\tau_{2}^{2}}\left(|\tau|^{2}\partial_{1}^{2}+\partial_{2}^{2}-2\tau_{1}\partial_{1}\partial_{2}\right)=e^{2\Omega}g^{\alpha\beta}\partial_{\alpha}\partial_{\beta}.\end{split}$
(20)

We turn to the TSt. Two vectors $L_{\alpha}$ specifying it can be written as:

${L^{a}}_{\alpha}=\begin{pmatrix}X&XT_{1}+YT_{2}\\\
-Y&-YT_{1}+XT_{2}\end{pmatrix}.$ (21)

They are constant vielbein in the TS. $X,Y$ contains the information about the
length and orientation of $L_{1}$.151515In the WSt, instead of the “Cartesian”
variables $X,Y$, polar variables $e^{\Omega},\phi$ were used for the object
$e^{a}_{\alpha}$. On the WSt $e^{a}_{\alpha}$ change locally, while on the
flat TSt they can be defined globally $T\equiv T_{1}+iT_{2}$ is the modular
parameter of the TSt.

The momenta $q$ appearing in the Fourier transform on the TSt, maintain the
periodicity $X^{a}\sim X^{a}+L^{a}_{\alpha}$:

$e^{q_{a}L^{a}_{\alpha}}=1\Rightarrow q_{a}L^{a}_{\alpha}=2\pi n_{\alpha},$
(22)

for some integers $n_{\alpha}$. Momenta satisfying this condition are given by
the reciprocal lattice vectors:

$\begin{split}{\left(L^{-1}\right)^{\alpha}}_{a}&=\frac{1}{\mathcal{A}}\begin{pmatrix}-YT_{1}+XT_{2}&-XT_{1}-YT_{2}\\\
Y&X\end{pmatrix},\\\
q_{a}&=n_{\alpha}\left(L^{-1}\right)^{\alpha}_{a}.\end{split}$ (23)

One can choose $L_{1}$ to be along the x-axis $X=L,Y=0$, leading to:

$q=\frac{2\pi}{L}\left(n_{1}\left(1,-\frac{T_{1}}{T_{2}}\right)+n_{2}\left(0,\frac{1}{T_{2}}\right)\right)=\frac{2\pi}{L}\left(n_{1},\frac{-n_{1}T_{1}+n_{2}}{T_{2}}\right).$
(24)

The transformation $X^{a}(\sigma)=L^{a}_{\alpha}\sigma^{\alpha}$ maps the WSt
to the TSt, wrapping the last once. Many such different mappings are generated
by this map by applying a modular transformation on $T$ or $\tau$, but not on
both. Applying it for one of them, although not changing the torus, changes
the two TS cycles into which the two WS cycles $\sigma^{1},\sigma^{2}$ are
mapped. Applying the same modular transformation to $\tau$ and $T$ gives an
identical map, and it is a symmetry of the action (9). We will fix one modular
parameter $T$, and allow all possible modular parameters $\tau$ in the upper
half plane. We refer the reader to appendix A for more details.

##  3 PI-Calculation

The major part of this section is technical, and it is similar to section 3 of
[27]. The main difference is in subsection 3.4 regarding the vielbein solution
and, consequently, the expression for the WS-area. The reader who is not
interested in the technical details can read only this subsection, and skip to
the next section.

###  3.1 Gauge-Fixing

Our starting point is (13):

$\int\frac{\mathcal{D}e\mathcal{D}X\mathcal{D}\psi}{V_{\textrm{diff}}}\int
d\sigma_{1}d^{2}\sigma_{2}\sqrt{g(\sigma_{1})}\sqrt{g(\sigma_{2})}O(\sigma_{1})O(\sigma_{2})e^{iq_{1}\cdot
X(\sigma_{1})+iq_{2}\cdot X(\sigma_{2})-S_{\textrm{T}\bar{\textrm{T}}}}.$ (25)

The first stage is to gauge-fix the diffeomorphism invariance. We choose the
orthogonal gauge for the vielbein, written in subsection 2.2. There are two
remaining transformations not fixed by this choice. These are translations of
$\sigma^{\alpha}$, which are isometries of the gauged-fixed metric. We fix
them further by placing one of the operators at $\sigma_{1}^{\alpha}=0$. The
FP-identity takes the form:

$1=\int\mathcal{D}V(\sigma)\mathcal{D}\Omega(\sigma)\mathcal{D}\phi(\sigma)d^{2}\tau
J(e,\sigma_{1})\delta\left(e^{(V)}-\bar{e}(\Omega(\sigma),\phi(\sigma),\tau)\right)\delta\left(\sigma_{1}^{(V)}-0\right).$
(26)

The determinant $J$ is hard to calculate; see appendix B. However, it will not
be relevant for the high momentum limit. We do not have proof for that, but it
seems reasonable, as the determinant is a sense a local property, and expected
to be similar to the determinant calculated for the plane. The latter, indeed,
was independent of the momentum at high momentum.

Inserting the FP-identity into (25) and performing the usual FP-manipulations
gives:

$\int\mathcal{D}X\mathcal{D}\psi\mathcal{D}\Omega\mathcal{D}\phi d^{2}\tau
d\bar{A}(\sigma_{1})d\bar{A}(\sigma_{2})\delta(\sigma_{1})J(\bar{e},0)O(\sigma_{1})O(\sigma_{2})e^{iq_{1}\cdot
X(\sigma_{1})+iq_{2}\cdot X(\sigma_{2})-S_{0}-S_{JT}}.$ (27)

###  3.2 Target-Space Coordinate PI

To evaluate the PI over $X^{a}(\sigma)$, we decompose it into a “one winding”
mode (see subsection 2.2) and all other modes orthogonal to it:

$X^{a}(\sigma)=L^{a}_{\alpha}\sigma^{\alpha}+Y^{a}(\sigma).$ (28)

$Y^{a}(\sigma)$ are periodic functions on the WSt. Plugging this decomposition
into the $X$-dependent terms in (27) reads:

$e^{in_{1\alpha}\sigma_{1}^{\alpha}+in_{2\alpha}\sigma_{2}^{\alpha}}\int\mathcal{D}Ye^{iq_{1}\cdot
Y(\sigma_{1})+iq_{2}\cdot Y(\sigma_{2})+\Lambda\int
d^{2}\sigma\epsilon^{\alpha\beta}\epsilon_{ab}\bar{e}^{a}_{\alpha}(\sigma)\partial_{\beta}Y^{b}(\sigma)},$
(29)

where $n_{i\alpha}\equiv\left(q_{i}\right)_{a}L^{a}_{\alpha}$ are integers.
The remaining terms in $e^{-S_{\textrm{T}\bar{\textrm{T}}}}$ which are
independent of $Y$ are:

$e^{|\Lambda|\bar{A}}e^{|\Lambda|\mathcal{A}}e^{\Lambda\int
d^{2}\sigma\epsilon^{\alpha\beta}\epsilon_{ab}L^{a}_{\alpha}\bar{e}^{b}_{\beta}}.$
(30)

The $\mathcal{D}Y$ integration is ill-defined because the exponent is not a
pure phase. It will become a pure phase by Wick rotating $\bar{e}$, i.e.
$e^{\Omega}\rightarrow ie^{\Omega}$. In terms of $\Omega,\phi$, it sends
$\Omega\rightarrow\Omega+i\pi/2$, and does not change the measure
$\mathcal{D}\Omega\mathcal{D}\phi$. The $\mathcal{D}Y$-integration becomes
trivial and gives a $\delta$-function. We use a discretized notation to take
care of the $\delta$-functions appearing, which include momentum conservation.
The discretized measure is
$\mathcal{D}Y=\det\left(e^{2\Omega(\sigma)}\tau_{2}\right)\displaystyle\prod_{b=1}^{2}\prod_{\sigma=1}^{N}dY^{b}(\sigma)$,
and the $\mathcal{D}Y$-integration yields:

$\begin{split}e^{|\Lambda|\mathcal{A}}\delta(q_{1}+q_{2})\int&\mathcal{D}\psi\mathcal{D}\Omega\mathcal{D}\phi
d^{2}\tau
d\bar{A}(\sigma_{1})d\bar{A}(\sigma_{2})\delta(\sigma_{1})J(\bar{e},0)\det\left(e^{2\Omega}\tau_{2}\right)\prod_{b=1}^{2}\prod_{\sigma=1}^{N-1}\delta(K_{b}(\sigma))\cdot\\\
&O(\sigma_{1})O(\sigma_{2})e^{in_{1\alpha}\sigma^{\alpha}_{1}+in_{2\alpha}\sigma^{\alpha}_{2}-S_{0}(\psi,\bar{e})-|\Lambda|\left(\bar{A}+i\int
d^{2}\sigma\epsilon^{\alpha\beta}\epsilon_{ab}L^{a}_{\alpha}\bar{e}^{b}_{\beta}\right)},\\\
K_{b}(\sigma)\equiv&q_{1b}\delta_{\sigma,\sigma_{1}}+q_{2b}\delta_{\sigma,\sigma_{2}}-\Lambda\epsilon^{\alpha\beta}\epsilon_{ab}\partial_{\beta}\bar{e}^{a}_{\alpha}(\sigma).\end{split}$
(31)

We show it for the circle of length $L$ for simplicity. The analog of our
integral is:

$\int_{0}^{L}dY(1)\int_{-\infty}^{\infty}\prod_{i=2}^{N}dY(i)e^{i\sum_{j=1}^{N}f_{j}Y(j)}.$
(32)

The periodicity of $Y$ is implemented by the integration domain of $Y(1)$,
from $0$ to $L$. All the rest are not limited anymore. $f_{j}$ contains the
derivative of the vielbein plus the momenta $\frac{2\pi n_{j}}{L}$ ($n_{j}=0$
for all $N-2$ locations without momentum insertions). By changing variables
$Y(i)=Y(1)+Z(i)$ for $2\leq i\leq N$, the integral (32) becomes:

$\int_{0}^{L}dY(1)e^{iY(1)\sum_{j=1}^{N}f_{j}}\int_{-\infty}^{\infty}\prod_{i=2}^{N}dZ(i)e^{i\sum_{j=2}^{N}f_{j}Z(j)}.$
(33)

Since the derivative of the vielbein vanishes upon summation,
$\sum_{j=1}^{N}f_{j}=\frac{2\pi}{L}\sum_{j=1}^{N}n_{j}$. The integral over
$Y(1)$, although not extended to infinity, gives a Kronecker-delta:

$L\delta_{0,\sum_{j=1}^{N}n_{j}}\prod_{i=1}^{N-1}2\pi\delta(f_{j})=\det(2\pi)\delta\left(\sum_{i}q_{i}\right)\prod_{j=1}^{N-1}\delta(f_{j}),$
(34)

which is the analog of (31). In the last line we used the $\delta$-functions,
$\sum_{i=1}^{N}f_{i}=0$ and $f_{i}=0$ for $2\leq i\leq N$, and a continuous
$\delta\left(\sum_{i}q_{i}\right)$ instead of the Kronecker-delta. The
infinite factor $\det(2\pi)^{2}$ was omitted.

###  3.3 Conformal-factor and Orientation PI

The integral $\mathcal{D}\Omega\mathcal{D}\phi$ is easily performed because of
the constraint $K_{b}(\sigma)=0$. The constraint almost completely restricts
the possible $\Omega(\sigma),\phi(\sigma)$. Given one solution
$\Omega_{0},\phi_{0}$, all other solutions $f_{\Omega},f_{\phi}$ are given by:

$\begin{split}e^{f_{\Omega}+\epsilon f_{\phi}}&\equiv
e^{\Omega_{0}+\epsilon\phi_{0}}+e^{\bar{}\Omega+\epsilon\bar{\phi}}\Rightarrow\\\
f_{\Omega}(\Omega_{0},\bar{\Omega},\phi_{0},\bar{\phi})&\equiv\frac{1}{2}\ln\left(e^{2\Omega_{0}}+e^{2\bar{\Omega}}+2e^{\Omega_{0}+\bar{\Omega}}\cos(\phi_{0}-\bar{\phi})\right),\\\
f_{\phi}(\Omega_{0},\bar{\Omega},\phi_{0},\bar{\phi})&\equiv\tan^{-1}\left(\frac{e^{\Omega_{0}}\sin(\phi_{0})+e^{\bar{\Omega}}\sin(\bar{\phi})}{e^{\Omega_{0}}\cos(\phi_{0})+e^{\bar{\Omega}}\cos(\bar{\phi})}\right),\end{split}$
(35)

where $\bar{}\Omega,\bar{\phi}$ are constants. We discretize the
$\mathcal{D}\Omega,\mathcal{D}\phi$ measure as
$\det\left(e^{2\Omega(\sigma)}\tau_{2}\right)\displaystyle\prod_{\sigma=1}^{N}d\Omega(\sigma)d\phi(\sigma)$.
Out of the infinite integrals over $\Omega(\sigma),\phi(\sigma)$, only two
over $\bar{}\Omega,\bar{\phi}$ remains. Separating them properly from the
measure, for a general functional $\mathcal{F}$, gives:

$\int\left(\prod_{\sigma=1}^{N}d\Omega(\sigma)d\phi(\sigma)\right)\prod_{b=1}^{2}\prod_{\sigma^{\prime}=1}^{N-1}\delta(K_{b}(\sigma^{\prime}))\mathcal{F}(\Omega(\sigma),\phi(\sigma))=\int
d\bar{\Omega}d\bar{\phi}\sqrt{G^{\bar{\Omega},\bar{\phi}}}\frac{\mathcal{F}(f_{\Omega},f_{\phi})}{\sqrt{{\det}^{\prime}(Q^{\dagger}Q)}}.$
(36)

In (36), $Q(f_{\Omega},f_{\phi})$ denotes the linearization of the constraints
$K_{a}(\sigma)$ around the solution $f_{\Omega},f_{\phi}$, $\det^{\prime}$
excludes the two zero modes left to be integrated (the kernel of $Q$), and
$\sqrt{G^{\bar{\Omega},\bar{\phi}}}$ is the volume element in this two
dimensional zero subspace. $\sqrt{G^{\bar{\Omega},\bar{\phi}}}$ and
$Q^{\dagger}$ are calculated in appendix B. They are given by:

$\displaystyle\sqrt{G^{\bar{\Omega},\bar{\phi}}}$ $\displaystyle=\int
d^{2}\sigma e^{2\bar{}\Omega-2f_{\Omega}},$ (37)
$\displaystyle\sqrt{{\det}^{\prime}(Q^{\dagger}Q)}$
$\displaystyle=\det(e^{2\Omega}\tau_{2})\frac{\int d^{2}\sigma
e^{-2f_{\Omega}}}{\tau_{2}}\left|{\det}^{\prime}\left(\square\right)\right|,$
(38) $\displaystyle\square$
$\displaystyle\equiv-\frac{1}{\tau_{2}}(\partial+\partial\ln(\bar{z}))(\bar{\partial}+\bar{\partial}\ln(\bar{z})).$
(39)

Using (36) inside (31) yields:

$\begin{split}e^{|\Lambda|\mathcal{A}}\delta(q_{1}+q_{2})\int&\mathcal{D}\psi
d^{2}\tau
d\bar{\Omega}d\bar{\phi}d\bar{A}(\sigma_{1})d\bar{A}(\sigma_{2})\delta(\sigma_{1})\frac{\det(e^{2f_{\Omega}}\tau_{2})J(\bar{e},0)e^{2\bar{}\Omega}\tau_{2}}{\left|{\det}^{\prime}\left(\square\right)\right|}\cdot\\\
&O(\sigma_{1})O(\sigma_{2})e^{in_{1\alpha}\sigma^{\alpha}_{1}+in_{2\alpha}\sigma^{\alpha}_{2}-S_{0}(\psi,\bar{e})-|\Lambda|\left(\bar{A}+i\int
d^{2}\sigma\epsilon^{\alpha\beta}\epsilon_{ab}L^{a}_{\alpha}\bar{e}^{b}_{\beta}\right)}.\end{split}$
(40)

Since we performed the $\mathcal{D}\Omega,\mathcal{D}\phi$ integration,
$f_{\Omega},f_{\phi}$ will be denoted simply by $\Omega,\phi$.

###  3.4 Vielbein solution

$K_{b}(\sigma)=0$ are two differential equations for the vielbein:

$\begin{split}\partial_{1}{e^{2}}_{2}-\partial_{2}{e^{2}}_{1}&=-\sum_{i=1}^{2}\frac{(q_{i})_{1}}{|\Lambda|}\delta(\sigma-\sigma_{i}),\\\
-\partial_{1}{e^{1}}_{2}+\partial_{2}{e^{1}}_{1}&=-\sum_{i=1}^{2}\frac{(q_{i})_{2}}{|\Lambda|}\delta(\sigma-\sigma_{i}).\end{split}$
(41)

Adding the second equation multiplied by $i$ to the first equation gives:

$\begin{split}(-i\tau\partial_{1}+i\partial_{2})\bar{z}&=-2\pi\sum_{i=1}^{2}Q_{i}\delta(\sigma-\sigma_{i})\Rightarrow\\\
\bar{\partial}\bar{z}&=-2\pi\sum_{i}Q_{i}\delta(w-w_{i}).\end{split}$ (42)

where:

$\begin{split}\left(Q_{i}\right)_{a}&\equiv\frac{(q_{i})_{a}}{2\pi|\Lambda|},\\\
Q_{i}&\equiv\left(Q_{i}\right)_{1}+i\left(Q_{i}\right)_{2},\\\ z&\equiv
e^{\Omega+i\phi}.\end{split}$ (43)

We solve this equation using the Green function for the torus:

$\displaystyle\partial\bar{\partial}G$
$\displaystyle=-2\pi\delta(w)+\frac{\pi}{\tau_{2}},$ (44) $\displaystyle G$
$\displaystyle=-\ln\left|\Theta\left(w,\tau\right)\right|^{2}+\frac{2\pi\imaginary(w)^{2}}{\tau_{2}},$
(45)

where $\Theta\left(w,\tau\right)$ is one of the Jacobi Theta
functions.161616The first equation can be written in a diffeomorphism
invariant way:
$g^{\alpha\beta}\partial_{\alpha}\partial_{\beta}G=-4\pi\delta_{g}+\frac{4\pi}{\bar{A}}$,
with $\delta_{g}\equiv\frac{\delta^{2}(\sigma)}{\sqrt{g}}$. The second term is
a uniform charge density, making the total charge vanish. This is a
consequence of the Gauss law used in compact spaces. Solving for each source
$\delta(w-w_{i})$, and multiplying by $Q_{i}$, gives $\bar{z}$:

$\begin{split}Q_{i}\partial\bar{\partial}G_{i}&=-2\pi
Q_{i}\delta(w-w_{i})+\frac{\pi Q_{i}}{\tau_{2}}\Rightarrow\\\
\bar{z}&=\sum_{i}Q_{i}\partial G_{i}+e^{\bar{}\Omega+i\bar{\phi}}.\end{split}$
(46)

where in the last equation we used momentum conservation
$\sum_{i}Q_{i}=0$.171717Since $\sigma,\tau,w$ are dimensionless,
$e^{\bar{}\Omega+\bar{\phi}}$ should have mass dimension $-1$, like $Q_{i}$.
One could either multiply it by $L$ or declare $R=e^{\bar{}\Omega}$ to be
dimensionful. The notation $e^{\bar{}\Omega}$ is not appropriate then, but we
will use it for convenience. In the terminology of subsection 3.3:

$e^{\Omega_{0}+i\phi_{0}}\equiv\sum_{i}Q_{i}\partial G_{i}.$ (47)

The solution $G$ is not unique; any function $f(\tau,\bar{}\tau)$ can be added
to it. Also, $G$ is not modular invariant - it is not invariant under
$\tau\to-1/\tau$; rather, it shifts by $-\ln|\tau|$. One can modify it by
using $\eta$-function:

$G^{\prime}\equiv
G+\ln|\eta(\tau)|^{2}=-\ln\left|\frac{\Theta\left(w,\tau\right)}{\eta(\tau)}\right|^{2}+\frac{2\pi\imaginary(w)^{2}}{\tau_{2}}.$
(48)

$G^{\prime}$ is modular invariant:

$G^{\prime}(w,\tau+1)=G^{\prime}(w,\tau),\
G^{\prime}\left(\frac{w}{\tau},-\frac{1}{\tau}\right)=G^{\prime}(w,\tau).$
(49)

However, momentum conservation implies
$\sum_{i=1}Q_{i}G^{\prime}_{i}=\sum_{i=1}Q_{i}G_{i}$, and the same for all of
the derivatives of $G^{\prime}$, so we will use $G$ instead. Basic definitions
and properties of $\Theta,\eta$-functions and $G$ are given in appendix C.

The WS-area is:

$\bar{A}=\int d^{2}\sigma\sqrt{\bar{g}}=\int d^{2}\sigma
e^{2\Omega}\tau_{2}=\int\frac{d^{2}w}{2}|z|^{2}.$ (50)

Using (46) one can write $|z|^{2}$:

$|z|^{2}=\sum_{ij}Q_{i}\bar{Q}_{j}\partial
G_{i}\bar{\partial}\bar{G}_{j}+\sum_{i}2\real(Q_{i}\partial
G_{i}e^{\bar{}\Omega-i\bar{\phi}})+e^{2\bar{}\Omega}.$ (51)

The integral over $|z|^{2}$ can be evaluated easily using integration by
parts, the EOM of $G$, and momentum conservation:

$\begin{split}\int d^{2}w\partial G_{i}=\int d^{2}w\bar{\partial}G_{i}=0&\
\text{(periodicity)},\\\ \sum_{ij}Q_{i}\bar{Q}_{j}\int
d^{2}wG_{i}=\sum_{j}\bar{Q}_{j}(\dots)=0&\ \text{(momentum
conservation)}.\end{split}$ (52)

Then:

$\begin{split}\bar{A}&=\pi\sum_{ij}Q_{i}\bar{Q}_{j}G_{i}(w_{j})+e^{2\bar{}\Omega}\tau_{2}\\\
&=\frac{1}{4\pi|\Lambda|^{2}}\sum_{i}q_{i}^{2}G_{ii}+\frac{1}{2\pi|\Lambda|^{2}}\sum_{i<j}G_{ij}q_{i}\cdot
q_{j}+e^{2\bar{}\Omega}\tau_{2},\end{split}$ (53)

In the last line we used
$\bar{Q}_{i}Q_{j}+\bar{Q}_{j}Q_{i}=\frac{2}{(2\pi|\Lambda|)^{2}}q_{i}\cdot
q_{j}$ and defined $G_{ij}\equiv G_{i}(w_{j})=G(w_{ij})$. We note that
$\tau_{2}$ is absent from the first two terms in the R.H.S. of (53), since
$\int d^{2}w\delta(w)=1$, and that the sum reduces to
$\frac{q^{2}}{2\pi|\Lambda|^{2}}\left(G_{11}-G_{12}\right)$.

The divergent pieces of $\bar{A}$, $G_{ii}$, arise from the divergence of $G$
at $w=0$. $G$ at $|w|<<1$ can be approximated by:

$G(w,\tau)\approx-\ln(2\pi)^{2}-\ln|\eta(\tau)|^{6}-\ln|w|^{2}.$ (54)

The logarithmic divergence appeared also in the plane. It is expected since
the solution of (44) at small $w$ will not depend on the global properties of
the WS. This divergence will be treated by point-splitting regularization.
However, it is not trivial on which coordinates this splitting should be
implied. There are two natural candidates - the Cartesian coordinates and the
complex coordinates. It turns out that one should take $|w|=\epsilon$, rather
than $|\sigma|=\epsilon$.181818For the plane, the complex coordinate was
$w=\sigma^{1}+i\sigma^{2}$. The analog of $G$ here, solved the equation
$\partial\bar{\partial}G=-2\pi\delta(w)$, with the solution
$-\ln\left(|w|^{2}\right)$. The same sum in (53) appeared for the plane ((97)
in [27]), with $G_{ij}\to-\ln|w_{ij}|^{2}$. There $|w|=|\sigma|$, and
$-\ln|w_{ii}|^{2}$ was simply regularized as $-\ln(\epsilon^{2})$. It has the
following advantages:

  * •

it makes the final result finite,

  * •

the plane’s result is recovered in the appropriate limit,

  * •

the normalization of the operators is the as for the plane,

  * •

and the regularized WS-area is always positive, in particular, when
$\tau_{2}\to\infty$. We illustrate this below.

One disadvantage is that the regularized WS-area is not modular invariant,
i.e. $\ln{|\eta(\tau)|^{6}}+G(w_{12},\tau)$ is not modular invariant (while
$\ln{|\eta(\tau)|^{2}}+G(w_{12},\tau)$ is).191919Since
$PSL(2,\mathbb{Z})_{WS}$ are part of the diffeomorphism invariance, they can
not be broken. The regularization is chosen for one $\tau$, and for other
$\tau$ obtained by $PSL(2,\mathbb{Z})_{WS}$ transformation, it will be chosen
in a way such that the WS-area remains the same. So the exact statement is
that the regularization transforms in a complicated way under
$PSL(2,\mathbb{Z})_{WS}$. However, the global $PSL(2,\mathbb{Z})_{TS}$ is
broken by this regularization; see appendix A.

Therefore, we use
$G_{ii}\to-\ln\left(\epsilon^{2}\right)-\ln\left(|\eta(\tau)|^{6}\right)$, and
the regularized area is:

$\bar{A}=-\frac{\ln\left(\epsilon^{2}\right)}{4\pi|\Lambda|^{2}}\sum_{i}q_{i}^{2}-\frac{\ln\left(|\eta(\tau)|^{6}\right)}{4\pi|\Lambda|^{2}}\sum_{i}q_{i}^{2}+\frac{1}{2\pi|\Lambda|^{2}}\sum_{i<j}G_{ij}q_{i}\cdot
q_{j}+e^{2\bar{}\Omega}\tau_{2}.$ (55)

The finite part for our two-pt correlator reads:

$-\frac{q^{2}}{2\pi|\Lambda|^{2}}\left(\ln{|\eta(\tau)|^{6}}+G_{12}\right)+e^{2\bar{}\Omega}\tau_{2},$
(56)

implementing momentum conservation.

The remaining $\tau_{2}$ integral is finite with this regularized area. It
would not be so without the additional $-\ln\left(|\eta(\tau)|^{6}\right)$. At
large momentum, the integrand in (40) will be dominated by
$e^{-|\Lambda|\bar{A}}$, and the integral should converge since $\bar{A}$
should be positive. However, when we regularize and pull out the divergent
piece $-\ln\left(\epsilon^{2}\right)$, the remaining piece might be negative,
as for $\tau_{2}\gtrsim 2$:

$G(w,\tau)\approx
2\pi\left(\sigma^{2}-\frac{1}{2}\right)^{2}\tau_{2}=\frac{\pi}{2}\left(1-\frac{\sigma^{2}}{\pi}\right)^{2}\tau_{2}.$
(57)

The finite piece $-\ln|\eta(\tau)|^{6}$ we added in $G_{ii}$, at
$\tau_{2}\gtrsim 2$ behaves as $\frac{\pi}{2}\tau_{2}$. Then the integrand
$e^{-|\Lambda|\bar{A}}$ is approximately:

$e^{-\frac{q^{2}}{2\pi|\Lambda|}(G_{11}-G_{12})}\underset{\tau_{2}\to\infty}{\approx}e^{-\frac{q^{2}}{2\pi|\Lambda|}\left(1-{\left(1-2\sigma^{2}\right)^{2}}\right)\frac{\pi\tau_{2}}{2}},$
(58)

and the integral is now convergent.202020However, since an analytic
continuation is made for the $\sigma$ integral in (40), it might be not
relevant, as will turn out to be true for the large $\tau_{2}$ behavior.

Another quantity that should be regularized at this stage is the area element
$\sqrt{g(\sigma_{i})}$. We use point-splitting again
$|w_{i}-w_{i}|\to\epsilon$, and therefore:

$\displaystyle e^{\Omega(\sigma_{i})}=|z(w_{i})|$
$\displaystyle\to\frac{|Q_{i}|}{\epsilon},$ (59)
$\displaystyle\sqrt{g(\sigma_{i})}=|z(w_{i})|^{2}\tau_{2}$
$\displaystyle\to\frac{|Q_{i}|^{2}}{\epsilon^{2}}\tau_{2}.$ (60)

The explicit form of the vielbein can be used to simplify the last integral in
the exponent of (40):

$-i|\Lambda|\int
d^{2}\sigma\epsilon^{\alpha\beta}\epsilon_{ab}L^{a}_{\alpha}\bar{e}^{b}_{\beta}.$
(61)

Using (46) the vielbein can be written as:

$\begin{split}e^{a}_{\alpha}&\equiv\left(e_{s}\right)^{a}_{\alpha}+\left(e_{c}\right)^{a}_{\alpha}\\\
&\equiv e^{\Omega_{0}+\epsilon\phi_{0}}\begin{pmatrix}1&\tau_{1}\\\
0&\tau_{2}\end{pmatrix}+e^{\bar{}\Omega+\epsilon\bar{\phi}}\begin{pmatrix}1&\tau_{1}\\\
0&\tau_{2}\end{pmatrix}.\end{split}$ (62)

The coordinate dependent part of the vielbein
$\left(e_{s}\right)^{a}_{\alpha}$ does not contribute to (61) since $G_{i}$ is
periodic:

$\int d^{2}\sigma e^{\Omega_{0}}cos(\phi_{0})=\frac{1}{2}\sum_{i}\int
d^{2}\sigma(Q_{i}\partial G_{i}+c.c.)=0.$ (63)

The same it true for $\int d^{2}\sigma e^{\Omega_{0}}sin(\phi_{0})$. Therefore
only the constant part $\left(e_{c}\right)^{a}_{\alpha}$ contributes, and the
integral is trivial:
$-i|\Lambda|\epsilon^{\alpha\beta}\epsilon_{ab}L^{a}_{\alpha}\left(e_{c}\right)^{b}_{\beta}$.

###  3.5 Undeformed Fields PI

Until now we did not use the fact that the undeformed theory is a CFT. From
here on, we will use this assumption. For a primary operator $O(\sigma)$ with
a conformal dimension $\Delta$, the PI over the undeformed fields is:

$\int\mathcal{D}\psi
O(\sigma_{1})O(\sigma_{2})e^{-S_{0}\left(\psi,e^{2\Omega}g_{0}\right)}=e^{-(\Omega(\sigma_{1})+\Omega(\sigma_{2}))\Delta+W}Z_{0}[g_{0}]C^{0}(|\sigma_{1}-\sigma_{2}|),$
(64)

where $Z_{0}[g_{0}],C^{0}(|\sigma_{1}-\sigma_{2}|)$ are the undeformed
partition function and correlator respectively. $e^{W}$ is the Weyl anomaly:

$W=\frac{c}{12}\int
d^{2}\sigma\sqrt{g_{0}}g_{0}^{\alpha\beta}\partial_{\alpha}\Omega\partial_{\beta}\Omega,$
(65)

where $c$ is the central charge of the undeformed theory.212121In [27] the
factor $\frac{1}{24\pi}$ was inconsistent with the definition of the energy-
momentum tensor used in (1), and should be $\frac{1}{12}$ as in (65). The
formula is derived in appendix D, and the anomaly itself is evaluated there:

$e^{W}=e^{\frac{cI_{0}}{24}-\frac{\pi
c}{6}}\prod_{i=1}^{2}\left(\frac{r}{\epsilon}\right)^{\frac{\pi c}{3}}.$ (66)

In this equation, $\epsilon$ is the point-splitting regulator, $r$ is some
number, and $I_{0}$ is some finite quantity that is independent of the
momentum at large momentum.

The correlator thus contains the divergent factors:

$\prod_{i=1}^{2}\left[\left(\frac{r}{\epsilon}\right)^{\frac{\pi
c}{3}}\left(\frac{|Q_{i}|}{\epsilon}\right)^{2-\Delta}\epsilon^{\frac{q_{i}^{2}}{2\pi|\Lambda|}}\right].$
(67)

It will be eliminated by a renormalization of the operators:

$\mathcal{O}_{i}(q_{i})\rightarrow\hat{\mathcal{O}}_{i}(q_{i})\equiv(\bar{\mu}\epsilon)^{-2\pi|\Lambda||Q^{i}|^{2}}\left(\frac{\epsilon}{|Q^{i}|}\right)^{2-\Delta_{i}}\left(\frac{\epsilon}{r}\right)^{\frac{\pi
c}{3}}\mathcal{O}_{i}(q_{i}).$ (68)

This is the same renormalization of the operators in [27].222222Since $w$ is
dimensionless, such are $\epsilon$ and $\bar{\mu}$. Therefore, we use
$\bar{\mu}$ to distinguish it from the dimensionful renormalization scale
$\mu$ used for the plane. In particular, it is important that the
renormalization does not depend on the modular parameter of the WSt. It does
depend on the TSt properties through $\bar{\mu}$, as will be shown in
subsection 4.2.1.

The correlator of the renormalized operators is the remaining finite part:

$\begin{split}&e^{|\Lambda|\mathcal{A}}\delta(q_{1}+q_{2})\int d^{2}\tau
d\bar{\Omega}d\bar{\phi}d^{2}\sigma_{1}d^{2}\sigma_{2}\delta(\sigma_{1})\frac{\det(e^{2f_{\Omega}}\tau_{2})J(\bar{e},0)e^{2\bar{}\Omega}\tau_{2}^{3}}{\left|{\det}^{\prime}\left(\square\right)\right|}\cdot\\\
Z_{0}[g_{0}]C^{0}(|\sigma_{1}-\sigma_{2}|)&e^{\frac{cI_{0}}{24}-\frac{\pi
c}{6}+in_{1\alpha}\sigma^{\alpha}_{1}+in_{2\alpha}\sigma^{\alpha}_{2}+\frac{q^{2}}{2\pi|\Lambda|}\left(\ln\left(\frac{|\eta(\tau)|^{6}}{\bar{\mu}^{2}}\right)+G_{12}\right)-|\Lambda|e^{2\bar{}\Omega}\tau_{2}-i|\Lambda|\epsilon^{\alpha\beta}\epsilon_{ab}L^{a}_{\alpha}\left(e_{c}\right)^{b}_{\beta}}\\\
=&e^{|\Lambda|\mathcal{A}}\delta(q_{1}+q_{2})\int d^{2}\tau
d\bar{\Omega}d\bar{\phi}d^{2}\sigma\frac{\det(e^{2f_{\Omega}}\tau_{2})J(\bar{e},0)e^{2\bar{}\Omega}\tau_{2}^{3}}{\left|{\det}^{\prime}\left(\square\right)\right|}\cdot\\\
Z_{0}[g_{0}]C^{0}(|\sigma|)&e^{\frac{cI_{0}}{24}-\frac{\pi
c}{6}+in_{\alpha}\sigma^{\alpha}+\frac{q^{2}}{2\pi|\Lambda|}\left(\ln\left(\frac{|\eta(\tau)|^{6}}{\bar{\mu}^{2}}\right)+G_{12}\right)-|\Lambda|e^{2\bar{}\Omega}\tau_{2}-i|\Lambda|\epsilon^{\alpha\beta}\epsilon_{ab}L^{a}_{\alpha}\left(e_{c}\right)^{b}_{\beta}}.\end{split}$
(69)

In the last line we evaluated the $\sigma^{1}$ integral, and denoted
$\sigma^{2},n_{2\alpha}$ by $\sigma,n_{\alpha}$.

##  4 Saddle-point Evaluation

In the following section, the high momentum limit of (69) will be derived. The
saddle point approximation (or just “saddle” from now on) will be used for the
remaining integrals. The $\bar{}\Omega,\bar{\phi}$ integrals will be evaluated
in the limit of large TSt-area, compared to the deformation scale. Only two
terms in the exponent in (69) contribute to the saddle:

$e^{-|\Lambda|e^{2\bar{}\Omega}\tau_{2}-i|\Lambda|\epsilon^{\alpha\beta}\epsilon_{ab}L^{a}_{\alpha}\left(e_{c}\right)^{b}_{\beta}}.$
(70)

The $\tau,\sigma_{2}$ integrals will be evaluated in the large momentum limit,
compared to the deformation scale $\frac{q^{2}}{|\Lambda|}$. The ratio between
the momentum $\frac{|q|}{|\Lambda|}$ and the TSt length scale $L$ will dictate
the result. The relevant terms in the exponent are:

$e^{in_{\alpha}\sigma^{\alpha}+\frac{q^{2}}{2\pi|\Lambda|}\left(\ln\left(\frac{|\eta(\tau)|^{6}}{\bar{\mu}^{2}}\right)+G_{12}\right)}.$
(71)

We will not be able to evaluate the Gaussian integrals around the saddle for
the $\tau,\sigma_{2}$ integrals, and they are subleading in the momentum
anyway, so we will not evaluate them at all.

The undeformed correlator $C^{0}(|\sigma|)$ and the partition function
$Z_{0}[g_{0}]$ are independent of the momentum and, therefore will not
contribute to the saddle point approximation. The determinants do depend on
the momentum, but in the large momentum limit they go to a constant. This is
easily seen for the denominator ${\det}^{\prime}\left(\square\right)$ in (69),
as $\partial\ln(\bar{z})\sim\partial\ln(\partial G)$. This should work in the
same way for the numerator (as happened for the plane, (22) in [27]).

###  4.1 Constant Conformal-factor and Orientation

The saddle is used with the parameter $l\equiv L\sqrt{|\Lambda|}>>1$, where
$L$ is the length scale in $L^{a}_{\alpha}$. Changing variables $x\equiv
e^{\bar{}\Omega}\cos{\bar{\phi}},\ y\equiv e^{\bar{}\Omega}\sin{\bar{\phi}}$
inside the exponent in (70) gives:

$\tau_{2}\left(x^{2}+y^{2}\right)+i\left(L^{1}_{1}\tau_{2}+L^{2}_{2}-L^{2}_{1}\tau_{1}\right)x+i\left(L^{1}_{1}\tau_{1}-L^{1}_{2}+L^{2}_{1}\tau_{2}\right)y.$
(72)

This function has a saddle point:

$\begin{split}x_{s}=-\frac{i\left(L^{1}_{1}\tau_{2}+L^{2}_{2}-L^{2}_{1}\tau_{1}\right)}{2\tau_{2}},\\\
y_{s}=-\frac{i\left(L^{1}_{1}\tau_{1}-L^{1}_{2}+L^{2}_{1}\tau_{2}\right)}{2\tau_{2}}.\end{split}$
(73)

The exponent at this point is:

$e^{-|\Lambda|\frac{\left(L^{1}_{1}\tau_{2}+L^{2}_{2}-L^{2}_{1}\tau_{1}\right)^{2}+\left(L^{1}_{1}\tau_{1}-L^{1}_{2}+L^{2}_{1}\tau_{2}\right)^{2}}{4\tau_{2}}}=e^{-|\Lambda|\mathcal{A}\frac{\left(\tau_{2}+T_{2}\right)^{2}+\left(\tau_{1}-T_{1}\right)^{2}}{4T_{2}\tau_{2}}}.$
(74)

where we wrote $L^{a}_{\alpha}$ as in (21), and $\mathcal{A}\equiv
T_{2}\left(X^{2}+Y^{2}\right)=L^{2}T_{2}$ is the TSt area.

The measure satisfies $e^{2\bar{}\Omega}d\bar{}\Omega d\bar{\phi}=dxdy$, and
the Gaussian integral around the saddle gives $\frac{\pi}{|\Lambda|\tau_{2}}$.
(69) is then modified to:

$\begin{split}\frac{\pi
e^{|\Lambda|\mathcal{A}}}{|\Lambda|}\delta(q_{1}+q_{2})\int d^{2}\tau
d^{2}\sigma\frac{\det(e^{2f_{\Omega}}\tau_{2})J(\bar{e},0)\tau_{2}^{2}}{\left|{\det}^{\prime}\left(\square\right)\right|}Z_{0}[g_{0}]C^{0}(|\sigma|)\cdot\\\
e^{\frac{cI_{0}}{24}-\frac{\pi
c}{6}}e^{in_{\alpha}\sigma^{\alpha}+\frac{q^{2}}{2\pi|\Lambda|}\left(\ln\left(\frac{|\eta(\tau)|^{6}}{\bar{\mu}^{2}}\right)+G_{12}\right)-|\Lambda|\mathcal{A}\frac{\left(\tau_{2}+T_{2}\right)^{2}+\left(\tau_{1}-T_{1}\right)^{2}}{4T_{2}\tau_{2}}}.\end{split}$
(75)

###  4.2 Modular-parameter and Coordinate

It remains to perform the integrals over $\tau$ and $\sigma$. Which integral
will be performed first depends on the relation between the different scales
involved in the correlator. The integral is:

$\int d^{2}\tau d^{2}\sigma
e^{in_{\alpha}\sigma^{\alpha}+\frac{q^{2}}{2\pi|\Lambda|}\left(\ln\left(\frac{|\eta(\tau)|^{6}}{\bar{\mu}^{2}}\right)+G_{12}\right)-|\Lambda|\mathcal{A}\frac{\left(\tau_{2}+T_{2}\right)^{2}+\left(\tau_{1}-T_{1}\right)^{2}}{4T_{2}\tau_{2}}}.$
(76)

Using $l$ defined in the last subsection, and $q\sim\frac{n}{L}$, the three
terms in the exponent can be written schematically as:

$e^{in\sigma+\frac{n^{2}}{l^{2}}f_{1}(\tau,\sigma)+l^{2}f_{2}(\tau)}.$ (77)

The large TS-area we used means $l^{2}>>1$. The ratio $r\equiv n/l^{2}$
dictates the relative coefficients of the three terms of (77):

$e^{in\sigma+nrf_{1}(\tau,\sigma)+\frac{n}{r}f_{2}(\tau)}.$ (78)

There will be different results for $r<<1$ and for $r>>1$. The first region is
where the TS-area is much bigger than the momentum scale. Then the plane’s
result is expected to be recovered. The second region is where the momentum is
much bigger than all scales in the problem, and it will give another result.
The intermediate region is hard to analyze analytically and should be some
interpolation between these two limits. We will evaluate it numerically for a
simple TSt.

####  4.2.1 Plane’s Limit

In the $r<<1$ limit, $\tau$ is integrated first, considering only the last
term of (78). The saddle satisfies:

$\displaystyle\frac{\partial}{\partial\tau_{1}}:\
\frac{\tau_{1}-T_{1}}{\tau_{2}}=0,$ (79)
$\displaystyle\frac{\partial}{\partial\tau_{2}}:\
\frac{\tau_{2}^{2}-T_{2}^{2}-\left(\tau_{1}-T_{1}\right)^{2}}{\tau_{2}^{2}}=0.$
(80)

It has the solution $\tau_{1}=T_{1},\tau_{2}=T_{2}$, i.e. $\tau=T$, and
therefore $x=-iX,y=-iY$.232323$x,y$ are imaginary because of the analytic
continuation done for $\Omega$.

The exponent at the saddle is:

$e^{-|\Lambda|\frac{X^{2}+Y^{2}}{T_{2}}}=e^{-|\Lambda|\mathcal{A}},$ (81)

and it cancels an opposite factor in (75). The integral over $\sigma$ is:

$\int d^{2}\sigma
e^{in_{\alpha}\sigma^{\alpha}+\frac{q^{2}}{2\pi|\Lambda|}\left(\ln\left(\frac{|\eta(T)|^{6}}{\bar{\mu}^{2}}\right)+G(w,T)\right)}.$
(82)

We expect to recover the plane’s result, where the saddle behaved as
$\sigma\sim\frac{q}{|\Lambda|}$ (see paragraph after (109) in [27]). Using the
parameters $n,r$, it reads $\sigma\sim r$. It is then a reasonable guess that
the saddle appears for small $\sigma$. In particular, we can approximate
$G_{12}$ as in (54), and cancelling the $\ln{|\eta(T)|^{6}}$ term. Then (82)
becomes:

$\int d^{2}\sigma
e^{in_{\alpha}\sigma^{\alpha}-\frac{q^{2}}{2\pi|\Lambda|}\ln{|w|^{2}}}.$ (83)

We consider the analytic continuation of
$|w|^{2}=\left(\sigma^{1}+T_{1}\sigma^{2}\right)^{2}+\left(T_{2}\sigma^{2}\right)^{2}$
as a function of $\sigma^{1},\sigma^{2}$. The equations for the saddle are:

$\displaystyle\frac{\partial}{\partial\sigma^{1}}:\
in_{1}-\frac{q^{2}}{\pi|\Lambda|}\frac{\sigma^{1}+T_{1}\sigma^{2}}{|w|^{2}}=0,$
(84) $\displaystyle\frac{\partial}{\partial\sigma^{2}}:\
in_{2}-\frac{q^{2}}{\pi|\Lambda|}\frac{\left(\sigma^{1}+T_{1}\sigma^{2}\right)T_{1}+T_{2}^{2}\sigma^{2}}{|w|^{2}}=0.$
(85)

The solution is242424In the plane, $\sigma$ was dimensionful, while here it is
dimensionless, scaled with $L$. However, this $L$ can be swallowed in the
dimensionless regulator $\bar{\mu}$ to give the mass scale
$\mu\equiv\frac{\bar{\mu}}{L}$, which is the renormalization scale, and has
nothing to do with $L$.:

$\begin{split}|w|^{2}&=-\frac{q^{2}}{\pi^{2}|\Lambda|^{2}L^{2}},\\\
\sigma^{1}+T_{1}\sigma^{2}&=-i\frac{q_{1}}{\pi|\Lambda|L},\\\
T_{2}\sigma^{2}&=-i\frac{q_{2}}{\pi|\Lambda|L}.\end{split}$ (86)

Substituting in (83):

$\left(\frac{\bar{\mu}|q|}{\pi
e|\Lambda|L}\right)^{-\frac{q^{2}}{\pi|\Lambda|}}=\left(\frac{r}{\pi
e}\right)^{-\frac{nr}{\pi}}.$ (87)

This function has a maximum: $r_{\rm max}=\frac{\pi\sqrt{e}}{\bar{\mu}}$,
which means that in the region of validity of this approximation, the
correlator might increase, depending on $\bar{\mu}$, unlike the decreasing
behavior found for the plane. To clarify this ambiguity, we pay attention to
the plane’s correlator ((114) in [27]):

$\left(\frac{\mu|q|}{\pi e|\Lambda|}\right)^{-\frac{q^{2}}{\pi|\Lambda|}}.$
(88)

Since any physical result should not depend on the renormalization scale, and
$|\Lambda|$ is the only scale in the plane, one should use
$\mu=\sqrt{|\Lambda|}$. Then the correlator becomes:

$\left(\frac{q^{2}}{\pi e|\Lambda|}\right)^{-\frac{q^{2}}{2\pi|\Lambda|}}.$
(89)

For $\frac{q^{2}}{|\Lambda|}>>1$, which is the condition for using the saddle
approximation in the plane, the function decreases.252525Including the
renormalization scale $\mu$, the maximum of (88) is at
$|q|_{max}=\frac{\pi\sqrt{e}|\Lambda|}{\mu}$. Then for
$\sqrt{|\Lambda|}<<|q|<\frac{|\Lambda|}{\mu}$ the correlator might increase.
If $\mu=\sqrt{|\Lambda|}$ this region is empty. Returning to the torus, there
are two mass scales $\frac{1}{L},\sqrt{|\Lambda|}$ in the problem, which are
relevant for the renormalization scale $\mu\equiv\frac{\bar{\mu}}{L}$ (and
also many others by using the dimensionless quantity $l$). They correspond to
$\bar{\mu}=1,l$. Since the plane’s result is expected to be recovered in this
region, the choice $\bar{\mu}=l$ is preferred, and (87) reads:

$\left(\frac{q^{2}}{\pi
e|\Lambda|}\right)^{-\frac{q^{2}}{2\pi|\Lambda|}}=\left(\frac{nr}{\pi
e}\right)^{-\frac{nr}{2\pi}}.$ (90)

The maximum of this function is $\sim\frac{1}{l}$. For using the saddle
approximation in this region, one needs $nr>>1$. Hence
$\frac{n}{l}>>1\Rightarrow r>>\frac{1}{l}$, and (90) decreases.

There is also an additional phase
$\left(-1\right)^{\frac{q^{2}}{2\pi|\Lambda|}}$ which can be swallowed in the
operators’ renormalization. This is the same as (124) in [27]. However, this
phase does not arise in the $r>>1$ limit.

####  4.2.2 Torus’ Limit

In the $r>>1$ limit, the leading term in the integrand of (78) depends both on
$\tau$ and $\sigma$ and so one can not perform the integrals one after the
other as before. Symmetric points are natural candidates for saddle points,
and so we pay our attention to the symmetries of $G$.

The $\Theta$-function satisfies (see appendix C):

$\begin{split}\Theta(-w,\tau)=-\Theta(w,\tau)&\Rightarrow\Theta(\sigma^{1},\sigma^{2},\tau_{1},\tau_{2})=\Theta(-\sigma^{1},-\sigma^{2},\tau_{1},\tau_{2}),\\\
\overline{\Theta(w,\tau)}=\Theta(\bar{w},-\bar{}\tau)&\Rightarrow\Theta(\sigma^{1},\sigma^{2},\tau_{1},\tau_{2})=\Theta(\sigma^{1},-\sigma^{2},-\tau_{1},\tau_{2}),\end{split}$
(91)

where
$\Theta(\sigma^{1},\sigma^{2},\tau_{1},\tau_{2})\equiv\Theta(\sigma^{1}+(\tau_{1}+i\tau_{2})\sigma^{2},\tau_{1}+i\tau_{2})$.
(91) implies that the first term of $G$, $-\ln|\Theta(w,\tau)|^{2}$ is
invariant under262626The last symmetry is a composition of the first two, and
in fact, each one of the three can be deduced from the other two. In complex
variables, the symmetries are:  $w\to-w,\ w\to-\bar{w}\And\tau\to-\bar{}\tau,\
w\to\bar{w}\And\tau\to-\bar{}\tau.$ (92) :

$\begin{split}\sigma^{1}\to-\sigma^{1}&\And\sigma^{2}\to-\sigma^{2},\\\
\sigma^{1}\to-\sigma^{1}&\And\tau_{1}\to-\tau_{1},\\\
\sigma^{2}\to-\sigma^{2}&\And\tau_{1}\to-\tau_{1}.\end{split}$ (93)

The second term of $G$, $2\pi\tau_{2}\left(\sigma^{2}\right)^{2}$, is also
invariant under these transformations. Hence, these are symmetries of $G$.
These are also symmetries of $|\eta(\tau)|$.272727$|q|=e^{-2\pi\tau_{2}}$ is
independent of $\tau_{1}$. The other factors are
$\left(1-q^{m}\right)\left(1-\bar{q}^{m}\right)$, and they are symmetric with
respect to $\tau_{1}\to-\tau_{1}$, which corresponds to $q\to\bar{q}$. The
last symmetry can be used to evaluate the saddle in the $\tau_{1},\sigma^{2}$
integrals by assuming that the saddle is at $\tau_{1}=\sigma^{2}=0$.

One is then left with the $\tau_{2},\sigma^{1}$ integrals. The integral over
$\sigma^{1}$ is ill-defined because of the divergence at $\sigma^{1}=0$. This
divergence results from inserting the two operators in the same place on the
WS. It should be avoided due to the point-splitting regularization. To
overcome this problem, we note that the function
$\Theta\left(\sigma^{1},i\tau_{2}\right)$ is real, hence
$|\Theta|^{2}=\Theta^{2}$, which is now an analytic function, defined on the
“complex cylinder”:

$\sigma^{1}=x+iy,\ x\sim x+1,\ -\infty<y<\infty,$ (94)

$x$ is periodic due to the periodicity properties of $\Theta$. We define the
integration contour in the complex plane, going in a small half circle
$\epsilon e^{i\phi},\pi\leq\phi\leq 2\pi$ below $\sigma^{1}=0$. The branch-cut
of the logarithm will be on the positive imaginary axis. The contour is
closed, wrapping this cylinder once. We can deform this contour, and the
saddle will be where the derivative vanishes:

$\frac{d}{d\sigma^{1}}\ln\left(\Theta\left(\sigma^{1},i\tau_{2}\right)^{2}\right)=0\Rightarrow\frac{d\Theta\left(\sigma^{1},i\tau_{2}\right)}{d\sigma^{1}}=0.$
(95)

This procedure is explained in more detail in appendix E.282828The same
divergence at $\sigma=0$ happened in the plane. The analysis of this case is
given in the appendix mentioned above. The set of solutions to (95) lies on
the imaginary axis. On the imaginary axis there are brunch points of the log
at $y=m\tau_{2}$ for any integer $m$, and besides of these
$-\ln\left(\Theta\left(\sigma^{1},i\tau_{2}\right)^{2}\right)$ does not have
any other singularities. Between any of these brunch points sits one point
when the derivative vanishes. Since we do not want the contour to cross them,
the first negative one $-\tau_{2}<y_{s}(\tau_{2})<0$ will be the saddle for
the $\sigma$ integral. The integration contour is deformed appropriately, now
going downwards from the saddle down to $y\to-\infty$, see figure 1.

Figure 1: Real (left) and imaginary (right) parts of
$-\ln\left(\Theta\left(\sigma^{1},i\tau_{2}\right)^{2}\right)$ around the
first saddle at large r ($6\pi$), with $\tau_{2}$ given by (108)
($T_{1}=0,T_{2}=10$). The saddle is located at the black point. Two contours
of constant phases emerge from the saddle. One of them is the imaginary axis;
the other is inside the orange region, which is the deformed contour. The
bright point in the real part is a singularity due to a zero of the
$\Theta$-function. The white lines in the imaginary part are discontinuities
due to the log.

The saddle’s location $y_{s}(\tau_{2})$ can not be obtained analytically, but
numerically, one finds the following simple behavior292929The coefficients
were obtained numerically, and the coefficients $\pi,2\pi$ appearing in (96)
are $3.12,\ 6.28$ respectively. Maybe one can prove that the $\pi$’s are the
exact values. It is not essential, and we use them for clarity.:

$y_{s}(\tau_{2})\approx\begin{cases}-\pi\tau_{2}-2.3\tau_{2}^{2}\
&\tau_{2}\lesssim 0.7\\\ 1.1-2\pi\tau_{2}\ &\tau_{2}\gtrsim 0.7\end{cases}.$
(96)

It remains to evaluate the $\tau_{2}$ integral. First, we want to verify that
it converges. For this aim, we check the small and large $\tau_{2}$ domains.
Plugging (96) into $G$, leads to an almost line303030For
$0.7\lesssim\tau_{2}\lesssim 6.5$ one has a slightly different behavior:
$G(iy_{s}(\tau_{2}),\tau_{2})\approx 1.95-4.73\tau_{2}$. Aside from the linear
growth, there is also a logarithmic piece, hence as $\tau_{2}$ increases the
slope of the line decreases slowly, and the constant decreases rapidly to
zero.:

$G(iy_{s}(\tau_{2}),\tau_{2})\approx 0.05-4.43\tau_{2},\ 6.5\lesssim\tau_{2}.$
(97)

Hence at large $\tau_{2}$ the integrand of (76) becomes:

$e^{\frac{q^{2}}{2\pi|\Lambda|}\left(-\ln\left(\bar{\mu}^{2}\right)-\frac{\pi\tau_{2}}{2}+(0.05-4.43\tau_{2})\right)-|\Lambda|\mathcal{A}\frac{\tau_{2}}{4T_{2}}}.$
(98)

It decays fast enough to zero, and the integral converges in this region.

The small $\tau_{2}$ region is more difficult to analyze. However, the modular
properties of the $\Theta,\eta$-functions can be used to extract the leading
behavior. We use:

$\displaystyle\Theta\left(\frac{w}{\tau},-\frac{1}{\tau}\right)$
$\displaystyle=-i\sqrt{-i\tau}e^{\frac{i\pi w^{2}}{\tau}}\Theta(w,\tau),$ (99)
$\displaystyle\eta\left(-\frac{1}{\tau}\right)$
$\displaystyle=\sqrt{-i\tau}\eta(\tau),$ (100)

and plug $\tau=i\tau_{2},w=iy$. Then:

$\displaystyle-\ln\left(\Theta(iy,i\tau_{2})^{2}\right)$
$\displaystyle=-\ln\left(\Theta\left(\frac{y}{\tau_{2}},\frac{i}{\tau_{2}}\right)^{2}\right)+\ln{\tau_{2}}-\frac{2\pi
y^{2}}{\tau_{2}},$ (101) $\displaystyle\ln|\eta(i\tau_{2})|^{6}$
$\displaystyle=\ln\left|\eta\left(\frac{i}{\tau_{2}}\right)\right|^{6}-3\ln\tau_{2}.$
(102)

For $1/\tau_{2}>>1$, the r.h.s. can be approximated
as313131$\frac{y_{s}(\tau_{2})}{\tau_{2}}\approx-\pi$, therefore
$\sin{iy_{s}(\tau_{2})}=\sinh{y_{s}(\tau_{2})}$ is finite and does not
contribute another $\tau_{2}^{-1}$ term. This is in contrast to the situation
when $\sigma^{2}\neq 0$, see remark 20.:

$\displaystyle-\ln{\Theta\left(\frac{y_{s}(\tau_{2})}{\tau_{2}},\frac{i}{\tau_{2}}\right)^{2}}$
$\displaystyle\approx\frac{\pi}{2\tau_{2}},$ (103)
$\displaystyle\ln{\left|\eta\left(\frac{i}{\tau_{2}}\right)\right|}^{6}$
$\displaystyle\approx-\frac{\pi}{2\tau_{2}}.$ (104)

The small $\tau_{2}$ limit is therefore:

$\displaystyle-\ln\left(\Theta\left(\frac{iy_{s}(\tau_{2})}{2\pi},i\tau_{2}\right)^{2}\right)$
$\displaystyle\approx\frac{\pi}{2\tau_{2}}+\ln{\tau_{2}}+f_{\Theta},$ (105)
$\displaystyle\ln|\eta(i\tau_{2})|^{6}$
$\displaystyle\approx-\frac{\pi}{2\tau_{2}}-3\ln\tau_{2}.$ (106)

where $f_{\Theta}$ is some number.323232The finite part $f_{\eta}$ for $\eta$
is smaller than $0.006$ for $\tau_{2}\lesssim 0.9$, so we did not write it.
The integrand of (76) reads:

$e^{-\frac{q^{2}}{\pi|\Lambda|}\ln\left(\bar{\mu}\tau_{2}\right)-|\Lambda|\mathcal{A}\frac{|T|^{2}}{4T_{2}\tau_{2}}},$
(107)

for which the integral converges.333333Leaving the $-\ln|\eta(\tau)|^{6}$ from
$G_{11}$ was crucial for the cancellation of the $\frac{\pi}{2\tau_{2}}$.

As $r$ increases, the maximum of the exponent is reached at small $\tau_{2}$,
for which the exponent behaves as in (107). The saddle is at:

$\tau_{2\text{s}}=\frac{\pi|\Lambda|^{2}\mathcal{A}|T|^{2}}{4T_{2}q^{2}}=\frac{\pi|T|^{2}}{4r^{2}}.$
(108)

Plugging in the integrand yields:

$\left(\frac{4q^{2}}{\pi
e\bar{\mu}|\Lambda|^{2}L^{2}|T|^{2}}\right)^{\frac{q^{2}}{\pi|\Lambda|}}=\left(\frac{4q^{2}}{\pi
e|\Lambda|^{\frac{5}{2}}L^{3}|T|^{2}}\right)^{\frac{q^{2}}{\pi|\Lambda|}}=\left(\frac{4r^{2}}{\pi
el|T|^{2}}\right)^{\frac{nr}{\pi}}.$ (109)

One should also consider the other saddle at $\sigma^{1}=\tau_{1}=0$ and look
for the saddle of $\sigma^{2},\tau_{2}$. Since our regularization transforms
non-trivially under WS modular transformation, the saddles should not
necessarily be the same, and we check which one is larger between the two. We
use the identity:

$G(i\sigma^{2}\tau_{2},i\tau_{2})=\ln(\tau_{2})+G\left(\sigma^{2},\frac{i}{\tau_{2}}\right).$
(110)

The function $\Theta\left(\sigma^{2},\frac{i}{\tau_{2}}\right)$ is real, hence
$G\left(\sigma^{2},\frac{i}{\tau_{2}}\right)=-\ln\left(\Theta\left(\sigma^{2},\frac{i}{\tau_{2}}\right)^{2}\right)$.
From here the same analysis goes as before, using $y_{s}(\tau_{2})$ found in
(96), and the small $\tau_{2}$ and large $\tau_{2}$ limits of:

$\ln|\eta(\tau_{2})|^{6}=\begin{cases}-\frac{\pi}{2\tau_{2}}-3\ln(\tau_{2})&\tau_{2}<<1\\\
-\frac{\pi}{2}\tau_{2}&\tau_{2}>>1\end{cases}$ (111)

and

$G\left(iy_{s}\left(\frac{1}{\tau_{2}}\right),\frac{i}{\tau_{2}}\right)=\begin{cases}\frac{\pi}{2\tau_{2}}+1.1-\frac{2\pi}{\tau_{2}}&\tau_{2}<<1\\\
\frac{\pi}{2}\tau_{2}-\ln(\tau_{2})-\ln(4)-\frac{\pi}{2\tau_{2}}&\tau_{2}>>1.\end{cases}$
(112)

At small $\tau_{2}$, there might be a saddle resulting from the competition of
the logarithm and the power law. But, this saddle is at $\tau_{2}\approx\pi$,
and so the approximation for small $\tau_{2}$ does not hold there. At large
$\tau_{2}$, there is a competition:

$e^{\frac{q^{2}}{2\pi|\Lambda|}\left(-\ln\left(\bar{\mu}^{2}\right)-\ln
4-\frac{\pi}{2\tau_{2}}\right)-|\Lambda|\mathcal{A}\frac{\tau_{2}}{4T_{2}}},$
(113)

where the dominant $\tau_{2}$ terms are kept. This has a saddle at
$\tau_{2s}=\frac{|q|}{|\Lambda|L^{2}}=r$, and the exponent becomes:

$e^{-\frac{q^{2}}{2\pi|\Lambda|}\ln\left(4\bar{\mu}^{2}\right)}=(4l)^{-\frac{nr}{2\pi}}.$
(114)

This saddle decreases with $r$ and is, therefore, less dominant than the first
one we found in (109). Therefore (109) is the correlator at the very large
momentum limit.

The correlator (109) has two interesting properties:

  * •

At this region, it increases with the momentum, unlike for the infinite plane
and the large torus’ limit, where the correlator decreases.

  * •

It depends on an IR quantities $L,|T|$, so it presents a UV-IR mixing.

The different behaviors for small and large $r$ are compatible with the
interpretation presented in the introduction. The operators are smeared over a
scale $|q|/|\Lambda|$. As the momentum grows, they are more and more smeared,
and the correlation between them decreases. This happens without any
disruption for the infinite plane. However, for the finite TSt, at some
momentum $r\sim 1$, the operators are smeared all over the torus, and the
correlator starts increasing.

###  4.3 Intermediate Region

For the special case of TSt with $T_{1}=0$, and momentum aligned along one of
the torus axes, say $n_{2}=0$, one can find the transition between the two
regions of small and large $r$. (76) takes the form:

$\int d^{2}\tau d^{2}\sigma
e^{in\sigma^{1}+\frac{nr}{2\pi}\left(\ln\left(\frac{|\eta(\tau)|^{6}}{\bar{\mu}^{2}}\right)+G(w,\tau)\right)-\frac{n}{4r}\frac{\left(\tau_{2}+T_{2}\right)^{2}+\tau_{1}^{2}}{\tau_{2}}},$
(115)

where $n\equiv n_{1},q^{2}=\frac{n^{2}}{L^{2}}$. The relations $T_{1}=n_{2}=0$
allow to take the saddle $\sigma^{2}=\tau_{1}=0$ for $n>>1,nr>>1$ and all
$r$.343434The condition $n>>1$ is needed for this exponent to be dominant for
the saddle approximation. It is also specifically needed for the $r\sim 1$
region. The condition $nr>>1$ is needed for the $r<<1$ region, where the
saddle of $\sigma^{2}$ comes from the second term alone. Then one finds
numerically $\sigma^{1}(\tau_{2})=iy_{s}(\tau_{2})$.353535The relation (96)
was for large $r$. Finally, the saddle of $\tau_{2}$ of the whole expression
is found as a function of $r$:

$e^{-ny_{s}(\tau_{2})+\frac{nr}{2\pi}\left(\ln\left(\frac{|\eta(i\tau_{2})|^{6}}{\bar{\mu}^{2}}\right)-\ln\left(\Theta(iy_{s}(\tau_{2}),i\tau_{2})^{2}\right)\right)-\frac{n}{4r}\frac{\left(\tau_{2}+T_{2}\right)^{2}}{\tau_{2}}}.$
(116)

We used the values:

$T_{2}=10,\ l=\bar{\mu}=100,\ 0.05\leq r\leq 100.$ (117)

where $T_{2}=10$ and not $1$, so its transition to $0$ will be clear. $l=100$
such that for the whole range of $r$ the conditions
$n=rl^{2}>>1,nr=(rl)^{2}>>1$ are satisfied. The final result is shown in
figure 2. The correlator decreases and afterward increases, as calculated
analytically.

Figure 2: Saddle point approximation of the two-point correlator at large
momentum as a function of $r\equiv\frac{|q|}{|\Lambda|L}$.

$\tau_{2s}(r)$ is shown in the left part of figure 3. At small $r$ the saddle
of $\tau_{2s}(r)$ is the same as $T_{2}$. At $r=2\pi$, there is a transition
because the second term in the exponent (116) becomes relevant. Eventually, it
controls the large $r$ behavior, taking $\tau_{2s}(r)\sim r^{-2}$ to zero as
predicted by (108).

$\sigma^{1}_{s}(\tau_{2s}(r))$ is shown in the right part of figure 3. At
small $r$, the absolute value of the saddle increases linearly with $r$. After
the transition point, it decreases with $\tau_{2}$, see (96). The sharp
transition can be understood as follows. Along the imaginary axis, there are
many singularities and extremum points between them due to the
$-\ln\Theta^{2}$ term. The extremum points are right to the singularities, or
in the middle between them, see the right part of figure 4. However, for small
$r$, the first are “plane’s like”, in the sense that the $-ny$ term (which at
large enough $y$ is negligible anyway) changes their location, now being left
to the singularity; see the left part of figure 4. In particular, the first
extremum point, which is picked by the $\sigma^{1}$ contour integral, is
exactly the same as for the plane $y\sim-r$. As $r$ increases, the “plane’s
like” saddles disappear gradually, where at $r\sim 2\pi$, even the first
negative extremum is dictated by $-nr\ln\Theta^{2}$ alone. From there on, it
decreases with $\tau_{2s}(r)$. See appendix E for more details.

Figure 3: $\tau_{2}$ (left) and $\sigma^{1}$ (right) saddle point as a
function of $r\equiv\frac{|q|}{|\Lambda|L}$. $T_{2}=10$.

Figure 4: Saddles for the torus at small r (left) and large r (right). The
saddles are located at the minima. The “spikes” are the zeros of $\Theta$,
located at $\pi\tau_{2s}(r)$, with $\tau_{2s}=5,\ 0.055$ respectively. We used
$T_{2}=5$ for clarity. Note the difference of scales in the $x$-axis.

## Acknowledgments

The author would like to thank Ofer Aharony for general guidance during the
work and many useful discussions. This work was supported in part by an Israel
Science Foundation (ISF) grant number 2159/22, by Simons Foundation grant
994296 (Simons Collaboration on Confinement and QCD Strings), by the Minerva
Foundation with funding from the Federal German Ministry for Education and
Research, by the German Research Foundation through a German-Israeli Project
Cooperation (DIP) grant “Holography and the Swampland”, and by a research
grant from Martin Eisenstein.

##  Appendix A Modular Transformations

In this appendix, we briefly describe the modular transformation. We refer the
reader also to [30], pp.338-340. Here, we also add comments relevant to this
work. The modular transformations are transformations of the modular parameter
$\tau$, having the structure of the $PSL(2,\mathbb{Z})$ group:

$\tau\to\frac{a\tau+b}{c\tau+d}.$ (118)

$a,b,c,d\in\mathbb{Z}$ and $ad-bc=1$, since we keep the orientation of the
torus. It is trivial that $\mathcal{R}:\tau\to\bar{}\tau$ represents the same
torus. Hence, each orientation preserving transformation $\mathcal{P}$ is
accompanied by a changing orientation transformation $\mathcal{RP}$ (for which
$ad-bc=-1$). The $PSL(2,\mathbb{Z})$ group is generated by:

$\displaystyle\mathcal{T}:$ $\displaystyle\tau\to\tau+1,$ (119)
$\displaystyle\mathcal{S}:$ $\displaystyle\tau\to-\frac{1}{\tau}.$ (120)

The modular parameter $\tau$ does not contain the whole information about the
torus - the length of one edge is missing. In general, a torus is described by
two independent vectors on a lattice. They can be described as two numbers in
the complex plane $\omega_{1,2}$. Then the modular parameter
$\tau\equiv\frac{\omega_{2}}{\omega_{1}}$ cares only about the angle between
them and the ratio between their lengths. Therefore, it is enough when
parameterizing a torus for a CFT. However, for a general QFT, one needs
$\omega_{1,2}$. In terms of these, the modular transformations
reads363636There is a mistake in Eq.(10.6) of [30] according to its definition
of $\tau$ in pp.337.:

$\begin{pmatrix}\omega_{2}\\\ \omega_{1}\end{pmatrix}\to\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\begin{pmatrix}\omega_{2}\\\ \omega_{1}\end{pmatrix}.$ (121)

The new vectors are another two independent vectors, as the matrix is
invertible; they are two lattice vectors as the matrix elements are integers;
and it preserves the area of the torus since the determinant of the matrix is
unity. The matrices for the two generators are:

$\displaystyle\mathcal{T}:\begin{pmatrix}1&1\\\ 0&1\end{pmatrix},\
\mathcal{S}:\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}.$ (122)

Without loss of generality, we take $\omega_{1}=L$ to be on the real axis.
When applying a modular transformation, $L$ should be modified accordingly, so
the torus’ area $L^{2}\tau_{2}$ remains the same. For example, $L\to L|\tau|$
for $\mathcal{S}$.

Using two dimensional vectors rather than complex numbers, one has:

$\begin{pmatrix}L_{2}\\\ L_{1}\end{pmatrix}\to\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\begin{pmatrix}L_{2}\\\ L_{1}\end{pmatrix}.$ (123)

The choice $\omega_{1}=L$ reads:

$L_{1}^{a}=L\begin{pmatrix}1\\\ 0\end{pmatrix},\
L_{2}^{a}=L\begin{pmatrix}\tau_{1}\\\ \tau_{2}\end{pmatrix}.$ (124)

The vielbein (17) and $L^{a}_{\alpha}$ of the TSt (21) are essentially these
two, without putting $\omega_{1}$ on the real axis, but allow it to be at any
direction, and $L=e^{\Omega}$ for the WSt. Its direction is not important for
the fixed TSt but must be included for the WSt, where all angles at all
$\sigma$ are integrated.

If considering integration over the modular parameter only, the measure
$\frac{d^{2}\tau}{\tau_{2}^{2}}$ is modular invariant. It is trivial for
$\mathcal{T}$, and for $\mathcal{S}$ one has
$d^{2}\tau^{\prime}=\frac{d^{2}\tau}{|\tau|^{4}}$.373737This form resembles
the measure $\frac{dx}{x}$ which is invariant under $x\to 1/x$ transformation.
The $|\tau|^{-4}$ is also expected since
$\tau^{\prime}\sim\frac{1}{\tau}\Rightarrow\frac{d\tau^{\prime}}{d\tau}\sim\frac{1}{\tau^{2}}\Rightarrow\det{\frac{d\tau^{\prime}}{d\tau}}\sim\frac{1}{\tau^{4}}$.
If considering integration over all possible tori, then one can integrate over
all $L^{a}_{\alpha}$ with $d^{4}L$ which is modular invariant.383838Using
(123), the Jacobian for the four components $L^{a}_{\alpha}$ reads $(ad-
bc)^{2}=1$. Alternatively one can use $d\Omega
e^{2\Omega}d\phi\frac{d^{2}\tau}{\tau_{2}}$, which is also modular
invariant.393939It equals $Ad\Omega d\phi\frac{d^{2}\tau}{\tau_{2}^{2}}$ with
$A=e^{2\Omega}\tau_{2}$ the torus’ area. Any other power of $A$ is equally
good, e.g. $d^{4}L=e^{4\Omega}d\Omega d\phi d^{2}\tau=A^{2}d\Omega
d\phi\frac{d^{2}\tau}{\tau_{2}^{2}}$. These two measures appeared in [26], see
(43) and (44).

In our theory, we have two tori: the TSt and the WSt. The correlator, which
depends on the TSt, should not depend on the vectors we choose to describe the
torus, hence it is expected to be invariant under $PSL(2,\mathbb{Z})_{TS}$. It
also implies that we can two representative vectors that are convenient for
us. Indeed this is a global symmetry of the action (9). This can be seen as
follows. $S_{0}$ does not depend on the TSt, and so we need to check $S_{JT}$
alone. We denote $E^{a}_{\alpha}\equiv\partial_{\alpha}X^{a}-e^{a}_{\alpha}$.
Then, using matrix notation, $S_{JT}$ can be written as:

$S_{JT}=\int d^{2}\sigma\Tr(E\epsilon E^{T}\epsilon^{T}).$ (125)

Modular transformations of the TSt acts as $E^{a}_{\alpha}\to
A^{a}_{b}E^{b}_{\alpha}$, with $A\in PSL(2,\mathbb{Z})$. It changes the
allowed domain for $X^{a}$ and the two TS vectors $e^{a}_{1,2}$. The
Lagrangian changes as $\Tr(E\epsilon E^{T}\epsilon^{T})\to\Tr(AE\epsilon
E^{T}A^{T}\epsilon^{T})=\Tr(E\epsilon E^{T}A^{T}\epsilon^{T}A)$. Now
$A^{T}\epsilon^{T}A=\epsilon^{T}$, since it is just the determinant of $A$,
which is unity.

Modular transformations of the WSt are not global symmetries. Since we
integrate over all possible WS tori, they are part of the diffeomorphism
invariance of the theory. When considering all diffeomorphisms preserving the
coordinates’ domain (14), they correspond to the large diffeomorphisms:

$\begin{pmatrix}\sigma^{1}\\\ \sigma^{2}\end{pmatrix}\to\begin{pmatrix}a&-b\\\
-c&d\end{pmatrix}\begin{pmatrix}\sigma^{1}\\\ \sigma^{2}\end{pmatrix}.$ (126)

The gauge-fixing of the diffeomorphism invariance done in subsection 3.1 only
deals with small diffeomorphisms. The large diffeomorphisms
$PSL(2,\mathbb{Z})_{WS}$ remaining appear now as modular transformations. The
two metrics with $\tau^{\prime}=\frac{a\tau+b}{c\tau+d},\
e^{2\Omega^{\prime}}=e^{2\Omega}|c\tau+d|^{2}$ describe the same WSt, as they
are related by the change of coordinates (126). They are residual gauge
transformations. One usually eliminates them by reducing the integration
domain of $\tau$ from the upper half plane to some fundamental region.

However, in our theory, the situation is slightly different. We integrate over
all possible mappings $X^{a}(\sigma)$ from the WSt to the TSt. The relation
(28) sends specific WSt cycles $C_{WS}$ to specific TSt cycles $C_{TS}$, and
so it is a global gauge-fixing. Large diffeomorphisms do not respect this
relation. From another perspective, the transformation
$\tau\to\tau^{\prime},\Omega\to\Omega^{\prime}$ described before, although
describing the same WSt, changes $C_{WS}$. According to this, one has two
options:

  * •

The first is to fix some cycles of the TSt $C_{TS}$, which means fixing
$L^{a}_{\alpha}$. The modular parameter $\tau$ is then integrated over the
upper half plane.

  * •

The second is to integrate $\tau$ only in the fundamental domain, which means
we fix some WSt $C_{WS}$. Then, one needs to sum over different
$L^{a}_{\alpha}$ related by $SL(2,\mathbb{Z})_{TS}$ transformations.

In this work, we chose the first option. We fixed some $L^{a}_{\alpha}$ (with
$T$ in the fundamental region) and integrated $\tau$ over the whole upper half
plane. We note that now the $PSL(2,\mathbb{Z})_{TS}$ “combines” with
$PSL(2,\mathbb{Z})_{WS}$, i.e. applying the same modular transformation to
both tori is still a symmetry. This is because the same WS cycles are mapped
to the same TS cycles. $PSL(2,\mathbb{Z})_{TS}$ transformation alone is not a
symmetry anymore since it does not preserve the gauge-fixing condition (28).
It is broken in the quantum theory; the regularization we choose breaks these
combined $PSL(2,\mathbb{Z})$ transformations. Since $PSL(2,\mathbb{Z})_{WS}$
is part of the diffeomorphism invariance of the theory, it can not be broken,
and therefore, the global $PSL(2,\mathbb{Z})_{TS}$ is broken.

##  Appendix B Determinants

###  B.1 P

The FP-determinant $J$, appearing in (26), is related to the operator $P$,
which is the linearization of the constraints:

$\begin{split}e^{(V)}-\bar{e}(\Omega(\sigma),\phi(\sigma),\tau)&=0\\\
\sigma_{1}^{(V)}-0&=0\end{split}$ (127)

around their solution $\bar{e},0$. The operator $P$ is given by:

$P\begin{pmatrix}\delta V^{\beta}\\\ \delta\Omega\\\ \delta\phi\\\
d\tau_{i}\end{pmatrix}=\begin{pmatrix}\partial_{\beta}\bar{e}^{a}_{\alpha}+\bar{e}^{a}_{\beta}\partial_{\alpha}&-\bar{e}^{a}_{\alpha}&-\epsilon^{a}_{b}\bar{e}^{b}_{\alpha}&-\partial_{\tau_{i}}\bar{e}^{a}_{\alpha}\\\
\delta^{\alpha}_{\beta}\int
d^{2}\sigma\delta(\sigma)&0&0&0\end{pmatrix}\begin{pmatrix}\delta V^{\beta}\\\
\delta\Omega\\\ \delta\phi\\\ \delta\tau_{i}\end{pmatrix},$ (128)

where

$\begin{split}\partial_{\beta}\bar{e}^{a}_{\alpha}&=\bar{e}^{a}_{\alpha}\partial_{\beta}\Omega+\epsilon^{a}_{b}\bar{e}^{b}_{\alpha}\partial_{\beta}\phi,\\\
\partial_{\tau_{1}}\bar{e}^{a}_{\alpha}&=e^{\Omega}\begin{pmatrix}0&\cos(\phi)\\\
0&-\sin(\phi)\end{pmatrix},\\\
\partial_{\tau_{2}}\bar{e}^{a}_{\alpha}&=e^{\Omega}\begin{pmatrix}0&\sin(\phi)\\\
0&\cos(\phi)\end{pmatrix}.\end{split}$ (129)

The determinant $J$ is related to the operator $P$ by:

$J(\bar{e},0)=\sqrt{\det(P^{\dagger}P)}.$ (130)

To evaluate $P^{\dagger}$, one needs inner products for the different vector
spaces involved. The inner products we use are:

$\begin{split}G_{s}(\delta f,\delta g)&\equiv|\Lambda|\int dA\delta
f(\sigma)\delta g(\sigma),\\\ G_{v}\left(\delta V^{\alpha},\delta
W^{\beta}\right)&\equiv\int dAg_{\alpha\beta}\delta V^{\alpha}(\sigma)\delta
W^{\beta}(\sigma),\\\ G_{e}\left(\delta e^{a}_{\alpha},\delta
f^{b}_{\beta}\right)&\equiv\int dAg^{\alpha\beta}\delta_{ab}\delta
e^{a}_{\alpha}(\sigma)\delta f^{b}_{\beta}(\sigma),\\\
G_{\tau}(\delta\tau_{i},\delta\rho_{j})&\equiv\frac{\delta^{ij}\delta\tau_{i}\delta\rho_{j}}{\tau_{2}^{2}},\\\
G_{\sigma}(\delta\sigma_{1}^{\alpha},\delta{\sigma^{\prime}}_{1}^{\beta})&\equiv
g_{\alpha\beta}(0)\delta\sigma_{1}^{\alpha}\delta{\sigma^{\prime}}_{1}^{\beta}.\end{split}$
(131)

$G_{s}$ is used for $\delta\Omega,\delta\phi$, and can be obtained from
$G_{e}$ by considering how the vielbein is changing due to a change in
$\Omega,\phi$: $\delta
e^{a}_{\alpha}=e^{a}_{\alpha}\delta\Omega+\epsilon^{a}_{b}e^{b}_{\alpha}\delta\phi$.
The same is true for $G_{\tau}$. If one writes $e=e_{p}M$ as in (17), then:

$\int
d^{2}\sigma\sqrt{g}g^{\alpha\beta}\partial_{\tau_{i}}e^{a}_{\alpha}\partial_{\tau_{j}}e^{a}_{\beta}\delta\tau_{i}\delta\rho_{j}=A\Tr\left(\partial_{\tau_{i}}MM^{-1}T^{-M}\partial_{\tau_{j}}M\right)\delta\tau_{i}\delta\rho_{j}=\frac{A\delta^{ij}\delta\tau_{i}\delta\rho_{j}}{\tau_{2}^{2}},$
(132)

where $A$ is the WS-area, and can be removed since it is an overall constant.
This inner product is also expected since it is modular invariant. It leads to
the invariant measure $\frac{d^{2}\tau}{\tau_{2}^{2}}$.404040It is different
from the simpler inner product used in [26], which is not modular invariant.

$P^{\dagger}$ is given by:

$P^{\dagger}\begin{pmatrix}\delta e_{a\gamma}\\\
\delta\sigma_{1}^{\alpha}\end{pmatrix}=\begin{pmatrix}\bar{g}^{\beta\delta}\bar{g}^{\alpha\gamma}(-\bar{e}^{a}_{\beta}\partial_{\alpha}-\partial_{\alpha}\bar{e}^{a}_{\beta}+\partial_{\beta}\bar{e}^{a}_{\alpha})&\frac{\delta(\sigma)}{\sqrt{\bar{g}(0)}}\delta_{\alpha}^{\delta}\\\
-\bar{g}^{\alpha\gamma}\bar{e}^{a}_{\alpha}&0\\\
-\bar{g}^{\alpha\gamma}\epsilon^{a}_{b}\bar{e}^{b}_{\alpha}&0\\\
-\delta_{kj}\tau_{2}^{2}\int
d^{2}\sigma\sqrt{g}g^{\gamma\beta}\partial_{\tau_{j}}\bar{e}^{a}_{\beta}&0\end{pmatrix}\begin{pmatrix}\delta
e_{a\gamma}\\\ \delta\sigma_{1}^{\alpha}\end{pmatrix}.$ (133)

Composing $P^{\dagger}P$ gives:

$P^{\dagger}P=\begin{pmatrix}{P_{11}}^{\delta}_{\beta}&\partial^{\delta}&\varepsilon^{\delta\alpha}\partial_{\alpha}+2\varepsilon^{\delta\alpha}\partial_{\alpha}\Omega-2\partial^{\delta}\phi&{P_{14}}^{\delta
i}\\\ -2\partial_{\beta}\Omega-\partial_{\beta}&2&0&a_{i}\\\
-2\partial_{\beta}\phi-{\varepsilon_{\beta}}^{\alpha}\partial_{\alpha}&0&2&b_{i}\\\
{P_{41}}_{k\beta}&\tau_{2}^{2}a_{k}\int
d^{2}\sigma\sqrt{g}&\tau_{2}^{2}b_{k}\int
d^{2}\sigma\sqrt{g}&\delta_{k}^{i}\tau_{2}^{2}\bar{A}\end{pmatrix},$ (134)

where

$\begin{split}{P_{11}}^{\delta}_{\beta}\equiv&-\delta^{\delta}_{\beta}g^{\alpha\gamma}\partial_{\alpha}\partial_{\gamma}-(\delta^{\delta}_{\alpha}\partial_{\beta}\Omega+2\delta^{\delta}_{\beta}\partial_{\alpha}\Omega+\varepsilon^{\delta}_{\alpha}\partial_{\beta}\phi-\varepsilon_{\beta\alpha}\partial^{\delta}\phi)\partial^{\alpha}+\partial^{\delta}\Omega\partial_{\beta}\\\
&+2\partial^{\delta}\phi\partial_{\beta}\phi-2\varepsilon^{\delta}_{\alpha}\partial^{\alpha}\Omega\partial_{\beta}\phi-\partial^{\delta}\partial_{\beta}\Omega-\varepsilon^{\delta}_{\alpha}\partial^{\alpha}\partial_{\beta}\phi+\frac{\delta(\sigma)}{\sqrt{\bar{g}(0)}}\delta^{\delta}_{\beta},\\\
{P_{14}}^{\delta
i}\equiv&\bar{g}^{\beta\delta}\bar{g}^{\alpha\gamma}(\bar{e}^{a}_{\beta}\partial_{\alpha}\partial_{\tau_{i}}\bar{e}^{a}_{\gamma}+(\partial_{\alpha}\bar{e}^{a}_{\beta}-\partial_{\beta}\bar{e}^{a}_{\alpha})\partial_{\tau_{i}}\bar{e}^{a}_{\gamma}),\\\
{P_{41}}_{k\beta}\equiv&-\delta_{kj}\tau_{2}^{2}\int
d^{2}\sigma\sqrt{g}g^{\gamma\delta}\partial_{\tau_{j}}\bar{e}^{a}_{\delta}\left(\partial_{\beta}\bar{e}^{a}_{\gamma}+\bar{e}^{a}_{\beta}\partial_{\gamma}\right),\\\
a_{i}\equiv&\bar{g}^{\alpha\gamma}\bar{e}^{a}_{\alpha}\partial_{\tau_{i}}\bar{e}^{a}_{\gamma}=\Tr\left(M^{-T}\partial_{\tau_{i}}M^{T}\right),\\\
b_{i}\equiv&\bar{g}^{\alpha\gamma}\epsilon^{a}_{b}\bar{e}^{b}_{\alpha}\partial_{\tau_{i}}\bar{e}^{a}_{\gamma}=\Tr\left(\epsilon
M^{-T}\partial_{\tau_{i}}M^{T}\right).\end{split}$ (135)

One expects $J$ to have the following properties:

  1. 1.

$J\propto\det\left(\frac{e^{-2\Omega}}{\tau_{2}}\right)$. This factor will
cancel the extra $\det\left(e^{2f_{\Omega}}\tau_{2}\right)$ in (69). The first
row of $P^{\dagger}P$ is multiplied by $e^{-2\Omega}$, and it will be pulled
out as $\det\left(e^{-2\Omega}\right)$ when taking the determinant.414141Let
$M$ be a $n\crossproduct n$ block matrix, and say its first row is
$m_{1j}=ea_{1j}$, each one being a matrix. Then $M=EM^{\prime}$, where
$e_{11}=e,\ e_{ii}=I,\ e_{ij}=0$, and $m^{\prime}_{1j}=a_{1j},\
m^{\prime}_{ij}=m_{ij}$. Taking determinant of both sides gives
$\det(M)=\det(E)\det(M^{\prime})=\det(e)\det(M^{\prime})$. The differential
diffeomorphism space is two dimensional, so
$\det(e)=\det\left(e^{-2\Omega}\right)^{2}$, and the square root leaves
$\det\left(e^{-2\Omega}\right)$. We expect $e^{-2\Omega}$ to be accompanied by
$\tau_{2}^{-1}$, as they appear together in $\sqrt{g}$.

  2. 2.

$J\propto\frac{1}{\tau_{2}^{2}}$. The correlation function should be modular
invariant by changing $\tau$ and $T$ simultaneously. The FP-identity (26)
comes without $T$ and, therefore, should be modular invariant with respect to
$\tau$ alone. Given the integration measure $d^{2}\tau$, the combination
$\frac{d^{2}\tau}{\tau_{2}^{2}}$ is modular invariant (it indeed happened for
the partition function [26]).424242It does not mean that it is the only
dependence of $\tau_{2}$ inside $J$, the WS-area $\bar{A}=\tau_{2}\int
d^{2}\sigma e^{\Omega}$ is also modular invariant. It just means that only in
this factor $\tau_{2}$ will appear alone.

###  B.2 Q

The operator $Q$ is the linearization of the constraint $K_{a}(\sigma)=0$
around its solution $f_{\Omega},f_{\phi}$. It is given by:

$Q\begin{pmatrix}\delta\Omega\\\
\delta\phi\end{pmatrix}\equiv\epsilon^{\alpha\beta}\left(\partial_{\alpha}\bar{e}^{c}_{\beta}+\bar{e}^{c}_{\beta}\partial_{\alpha}\right)\begin{pmatrix}\delta^{b}_{c}&\epsilon^{b}_{c}\end{pmatrix}\begin{pmatrix}\delta\Omega\\\
\delta\phi\end{pmatrix}.$ (136)

It has two zero modes, as the general solution has two free parameters. It is
easy to verify that $Q$ indeed annihilates the two vectors:

$e^{\bar{}\Omega-
f_{\Omega}}\begin{pmatrix}\cos\left(f_{\phi}-\bar{\phi}\right)\\\
-\sin\left(f_{\phi}-\bar{\phi}\right)\end{pmatrix},\ e^{\bar{}\Omega-
f_{\Omega}}\begin{pmatrix}\sin\left(f_{\phi}-\bar{\phi}\right)\\\
\cos\left(f_{\phi}-\bar{\phi}\right)\end{pmatrix}.$ (137)

We use the inner product out of which the discrete measure
$\prod_{\sigma=1}^{N}d\Omega(\sigma)d\phi(\sigma)$ is naturally defined (see
subsection 3.3), which is
$\sum_{\sigma=1}^{N}\left(d\Omega(\sigma)d\Omega^{\prime}(\sigma)+d\phi(\sigma)d\phi^{\prime}(\sigma)\right)$.
Then $Q^{\dagger}$ is:

$Q^{\dagger}\delta K_{b}\equiv-\begin{pmatrix}\delta^{b}_{c}\\\
\epsilon^{b}_{c}\end{pmatrix}\epsilon^{\alpha\beta}\bar{e}^{c}_{\beta}\partial_{\alpha}\delta
K_{b}.$ (138)

The composition $Q^{\dagger}Q$ gives:

$\begin{split}Q^{\dagger}Q&=\begin{pmatrix}\ A&B\ \\\ -B&A\ \end{pmatrix},\\\
A&\equiv-
gg^{\alpha\beta}\left(\partial_{\alpha}\partial_{\beta}+\partial_{\alpha}\Omega\partial_{\beta}+\partial_{\beta}\Omega\partial_{\alpha}+\partial_{\alpha}\Omega\partial_{\beta}\Omega-\partial_{\alpha}\phi\partial_{\beta}\phi+\partial_{\alpha}\partial_{\beta}\Omega\right),\\\
B&\equiv
gg^{\alpha\beta}\left(\partial_{\alpha}\phi\partial_{\beta}+\partial_{\beta}\phi\partial_{\alpha}+\partial_{\alpha}\phi\partial_{\beta}\Omega+\partial_{\beta}\phi\partial_{\alpha}\Omega+\partial_{\alpha}\partial_{\beta}\phi\right),\end{split}$
(139)

where $g=\det(g_{\alpha\beta})=e^{4\Omega}\tau_{2}^{2}$.434343In [27] another
term appeared in $A$:
$2g\varepsilon^{\alpha\beta}\partial_{\alpha}\Omega\partial_{\beta}\phi,$. It
was a mistake, reminiscent of old versions. By deriving this result, the
following identities were used:

$\epsilon^{\gamma\delta}\epsilon^{\alpha\beta}e^{b}_{\beta}e^{c}_{\delta}\begin{pmatrix}\delta_{cb}\\\
\epsilon_{cb}\end{pmatrix}=g\begin{pmatrix}g^{\gamma\alpha}\\\
\varepsilon^{\gamma\alpha}\end{pmatrix},$ (140)

which are valid in general, particularly both for the plane and for the torus.
The diagonal form of $Q^{\dagger}Q$ is:

$\begin{split}\begin{pmatrix}\ A&B\ \\\ -B&A\
\end{pmatrix}&=U\begin{pmatrix}A+iB&0\\\ 0&A-iB\ \end{pmatrix}U^{\dagger},\\\
U&\equiv\frac{1}{\sqrt{2}}\begin{pmatrix}I&I\ \\\ iI&-iI\
\end{pmatrix}.\end{split}$ (141)

The explicit form of the operators $A\pm iB$ can be simplified using
$\bar{z}=e^{\Omega-i\phi}$:

$\sqrt{{\det}^{\prime}(Q^{\dagger}Q)}=\left|{\det}^{\prime}\left(-gg^{\alpha\beta}(\partial_{\alpha}+\partial_{\alpha}\ln(\bar{z}))(\bar{\partial}_{\beta}+\bar{\partial}_{\beta}\ln(\bar{z}))\right)\right|.$
(142)

The form of the operator inside the determinant looks nicer when using complex
coordinates:

$gg^{\alpha\beta}(\partial_{\alpha}+\partial_{\alpha}\ln(\bar{z}))(\bar{\partial}_{\beta}+\bar{\partial}_{\beta}\ln(\bar{z}))=e^{2\Omega}(\partial+\partial\ln(\bar{z}))(\bar{\partial}+\bar{\partial}\ln(\bar{z})).$
(143)

However, the form of the constraints $K_{b}(\sigma)$ was obtained using the
$\sigma$ coordinates, and the complex coordinates $w$ depend explicitly on
$\tau$, which is integrated, so this determinant should be thought of in terms
of $\sigma$ coordinates only.

(143) suggest to take $\det(e^{2\Omega}\tau_{2})$ outside of (142). But,
because of the zero modes exclusion, we can not naively split
${\det}^{\prime}\left(e^{2\Omega}\tau_{2}\square\right)\rightarrow{\det}\left(e^{2\Omega}\tau_{2}\right){\det}^{\prime}(\square)$,
where
$\square\equiv\frac{1}{\tau_{2}}(\partial+\partial\ln(\bar{z}))(\bar{\partial}+\bar{\partial}\ln(\bar{z}))$.
In order to “resolve” the zero modes, the operator $D\equiv
e^{2\Omega}\tau_{2}\square$ is regulated: $D_{\epsilon}\equiv
e^{2\Omega}\tau_{2}(\square+\epsilon)$, where $\epsilon$ is a small number.
The eigenvalues $\lambda^{\epsilon}_{n}$ of $D_{\epsilon}$ are slightly
shifted from $\lambda_{n}$ of $D$:
$\lambda^{\epsilon}_{n}\approx\lambda_{n}+\epsilon\alpha_{n}$. The eigenvalue
$\lambda_{0}=0$ was excluded in ${\det}^{\prime}(D)$, and therefore
$\lambda^{\epsilon}_{0}\approx 0+\epsilon\alpha_{0}$ will be excluded in the
regularized version. Using these definitions, we evaluate:

$\begin{split}{\det}^{\prime}(D_{\epsilon})&=\prod_{n\neq
0}\lambda^{\epsilon}_{n}=\frac{\prod_{n}\lambda^{\epsilon}_{n}}{\lambda^{\epsilon}_{0}}=\frac{\det(D_{\epsilon})}{\epsilon\alpha_{0}}=\frac{\det(e^{2\Omega}\tau_{2})\det(\square+\epsilon)}{\epsilon\alpha_{0}}=\det(e^{2\Omega}\tau_{2})\frac{{\det}^{\prime}(\square+\epsilon)\epsilon}{\epsilon\alpha_{0}}\\\
&\xrightarrow[\epsilon\rightarrow
0]{}{\det}^{\prime}(D)=\frac{1}{\alpha_{0}}\det(e^{2\Omega}\tau_{2}){\det}^{\prime}(\square).\end{split}$
(144)

The finite factor $\alpha_{0}^{-1}$ can be calculated as follows. We write the
eigenfunction $f^{\epsilon}_{0}\approx\frac{1}{\bar{z}}+\epsilon f_{0}$, and
solve the eigenvalue equation:

$e^{2\Omega}\left(\square+\epsilon\right)f^{\epsilon}_{0}=\lambda^{\epsilon}_{0}f^{\epsilon}_{0}\Rightarrow\partial\bar{\partial}(\bar{z}f_{0})=\frac{\alpha_{0}}{e^{2\Omega}\tau_{2}}-1.$
(145)

$f_{0}$ and $\bar{z}$ satisfy periodic boundary conditions, therefore:

$\int
d^{2}\sigma\partial\bar{\partial}(\bar{z}f_{0})=0\Rightarrow\alpha_{0}=\frac{\tau_{2}}{\int
d^{2}\sigma e^{-2\Omega}}.$ (146)

Finally, the determinant can be written as:

$\sqrt{{\det}^{\prime}(Q^{\dagger}Q)}=\det(e^{2\Omega}\tau_{2})\frac{\int
d^{2}\sigma
e^{-2\Omega}}{\tau_{2}}\left|{\det}^{\prime}\left(-\frac{1}{\tau_{2}}(\partial+\partial\ln(\bar{z}))(\bar{\partial}+\bar{\partial}\ln(\bar{z}))\right)\right|.$
(147)

##  Appendix C Special Functions

###  C.1 Theta-function

The $\Theta(w,\tau)$-function is defined as:

$\begin{split}\Theta(w,\tau)&\equiv
i\sum_{n=-\infty}^{\infty}(-1)^{n}q^{\frac{(n-\frac{1}{2})^{2}}{2}}z^{n-\frac{1}{2}},\\\
&=2q^{\frac{1}{8}}\sin(\pi
w)\prod_{m=1}^{\infty}(1-q^{m})(1-zq^{m})(1-z^{-1}q^{m}),\\\ \end{split}$
(148)

where:

$q\equiv e^{2\pi i\tau},\ z\equiv e^{2\pi iw}.$ (149)

$\Theta(w,\tau)$ here is $EllipticThetaPrime[1,z^{\prime},q^{\prime}]$ in
Mathematica, with $q^{\prime}=\sqrt{q}=e^{\pi i\tau}$ and
$z^{\prime}=\frac{-i\ln(z)}{2}=\pi w$. It was used in the numerical
calculation done in section 4. In the literature it is called
$-\Theta_{11}(w,\tau)=-\Theta\genfrac{[}{]}{0.0pt}{2}{\frac{1}{2}}{\frac{1}{2}}(w,\tau)$.
There are various conventions for the variables and definitions of $z,q$, so
one has to be careful when comparing expressions from different sources. We
list its important properties:

$\displaystyle\Theta(0,\tau)$ $\displaystyle=0,$ (150)
$\displaystyle\Theta(-w,\tau)$ $\displaystyle=-\Theta(w,\tau),$ (151)
$\displaystyle\overline{\Theta(w,\tau)}$
$\displaystyle=-\Theta(-\bar{w},-\bar{}\tau)=\Theta(\bar{w},-\bar{}\tau),$
(152) $\displaystyle\Theta(w,\tau)$ $\displaystyle\approx 2\eta(\tau)^{3}\pi
w\ |w|<<1.$ (153)

It is not periodic:

$\displaystyle\Theta(w+1,\tau)$ $\displaystyle=-\Theta(w,\tau),$ (154)
$\displaystyle\Theta(w+\tau,\tau)$ $\displaystyle=-e^{-\pi i\tau-2\pi
iw}\Theta(w,\tau).$ (155)

Its modular properties are:

$\begin{split}\Theta(w,\tau+1)&=e^{\frac{i\pi}{4}}\Theta(w,\tau),\\\
\Theta\left(\frac{w}{\tau},-\frac{1}{\tau}\right)&=-i\sqrt{-i\tau}e^{\frac{i\pi
w^{2}}{\tau}}\Theta(w,\tau).\end{split}$ (156)

###  C.2 Eta-function

The $\eta(\tau)$-function is defined as:

$\eta(\tau)\equiv q^{\frac{1}{24}}\prod_{m=1}^{\infty}(1-q^{m}),$ (157)

with $q$ as before. $\eta(\tau)$ here is $DedekindEta[\tau]$ in Mathematica.
It was used in the numerical calculation done section 4. Its modular
properties are:

$\begin{split}\eta(\tau+1)&=e^{\frac{i\pi}{12}}\eta(\tau),\\\
\eta\left(-\frac{1}{\tau}\right)&=(-i\tau)^{\frac{1}{2}}\eta(\tau).\end{split}$
(158)

###  C.3 G-function

$G$ is naturally defined as a function of $w$ and $\tau$ as in (44):

$G(w,\tau)\equiv-\ln\left|\Theta\left(w|\tau\right)\right|^{2}+\frac{2\pi\imaginary(w)^{2}}{\tau_{2}}.$
(159)

One can replace $w$ by the Cartesian coordinates $\sigma^{1,2}$:

$G(\sigma^{1},\sigma^{2},\tau)\equiv G(\sigma^{1}+\tau\sigma^{2},\tau).$ (160)

It satisfies:

$\displaystyle G(-w,\tau)$ $\displaystyle=G(w,\tau),$ (161)
$\displaystyle\overline{G(w,\tau)}$ $\displaystyle=G(w,\tau).$ (162)

The second term in (159) “fixes” the non-periodicity of the $\Theta$-function:

$\displaystyle G(w+1,\tau)=G(w,\tau)$ $\displaystyle\Rightarrow
G(\sigma^{1}+1,\sigma^{2},\tau)=G(\sigma^{1},\sigma^{2},\tau),$ (163)
$\displaystyle G(w+\tau,\tau)=G(w,\tau)$ $\displaystyle\Rightarrow
G(\sigma^{1},\sigma^{2}+1,\tau)=G(\sigma^{1},\sigma^{2},\tau).$ (164)

The modular properties of $G$ are dictated by those of $\Theta$ and $\eta$:

$\begin{split}G(w,\tau)=G(w,\tau+1)&\Rightarrow
G(\sigma^{1},\sigma^{2},\tau)=G(\sigma^{1}-\sigma^{2},\sigma^{2},\tau+1),\\\
G\left(\frac{w}{\tau},-\frac{1}{\tau}\right)=-\ln{|\tau|}+G(w,\tau)&\Rightarrow
G\left(\sigma^{2},-\sigma^{1},-\frac{1}{\tau}\right)=-\ln{|\tau|}+G(\sigma^{1},\sigma^{2},\tau).\end{split}$
(165)

When changing the second argument of $G(w,\tau)$ to $f_{3}(\tau)$, the first
argument should be in the form
$f_{1}(\sigma^{1,2})+f_{2}(\sigma^{1,2})f_{3}(\tau)$, and then identify the
dependence in Cartesian coordinates
$G(f_{1}(\sigma^{1,2}),f_{2}(\sigma^{1,2}),f_{3}(\tau))$.

The divergences of $G$ come solely from zeros of $\Theta$. The latter has a
unique zero at $\sigma^{1}=\sigma^{2}=0$, so this is the only divergence of
$G$. The value $\tau_{2}=0$ is also problematic, but it is not allowed.

At $\tau_{2}\gtrsim 2$, $G$ grow linearly with $\tau_{2}$. Three terms
contribute to the slope: $q^{\frac{1}{8}}$ and $\sinh(\pi w)$ from the first
term in the r.h.s. of (159), and $2\pi\tau_{2}(\sigma^{2})^{2}$ from the
second term:

$G(\sigma^{1},\sigma^{2},\tau)\approx
2\pi\left(\sigma^{2}-\frac{1}{2}\right)^{2}\tau_{2}.$ (166)

The small $\tau_{2}$ behavior is more subtle and can be derived by the modular
properties of $G$.

##  Appendix D Weyl Anomaly

In subsection 3.5, the Weyl anomaly for a conformal change
$g=e^{2\Omega}g_{0}$ of a given initial metric $g_{0}$ was needed. It can be
calculated in similar steps to those presented in [31]. The following
differential equation for a partition function $Z[g]$ of a QFT defined on a
curved space with the metric $g$ holds:

$\frac{\delta Z\left[g\right]}{\delta
g_{\alpha\beta}(\sigma)}=-\frac{1}{2}\sqrt{g(\sigma)}Z[g]\langle
T^{\alpha\beta}(\sigma)\rangle.$ (167)

Using $\delta g_{\alpha\beta}=2g_{\alpha\beta}\delta\Omega$ gives:

$\frac{\delta
Z\left[e^{2\Omega}g_{0}\right]}{\delta\Omega(\sigma)}=-e^{2\Omega}\sqrt{g_{0}(\sigma)}Z\left[e^{2\Omega}g_{0}\right]\langle
T^{\alpha}_{\alpha}(\sigma)\rangle.$ (168)

For a manifold without a boundary, the trace of the energy-momentum tensor is
$-\frac{c}{12}R$. If $g_{0}$ is a flat metric, the Ricci scalar for $g$ is:

$R=-2e^{-2\Omega}\nabla^{2}_{g_{0}}\Omega.$ (169)

Inserting these two relations into (168) yields:

$\frac{\delta
Z\left[e^{2\Omega}g_{0}\right]}{\delta\Omega(\sigma)}=-\frac{c}{6}\sqrt{g_{0}(\sigma)}Z\left[e^{2\Omega}g_{0}\right]\nabla^{2}_{g_{0}}\Omega.$
(170)

This equation can be integrated:

$Z\left[e^{2\Omega}g_{0}\right]=Z\left[g_{0}\right]e^{-\frac{c}{12}\int
d^{2}\sigma\sqrt{g_{0}(\sigma)}\Omega\nabla^{2}_{g_{0}}\Omega}\equiv
Z\left[g_{0}\right]e^{W}.$ (171)

Integration by parts gives (65):

$W=\frac{c}{12}\int
d^{2}\sigma\sqrt{g_{0}}g_{0}^{\alpha\beta}\partial_{\alpha}\Omega\partial_{\beta}\Omega.$
(172)

In complex coordinates, it reads:

$W=\frac{c}{6}\int
d^{2}w\sqrt{g_{0}(\sigma)}\partial\Omega\bar{\partial}\Omega.$ (173)

The measure $d^{2}w$ is nothing than the Cartesian measure, it is convenient
to use it when integrating $\delta(w)$.

We evaluate it for $\Omega=\frac{1}{2}(\ln(z)+\ln(\bar{z}))$. The symmetric
form $\partial\Omega\bar{\partial}\Omega$ is preferred from
$\Omega\partial\bar{\partial}\Omega$, since these two derivatives is ambiguous
for the $\ln(w),\ln(\bar{w})$ inside $\Omega$.

(173) yields three terms, all multiplied by $\frac{c}{24}$.
The first is:

$\begin{split}\int
d^{2}w\frac{\bar{\partial}\bar{z}\partial\bar{z}}{\bar{z}^{2}}+c.c.&=-2\pi\sum_{i=1}^{2}Q^{i}\frac{\partial\bar{z}(w_{i})}{\bar{z}^{2}(w_{i})}+c.c.\\\
&=-\sum_{i=1}^{2}Q^{i}\int_{0}^{2\pi}d\alpha\frac{Q^{i}/(\epsilon
e^{i\alpha})^{2}}{(Q^{i})^{2}/(\epsilon
e^{i\alpha})^{2}}+c.c.=-\sum_{i=1}^{2}4\pi.\end{split}$ (174)

The second is:

$\int d^{2}w\frac{\partial z\bar{\partial}\bar{z}}{|z|^{2}}=\int
d^{2}w\frac{4\pi^{2}}{|z|^{2}}\sum_{i,j=1}Q^{i}\bar{Q}^{j}\delta(w-w_{i})\delta(w-w_{j})=0.$
(175)

For $i\neq j$, it vanishes immediately, but it also vanishes for $i=j$ because
of the point-splitting regularization.
The third is:

$\int d^{2}w\frac{\bar{\partial}z\partial\bar{z}}{|z|^{2}}=\int
d^{2}w\frac{1}{|z|^{2}}\sum_{i,j=1}^{2}Q^{i}\bar{Q}^{j}\partial^{2}G_{i}(w)\bar{\partial}^{2}G_{j}(w).$
(176)

The factor
$\sum_{i,j=1}^{2}Q^{i}\bar{Q}^{j}\partial^{2}G_{i}(w)\bar{\partial}^{2}G_{j}(w)$,
gives poles at $w_{i}$ for all $i$. As $w$ approaches $w_{i}$,
$\partial^{2}G_{i}(w)\sim\frac{1}{(w-w_{i})^{2}}$. Hence the integrand becomes
$|w-w_{i}|^{-2}$, which gives a logarithmic divergence $-2\pi\ln(\epsilon)$
(the integral of terms with $j\neq i$ near $w_{i}$ vanishes due to the point-
splitting). The factor $\frac{1}{|z|^{2}}$ gives poles at the zeros of
$\bar{z}$, which we denote as $p_{i}$. The residue at these poles is
$1$.444444The sum of all residues of $\frac{\partial\bar{z}}{\bar{z}}$
vanishes, consistent with the fact that it is periodic. Each of these poles
will contribute another logarithmic divergence $-2\pi\ln(\epsilon)$.
Therefore, using some arbitrary cut $\epsilon<|w-w_{i}|,|w-p_{i}|<r$ around
each pole, the integral of $\left|\frac{\partial\bar{z}}{\bar{z}}\right|^{2}$
is $2\sum_{i=1}^{2}4\pi\ln\left(\frac{r}{\epsilon}\right)+I_{0}$, where
$I_{0}$ is finite and depends on the momenta $Q^{i}$ and the positions
$w_{i}$. The divergent part can be deduced in the following way. Since the
point splitting is imposed for $|w-w_{i}|,|w-p_{i}|$ around each pole, it is
convenient to change variables:

$\displaystyle{\sigma^{1}}^{\prime}=\sigma^{1}+\tau_{1}\sigma^{2},\
{\sigma^{2}}^{\prime}=\tau_{2}\sigma^{2},\
d{\sigma^{1}}^{\prime}d{\sigma^{2}}^{\prime}=\tau_{2}d\sigma^{1}d\sigma^{2},$
(177) $\displaystyle
r^{\prime}=\sqrt{\left({\sigma^{1}}^{\prime}\right)^{2}+\left({\sigma^{2}}^{\prime}\right)^{2}},\
\phi^{\prime}=\tan^{-1}\left(\frac{{\sigma^{2}}^{\prime}}{{\sigma^{1}}^{\prime}}\right),\
r^{\prime}dr^{\prime}d\phi^{\prime}=d{\sigma^{1}}^{\prime}d{\sigma^{2}}^{\prime},$
(178)

and then the integral reads:

$\int_{\epsilon<|w|<r}\frac{d^{2}w}{|w|^{2}}=\int_{\epsilon<r^{\prime}<r}\frac{2\tau_{2}d\sigma^{1}d\sigma^{2}}{r^{\prime
2}}=2\int_{\epsilon<r^{\prime}<r}\frac{r^{\prime}dr^{\prime}d\phi^{\prime}}{r^{\prime
2}}=4\pi\ln\left(\frac{r}{\epsilon}\right)$ (179)

In conclusion, $W$ takes the form:

$W=\frac{c}{24}\left(\sum_{i=1}^{2}8\pi\ln\left(\frac{r}{\epsilon}\right)+I_{0}-4\pi\right).$
(180)

Upon exponentiation $e^{W}$ gives:

$e^{\frac{cI_{0}}{24}-\frac{\pi
c}{6}}\prod_{i=1}^{2}\left(\frac{r}{\epsilon}\right)^{\frac{\pi c}{3}}.$ (181)

A divergent part $\epsilon^{-\frac{\pi c}{3}}$ is present for each operator.
$I_{0}$ is independent of the momentum at large momentum and, therefore, is
not relevant for the saddle approximation.454545The argument given in [27]
pp.25 why $I_{0}$ is not relevant for the saddle is wrong, since $I_{0}$ is
inside an exponent. The right argument is given here.

##  Appendix E Saddle on The Complex Plane

In this appendix, we briefly explain how the saddle point approximation is
used for complex integration. We illustrate it for a simple example of a
Gaussian integral. Then we show how it works for the high momentum limit of
the plane and the torus.

###  E.1 contour integration

The integral $\int_{-\infty}^{\infty}dxe^{Nf(x)}$ can be evaluated by analytic
continuing $f(x)\to f(w),\ w\equiv x+iy$ to the complex plane. Then, assuming
$e^{Nf}$ decays fast enough, one can turn the line into a closed contour and
deform it as wanted. If it crosses any poles / branch-cuts, then their
contribution should be collected.

For using the saddle point approximation at large $N$, one wants to avoid
integrating over a changing phase. It is preferred that the phase of the
integrand will be constant along the integration contour. The way to find a
point $w_{0}$ suitable for a saddle is by solving $\frac{df(w)}{dw}=0$. Since
$f(w)$ is an analytic function, $\frac{df(w)}{dw}=0$ guarantees that as two
functions of two variables $\real f(x,y),\imaginary f(x,y)$, all of the
directional derivatives vanish at $w_{0}$. In particular, this is a stationary
point of the phase and the modulus of $e^{Nf}$.

For any analytic function with the expansion $g(w_{0}+dw)\approx
g(w_{0})+\frac{g^{(n)}(w_{0})}{n!}\cdot dw^{n}$ where the first $n-1$
derivatives vanish at $w_{0}$, one can scan the direction $\phi$ around
$w_{0}$ with $dw=\epsilon e^{i\phi}$ for small $\epsilon$. Then:

$g(w_{0},\phi)\approx
g(w_{0})+\frac{g^{(n)}(w_{0})}{n!}\epsilon^{n}e^{in\phi}.$ (182)

The phase of $g$ is:

$\arg(g(w_{0}+dw))\approx\arg(g(w_{0}))+\epsilon^{n}\left(c_{1}\cos{n\phi}+c_{2}\sin{n\phi}\right),$
(183)

where $c_{a,2}$ are constants. Hence, the phase will change a bit, and as $dw$
makes a full round $g(w_{0}+dw)$ makes $n$-rounds. In particular, for $n=1$,
which is a regular point, each allowed phase (which is $\epsilon$ close to
$\arg(g(w_{0})$) will appear twice (except the two extreme phases that will
appear once). For $n=2$, when the first derivative vanishes, each phase will
appear four times (except the two extreme phases, which will appear twice).
Therefore, around the saddle, which in general has $n=2$, there will be two
lines (four directions) going out $g(w_{0})$ with the same phase. The modulus
will also be extremized at $w_{0}$ along each line, and we want a line along
which it is a maximum.

###  E.2 Gaussian Integral

As a simple example, we consider the Gaussian integral $\int dqe^{iqx-q^{2}}$
at large $x$. We expect from naive considerations to find saddle at $q_{0}\sim
x$. Indeed:

$\frac{d}{dq}\left(iqx-q^{2}\right)=0\Rightarrow q_{0}=\frac{ix}{2}.$ (184)

Plugging in the integral gives $e^{-\frac{x^{2}}{4}}$, which is the saddle
approximation of this integral. To show this through the line of reasoning
presented above, we find the lines of constant phase emerging from the saddle.
The imaginary part at $\frac{ix}{2}$ is zero. Setting $q=k+il$, one solves the
following equation:

$\Im{i(k+il)x-(k+il)^{2})}=0.$ (185)

There are two solutions. The first is $k=0$, and the second is
$l=\frac{x}{2}$. The two contours of constant phase are $q=il$ and
$q=k+i\frac{x}{2}$. Along the first contour, the function is minimized at
$\frac{x}{2}$, while along the second contour, it is maximized. Therefore we
choose the second contour. The full contour can be completed with two segments
of finite imaginary part and real part going to infinity464646The contour can
not be completed as usual with great upper / lower half circles.:

$\begin{split}C_{1}&=\\{k,\ -r\leq k\leq r\\},\\\ C_{2}&=\\{r+ik,\ 0\leq
k\leq\frac{x}{2}\\},\\\ C_{3}&=\\{(-k+i\frac{x}{2}),\ -r\leq k\leq r\\},\\\
C_{4}&=\\{-r-i\left(\frac{x}{2}-k\right),\ 0\leq
k\leq\frac{x}{2}\\}.\end{split}$ (186)

As $r\to\infty$ the contribution of $C_{2,4}$ vanishes. Since the function
does not have any poles inside the contour, the sum of $C_{1,3}$ also
vanishes. They go in opposite directions, so:

$\int_{-\infty}^{\infty}dqe^{iqx-q^{2}}=\int_{-\infty}^{\infty}dke^{i(k+ix/2)x-(k+ix/2)^{2}}=e^{i0}\int_{-\infty}^{\infty}dke^{-\frac{x^{2}}{4}-k^{2}}.$
(187)

The last integral, now over a real function, can be evaluated using the saddle
approximation (in this case, it is trivial, but in general, it can not be
solved exactly).

###  E.3 Plane

The high momentum limit of the two-pt function defined on the plane takes the
form (see (109) in [27]):

$I=\int d^{2}\sigma
e^{iq\sigma-\frac{q^{2}}{\pi|\Lambda|}\ln\left(|\sigma|\right)}.$ (188)

Without loss of generality, we choose $q>0$ to be in the $\sigma^{1}$-axis.
Therefore, the saddle should be with $\sigma^{2}=0$. For
$\frac{q^{2}}{|\Lambda|}>>1$, the integral over $\sigma^{2}$ can be evaluated
using saddle point approximation (without any modification as the integrand is
real)474747In this case, it can be evaluated exactly by:  $\int
d\sigma^{2}\frac{1}{\left(\left(\sigma^{1}\right)^{2}+\left(\sigma^{2}\right)^{2}\right)^{a}}=\frac{\sqrt{\pi}\Gamma\left(a-\frac{1}{2}\right)}{\Gamma(a)}\frac{|\sigma^{1}|}{{\sigma^{1}}^{2(1-a)}}.$
(189) :

$I=\int
d\sigma^{2}e^{-\frac{q^{2}}{2\pi|\Lambda|}\ln\left(\left(\sigma^{1}\right)^{2}+\left(\sigma^{2}\right)^{2}\right)}\approx\sqrt{\frac{2\pi^{2}|\Lambda|\left(\sigma^{1}\right)^{2}}{q^{2}}}e^{-\frac{q^{2}}{2\pi|\Lambda|}\ln\left(\left(\sigma^{1}\right)^{2}\right)}.$
(190)

It remains to perform the integral over $\sigma^{1}$ is:

$I=\int
d\sigma^{1}\sqrt{\frac{2\pi^{2}|\Lambda|}{q^{2}}}e^{iq\sigma^{1}+\left(-\frac{q^{2}}{2\pi|\Lambda|}+\frac{1}{2}\right)\ln\left(\left(\sigma^{1}\right)^{2}\right)}.$
(191)

We analytic continue $\sigma^{1}\to w\equiv x+iy$. The singularity at
$\sigma^{1}=0$ will be skipped by a small half circle going below it:

$C_{\epsilon}=\\{\epsilon e^{i\phi},\ \pi\leq\phi\leq 2\pi\\}.$ (192)

This is an implementation of the point-splitting regularization. The part on
the real axis is:

$C_{L}=\\{x,\ \epsilon\leq|x|\leq R\\}.$ (193)

Another large half circle, on which the integrand decays, makes the
integration contour closed:

$C_{R}=\\{Re^{i\phi},\ 0\leq\phi\leq\pi,\ R\to\infty\\}.$ (194)

The branch-cut of the logarithm is chosen, for reasons that will be clear
during the calculation, to be along the positive imaginary axis. Altogether
the integral can be evaluated as:

$I=\sqrt{\frac{2\pi^{2}|\Lambda|}{q^{2}}}\displaystyle\lim_{R\to\infty}\int_{C_{L}+C_{\epsilon}+C_{R}}dwe^{iqw+\left(-\frac{q^{2}}{2\pi|\Lambda|}+\frac{1}{2}\right)\ln\left(w^{2}\right)}.$
(195)

Once we have a closed contour, we can change its shape, and as long as the
contour does not cross the pole at $w=0$ or the branch-cut, the integral
remains the same.

The saddle of $f(w)=iqw-\frac{q^{2}}{2\pi|\Lambda|}\ln\left(w^{2}\right)$ is
at $w_{0}=-i\frac{q}{\pi|\Lambda|}$484848At $\frac{q^{2}}{|\Lambda|}>>1$, one
can use
$\frac{q^{2}}{2\pi|\Lambda|}+\frac{1}{2}\approx\frac{q^{2}}{2\pi|\Lambda|}$.
The extra $\frac{1}{2}$ will give a subleading contribution when substituting
the saddle in the integrand.. The phase of the integrand at the saddle is
$e^{-i\frac{q^{2}}{2|\Lambda|}}$. Two lines of this constant phase pass
through the saddle, obeying the equation:

$qx-\frac{q^{2}}{\pi|\Lambda|}\tan^{-1}\left(\frac{y}{x}\right)=-\frac{q^{2}}{2|\Lambda|}.$
(196)

The first solution is the imaginary axis: $x=0\Rightarrow w=iy,\ y<0$. Along
this line, however, the function:

$\real f(0+iy)=-qy-\frac{q^{2}}{\pi|\Lambda|}\ln|y|$ (197)

has a minimum at $y=-\frac{q}{\pi|\Lambda|}$ and not a maximum. The second
solution is given by
$y(x)=x\tan\left(\frac{\pi}{2}+\frac{\pi|\Lambda|x}{q}\right)$. Along this
line:

$\real
f(x+iy(x))=qx\cot\left(\frac{\pi|\Lambda|x}{q}\right)+\frac{q^{2}}{\pi|\Lambda|}\ln\left|\frac{\sin\left(\frac{\pi|\Lambda|x}{q}\right)}{x}\right|.$
(198)

It is manifestly symmetric with respect to reflection of $x$ and gets a
maximum at $x=0$. Starting from $x=0,y=-\frac{q}{\pi|\Lambda|}$, $y$ grows to
infinity as $x$ arrives to $\pm\frac{q}{|\Lambda|}$. $\real f$ and $\imaginary
f$ are shown in figure 5. These two contours of constant imaginary part
emerging from the saddle are shown. It can be seen graphically that on the
vertical line, the real part is minimized at the saddle, while on the other,
it is maximized. Hence we deform the contour to go along $(x,y(x))$, as shown
in figure 6.

Figure 5: The real (left) and imaginary (right) parts of $f(w)$, moded by
$2\pi$ ($0$ is identified with $2\pi$). The saddle is shown as a black point.
The two contours of constant imaginary part, equals to the imaginary part at
the saddle and emerge from it, are shown in black lines. Along one line the
saddle is a minimum of the real part, along the other it is a maximum. The
latter is the deformed contour. Figure 6: Deformation of the integration
contour for the plane’s correlator. The blue line represents the initial
contour, and the red line the deformed contour. The black arrows indicate how
the contour is deformed. The green arrows indicate that the radius of the blue
contour, as well as the upper part of the red contour, should be taken to
infinity. On the deformed contour, the integrand has a constant phase, and its
absolute value reaches a maximum at its bottom.

The integral along the deformed contour can be evaluated using the saddle
approximation (including the determinant):

$\int
dwe^{iqw+\left(-\frac{q^{2}}{2\pi|\Lambda|}+\frac{1}{2}\right)\ln\left(w^{2}\right)}\approx
e^{-i\frac{q^{2}}{2|\Lambda|}}\left(\frac{|q|}{\pi
e|\Lambda|}\right)^{-\frac{q^{2}}{\pi|\Lambda|}}\frac{|q|}{\pi|\Lambda|}\sqrt{\frac{2}{|\Lambda|}}.$
(199)

Altogether the full integral is:

$\frac{2}{|\Lambda|}\left(\frac{|q|}{\pi
e|\Lambda|}\right)^{-\frac{q^{2}}{\pi|\Lambda|}}e^{-i\frac{q^{2}}{2|\Lambda|}},$
(200)

as the naive evaluation in (111) of [27].

###  E.4 Torus

We show here the two constant phase contours for the $\sigma^{1}$ integral
((115), after setting $\tau_{1}=\sigma^{2}=0$). One is along the imaginary
axis, as can be seen immediately, and the real part is minimized at the saddle
(see figure 4). The other is orthogonal to the imaginary axis, and the real
part is maximized at the saddle. For $r<<1$, the deformed contour will look
like the deformed contour for the plane, going upwards (see figure 5).
However, for $r>>1$, the deformed contour goes downwards.

The function $f$ for the torus, for $T_{1}=n_{2}=0$, was given in (115). The
saddle $\sigma^{2}=\tau_{1}=0$ holds for all $r$, and one remains with:

$f(w)=inw+\frac{nr}{2\pi}\left(\ln{|\eta(\tau_{2})|^{6}}-\ln\left(\Theta(w,\tau_{2})^{2}\right)\right)-\frac{n}{4r}\frac{\left(\tau_{2}+T_{2}\right)^{2}}{\tau_{2}},$
(201)

where $w$ is not $\sigma^{1}+\tau\sigma^{2}$, but rather the analytic
continuation of $\sigma^{1}$ to the complex cylinder. The part relevant for
$w$ is:

$f(w)=iw-\frac{r}{2\pi}\ln\left(\Theta(w,\tau)^{2}\right).$ (202)

We removed $n$ since it is common for all factors. The saddles are the minima
of $f$ along the imaginary axes. To present the different behavior for small
and large $r$, we consider $T_{2}=5$, and $\frac{r}{2\pi}=0.3,3$. For these
values, according to the analysis in section 4, one has $\tau_{2s}\approx
T_{2},\frac{\pi T_{2}^{2}}{4r^{2}}$. The minima were shown in figure 4. The
“spikes” are the zeros of $\Theta$, located at $\tau_{2s}(r)$, with
$\tau_{2s}=5,\ 0.055$ respectively. For large $r$, one sees the minima due to
the $\Theta$-function only, which are approximately equally spaced, and the
function presents an overall decrease. For small $r$, the first term $inw$
changes the location of the first two minima with respect to the others, being
close to the right singularity rather than to the left singularity.

We show the real part and imaginary part of $f$ near its first three saddles
for small $r$ in figure 7. The first will be relevant for the integral, but
the third one, being dominated by the $\Theta$-function, presents the large
$r$ saddle. One sees that the constant phase contours of the first two are
going up while the third one is going down. The contours for large $r$ were
shown in figure 1 (for clarity, we used $T_{2}=10$), which resembles the third
saddle of small $r$.

Figure 7: Real (left) and imaginary (right) parts of $f$ around the first
three saddles at small r (left). The saddles are located at the black point.
One of the constant phase contour is the imaginary axis, the other can be seen
on the figures. The bright points in the real part are the singularities due
to the zeros of the $\Theta$-function. The white lines in the imaginary parts
are discontinuities due to the log. Note the different regions of the
$y$-axis.

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