More on Torus Wormholes in 3d Gravity

Cynthia Yan

Stanford Institute for Theoretical Physics,

Stanford University, Stanford, CA 94305

Abstract

> We study further the duality between semiclassical AdS3 and formal CFT2
> ensembles. First, we study torus wormholes (Maldacena-Maoz wormholes with
> two torus boundaries) with one insertion or two insertions on each boundary
> and find that they give non-decaying contribution to the product of two
> torus one-point or two-point functions at late-time. Second, we study the
> $\mathbb{Z}_{2}$ quotients of a torus wormhole such that the outcome has one
> boundary. We identify quotients that give non-decaying contributions to the
> torus two-point function at late-time.
>
> We comment on reflection (R) or time-reversal (T) symmetry v.s. the
> combination RT that is a symmetry of any relativistic field theory. RT
> symmetry itself implies that to the extent that a relativistic quantum field
> theory exhibits random matrix statistics it should be of the GOE type for
> bosonic states and of the GSE type for fermionic states. We discuss related
> implications of these symmetries for wormholes.

###### Contents

  1. 1 Introduction
  2. 2 Torus one-point function wormhole
    1. 2.1 $\Delta\rightarrow 0$ limit
    2. 2.2 Lorentzian torus limit
  3. 3 Torus two-point function wormholes
    1. 3.1 Large time-separation limit
    2. 3.2 Single-boundary quotients of a torus wormhole
      1. 3.2.1 Semi-classical gravity calculation
      2. 3.2.2 CFT calculation
  4. 4 Comments on RT symmetry
  5. A JT gravity
    1. A.1 Introduction
    2. A.2 Product of two one-point functions
    3. A.3 Product of two two-point functions
      1. A.3.1 crosscap
  6. B Calculation details
    1. B.1 Calculating product of two torus one-point functions
    2. B.2 Calculating product of two torus two-point functions
    3. B.3 Averaged product of two torus one-point function in $\Delta\rightarrow 0$ limit
    4. B.4 Saddle-point approximation in 3d

##  1 Introduction

It was conjectured that there is an approximate duality between semiclassical
3d gravity and an ensemble of formal CFT2s with large central charge $c$ and a
sparse low-energy spectrum [1, 2, 3] 111This is not a true microscopic
ensemble. For a free toy model of CFT2 ensemble microscopically defined see
[4, 5]. This is analogous to the duality between Jackiw-Teitelboim (JT)
gravity [6, 7, 8] and random matrix theory [9, 10]. These dualities are both
different from earlier examples of AdS/CFT and their boundary ensembles
satisfy the Eigenstate Thermalization Hypothesis (ETH) [11, 12].

This proposed duality provides a playground to study various quantities. One
thing is a BTZ black hole [13] in AdS3 with light operator insertions. Here
light and heavy are respectively distinguished by whether the operator
dimension $(h,\bar{h})$ is below or above the black hole threshold
$\frac{c}{24}$ where $c=\frac{3}{2G}$ is the central charge. In particular, we
study averaged one-point and two-point correlators on a torus on the CFT2
ensemble side. Their variances involve a torus wormhole on the 3d gravity
side. More precisely, the averaged product of two torus correlators are dual
to a Maldacena-Maoz wormhole [14] with insertions. A Maldacena-Maoz wormhole
has two asymptotic boundaries with identical topology and constant moduli. The
boundaries are Riemann surfaces, with the simplest options Riemann spheres or
tori. In order for a Riemann sphere to be stable, there need to be at least
three insertions. For torus, at least one insertion. If we take the operator
dimension to zero for a torus wormhole, we can connect back to [15, 16] which
calculates the torus wormhole partition function in 3d pure gravity [17].

The BTZ black hole can be reduced to JT gravity by Kaluza-Klein mechanism [18,
19]. In this paper we study 3d gravity using methods inspired by 2d gravity.
In particular, there is an analogy between a solid torus in 3d and a disk in
2d as shown in figure 1(a). This can be extended to the analogy between a
torus wormhole in 3d and a cylinder (or double-trumpet) in 2d as shown in
figure 1(b).

Figure 1: (a) Analogy between a 3d solid torus and a 2d disk (b) Analogy
between a 3d torus wormhole and a 2d cylinder

We can take the analogy further by taking a $\mathbb{Z}_{2}$ quotient of a
torus wormhole in 3d and compare that to a $\mathbb{Z}_{2}$ quotient of a
cylinder in 2d. In 2d, identifying antipodal points on a cylinder gives a disk
with a crosscap inserted as shown in figure 2(a). Similarly in 3d, we identify
the torus wormhole as shown in figure 2(b) and get a solid torus with a
thinner solid torus carved out in the middle. Every point on the blue torus is
identified with another point on that blue torus. We find that there are two
ways of doing the $\mathbb{Z}_{2}$ quotient; one orientable and one non-
orientable. Each configuration gives a non-decaying contribution to torus two-
point function reminiscent to that a disk with a crosscap gives a non-decaying
contribution to the thermal two-point function in 1d [20].

Figure 2: (a) $\mathbb{Z}_{2}$ quotient of a cylinder in 2d (b)
$\mathbb{Z}_{2}$ quotients of a torus wormhole in 3d

We can understand these two contributions using Random Matrix Theory (RMT)
from the boundary side perspective. In particular, the orientable geometry
comes from RT symmetry contained in the proposed formal CFT2 ensemble. In 2d,
CPT symmetry is equivalent to reflection+time reversal (RT). While the non-
orientable geometry arises from adding time reversal (T) symmetry to the
formal CFT2 ensemble. Noticing this RT symmetry has bigger consequences, one
is that it tells us that the torus wormhole partition function calculated in
[15, 16] should be multiplied by a factor of two. The other is that we notice
that actually any relativistic quantum field theory with random matrix
statistics has energy eigenvalue distribution a GOE ensemble for bosonic
states and a GSE ensemble for fermionic states.

We show the 3d calculations in the main text and the parallel 2d calculations
in Appendix A. In section 2, we consider one insertion on each boundary torus.
First in section 2.1, we find that if we take the mass of the insertions to
zero, we get a result similar to the partition function found in [15, 16] but
with a factor off. Second in section 2.2, we find that if we analytically
continue the two boundary tori, the average product of one-point functions do
not decay over time. In section 3, we consider two insertions on each boundary
torus. First in section 3.1, we find that if we analytically continue the
location of one of the insertions on each boundary, the average product of
two-point functions do not decay over time. Second in section 3.2 we discuss
the $\mathbb{Z}_{2}$ quotients of a torus wormhole and their contributions to
torus two-point function both from a bulk perspective and from a boundary
perspective, and examine the effect of RT and T symmetry. Finally, in section
4 we look again at the partition function found by [15, 16] and also examine a
generic relativistic quantum field theory with random matrix statistics.

### Review: CFT2 ensemble

We review some key properties of unitary compact CFT2 following the
presentation of [3] and state the conjectures of [1] that we will also assume
in this paper. A CFT2 is defined by its central charge $c$ and a list of
primary operators with known scaling dimensions $(h,\bar{h})$ and operator
product expansion (OPE) coefficients $c_{ijk}$. We reparametrize in terms of
“Liouville parameters”

$c=1+6Q^{2}=1+6(b+\frac{1}{b})^{2}\quad\quad
h=\alpha(Q-\alpha)\quad\quad\alpha=\frac{Q}{2}+iP$ (1.1)

Also denote the spin as $s=h-\bar{h}$. Viewed as an ensemble, CFT2’s have some
universal features depending only on the central charge, we list two such
features below. Both can be formulated in two ways which are equivalent to
each other because of modular invariance.

  1. 1.

    1. (a)

The identity operator has

$h_{\mathbb{1}}=0=\bar{h}_{\mathbb{1}}$ (1.2)

    2. (b)

Cardy’s formula for the density of primary states

$\rho(h,\bar{h})\approx\exp\\{4\pi(\sqrt{\frac{(c-1)h}{24}}+\sqrt{\frac{(c-1)\bar{h}}{24}})\\}\quad\text{as}\quad
h,\bar{h}\rightarrow\infty$ (1.3)

  2. 2.

    1. (a)

For any operator $O_{i}$, the OPE coefficient of two $O_{i}$’s and the
identity is

$c_{ii\mathbb{1}}=1$ (1.4)

    2. (b)

Averaged value of the OPE coefficients over all primary operators is given by

$\overline{|c_{abc}|^{2}}=C_{0}(h_{a},h_{b},h_{c})C_{0}(\bar{h}_{a},\bar{h}_{b},\bar{h}_{c})$
(1.5)

when at least one of the three operator is heavy. Here $C_{0}$ is given by

$C_{0}(h_{1},h_{2},h_{3})=C_{0}(P_{1},P_{2},P_{3})=\frac{\Gamma_{b}(2Q)}{\sqrt{2}\Gamma_{b}(Q)^{3}}\frac{\Gamma_{b}(\frac{Q}{2}\pm
iP_{1}\pm iP_{2}\pm iP_{3})}{\prod_{k=1}^{3}\Gamma_{b}(Q\pm 2iP_{k})}$ (1.6)

where

$\Gamma_{b}(x)=\frac{\Gamma_{2}(x|b,b^{-1})}{\Gamma_{2}(Q/2|b,b^{-1})}$ (1.7)

From now on, following the conventions of [1], we assume that our CFTs are
holographic, i.e. with a large central charge and a sparse spectrum of low-
lying operators. We also extend 1(b) to hold above the black hole threshold
and 2(b) to hold for all primaries. In addition assume

$\overline{c_{abc}}=0$ (1.8)

and

$\overline{c_{abc}c^{*}_{def}}=C_{0}(h_{a},h_{b},h_{c})C_{0}(\bar{h}_{a},\bar{h}_{b},\bar{h}_{c})(\delta_{ad}\delta_{be}\delta_{cf}+(-1)^{s_{a}+s_{b}+s_{c}}\delta_{ad}\delta_{bf}\delta_{ce}+\text{4
more terms})$ (1.9)

where the remaining four terms are cyclic permutations of the first two terms.
In section 3.2.2, we discuss a way to understand this ensemble from the
prospective of Random Matrix Theory.

We introduce a graphical way of representing the OPE coefficients. Consider a
CFT2 on a Riemann sphere. Let $O_{1}$, $O_{2}$, and $O_{3}$ be three
insertions. Instead of thinking of these as point insertions we can expand the
point out to form cycles and the Riemann sphere with three insertions now look
like a pair of pants. This just gives the OPE coefficient $c_{123}$.

$\braket{O_{1}O_{2}O_{3}}=\includegraphics[valign={c},width=65.04034pt]{figures/opecoeff.png}=c_{123}$
(1.10)

[1] showed that the averaged product of two three-point-functions of two
CFT2’s on Riemann spheres matches the semiclassical action of the sphere
wormhole shown in figure 3 on the 3d gravity side.

Figure 3: A Maldacena-Maoz wormhole with two boundaries both Riemann spheres
with three light insertions

They are both equal to the universal asymptotic formula for OPE coefficients
i.e.

$\overline{\braket{O_{1}O_{2}O_{3}}_{S^{2}}\braket{O_{1^{\prime}}O_{2^{\prime}}O_{3^{\prime}}}}_{S^{2}}=\overline{c_{123}c_{1^{\prime}2^{\prime}3^{\prime}}}=|C_{0}(h_{1},h_{2},h_{3})|^{2}$
(1.11)

##  2 Torus one-point function wormhole

In this section, we focus on studying a torus wormhole with one insertion on
each boundary. In section 2.1 we take the mass of the insertions to zero and
compare with partition function in [15, 16], in section 2.2 we show that the
averaged product of two torus one-point functions does not decay over time. We
consider the averaged product of two one-point functions.

$\overline{\braket{O_{1}}_{T^{2}(\tau,\bar{\tau})}\braket{O_{1}}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}}$
(2.1)

Note that pictorially, for each torus one-point function we can think of it as
a torus with a hole. Then this can be decomposed as a sum of the product of
OPE coefficient $c_{1pp}$ and conformal block
$|\mathcal{F}_{1}^{g=1}(h_{p};\tau)|^{2}$ over primary operators $p$. The sum
over descendents of each primary operator is encoded in the conformal block.

$\displaystyle\braket{O_{1}}_{T^{2}(\tau,\bar{\tau})}$
$\displaystyle=\includegraphics[valign={c},width=43.36464pt]{figures/torus1pt.png}$
(2.2)
$\displaystyle=\sum_{p}\includegraphics[valign={c},width=43.36464pt]{figures/ope1pp.png}\left|\includegraphics[valign={c},width=43.36464pt]{figures/toruscb.png}\right|^{2}$
(2.3) $\displaystyle=\sum_{p}c_{1pp}|\mathcal{F}_{1}^{g=1}(h_{p};\tau)|^{2}$
(2.4)
$\displaystyle=\sum_{p}c_{1pp}\mathcal{F}_{1}^{g=1}(h_{p};\tau)\overline{\mathcal{F}}_{1}^{g=1}(\bar{h}_{p};\bar{\tau})$
(2.5)

The proposed ensemble of formal CFT2 (1.11) predicts that the averaged product
of two torus one-point functions is given by

$\displaystyle\overline{\braket{O_{1}}_{T^{2}(\tau,\bar{\tau})}\braket{O_{1}}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}}$
$\displaystyle\approx\left(1+(-1)^{s_{1}}\right)\big{|}\int
dh_{p}\rho_{0}(h_{p})C_{0}(h_{1},h_{p},h_{p})\mathcal{F}_{1}^{g=1}(h_{p};\tau)\overline{\mathcal{F}}_{1}^{g=1}(h_{p};-\tau^{\prime})\big{|}^{2}$
(2.6)
$\displaystyle=\left(1+(-1)^{s_{1}}\right)\braket{O_{1}}^{L}_{T^{2}(\tau,-\tau^{\prime})}\braket{O_{1}}^{L}_{T^{2}(-\bar{\tau}^{\prime},\bar{\tau})}$
(2.7)

where $\braket{O_{1}}^{L}_{T^{2}(\tau,-\tau^{\prime})}$ and
$\braket{O_{1}}^{L}_{T^{2}(-\bar{\tau}^{\prime},\bar{\tau})}$ are torus
Liouville one-point functions. For a derivation of this result see Appendix
B.1.

It was shown in [1] that in the large $c$ limit (to one-loop order), (2.7)
agrees with the semiclassical limit of the 3d gravity calculation of a torus
wormhole with a defect insertion. It is interesting to contemplate that the
RHS of (2.6) could actually be the exact answer for this wormhole in 3d
gravity. Our understanding is that this has indeed been established as part of
the work of [21]. This motivates us to take the formula seriously enough to
consider a non-semiclassical limit where $\Delta\to 0$.

###  2.1 $\Delta\rightarrow 0$ limit

In this section, we take the mass of the insertions to zero $\Delta\rightarrow
0$. There are a couple of interesting features that we want to point out about
this limit. First, the expression is simple and given by

$\overline{\braket{O_{1}}_{T^{2}(\tau,\bar{\tau})}\braket{O_{1}}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}}=\frac{4\left(1+(-1)^{s_{1}}\right)Q^{2}}{\Delta^{2}\pi^{2}}\int
dkd\bar{k}\chi_{k}(\tau)\chi_{k}(\tau^{\prime})\chi_{\bar{k}}(\bar{\tau})\chi_{\bar{k}}(\bar{\tau}^{\prime})$
(2.8)

where the character is given by

$\chi_{k}(\tau)=\frac{q^{\frac{k^{2}}{4}}}{\eta(\tau)}=\frac{e^{\frac{i\pi\tau
k^{2}}{2}}}{\eta(\tau)}\quad\quad k=2P=\sqrt{4h_{p}-Q^{2}}$ (2.9)

Here $\eta(\tau)$ is the Dedekind eta function. Second, naively we would guess
that in the $\Delta\rightarrow 0$ limit, the averaged product of two one-point
functions $\overline{\braket{O}\braket{O}}$ should reproduce the partition
function predicted by Cotler-Jensen [15, 16]. This turns out to not to be the
case, and the discrepancy is simple, just a factor inside our integral over
$k$ and $\bar{k}$. The same kind of discrepancy also appears in 2d because
even though we take $\Delta\rightarrow 0$ there is a still a sum over windings
of the particle around the cylinder which contributes a divergent overall
factor (see Appendix A.2 for a detailed 2d analysis).

Let us begin with our formula of the averaged product of two one-point
functions

$\overline{\braket{O_{1}}_{T^{2}(\tau,\bar{\tau})}\braket{O_{1}}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}}\approx\left(1+(-1)^{s_{1}}\right)\big{|}\int
dh_{p}\rho_{0}(h_{p})C_{0}(h_{1},h_{p},h_{p})\mathcal{F}_{1}^{g=1}(h_{p};\tau)\overline{\mathcal{F}}_{1}^{g=1}(h_{p};-\tau^{\prime})\big{|}^{2}$
(2.10)

where $C_{0}$ is given by (1.6). Now we take the weight of $O_{1}$ to zero,
i.e. $\Delta=2h_{1}\rightarrow 0$. In this limit $O_{1}$ becomes the identity
operator. Then the above equation becomes

$\overline{\braket{O_{1}}_{T^{2}(\tau,\bar{\tau})}\braket{O_{1}}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}}\approx\frac{4\left(1+(-1)^{s_{1}}\right)Q^{2}}{\Delta^{2}\pi^{2}}\left|\int
dP\,\mathcal{F}_{\mathbb{1}}^{g=1}(h_{p};\tau)\overline{\mathcal{F}}_{\mathbb{1}}^{g=1}(h_{p};-\tau^{\prime})\right|^{2}$
(2.11)

for derivation see B.3.

From (2.5) we know that

$Z_{T^{2}}(\tau)=\sum_{p}\mathcal{F}_{\mathbb{1}}^{g=1}(h_{p};\tau)\overline{\mathcal{F}}_{\mathbb{1}}^{g=1}(\bar{h}_{p};\bar{\tau})$
(2.12)

But we also know that

$Z_{T^{2}}(\tau)=\sum_{p}\chi_{k}(\tau)\bar{\chi}_{k}(\bar{\tau})$ (2.13)

so

$\mathcal{F}_{\mathbb{1}}^{g=1}(h_{p};\tau)\overline{\mathcal{F}}_{\mathbb{1}}^{g=1}(\bar{h}_{p};\bar{\tau})=\chi_{k}(\tau)\bar{\chi}_{k}(\bar{\tau})$
(2.14)

and the averaged product of two torus one-point function in the limit
$\Delta\rightarrow 0$ is given by

$\overline{\braket{O_{1}}_{T^{2}(\tau,\bar{\tau})}\braket{O_{1}}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}}=\frac{4\left(1+(-1)^{s_{1}}\right)Q^{2}}{\Delta^{2}\pi^{2}}\int
dkd\bar{k}\,\chi_{k}(\tau)\chi_{k}(\tau^{\prime})\chi_{\bar{k}}(\bar{\tau})\chi_{\bar{k}}(\bar{\tau}^{\prime})$
(2.15)

On the other hand, the partition function given in Cotler&Jensen [15, 16] is

$Z_{T^{2}\times
I}(\tau_{1},\bar{\tau}_{1},\tau_{2},\bar{\tau}_{2})=\sum_{\gamma\in\mathrm{PSL}(2;\mathbb{Z})}\widetilde{Z}(\tau_{1},\bar{\tau}_{1},\gamma\tau_{2},\gamma\bar{\tau}_{2})$
(2.16)

where

$\widetilde{Z}(\tau_{1},\bar{\tau}_{1},\tau_{2},\bar{\tau}_{2})=\sqrt{\mathrm{Im}(\tau_{1})\mathrm{Im}(\tau_{2})}\int_{0}^{\infty}dkd\bar{k}\chi_{k}(\tau_{1})\chi_{k}(\tau_{2})\bar{\chi}_{\bar{k}}(\bar{\tau}_{1})\bar{\chi}_{\bar{k}}(\bar{\tau}_{2})k\bar{k}$
(2.17)

so the $\lim_{\Delta\rightarrow 0}\overline{\braket{O}\braket{O}}$ differ by a
factor $k\bar{k}$ inside the integral compared to the partition function
$\widetilde{Z}$ before the sum over $\mathrm{PSL}(2,\mathbb{Z})$. However, not
like in 2d, in 3d we do not know a precise way to calculate this factor yet.

We can also compare the expressions after integration which are

$\overline{\braket{O_{1}}_{T^{2}(\tau,\bar{\tau})}\braket{O_{1}}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}}=\frac{4\left(1+(-1)^{s_{1}}\right)Q^{2}}{\Delta^{2}\pi^{2}}\frac{1}{|\eta(\tau_{1})|^{2}|\eta(\tau_{2})|^{2}}\frac{1}{|\tau_{1}+\tau_{2}|}$
(2.18)

v.s.

$\widetilde{Z}(\tau_{1},\bar{\tau}_{1},\tau_{2},\bar{\tau}_{2})=\frac{\sqrt{\mathrm{Im}(\tau_{1})\mathrm{Im}(\tau_{2})}}{2\pi^{2}|\eta(\tau_{1})|^{2}|\eta(\tau_{2})|^{2}}\frac{1}{|\tau_{1}+\tau_{2}|^{2}}$
(2.19)

###  2.2 Lorentzian torus limit

In this section, we examine the late-time behavior of the product of two torus
one-point functions. In 2d, the simplest quantity to probe the late-time
behavior is the spectral form factor $\braket{Z(\beta+it)Z(\beta-it)}$ [22].
This is two thermal partition functions, one with $\beta$ analytically
continued to $\beta+it$ and the other to $\beta-it$. We can do a similar
analytical continuation in 3d.

Writing the torus one-point function in operator form, we get

$\braket{O_{1}}_{T^{2}(\tau,\bar{\tau})}=\mathrm{tr}(O_{1}e^{-\beta
H+isP})=\mathrm{tr}(O_{1}e^{i\tau(h-c/24)}e^{-i\bar{\tau}(\bar{h}-c/24)})$
(2.20)

where $\tau=i\beta+s$ and $\bar{\tau}=-i\beta+s$ are complex conjugates since
the torus $T^{2}$ is a Euclidean torus. Also by definition $H=h+\bar{h}-c/12$
and $P=h-\bar{h}$.

The above is one boundary of the Maldacena-Maoz wormhole. The other side
$T^{2}(\tau^{\prime},\bar{\tau}^{\prime})$ is another torus that look like the
reflection of $T^{2}(\tau,\bar{\tau})$. Thus $\tau^{\prime}=i\beta-s$ and
$\bar{\tau}^{\prime}=-i\beta-s$. Then the torus one-point function is

$\braket{O_{1}}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}=\mathrm{tr}(O_{1}e^{-\beta
H-isP})$ (2.21)

For simplicity, we take $s=0$ so the product of the two torus one-point
function becomes

$\braket{O_{1}}_{T^{2}(\tau,\bar{\tau})}\braket{O_{1}}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}=\mathrm{tr}(O_{1}e^{-\beta
H})\mathrm{tr}(O_{1}e^{-\beta H})$ (2.22)

Now we can do analytical continuation similar to $\braket{Z(\beta+it)Z(\beta-
it)}$ [22], and make the two tori into Lorentzian tori by taking
$\beta\mapsto\beta+it$ for $T^{2}(\tau,\bar{\tau})$ while taking
$\beta\mapsto\beta-it$ for $T^{2}(\tau,\bar{\tau})$. But this is just saying
now

$\displaystyle\tau$ $\displaystyle=i\beta-t\quad\quad\tau^{\prime}=i\beta+t$
(2.23) $\displaystyle\bar{\tau}$
$\displaystyle=-i\beta+t\quad\quad\bar{\tau}^{\prime}=-i\beta-t$ (2.24)

so we are calculating the quantity

$\braket{O_{1}}_{T^{2}(\tau,\bar{\tau})}\braket{O_{1}}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}=\braket{O_{1}}_{T^{2}(i\beta-t,-i\beta+t)}\braket{O_{1}}_{T^{2}(i\beta+t,-i\beta-t)}=\mathrm{tr}(O_{1}e^{-(\beta+it)H})\mathrm{tr}(O_{1}e^{-(\beta-
it)H})$ (2.25)

(2.7) then becomes

$\overline{\braket{O_{1}}_{T^{2}(\tau,\bar{\tau})}\braket{O_{1}}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}}=\left(1+(-1)^{s_{1}}\right)\braket{O_{1}}^{L}_{T^{2}(i\beta-t,-i\beta-t)}\braket{O_{1}}^{L}_{T^{2}(i\beta+t,-i\beta+t)}$
(2.26)

But now we can see that the Liouville one-point functions on the right-hand-
side are on Euclidean tori instead of Lorentzian tori. Another way to present
the answer is

$\overline{\mathrm{tr}(O_{1}e^{-(\beta+it)H})\mathrm{tr}(O_{1}e^{-(\beta-
it)H})}=\left(1+(-1)^{s_{1}}\right)\mathrm{tr}_{L}(O_{1}e^{-\beta
H-itP})\mathrm{tr}_{L}(O_{1}e^{-\beta H+itP})$ (2.27)

On the LHS we start with an average over the formal ensemble of CFTs of a
quantity similar to the spectral form factor, but with an operator inserted in
each factor. The RHS is the answer for this ensemble average, or equivalently,
it is the wormhole contribution in 3d gravity – either way it boils down to a
computation in Liouville theory involving an operator insertion with the same
dimension. The key point is that on the RHS, the large time parameter $t$
multiplies the momentum operator $P$, not the Hamiltonian $H$. Now, $P$ is
quantized so the RHS is periodic in time with a short period, and in
particular does not decay for large $t$.

##  3 Torus two-point function wormholes

In this section we focus on studying a torus wormhole with two insertions on
each boundary. In section 3.1, we show that the averaged product of two torus
two-point functions does not decay over time and in section 3.2 we study
$\mathbb{Z}_{2}$ quotients of this torus wormhole. These quotients give non-
decaying contributions to the torus two-point function. We calculate these
contributions from the bulk in section 3.2.1 and from the boundary in section
3.2.2 where we justify and extend the proposed formal CFT2 ensemble (1.9)
using RMT.

We consider the averaged product of two torus two-point functions

$\overline{\braket{O_{1}(v,\bar{v})O_{2}(0)}_{T^{2}(\tau,\bar{\tau})}\braket{O_{1}(v^{\prime},\bar{v}^{\prime})O_{2}(0)}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}}$
(3.1)

Each two-point function is given by

$\displaystyle\braket{O_{1}(v,\bar{v})O_{2}(0)}_{T^{2}(\tau,\bar{\tau})}$
$\displaystyle=\includegraphics[valign={c},width=65.04034pt]{figures/torus2pt.png}$
(3.2)
$\displaystyle=\sum_{p,q}\includegraphics[valign={c},width=43.36464pt]{figures/ope1pq.png}\includegraphics[valign={c},width=43.36464pt]{figures/ope2pq.png}\left|\includegraphics[valign={c},width=43.36464pt]{figures/torus2ptcb.png}\right|^{2}$
(3.3)
$\displaystyle=\sum_{p}c_{1pq}c_{2pq}|\mathcal{F}_{12}^{g=1}(h_{p},h_{q};\tau,v)|^{2}$
(3.4)
$\displaystyle=\sum_{p}c_{1pq}c_{2pq}\mathcal{F}_{12}^{g=1}(h_{p},h_{q};\tau,v)\overline{\mathcal{F}}_{12}^{g=1}(\bar{h}_{p},\bar{h}_{q};\bar{\tau},\bar{v})$
(3.5)

Thus the averaged product of two torus two-point functions is given by

$\displaystyle\overline{\braket{O_{1}(v,\bar{v})O_{2}(0)}_{T^{2}(\tau,\bar{\tau})}\braket{O_{1}(v^{\prime},\bar{v}^{\prime})O_{2}(0)}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}}$
$\displaystyle\approx$ $\displaystyle 2\big{|}\int
dh_{p}dh_{q}\rho_{0}(h_{p})\rho_{0}(h_{q})C_{0}(h_{1},h_{p},h_{q})C_{0}(h_{2},h_{p},h_{q})\mathcal{F}_{12}^{g=1}(h_{p},h_{q};\tau,v)\overline{\mathcal{F}}_{12}^{g=1}(h_{p},h_{q};-\tau^{\prime},-v^{\prime})\big{|}^{2}$
(3.6) $\displaystyle\approx$ $\displaystyle
2\braket{O_{1}(v,-v^{\prime})O_{2}(0)}^{L}_{T^{2}(\tau,-\tau^{\prime})}\braket{O_{1}(-\bar{v}^{\prime},\bar{v})O_{2}(0)}^{L}_{T^{2}(-\bar{\tau}^{\prime},\bar{\tau})}$
(3.7)

where $\braket{O_{1}(v,-v^{\prime})O_{2}(0)}^{L}_{T^{2}(\tau,-\tau^{\prime})}$
and
$\braket{O_{1}(-\bar{v}^{\prime},\bar{v})O_{2}(0)}^{L}_{T^{2}(-\bar{\tau}^{\prime},\bar{\tau})}$
are torus Liouville two-point functions. For details, see Appendix B.2.

###  3.1 Large time-separation limit

In this section, we examine the late-time behavior of the product of two torus
two-point functions.

$\braket{O_{1}(v,\bar{v})O_{2}(0)}_{T^{2}(\tau,\bar{\tau})}\braket{O_{1}(v^{\prime},\bar{v}^{\prime})O_{2}(0)}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}=\mathrm{tr}(O_{1}(v,\bar{v})O_{2}(0)e^{-\beta
H})\mathrm{tr}(O_{1}(v,^{\prime}\bar{v}^{\prime})O_{2}(0)e^{-\beta H})$ (3.8)

Note that here the notion of late-time is different from section 2.2. In
section 2.2, at late time the two tori on the boundary becomes large in time
direction. Here two operators inserted on each torus becomes far away from
each other. We take the twist of the Euclidean torus $s=0$ but we do not
analytically continue $\beta$. Instead, we just analytically continue the
location of the insertion $O_{1}$ on both boundaries

$\displaystyle v$
$\displaystyle=i(\tfrac{\beta}{2}+it)=i\tfrac{\beta}{2}-t\quad\quad
v^{\prime}=i(\tfrac{\beta}{2}-it)=i\tfrac{\beta}{2}+t$ (3.9)
$\displaystyle\bar{v}$
$\displaystyle=-i(\tfrac{\beta}{2}+it)=-i\tfrac{\beta}{2}+t\quad\quad\bar{v}^{\prime}=-i(\tfrac{\beta}{2}-it)=-i\tfrac{\beta}{2}-t$
(3.10)

plugging these into (3.7) we get

$\overline{\braket{O_{1}(v,\bar{v})O_{2}(0)}_{T^{2}(\tau,\bar{\tau})}\braket{O_{1}(v^{\prime},\bar{v}^{\prime})O_{2}(0)}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}}\\\
=2\braket{O_{1}(i\tfrac{\beta}{2}-t,-i\tfrac{\beta}{2}-t)O_{2}(0)}^{L}_{T^{2}(i\beta,-i\beta)}\braket{O_{1}(i\tfrac{\beta}{2}+t,-i\tfrac{\beta}{2}+t)O_{2}(0)}^{L}_{T^{2}(i\beta,-i\beta)}$
(3.11)

Observe that one torus has one insertion at $0$ and another insertion at
$i\frac{\beta}{2}-\lfloor t\rfloor$. The other torus has one insertion at $0$
and another insertion at $i\frac{\beta}{2}+\lfloor t\rfloor$. Here
$\lfloor\cdot\rfloor$ means the fractional part of $t$. Thus, as $t$ becomes
large, the arguments don’t change much, so
$\overline{\braket{O_{1}(v,\bar{v})O_{2}(0)}_{T^{2}(\tau,\bar{\tau})}\braket{O_{1}(v^{\prime},\bar{v}^{\prime})O_{2}(0)}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}}$
does not decay at late time.

###  3.2 Single-boundary quotients of a torus wormhole

So far we have discussed two-boundary configurations with large Lorentzian
separation that do not decay with time. In this section we calculate
$\mathbb{Z}_{2}$ quotients of a torus wormhole from both the bulk side
(section 3.2.1) and the boundary side (section 3.2.2). These configurations
each has one boundary which is a torus, so they contribute to the torus two-
point function. And then using the result from 3.1, we observe that these
contributions do not decay at late time.

####  3.2.1 Semi-classical gravity calculation

Let us classify ways of doing $\mathbb{Z}_{2}$ quotient of a torus wormhole
that gives a smooth geometry with one torus boundary. In other words, we
classify $\mathbb{Z}_{2}$ symmetries of the torus wormhole such that the two
boundaries are mapped to each other. Our trick of doing the classification is
to focus on the zero-curvature slice in the middle of the torus wormhole. This
zero-curvature slice is a torus that is mapped to itself under the
$\mathbb{Z}_{2}$ symmetry. We represent this torus as a square with sides
identified as shown in figure 4, i.e. $(x,y)\sim(x+\mathbb{Z},y+\mathbb{Z})$.

$(x,y)\sim(x+\mathbb{Z},y+\mathbb{Z})$

Figure 4: A torus represented as a square with sides identified.

There are two inequivalent $\mathbb{Z}_{2}$ symmetries and doing a quotient
results in either a torus $T^{2}$ or a Klein bottle $K^{2}$ as shown in table
1. Without loss of generality, the fundamental region is the part to the left
of the blue dashed line.

$T^{2}$ | $K^{2}$
---|---
|
$(x,y)\sim(x+1/2,y)$ | $(x,y)\sim(x+1/2,-y)$
Table 1: Ways of doing a $\mathbb{Z}_{2}$ quotient of a torus

Going back to 3d, the above analysis tells us that there are two ways of doing
$\mathbb{Z}_{2}$ quotient of a torus wormhole. Let us use the coordinate
$(x,y,z)\in T^{2}\times[-1,1]$ for the torus wormhole. Thus these two
$\mathbb{Z}_{2}$ quotient corresponds to (1) $(x,y,z)\sim(x+1/2,y,-z)$ and (2)
$(x,y,z)\sim(x+1/2,-y,-z)$ respectively.222$(x,y,z)\sim(x+1/2,y,-z)$,
$(x,y,z)\sim(x,y+1/2,-z)$, and $(x,y,z)\sim(x+1/2,y+1/2,-z)$ are equivalent
identifications because we can change the ways we represent a torus on a
$\mathbb{R}^{2}$ plane. We should note that both these configurations contain
a Mobius band $(x,0,z)\sim(x+1/2,0,-z)$. Now let us examine whether our two
kinds of $\mathbb{Z}_{2}$ quotients are orientable or not. To do that we go
back to the torus wormhole which is a 3d smooth manifold embedded in
$\mathbb{R}^{3}$. We can think of the identifications as transition maps
between two different elements in the atlas of the torus wormhole. The
$\mathbb{Z}_{2}$ quotient is orientable iff the atlas is oriented. The atlas
is oriented (unoriented) if the Jacobi determinant is positive (negative).

  1. (1)

$(x,y,z)\sim(x+1/2,y,-z)$. This has Jacobi det $=-1$, which means it’s a non-
orientable 3d geometry. In particular, this configuration is equivalent to a
Mobius strip $\times\,S^{1}$.

  2. (2)

$(x,y,z)\sim(x+1/2,-y,-z)$. This has Jacobi det $=1$, which means it’s an
orientable 3d geometry. Intuitively, it has a twist in the $z$ direction and
another twist in the $y$-direction, they cancel each other.

Notice that there is an analogy between $\mathbb{Z}_{2}$ quotients of a 3d
torus wormhole and the $\mathbb{Z}_{2}$ quotient of a 2d cylinder as shown in
figure 5. $\mathbb{Z}_{2}$ quotient of a 2d cylinder is the same as a disk
with a crosscap inserted as shown in figure 5(a). $\mathbb{Z}_{2}$ quotients
of a 3d torus wormhole are also equivalent to carving out a solid torus and
then identify points on this torus as shown in figure 5(b). We should note
that if we insert operators on the boundaries of torus wormhole and then take
the $\mathbb{Z}_{2}$ quotient, the insertions need to be compatible with the
corresponding $\mathbb{Z}_{2}$ symmetry.

Figure 5: (a) $\mathbb{Z}_{2}$ quotient of a cylinder in 2d (b)
$\mathbb{Z}_{2}$ quotients of a torus wormhole in 3d

Before analyzing the insertions in 3d torus wormhole, let us recall a similar
situation in 2d where we insert four operators on the cylinder that respect
the antipodal map so that we can do the antipodal identification for the
cylinder later. The configuration of insertions compatible with the
$\mathbb{Z}_{2}$ symmetry is shown in figure 6. The two $V$’s are antipodal of
each other and the two $W$’s are also antipodal of each other. We analytically
continue the Euclidean distance between $V$ and $W$ to $\frac{\beta}{2}+it$
and $\frac{\beta}{2}-it$. In particular, we can check that the configuration
shown in figure 6 makes sense by observing that the left boundary and right
boundary of the cylinder gives the same two-point correlator.

$\displaystyle\mathrm{LHS}$
$\displaystyle=\mathrm{tr}(e^{-(\frac{\beta}{2}-it)H}Ve^{-(\frac{\beta}{2}+it)H}W)$
(3.12) $\displaystyle\mathrm{RHS}$
$\displaystyle=\mathrm{tr}(e^{-(\frac{\beta}{2}+it)H}We^{-(\frac{\beta}{2}-it)H}V)$
(3.13)

so LHS=RHS.

Figure 6: In 2d, we get a cylinder from the hyperbolic disk by identifying the
two brown geodesics. Then we insert two pairs of operators $V$ and $W$ on the
boundary.

If we use the $(x,z)$ coordinate to denote $S^{1}\times[-1,1]$, the insertions
are at

$\displaystyle(x_{V},z_{V})$ $\displaystyle=(0,-1)$
$\displaystyle(x^{\prime}_{V},z^{\prime}_{V})=(\frac{\beta}{2},1)$ (3.14)
$\displaystyle(x_{W},z_{W})$ $\displaystyle=(\frac{\beta}{2}+it,-1)$
$\displaystyle(x^{\prime}_{W},z^{\prime}_{W})=(\beta+it,1)$ (3.15)

These satisfy $\mathbb{Z}_{2}$ symmetry because
$(x^{\prime}_{v},z^{\prime}_{v})=(x_{v}+\beta/2,-z_{v})$ and
$(x^{\prime}_{w},z^{\prime}_{w})=(x_{w}+\beta/2,-z_{w})$.

Analogously in 3d we again insert four operators so that they satisfy the
$\mathbb{Z}_{2}$ symmetries, i.e.
$(x^{\prime}_{v},y^{\prime}_{v},z^{\prime}_{v})=(x_{v}+\beta/2,\pm
y_{v},-z_{v})$ and
$(x^{\prime}_{w},y^{\prime}_{w},z^{\prime}_{w})=(x_{w}+\beta/2,\pm
y_{w},-z_{w})$

$\displaystyle(x_{V},y_{V},z_{V})$ $\displaystyle=(0,0,-1)$
$\displaystyle(x^{\prime}_{V},y^{\prime}_{V},z^{\prime}_{V})=(\frac{\beta}{2},0,1)$
(3.16) $\displaystyle(x_{W},y_{W},z_{W})$
$\displaystyle=(\frac{\beta}{2}+it,0,-1)$
$\displaystyle(x^{\prime}_{W},y^{\prime}_{W},z^{\prime}_{W})=(\beta+it,0,1)$
(3.17)

Note that both (1) and (2) $\mathbb{Z}_{2}$ quotients are compatible with
these operator insertions because the $y$-coordinates of all four insertions
are zero and the only difference between (1) and (2) is from the
$y$-coordinates. Again the distance between $V$ and $W$ are
$\frac{\beta}{2}+it$ and $\frac{\beta}{2}-it$. If the tori are given by
$T^{2}(\tau,\bar{\tau})=T^{2}(i\beta,-i\beta)$ and
$T^{2}(\tau^{\prime},\bar{\tau}^{\prime})=T^{2}(i\beta,-i\beta)$ respectively.
The insertions should be $V$ at $0$ and $W$ at
$(v,\bar{v})=\big{(}i(\frac{\beta}{2}+it),-i(\frac{\beta}{2}+it)\big{)}$ for
the left torus, and should be $W$ at $0$ and $V$ at
$(v^{\prime},\bar{v}^{\prime})=\big{(}i(\frac{\beta}{2}-it),-i(\frac{\beta}{2}-it)\big{)}$
for the right torus.

Figure 7: We insert operators the same way for a 3d torus wormhole as for a 2d
cylinder.

There is one more subtlety to the configuration. For the operators to satisfy
the $\mathbb{Z}_{2}$ symmetries, we should actually insert $TVT^{-1}$ and
$TWT^{-1}$ for (1) and $(RT)V(RT)^{-1}$ and $(RT)W(RT)^{-1}$ for (2) where $T$
denotes time-reversal symmetry and $R$ denotes reflection.

Thus the torus wormhole contribution to the averaged product of two torus two-
point functions we are calculating is

  1. (1)

$\overline{\braket{W(v,\bar{v})V(0)}_{T^{2}(\tau,\bar{\tau})}\braket{TV(v^{\prime},\bar{v}^{\prime})W(0)T^{-1}}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}}$
(3.18)

  2. (2)

$\overline{\braket{W(v,\bar{v})V(0)}_{T^{2}(\tau,\bar{\tau})}\braket{(RT)V(v^{\prime},\bar{v}^{\prime})W(0)(RT)^{-1}}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}}$
(3.19)

Now we take $V=O_{1}$ and discuss the two cases separately

  1. (1)

(3.18) is only nonzero when

$W=TO_{1}T^{-1}$ (3.20)

Using (3.7), we get

$\overline{\braket{TO_{1}(v,\bar{v})T^{-1}O_{1}(0)}_{T^{2}(\tau,\bar{\tau})}\braket{TO_{1}(v^{\prime},\bar{v}^{\prime})T^{-1}O_{1}(0)}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}}\\\
=2\braket{TO_{1}(i\tfrac{\beta}{2}-t,-i\tfrac{\beta}{2}-t)T^{-1}O_{1}(0)}^{L}_{T^{2}(i\beta,-i\beta)}\braket{TO_{1}(i\tfrac{\beta}{2}+t,-i\tfrac{\beta}{2}+t)T^{-1}O_{1}(0)}^{L}_{T^{2}(i\beta,-i\beta)}$
(3.21)

This is given by two quotient wormhole contributions glued together and
semiclassically we can write

$e^{-S_{\text{wormhole}}}=e^{-2S_{\text{quotient}}}$ (3.22)

Thus quotient (1) contributes to the torus two-point function by

$\displaystyle\overline{\braket{TO_{1}T^{-1}O_{1}}}_{(1)}=\includegraphics[valign={c},width=65.04034pt]{figures/torus2pt2.png}$
(3.23)
$\displaystyle\approx_{\text{classical}}\sqrt{\overline{\braket{TO_{1}(v,\bar{v})T^{-1}O_{1}(0)}_{T^{2}(\tau,\bar{\tau})}\braket{TO_{1}(v^{\prime},\bar{v}^{\prime})T^{-1}O_{1}(0)}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}}}$
(3.24)
$\displaystyle=\sqrt{2\braket{TO_{1}(i\tfrac{\beta}{2}-t,-i\tfrac{\beta}{2}-t)T^{-1}O_{1}(0)}_{T^{2}(i\beta,-i\beta)}\braket{TO_{1}(i\tfrac{\beta}{2}+t,-i\tfrac{\beta}{2}+t)T^{-1}O_{1}(0)}_{T^{2}(i\beta,-i\beta)}}$
(3.25)

and this is non-decaying over time. We should note that this equation is only
valid in the leading (zero loop) classical approximation, because we are just
using the relationship between the classical actions (3.22). Also since

$\overline{\braket{O_{1}(v,\bar{v})O_{2}(0)}_{T^{2}(\tau,\bar{\tau})}\braket{O_{1}(v^{\prime},\bar{v}^{\prime})O_{2}(0)}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}}=\\\
2\big{|}\int
dh_{p}dh_{q}\rho_{0}(h_{p})\rho_{0}(h_{q})C_{0}(h_{1},h_{p},h_{q})C_{0}(h_{2},h_{p},h_{q})\mathcal{F}_{11}^{g=1}(h_{p},h_{q};\tau,v)\mathcal{F}_{11}^{g=1}(h_{p},h_{q};\tau^{\prime},v^{\prime})\big{|}^{2}$
(3.26)

we can use saddle-point approximation (for a justification see Appendix B.4)
to get

$\overline{\braket{TO_{1}T^{-1}O_{1}}}_{(1)}\approx_{\text{classical}}|\int
dh_{p}\rho_{0}(h_{p})C_{0}(h_{1},h_{p},h_{p})\mathcal{F}_{11}^{g=1}(h_{p},h_{p};\tau,v)|^{2}$
(3.27)

  2. (2)

(3.19) is only nonzero when

$W=(RT)O_{1}(RT)^{-1}$ (3.28)

Using (3.7), we get

$\overline{\braket{(RT)O_{1}(v,\bar{v})(RT)^{-1}O_{1}(0)}_{T^{2}(\tau,\bar{\tau})}\braket{(RT)O_{1}(v^{\prime},\bar{v}^{\prime})(RT)^{-1}O_{1}(0)}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}}\\\
=2\braket{(RT)O_{1}(i\tfrac{\beta}{2}-t,-i\tfrac{\beta}{2}-t)(RT)^{-1}O_{1}(0)}^{L}_{T^{2}(i\beta,-i\beta)}\braket{(RT)O_{1}(i\tfrac{\beta}{2}+t,-i\tfrac{\beta}{2}+t)(RT)^{-1}O_{1}(0)}^{L}_{T^{2}(i\beta,-i\beta)}$
(3.29)

Quotient (2) contributes to the torus two-point function by

$\displaystyle\overline{\braket{(RT)O_{1}(RT)^{-1}O_{1}}}_{(2)}=\includegraphics[valign={c},width=65.04034pt]{figures/torus2pt2.png}$
(3.30)
$\displaystyle\approx_{\text{classical}}\sqrt{\overline{\braket{(RT)O_{1}(v,\bar{v})(RT)^{-1}O_{1}(0)}_{T^{2}(\tau,\bar{\tau})}\braket{(RT)O_{1}(v^{\prime},\bar{v}^{\prime})(RT)^{-1}O_{1}(0)}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}}}$
(3.31)
$\displaystyle=\sqrt{2\braket{(RT)O_{1}(i\tfrac{\beta}{2}-t,-i\tfrac{\beta}{2}-t)(RT)^{-1}O_{1}(0)}_{T^{2}(i\beta,-i\beta)}\braket{(RT)O_{1}(i\tfrac{\beta}{2}+t,-i\tfrac{\beta}{2}+t)(RT)^{-1}O_{1}(0)}_{T^{2}(i\beta,-i\beta)}}$
(3.32)

and this is non-decaying over time. Again this is only valid in the leading
(zero loop) classical approximation. And again using saddle-point
approximation

$\overline{\braket{(RT)O_{1}(RT)^{-1}O_{1}}}_{(2)}\approx_{\text{classical}}|\int
dh_{p}\rho_{0}(h_{p})C_{0}(h_{1},h_{p},h_{p})\mathcal{F}_{11}^{g=1}(h_{p},h_{p};\tau,v)|^{2}$
(3.33)

But $RT$ acting on $O_{1}$ is equivalent to a $180^{\circ}$ rotation so

$\displaystyle\overline{\braket{O_{1}O_{1}}}_{(2)}$
$\displaystyle=(-1)^{s_{1}}\overline{\braket{(RT)O_{1}(RT)^{-1}O_{1}}}_{(2)}$
(3.34) $\displaystyle\approx_{\text{classical}}(-1)^{s_{1}}|\int
dh_{p}\rho_{0}(h_{p})C_{0}(h_{1},h_{p},h_{p})\mathcal{F}_{11}^{g=1}(h_{p},h_{p};\tau,v)|^{2}$
(3.35)

(3.27) and (3.35) are results of a classical approximation. However, it’s
interesting to contemplate that whether they are actually exact equations.

####  3.2.2 CFT calculation

Before calculating the $\mathbb{Z}_{2}$ quotients of a torus wormhole from the
boundary side, let us examine the boundary ensemble in [1] more closely.

$\overline{c_{abc}c^{*}_{def}}=C_{0}(h_{a},h_{b},h_{c})C_{0}(\bar{h}_{a},\bar{h}_{b},\bar{h}_{c})(\delta_{ad}\delta_{be}\delta_{cf}+(-1)^{s_{a}+s_{b}+s_{c}}\delta_{ad}\delta_{bf}\delta_{ce}+\text{4
more terms})$ (3.36)

where the remaining four terms are cyclic permutations of the first two terms.
Let’s start by writing the OPE coefficients of two heavy operators
$O_{i},O_{j}$ and one light operator $O_{\alpha}$ in operator form

$c_{i\alpha j}=\braket{i}{O_{\alpha}}{j}=\braket{O_{i}O_{\alpha}O_{j}}$ (3.37)

Here all operators that are Hermitian. We should note that

$c_{i\alpha
j}=\braket{i}{O_{\alpha}}{j}=\braket{i}{O_{\alpha}^{\dagger}}{j}=\braket{j}{O_{\alpha}}{i}^{*}=c_{j\alpha
i}^{*}$ (3.38)

We now show that an OPE coefficient $c_{123}$ is real if $s_{1}+s_{2}+s_{3}$
is even and purely imaginary if $s_{1}+s_{2}+s_{3}$ is odd. If the operator
$O$ is a symmetric traceless tensor of spin $s=h-\bar{h}$, we can write it in
component form as

$O(x,J)=O_{\mu_{1},\cdots,\mu_{s}}J^{\mu_{1}}\cdots J^{\mu_{s}}$ (3.39)

Let $\mathrm{Rot}$ denote rotation by $180^{\circ}$, define
$\ket{1}=O_{1}(x_{1},J_{1})\ket{0}$ and $\ket{3}=O_{3}(x_{3},J_{3})\ket{0}$,
then three-point function is given by

$\braket{1}{O_{2}(x_{2},J_{2})}{3}=\mathrm{Rot}\braket{1}{O_{2}(x_{2},J_{2})}{3}\\\
=(-1)^{s_{1}+s_{2}+s_{3}}\braket{3}{O_{2}(x_{2},J_{2})}{1}=(-1)^{s_{1}+s_{2}+s_{3}}\braket{1}{O_{2}(x_{2},J_{2})}{3}^{*}$
(3.40)

where in the second line the prefactor comes from rotating the spin while the
braket comes from rotating operators keeping spin fixed. Thus

$c_{i\alpha j}^{*}=(-1)^{s_{i}+s_{\alpha}+s_{j}}c_{i\alpha j}$ (3.41)

The ensemble of OPE coefficients can be justified by considering a simple
model of Haar random matrices. We introduce a change of basis by unitary
matrix $u$ and write

$c_{i\alpha j}=(u^{\dagger}O_{\alpha}u)_{ij}$ (3.42)

where $u$ is a unitary matrix and they form a Gaussian unitary ensemble (GUE)
[23]. We can justify writing the change of basis $u$ by thinking that there
are two sets of basis: local basis and energy eigenbasis. In local basis, the
operator $O_{\alpha}$ is simple but in the energy eigenbasis $O_{\alpha}$ need
to be transformed by some rather complicated change of basis, so we can take
those complicated matrix $u$ to be random. We can then model averaging over
ensemble of OPE coefficients as integral over random unitary matrices. Recall
we have the leading order contribution to Weingarten’s formula [24]

$\int
du\,u_{i_{1}}^{\phantom{i_{1}}j_{1}}u_{i_{2}}^{\phantom{i_{2}}j_{2}}(u^{\dagger})_{k_{1}}^{\phantom{k_{1}}l_{1}}(u^{\dagger})_{k_{2}}^{\phantom{k_{2}}l_{2}}\approx\frac{1}{L^{2}}\big{(}\delta_{i_{1}}^{l_{1}}\delta_{i_{2}}^{l_{2}}\delta_{k_{1}}^{j_{1}}\delta_{k_{2}}^{j_{2}}+\delta_{i_{1}}^{l_{2}}\delta_{i_{2}}^{l_{1}}\delta_{k_{1}}^{j_{2}}\delta_{k_{2}}^{j_{1}}\big{)}$
(3.43)

Now let us review how to derive this formula. The general idea is we observe
that $u$ is a projector onto invariants. Thus we can first find the invariant
states then we can just write the integral formula immediately. More
specifically, denote our Hilbert space by $\mathcal{H}$, then our random
matrices as maps $u,u^{\dagger}:\mathcal{H}\rightarrow\mathcal{H}$ and
$u^{*},u^{T}:\mathcal{H}^{*}\rightarrow\mathcal{H}^{*}$.

$\ket{i}\mapsto
u_{i}^{\phantom{i}j}\ket{j}\quad\quad\bra{i}\mapsto(u^{*})^{i}_{\phantom{i}j}\bra{j}$
(3.44)

Let us denote elements of
$\\{\mathcal{H},\mathcal{H},\mathcal{H}^{*},\mathcal{H}^{*}\\}$ using the
indices $1,2,3,4$ respectively. Then the following state is invariant under
evolutions by Haar random matrices333The states $\ket{i}$ are orthogonal in
the leading order approximation of Weingarten’s formula.

$\ket{13}=\sum_{i}\ket{i}\bra{i}$ (3.45)

because

$u\otimes
u^{*}\ket{13}=\sum_{i}(u\ket{i})(u^{*}\bra{i})=\sum_{i,j,k}\ket{j}\braket{j}{u}{i}\braket{i}{u^{\dagger}}{k}\bra{k}=\sum_{j,k}\ket{j}\braket{j}{uu^{\dagger}}{k}\bra{k}=\sum_{j}\ket{j}\bra{j}=\ket{13}$
(3.46)

Note in particular that $\ket{13}$ is the identity operator which is basis-
independent, and so well-defined. The same hold for
$\ket{24},\ket{14},\ket{23}$ so

$\displaystyle\int
du\,u_{i_{1}}^{\phantom{i_{1}}j_{1}}u_{i_{2}}^{\phantom{i_{2}}j_{2}}(u^{\dagger})_{k_{1}}^{\phantom{k_{1}}l_{1}}(u^{\dagger})_{k_{2}}^{\phantom{k_{2}}l_{2}}$
$\displaystyle=\int
du\,u_{i_{1}}^{\phantom{i_{1}}j_{1}}u_{i_{2}}^{\phantom{i_{2}}j_{2}}(u^{*})_{\phantom{l_{1}}k_{1}}^{l_{1}}(u^{*})_{\phantom{l_{2}}k_{2}}^{l_{2}}$
(3.47)
$\displaystyle\approx\frac{1}{L^{2}}\big{(}\delta_{i_{1}}^{l_{1}}\delta_{i_{2}}^{l_{2}}\delta_{k_{1}}^{j_{1}}\delta_{k_{2}}^{j_{2}}+\delta_{i_{1}}^{l_{2}}\delta_{i_{2}}^{l_{1}}\delta_{k_{1}}^{j_{2}}\delta_{k_{2}}^{j_{1}}\big{)}$
(3.48)

For two boundary dimensions, we should modify (3.43) by adding reflection +
time-reversal, i.e. RT symmetry.444Relativistic theories always have this
symmetry. It’s sometimes called CPT symmetry. We want to know how the ensemble
of OPE coefficients would change if we add RT symmetry (i.e. $RT$ commuting
with $u$ or $u^{\dagger}RTu=RT$) to GUE. Recall that Lorentzian time-reversal
$T$ is antilinear and antiunitary. Since we know that complex conjugate $K$ is
antilinear and antiunitary 555Complex conjugate operator is an antilinear and
antiunitary operator $K:\mathcal{H}\rightarrow\mathcal{H}$ because
$K\sum\alpha_{i}\ket{i}=\sum\alpha_{i}^{*}\ket{i}$ and if we write
$\ket{\psi}=\sum\alpha_{i}\ket{i},\quad\ket{\chi}=\sum\beta_{i}\ket{i}$ then
we have the inner product
$\braket{K\chi}{K\psi}=\sum_{i,j}(\bra{j}\beta_{j})(\alpha^{*}_{i}\ket{i})=\sum_{i}\alpha_{i}^{*}\beta_{i}\braket{i}{i}=\sum_{i,j}(\bra{j}\alpha^{*}_{j})(\beta_{i}\ket{i})=\braket{\psi}{\chi}$.,
it is natural to model $T$ with a factor of $K$ in it. With $RT$ symmetry,
there are more invariant states

$\ket{12}^{\prime}=\sum_{i}\ket{i}RTK\ket{i}$ (3.49)

since

$u\otimes
u\ket{12}^{\prime}=\sum_{i}\ket{i}uRTKu^{T}\ket{i}=\sum_{i}\ket{i}uRTu^{\dagger}K\ket{i}=\sum_{i}\ket{i}RTK\ket{i}=\ket{12}^{\prime}$
(3.50)

Therefore, time-reversal symmetry adds a term to the Weingarten’s formula

$\int
du\,u_{i_{1}}^{\phantom{i_{1}}j_{1}}u_{i_{2}}^{\phantom{i_{2}}j_{2}}(u^{\dagger})_{k_{1}}^{\phantom{k_{1}}l_{1}}(u^{\dagger})_{k_{2}}^{\phantom{k_{2}}l_{2}}\approx\frac{1}{L^{2}}\big{(}\delta_{i_{1}}^{l_{1}}\delta_{i_{2}}^{l_{2}}\delta_{k_{1}}^{j_{1}}\delta_{k_{2}}^{j_{2}}+\delta_{i_{1}}^{l_{2}}\delta_{i_{2}}^{l_{1}}\delta_{k_{1}}^{j_{2}}\delta_{k_{2}}^{j_{1}}\\\
+(RTK)^{-1}_{i_{1},i_{2}}(RTK)^{j_{1},j_{2}}(RTK)^{-1}_{k_{1},k_{2}}(RTK)^{l_{1},l_{2}}\big{)}$
(3.51)

Thus the averaged product of two OPE coefficients with one light and two heavy
operators is

$\displaystyle\overline{c_{i\alpha j}c_{k\beta l}^{*}}$ $\displaystyle=\int
du(u^{\dagger}O_{\alpha}u)_{ij}(u^{\dagger}O_{\beta}u)_{lk}$ (3.52)
$\displaystyle=\frac{1}{L^{2}}\left(\text{Tr}(O_{\alpha}O_{\beta})\delta_{ik}\delta_{jl}+\mathrm{Tr}\big{(}(RTK)O^{T}_{\alpha}(RTK)^{-1}O_{\beta}\big{)}(RTK)^{-1}_{il}(RTK)_{jk}\right)$
(3.53)
$\displaystyle=\frac{1}{L^{2}}\left(\text{Tr}(O_{\alpha}O_{\beta})\delta_{ik}\delta_{jl}+\mathrm{Tr}\big{(}(RT)O^{\dagger}_{\alpha}(RT)^{-1}O_{\beta}\big{)}(RTK)^{-1}_{il}(RTK)_{jk}\right)$
(3.54)
$\displaystyle=\frac{1}{L}\left(\delta_{\alpha\beta}\delta_{ik}\delta_{jl}+(-1)^{s_{i}+s_{\alpha}+s_{j}}\delta_{\alpha\beta}\delta_{il}\delta_{jk}\right)$
(3.55)

where in the last line we used two facts: First, $RT$ acting on $O_{\alpha}$
by conjugation is equivalent to rotation by $180^{\circ}$ so

$\mathrm{Tr}\big{(}(RT)O^{\dagger}_{\alpha}(RT)^{-1}O_{\beta}\big{)}=(-1)^{s_{\alpha}}\mathrm{Tr}(O_{\alpha}^{\dagger}O_{\beta})=(-1)^{s_{\alpha}}\delta_{\alpha\beta}$
(3.56)

Second, we can write $(RTK)^{-1}_{il}$ and $(RTK)_{jk}$ in braket notation as

$\displaystyle(RTK)^{-1}_{il}$
$\displaystyle=\left(\sum_{n}\bra{n}(RTK)^{-1}\bra{n}\right)\ket{i}\ket{l}=\sum_{n}\braket{i}{(RT)^{-1}}{n}\braket{n}{l}=\braket{0}{O_{i}^{\dagger}(RT)^{-1}O_{l}}{0}=(-1)^{s_{i}}\delta_{il}$
(3.57) $\displaystyle(RTK)_{jk}$
$\displaystyle=\bra{j}\bra{k}\left(\sum_{n}\ket{n}RTK\ket{n}\right)=\sum_{n}\braket{j}{n}\braket{n}{RT}{k}=\braket{0}{O^{\dagger}_{j}RTO_{k}}{0}=(-1)^{s_{j}}\delta_{jk}$
(3.58)

Thus we have reproduced (3.36).

Now we want to know how the ensemble of OPE coefficients would change if we
add time-reversal symmetry without reflection $R$ (i.e. the time-reversal
operator $T$ commuting with $u$ i.e. $u^{\dagger}Tu=T$) to GUE. With $T$
symmetry, we have invariant state

$\ket{12}=\sum_{i}\ket{i}TK\ket{i}$ (3.59)

because

$u\otimes
u\ket{12}=\sum_{i}\ket{i}uTKu^{T}\ket{i}=\sum_{i}\ket{i}uTu^{\dagger}K\ket{i}=\sum_{i}\ket{i}TK\ket{i}=\ket{12}$
(3.60)

Therefore, time-reversal symmetry adds a term to the Weingarten’s formula

$\int
du\,u_{i_{1}}^{\phantom{i_{1}}j_{1}}u_{i_{2}}^{\phantom{i_{2}}j_{2}}(u^{\dagger})_{k_{1}}^{\phantom{k_{1}}l_{1}}(u^{\dagger})_{k_{2}}^{\phantom{k_{2}}l_{2}}\supset(TK)^{-1}_{i_{1},i_{2}}(TK)^{j_{1},j_{2}}(TK)^{-1}_{k_{1},k_{2}}(TK)^{l_{1},l_{2}}$
(3.61)

This would add an additional term to the averaged product of OPE coefficients

$\displaystyle\overline{c_{i\alpha j}c_{k\beta l}^{*}}$
$\displaystyle\supset\frac{1}{L^{2}}\mathrm{Tr}\big{(}(TK)O^{T}_{\alpha}(TK)^{-1}O_{\beta}\big{)}(TK)^{-1}_{il}(TK)_{jk}$
(3.62)
$\displaystyle=\frac{1}{L^{2}}\mathrm{Tr}(TO_{\alpha}^{\dagger}T^{-1}O_{\beta})(TK)^{-1}_{il}(TK)_{jk}$
(3.63)
$\displaystyle=\begin{cases}\frac{1}{L}(TK)^{-1}_{il}(TK)_{jk}&O_{\alpha}=TO_{\beta}T^{-1}\\\
0&\text{otherwise}\end{cases}$ (3.64)

There are two kinds of anomalies $T^{2}=\pm 1$. For $T^{2}=1$, we can just
take $T=K$ and the condition $TuT^{-1}=u$ reduces $u^{\dagger}u=1$ to
$u^{T}u=1$, which is equivalent to saying $u$ is orthogonal, and the ensemble
becomes GOE. Thus the added term to ensemble simplifies to

$\overline{c_{i\alpha j}c_{k\beta
l}^{*}}\supset\begin{cases}\frac{1}{L}\delta_{il}\delta_{jk}&O_{\alpha}=TO_{\beta}T^{-1}\\\
0&\text{otherwise}\end{cases}$ (3.65)

For $T^{2}=-1$, we can take $T=K\omega$ where $\omega=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}$. The condition $TuT^{-1}=u$ reduces $u^{\dagger}u=1$ to
$u^{T}\omega u=\omega$, which is equivalent to saying $u$ is symplectic, and
the ensemble becomes GSE. Thus the added term to ensemble simplifies to

$\overline{c_{i\alpha j}c_{k\beta
l}^{*}}\supset\begin{cases}\frac{1}{L}\omega^{-1}_{il}\omega_{jk}&O_{\alpha}=TO_{\beta}T^{-1}\\\
0&\text{otherwise}\end{cases}$ (3.66)

Now we can use results from last subsection to calculate torus two-point
functions. Note that on our 2d boundary, we always have $RT$ symmetry so the
ensemble is given by (3.55), which we now use to contract indices of the OPE
coefficients

$\displaystyle\overline{\braket{O_{1}(v,\bar{v})O_{1}(0)}_{T^{2}(\tau,\bar{\tau})}}$
$\displaystyle=\sum_{p,q}\overline{c_{1pq}c_{1pq}^{*}}\left|\includegraphics[valign={c},width=52.03227pt]{figures/torus2pticb.png}\right|^{2}=\sum_{p,q}\overline{c_{1pq}c_{1pq}^{*}}\big{|}\mathcal{F}_{11}^{(N)}(h_{p},h_{q};\tau,v)\big{|}^{2}$
(3.67)
$\displaystyle=\sum_{p,q}\left(1+(-1)^{s_{1}+s_{p}+s_{q}}\delta_{pq}\right)\big{|}C_{0}(h_{1},h_{p},h_{q})\big{|}^{2}\big{|}\mathcal{F}_{11}^{(N)}(h_{p},h_{q};\tau,v)\big{|}^{2}$
(3.68)
$\displaystyle=\sum_{p,q}\big{|}C_{0}(h_{1},h_{p},h_{q})\mathcal{F}_{11}^{(N)}(h_{p},h_{q};\tau,v)\big{|}^{2}$
$\displaystyle\quad\quad\quad+(-1)^{s_{1}}\sum_{p}\big{|}C_{0}(h_{1},h_{p},h_{p})\mathcal{F}_{11}^{(N)}(h_{p},h_{p};\tau,v)\big{|}^{2}$
(3.69) $\displaystyle=\big{|}\int
dh_{p}dh_{q}\rho_{0}(h_{p})\rho_{0}(h_{q})C_{0}(h_{1},h_{p},h_{q})\mathcal{F}_{11}^{g=1}(h_{p},h_{q};\tau,v)\big{|}^{2}$
$\displaystyle\quad\quad\quad+(-1)^{s_{1}}\big{|}\int
dh_{p}\rho_{0}(h_{p})C_{0}(h_{1},h_{p},h_{p})\mathcal{F}_{11}^{g=1}(h_{p},h_{p};\tau,v)\big{|}^{2}$
(3.70)

If we also add time-reversal symmetry, the ensemble becomes (3.65). This would
add another term to the two-point function

$\displaystyle\overline{\braket{TO_{1}(v,\bar{v})T^{-1}O_{1}(0)}_{T^{2}(\tau,\bar{\tau})}}$
$\displaystyle\supset\sum_{p}\overline{c_{1pp}^{2}}\left|\includegraphics[valign={c},width=52.03227pt]{figures/cccb.png}\right|^{2}=\sum_{p}\overline{c_{1pp}^{2}}|\mathcal{F}_{11}^{g=1}(h_{p},h_{p};\tau,v)|^{2}$
(3.71) $\displaystyle=|\int
dh_{p}\rho_{0}(h_{p})C_{0}(h_{1},h_{p},h_{p})\mathcal{F}_{11}^{g=1}(h_{p},h_{p};\tau,v)|^{2}$
(3.72)

Thus we see that the first term of (3.70) is the original decaying result of
[1], the second term of (3.70) matches contribution from (2) orientable
$\mathbb{Z}_{2}$ quotient of torus wormhole on the gravity side (3.35), and
(3.72) matches contribution from (1) non-orientable $\mathbb{Z}_{2}$ quotient
(3.27).

##  4 Comments on RT symmetry

In this section, we first show that a generic relativistic quantum field
theory with random matrix statistics should be of the GOE type for bosonic
states and GSE for fermionic states, then we point out that the partition
function of a torus wormhole calculated in [15, 16] needs another
multiplicative factor of $2$.

Section 3.2.2 tells us that the CFT2 ensemble proposed by [1] is inherently
GOE for bosonic states and GSE for fermionic states since it contains $RT$
symmetry, and $RT$ symmetry is an anti-linear, anti-unitary symmetry that
squares to $(-1)^{F}$ [25]. We should note that in 2d, $RT$ symmetry always
exists (this can be understood as coming from the CPT theorem 666C=charge
conjugation, P=parity, T=time-reversal. In 2d, P=R and CT together is the
time-reversal we are considering in RT.). Inspired by the above observation,
we claim that to the extent that a relativistic quantum field theory exhibits
random matrix statistics it should be of the GOE type for bosonic states and
of the GSE type for fermionic states. To start with, a relativistic quantum
field theory exhibits random matrix statistics is of GUE, if the system has no
additional symmetry, adding RT symmetry would make it into GOE or GSE. One
caveat is that if the Hamiltonian has an additional symmetry which block
diagonalize it, the RT symmetry could potentially exchange different blocks
instead of acting on each individual block. We now show that the above
situation does not happen. We can block diagonalize the Hamiltonian into
different momentum blocks

$H=\begin{pmatrix}\framebox{$H_{p_{1}}$}&0&0\\\ 0&\framebox{$H_{p_{2}}$}&0\\\
0&0&\ddots\\\ \end{pmatrix}$ (4.1)

In order to show the energy eigenvalue distribution is GOE or GSE, we need to
make sure that each individual subblock of $H$ commutes with RT. This is
equivalent to showing that momentum commutes with RT. Let $T_{\mu\nu}$ be the
stress-energy tensor then the momentum is given by

$p(t)=\int dx\,T_{01}(x,t)$ (4.2)

From here, we can show that

$RT\,p(0)=p(0)$ (4.3)

Therefore, momentum commutes with CPT and we conclude that any chaotic CFT2
has energy eigenvalue distribution a GOE for bosonic states and a GSE for
fermionic states.

RT symmetry has implications for torus wormhole partition function studied in
[15, 16]. In Euclidean AdS3 bulk, having $RT$ symmetry means that in addition
to the usual torus wormhole that can be obtained by gluing together two torus
trumpets as shown in figure 8(a), there should always exists a configuration
that first act on the right torus trumpet with RT, i.e. rotate the right torus
trumpet by $180^{\circ}$, and then glue it to the left torus trumpet as shown
in figure 8(b). This implies that the partition function of a torus wormhole
calculated in [15, 16] needs another multiplicative factor of $2$.

Figure 8: (a) gluing two torus trumpets to get a torus wormhole (b) act on the
right torus trumpet with RT and then glue to the left trumpet

Now we explain another way of understanding this which is through the Mapping
Class Group. The Mapping Class Group (MCG) of a torus $T^{2}$ is given by the
automorphisms of $T^{2}$, $\mathrm{Aut}(T^{2})$, modding out by the path
component of the identity in $\mathrm{Aut}(T^{2})$, i.e.

$\mathrm{MCG}(T^{2})=\mathrm{Aut}(T^{2})/\mathrm{Aut}_{0}(T^{2})$ (4.4)

It turns out that

$\mathrm{MCG}(T^{2})=\mathrm{SL}(2,\mathbb{Z})$ (4.5)

Intuitively, this can be understood by focusing on how the two cycles of a
torus $\alpha$ and $\beta$ get mapped. Recall that a torus is a quotient of
the complex plane by a 2d lattice as shown in figure 9.

Figure 9: $T^{2}$ is a quotient of the complex plane by a 2d lattice

The lattice has $\alpha$ and $\beta$ as basis vectors, and we think of them as
complex numbers. We want to determine the number of ways this torus can be
mapped to itself ignoring simply zooming in or out. $(\alpha,\beta)$ gets
mapped to $(p\alpha,q\beta)$ where $p$ and $q$ are relatively prime integers
with the same sign. We know that any such pair $(p,q)$ can be mapped from
$(1,1)$ by a unique element of $\mathrm{SL}(2,\mathbb{Z})$

$\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\begin{pmatrix}1\\\
1\end{pmatrix}=\begin{pmatrix}p\\\ q\end{pmatrix}$ (4.6)

Ignoring the sign of $(p,q)$. The complex structure of the torus is given by

$\tau=\frac{\alpha}{\beta}$ (4.7)

and it get mapped by
$\mathrm{PSL}(2,\mathbb{Z})=\mathrm{SL}(2,\mathbb{Z})/\\{\pm 1\\}$ to

$\tau\mapsto\frac{a\tau+b}{c\tau+d}$ (4.8)

However, we should note that this representation of $\mathrm{MCG}$ ignores the
case where $(\alpha,\beta)$ gets mapped to $(-\alpha,-\beta)$. This is the
case where the parallelogram formed by $(\alpha,\beta)$ get rotated by
$180^{\circ}$. This $180^{\circ}$ rotation is not included in
$\mathrm{PSL}(2,\mathbb{Z})$ because it is identified with the identity
element. However, with $RT$ symmetry, the directions of $\alpha$ and $\beta$
are important as shown in figure 10.

Figure 10: A $180^{\circ}$ rotation flips the directions of both $\alpha$ and
$\beta$ cycles of a torus.

Thus we should add in this element back, i.e. we should modify (2.16) to

$Z_{T^{2}\times
I}(\tau_{1},\bar{\tau}_{1},\tau_{2},\bar{\tau}_{2})=\sum_{\gamma\in\mathrm{SL}(2;\mathbb{Z})}\widetilde{Z}(\tau_{1},\bar{\tau}_{1},\gamma\tau_{2},\gamma\bar{\tau}_{2})$
(4.9)

This gives an additional multiplicative factor of two in the result. In large
spin, this result would again reduce to a double-trumpet in JT but with an
additional factor of two, which is consistent with JT with time-reversal
symmetry added777Time-reversal symmetry adds the double-trumpet glued from two
trumpets but with a reflection on one of the interfaces being glued. [10].
Also note that this is more consistent with (2.15), if we take $s_{1}$ even
and large.

## Acknowledgements

I want to give special thanks to Douglas Stanford for patient guidance,
extensive discussions, and inspiring comments throughout this project. I am
also grateful to Eleny Ionel, Steven Kerckhoff, Raghu Mahajan, Henry Maxfield,
and Xiaoliang Qi for discussions.

##  Appendix A JT gravity

Calculations in the main text parallel those in JT gravity. In this Appendix,
we give analogous results in JT gravity following the order of presentation of
the main text. For a more detailed presentation see [20].

In 2d, correlators are studied in the context of Jackiw-Teitelboim (JT)
gravity [6, 7, 8] which is dual to an ensemble of quantum mechanical systems
on the boundary [9, 10] and can be described by Random Matrix Theory (RMT).
Saad [26] 888continuing the idea of [27] computed bosonic two-point
correlation functions using the techniques developed by Yang [28] 999for other
approaches see [29, 30, 31, 27, 32] on the bulk side and compared with RMT
predictions for operators satisfying Eigenstate Thermalization Hypothesis
(ETH) [12, 11] on the boundary side.

###  A.1 Introduction

The 2d gravity theory we will study consists of the Einstein-Hilbert action +
JT gravity action + action from matter. JT gravity on a 2d manifold $M$ has
Euclidean action

$I_{JT}=-\frac{1}{2}\left(\int_{M}\phi(R+2)+2\int_{\partial
M}\phi_{b}(K-1)\right)$ (A.1)

Classically, the equation of motion fixes the bulk geometry to be AdS2 with
$R=-2$ and the action reduces to a Schwarzian action on the boundary [33]. In
2d, the Einstein-Hilbert action is purely topological and can be written as

$I_{EH}=-\chi S_{0}$ (A.2)

where $\chi=2-2g-n$ is the Euler character for manifold $M$ with $g$ the genus
and $n$ the number of boundaries, and $S_{0}$ is the zero-temperature bulk
entropy which is a constant. The Einstein-Hilbert action then contributes an
overall factor $e^{\chi S_{0}}$ to the partition function. In all of our
figures the orange disks represent infinite hyperbolic space (or its quotient)
and yellow geometries inside represent the physical Euclidean spacetimes, with
wiggly regularized boundaries described by the Schwarzian theory [33].

The two main shapes of Euclidean AdS we consider in this review are a
hyperbolic disk which has one asymptotically boundary with renormalized length
$\beta$, and a hyperbolic trumpet which has one asymptotic boundary with
renormalized length $\beta$ and one geodesic boundary with length $b$ (see
figure 11). That is because a disk is the simplest hyperbolic geometry with
one asymptotic boundary and a trumpet can be thought of as a building block of
more complicated geometries via attaching a Riemann surface with one geodesic
boundary to the geodesic boundary of the trumpet.

JT path integrals without operator insertions can be computed directly by
doing the path integral over the wiggly boundary of the disk and the trumpet
explicitly. Disk [34, 35, 36, 37, 38, 39, 40, 28, 41] and trumpet partition
functions [37, 26] are given respectively by

$Z_{\text{Disk}}(\beta)=e^{S_{0}}\frac{e^{\frac{2\pi^{2}}{\beta}}}{\sqrt{2\pi}\beta^{3/2}}=e^{S_{0}}\int_{0}^{\infty}dE\,\underbrace{\frac{\sinh(2\pi\sqrt{2E})}{2\pi^{2}}}_{\rho_{0}(E)}e^{-\beta
E}$ (A.3)

and

$Z_{\text{Trumpet}}(\beta,b)=\frac{e^{-\frac{b^{2}}{2\beta}}}{\sqrt{2\pi\beta}}=\int_{0}^{\infty}dE\,\frac{\cos(b\sqrt{2E})}{\pi\sqrt{2E}}e^{-\beta
E}$ (A.4)

where $\rho_{0}(E)$ denotes the density of state.

Figure 11: (a) disk partition function (b) trumpet partition function

To compute path integrals with operator insertions we need more tools. Before
we do that, we should note that there is another way of computing the disk
partition function. A disk can be decomposed into two Hartle-Hawking
wavefunctions by the following procedure

$\displaystyle Z_{\text{Disk}}(\beta)$
$\displaystyle=\includegraphics[valign={c},width=86.72267pt]{figuresjt/diskcopy.png}$
(A.5)
$\displaystyle=\int\,e^{\ell}d\ell\includegraphics[valign={c},width=86.72267pt]{figuresjt/splitcopy.png}$
(A.6)
$\displaystyle=e^{S_{0}}\int\,e^{\ell}d\ell\,\varphi_{\text{Disk},\tau}(\ell)\varphi_{\text{Disk},\beta-\tau}(\ell)$
(A.7)

This decomposition may seem redundant since we already know how to calculate
$Z_{\mathrm{disk}}$ but this procedure teaches us how to calculate two-point
correlation functions. To do that, we just need another factor of
$e^{-\Delta\ell}$ in the integral, which is the QFT two-point correlation
function of two boundary operators $V$ of conformal weight $\Delta$ with
renormalized geodesic distance $\ell$ apart. Disk contribution to two-point
correlation functions at time $t=-i\tau$ is then given by

$\displaystyle\braket{V(t=-i\tau)V(0)}_{\chi=1}$
$\displaystyle=\includegraphics[valign={c},width=86.72267pt]{figuresjt/diskcorrelatorcopy.png}$
(A.8)
$\displaystyle=\int\,e^{\ell}d\ell\includegraphics[valign={c},width=86.72267pt]{figuresjt/splitcopy.png}e^{-\Delta\ell}$
(A.9)
$\displaystyle=e^{S_{0}}\int\,e^{\ell}d\ell\,\varphi_{\text{Disk},\tau}(\ell)\varphi_{\text{Disk},\beta-\tau}(\ell)e^{-\Delta\ell}$
(A.10)

Note that this two-point correlator is not normalized by dividing out the disk
partition function which is of order $e^{S_{0}}$. Hartle-Hawing wavefunctions
can be written in a simple closed form by first writing the wavefunctions with
fixed energy boundary conditions given by

$\varphi_{E}(\ell)=\braket{\ell}{E}=4e^{-\ell/2}K_{i\sqrt{8E}}(4e^{-\ell/2})$
(A.11)

where $K$ is a Bessel-K function. Hartle-Hawking wavefunctions i.e.
wavefunctions with fixed length boundary condition are given by [28, 26]

$\varphi_{\text{Disk},\tau}(\ell)=\int_{0}^{\infty}dE\,\rho_{0}(E)e^{-\tau
E}\varphi_{E}(\ell)$ (A.12)

Now we review two important relations that the Hartle-Hawking wavefunctions
satisfy:

$\displaystyle\int_{-\infty}^{\infty}e^{\ell}d\ell\,\varphi_{E}(\ell)\varphi_{E^{\prime}}(\ell)$
$\displaystyle=\frac{\delta(E-E^{\prime})}{\rho_{0}(E)}$ (A.13)
$\displaystyle\int_{-\infty}^{\infty}e^{\ell}d\ell\,\varphi_{E}(\ell)\varphi_{E^{\prime}}(\ell)e^{-\Delta\ell}$
$\displaystyle=|V_{E,E^{\prime}}|^{2}=\frac{\left|\Gamma\left(\Delta+i(\sqrt{2E}+\sqrt{2E^{\prime}})\right)\Gamma\left(\Delta+i(\sqrt{2E}-\sqrt{2E^{\prime}})\right)\right|^{2}}{2^{2\Delta-1}\Gamma(2\Delta)}$
(A.14)

In particular using (A.13) we can verify

$Z_{\text{Disk}}(\beta)=e^{S_{0}}\int
e^{\ell}d\ell\,\varphi_{\text{Disk},\tau}(\ell)\varphi_{\text{Disk},\beta-\tau}(\ell)$
(A.15)

by plugging in (A.3, A.12). In addition to partition functions and Hartle-
Hawking states, we review a final and important tool we use: propagators i.e.
time evolution operators of Hartle-Hawking wavefunctions such that

$\varphi_{\text{Disk},\beta+\beta_{1}+\beta_{2}}(\ell)=\int
e^{\ell^{\prime}}d\ell^{\prime}\,P_{\text{Disk}}(\beta_{1},\beta_{2},\ell,\ell^{\prime})\varphi_{\text{Disk},\beta}(\ell^{\prime})$
(A.16)

we can check [26] that the above relations are solved by

$P_{\text{Disk}}(\beta_{1},\beta_{2},\ell,\ell^{\prime})=\int
dE\,\rho_{0}(E)e^{-(\beta_{1}+\beta_{2})E}\varphi_{E}(\ell)\varphi_{E}(\ell^{\prime})$
(A.17)

Figure 12: (a) Hartle-Hawking wavefunction (b) disk propagator

###  A.2 Product of two one-point functions

In this section, we focus on the product of two one-point functions, which on
the gravity side corresponds to a cylinder with one operator insertion on each
side as shown in figure 13.

Figure 13: (a) a cylinder with one operator insertion on each side (b)
embedded on a hyperbolic disk

As shown in figure 13(b), this is given by a propagator so

$\displaystyle\braket{V}_{\beta_{1}}\braket{V}_{\beta_{2}}$
$\displaystyle=\int
e^{\ell}d\ell\,P_{\text{Disk}}(\beta_{1},\beta_{2},\ell,\ell)e^{-\Delta\ell}$
(A.18) $\displaystyle=\int
dE\,\rho_{0}(E)e^{-(\beta_{1}+\beta_{2})E}|V_{E,E}|^{2}$ (A.19)
$\displaystyle=\int
ds\,s\rho_{0}(s)e^{-(\beta_{1}+\beta_{2})s^{2}/2}\frac{\Gamma(\Delta\pm
i2s)\Gamma(\Delta)^{2}}{2^{2\Delta-1}\Gamma(2\Delta)}$ (A.20)

where $s=\sqrt{2E}$.

Take the limit $\Delta\rightarrow 0$ we get

$\displaystyle\braket{V}_{\beta_{1}}\braket{V}_{\beta_{2}}$
$\displaystyle=4\int
ds\,s\rho_{0}(s)e^{-(\beta_{1}+\beta_{2})s^{2}/2}\frac{\Gamma(\pm
2is)}{\Delta}$ (A.21) $\displaystyle=\frac{4}{\Delta}\int
ds\,s\rho_{0}(s)e^{-(\beta_{1}+\beta_{2})s^{2}/2}\frac{\pi}{2s\sinh(2\pi s)}$
(A.22)

where we have used the fact that when $\Delta\rightarrow 0$

$\Gamma(\Delta)=\frac{\Gamma(1+\Delta)}{\Delta}=\frac{1}{\Delta}-\gamma+O(\Delta)$
(A.23)

and that

$\Gamma(z)\Gamma(-z)=\frac{\Gamma(z)\Gamma(1-z)}{-z}=-\frac{\pi}{z\sin\pi z}$
(A.24)

But now we should recall that the density of states is given by

$\rho_{0}(s)=\frac{\sinh(2\pi s)}{2\pi^{2}}$ (A.25)

so

$\displaystyle\braket{V}_{\beta_{1}}\braket{V}_{\beta_{2}}$
$\displaystyle=\frac{1}{\pi\Delta}\int ds\,e^{-(\beta_{1}+\beta_{2})s^{2}/2}$
(A.26)
$\displaystyle=\frac{1}{\pi\Delta}\sqrt{\frac{\pi}{2(\beta_{1}+\beta_{2})}}$
(A.27)

But on the other hand, we can directly calculate the partition function of a
cylinder by using the partition function of trumpet

$\displaystyle Z_{\text{cylinder}}$
$\displaystyle=\includegraphics[valign={c},width=73.7146pt]{figuresjt/cylindercom.png}$
(A.28) $\displaystyle=\int
db\,b\includegraphics[valign={c},width=86.72267pt]{figuresjt/cylindersevered.png}$
(A.29) $\displaystyle=\int
db\,b\,Z_{\text{Trumpet}}(\beta_{1},b)Z_{\text{Trumpet}}(\beta_{2},b)$ (A.30)
$\displaystyle=\frac{\sqrt{\beta_{1}\beta_{2}}}{2\pi(\beta_{1}+\beta_{2})}$
(A.31)

This disagrees with equation (A.27), but we can explain the reason as follows.

We can also calculate the product of two one-point functions using the trumpet
partition function, but when doing that we need to include an additional
factor $e^{-\Delta\ell}$. This means that we need to sum over different
windings which contributes a factor

$\sum_{n=-\infty}^{\infty}e^{-\Delta bn}\approx\frac{2}{b\Delta}$ (A.32)

Now we can calculate product of two one-point functions again including the
winding factor with weight $\Delta\rightarrow 0$

$\displaystyle\braket{V}_{\beta_{1}}\braket{V}_{\beta_{2}}(\Delta\rightarrow
0)$ $\displaystyle=\int db\,b\,\sum_{n=-\infty}^{\infty}e^{-\Delta
bn}\,Z_{\text{Trumpet}}(\beta_{1},b)Z_{\text{Trumpet}}(\beta_{2},b)$ (A.33)
$\displaystyle=\frac{2}{\Delta}\int
db\,Z_{\text{Trumpet}}(\beta,b)Z_{\text{Trumpet}}(\beta^{\prime},b)$ (A.34)
$\displaystyle=\frac{1}{\pi\Delta}\sqrt{\frac{\pi}{2(\beta_{1}+\beta_{2})}}$
(A.35)

This now matches equation (A.27) so we have the discrepancy fixed.

###  A.3 Product of two two-point functions

In this section, we focus on the product of two two-point functions, which on
the gravity side corresponds to a cylinder with two operator insertions on
each side as shown in figure 14.

Figure 14: (a) a cylinder with two operator insertions on each side (b)
embedded on a hyperbolic disk

As shown in figure 14(b), this is given by two propagators glued together so

$\displaystyle\braket{V(\tau_{1})W(0)}\braket{V(\beta_{2}-\tau_{2})W(0)}$
(A.36) $\displaystyle=$ $\displaystyle\int e^{\ell}d\ell
e^{\ell^{\prime}}d\ell^{\prime}P_{\text{Disk}}(\tau_{1},\beta_{1}-\tau_{1},\ell,\ell^{\prime})P_{\text{Disk}}(\beta_{2}-\tau_{2},\tau_{2},\ell^{\prime},\ell)e^{-\Delta_{V}\ell}e^{-\Delta_{W}\ell^{\prime}}$
(A.37) $\displaystyle=$ $\displaystyle\int
dEdE^{\prime}\rho_{0}(E)\rho_{0}(E^{\prime})e^{-(\beta_{2}+\tau_{1}-\tau_{2})E-(\beta_{1}-\tau_{1}+\tau_{2})E^{\prime}}|V_{E,E^{\prime}}|^{2}|W_{E,E^{\prime}}|^{2}$
(A.38)

In particular, if we take $V=W$ and $\tau_{1}=\tau_{2}$, the above expression
becomes

$\displaystyle\braket{V(\tau_{1})V(0)}\braket{V(\beta_{2}-\tau_{2})V(0)}$
$\displaystyle=\int
dEdE^{\prime}\rho_{0}(E)\rho_{0}(E^{\prime})e^{-\beta_{2}E-\beta_{1}E^{\prime}}|V_{E,E^{\prime}}|^{4}$
(A.39) $\displaystyle=\int
dsds^{\prime}ss^{\prime}\rho_{0}(s)\rho_{0}(s^{\prime})e^{-\frac{\beta_{1}s^{\prime
2}}{2}-\frac{\beta_{2}s^{2}}{2}}\left(\frac{\prod_{\pm_{1,2}}\Gamma\left(\Delta\pm_{1}is\pm_{2}is^{\prime}\right)}{2^{2\Delta-1}\Gamma(2\Delta)}\right)^{2}$
(A.40)

where $s=\sqrt{2E}$ and $s^{\prime}=\sqrt{2E^{\prime}}$.

####  A.3.1 crosscap

In this section we give a way of calculating the disk with a crosscap
contribution to the 2-point correlator by first doubling the crosscap
configuration to give a cylinder and then take the square root. For a review
of JT gravity framework and 2d crosscap see Appendix LABEL:2dcc.

We start from the contribution to the averaged product of two 2pt functions.
This comes from the cylinder

$\overline{\braket{V(\tau)V(0)}\braket{V(\beta-\tau)V(0)}}=\int
dEdE^{\prime}\underbrace{\frac{\sinh(2\pi\sqrt{2E})}{2\pi^{2}}}_{\rho_{0}(E)}\underbrace{\frac{\sinh(2\pi\sqrt{2E^{\prime}})}{2\pi^{2}}}_{\rho_{0}(E^{\prime})}e^{-\beta(E+E^{\prime})}|V_{E,E^{\prime}}|^{4}$
(A.41)

where

$|V_{E,E^{\prime}}|^{2}=\frac{\left|\Gamma\left(\Delta+i(\sqrt{2E}+\sqrt{2E^{\prime}})\right)\Gamma\left(\Delta+i(\sqrt{2E}-\sqrt{2E^{\prime}})\right)\right|^{2}}{2^{2\Delta-1}\Gamma(2\Delta)}$
(A.42)

Let $s=\sqrt{2E}$ and $s^{\prime}=\sqrt{2E^{\prime}}$ we can rewrite the above
equation as

$\overline{\braket{V(\tau)V(0)}\braket{V(\beta-\tau)V(0)}}=\int
dsds^{\prime}ss^{\prime}\frac{\sinh(2\pi s)}{2\pi^{2}}\frac{\sinh(2\pi
s^{\prime})}{2\pi^{2}}e^{-\frac{\beta}{2}(s^{2}+s^{\prime
2})}\left(\frac{\prod_{\pm_{1,2}}\Gamma\left(\Delta\pm_{1}is\pm_{2}is^{\prime}\right)}{2^{2\Delta-1}\Gamma(2\Delta)}\right)^{2}$
(A.43)

Now make a change of variables

$s_{1}+s_{2}=s_{+}\quad\quad s_{1}-s_{2}=s_{-}$ (A.44)

Then the above expression becomes

$\overline{\braket{V(\tau)V(0)}\braket{V(\beta-\tau)V(0)}}\propto\\\ \int
ds_{+}ds_{-}(s_{+}^{2}-s_{-}^{2})(e^{2\pi s_{+}}+e^{-2\pi s_{+}}-e^{2\pi
s_{-}}-e^{-2\pi
s_{-}})e^{-\frac{\beta}{4}(s_{+}^{2}+s_{-}^{2})}\left(\frac{\prod_{\pm_{1,2}}\Gamma\left(\Delta\pm_{1}is_{+}\right)\Gamma\left(\Delta\pm_{2}is_{-}\right)}{2^{2\Delta-1}\Gamma(2\Delta)}\right)^{2}$
(A.45)

Using the Stirling approximation

$\prod_{\pm}\Gamma(\Delta\pm is_{+})\approx e^{-\pi s_{+}}s_{+}^{2\Delta-1}$
(A.46)

So we pick the term $s_{+}^{2}$ and $e^{2\pi s_{+}}$ in the expansion

$\overline{\braket{V(\tau)V(0)}\braket{V(\beta-\tau)V(0)}}\propto\int
ds_{+}ds_{-}s_{+}^{4\Delta}e^{-\frac{\beta}{4}(s_{+}^{2}+s_{-}^{2})}\left(\frac{\prod_{\pm}\Gamma\left(\Delta\pm
is_{-}\right)}{2^{2\Delta-1}\Gamma(2\Delta)}\right)^{2}$ (A.47)

Saddle point of $s_{-}$ is at $s_{-}=0$, so (A.41) can be approximated by

$\overline{\braket{V(\tau)V(0)}\braket{V(\beta-\tau)V(0)}}\approx\int
dE\underbrace{\frac{\sinh(2\pi\sqrt{2E})}{2\pi^{2}}}_{\rho_{0}(E)}\underbrace{\frac{\sinh(2\pi\sqrt{2E})}{2\pi^{2}}}_{\rho_{0}(E)}e^{-\beta
2E}|V_{E,E}|^{4}$ (A.48)

We should note that here we only care about the saddle point (i.e. only the
integrand of the above integral). Thus we get

$\sqrt{\overline{\braket{V(\tau)V(0)}\braket{V(\beta-\tau)V(0)}}}\approx\int
dE\underbrace{\frac{\sinh(2\pi\sqrt{2E})}{2\pi^{2}}}_{\rho_{0}(E)}e^{-\beta
E}|V_{E,E}|^{2}$ (A.49)

In [20], crosscap contribution to the 2-point function was calculated directly
using propagator

$\displaystyle\braket{V(t=-i\tau)V(0)}_{\text{cc},0}$ $\displaystyle=\int
e^{\ell}d\ell\,P_{\text{Disk}}(\tau,\beta-\tau,\ell,\ell)e^{-\Delta\ell}$
(A.50)
$\displaystyle=\includegraphics[valign={c},width=108.405pt]{figures/2dcc}$
(A.51) $\displaystyle=\int e^{\ell}d\ell\,\int dE\,\rho_{0}(E)e^{-\beta
E}\varphi_{E}(\ell)^{2}e^{-\Delta\ell}$ (A.52) $\displaystyle=\int
dE\,\rho_{0}(E)e^{-\beta E}|V_{E,E}|^{2}$ (A.53)

Therefore (A.49) is exactly the crosscap contribution, so we just showed that

$\sqrt{\overline{\braket{V(\tau)V(0)}\braket{V(\beta-\tau)V(0)}}}\approx\overline{\braket{V(t)V(0)}}_{cc}$
(A.54)

##  Appendix B Calculation details

###  B.1 Calculating product of two torus one-point functions

We start from the torus partition function

$Z(\tau,\bar{\tau})=(q\bar{q})^{-c/24}\mathrm{Tr}(q^{L_{0}}\bar{q}^{\bar{L}_{0}})$
(B.1)

where $q=e^{2\pi i\tau}$. Based on this the one-point function can be written
as

$\displaystyle\braket{O_{1}}_{T^{2}(\tau,\bar{\tau})}$
$\displaystyle=(q\bar{q})^{-c/24}\mathrm{Tr}(q^{L_{0}}\bar{q}^{\bar{L}_{0}}O_{1})$
(B.2)
$\displaystyle=(q\bar{q})^{-c/24}\sum_{h}\braket{h,\bar{h}}{O_{1}}{h,\bar{h}}q^{L_{0}}\bar{q}^{\bar{L}_{0}}$
(B.3)
$\displaystyle=\sum_{p}\braket{h_{p},\bar{h}_{p}}{O_{1}}{h_{p},\bar{h}_{p}}\sum_{h\in\mathcal{V}_{h_{p}}}\frac{\braket{h,\bar{h}}{O_{1}}{h,\bar{h}}}{\braket{h_{p},\bar{h}_{p}}{O_{1}}{h_{p},\bar{h}_{p}}}e^{i\tau(h-c/24)}e^{-i\bar{\tau}(\bar{h}-c/24)}$
(B.4)
$\displaystyle=\sum_{p}c_{1pp}\sum_{h\in\mathcal{V}_{h_{p}}}\frac{\braket{h,\bar{h}}{O_{1}}{h,\bar{h}}}{\braket{h_{p},\bar{h}_{p}}{O_{1}}{h_{p},\bar{h}_{p}}}e^{i\tau(h-c/24)}e^{-i\bar{\tau}(\bar{h}-c/24)}$
(B.5)

where we decompose a sum over all operators into a sum over primaries and a
sum over all descendents of a primary. Thus, we get an expression for the
conformal block

$\mathcal{F}_{1}^{g=1}(h_{p};\tau)\overline{\mathcal{F}}_{1}^{g=1}(\bar{h}_{p};\bar{\tau})=\sum_{h\in\mathcal{V}_{h_{p}}}\frac{\braket{h,\bar{h}}{O_{1}}{h,\bar{h}}}{\braket{h_{p},\bar{h}_{p}}{O_{1}}{h_{p},\bar{h}_{p}}}e^{i\tau(h-c/24)}e^{-i\bar{\tau}(\bar{h}-c/24)}$
(B.6)

In particular

$\mathcal{F}_{1}^{g=1}(h_{p};\tau^{\prime})\overline{\mathcal{F}}_{1}^{g=1}(\bar{h}_{p};\bar{\tau}^{\prime})=\overline{\mathcal{F}}_{1}^{g=1}(h_{p};-\tau^{\prime})\mathcal{F}_{1}^{g=1}(\bar{h}_{p};-\bar{\tau}^{\prime})$
(B.7)

since complex conjugation gives a minus sign while $\tau\mapsto-\tau$ gives
another minus sign, these two minus signs cancel.

Thus the averaged product of two torus one-point functions is given by

$\displaystyle\overline{\braket{O_{1}}_{T^{2}(\tau,\bar{\tau})}\braket{O_{1}}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}}$
$\displaystyle=\sum_{p,q}\overline{c_{1pp}c_{1qq}}\mathcal{F}_{1}^{g=1}(h_{p};\tau)\overline{\mathcal{F}}_{1}^{g=1}(\bar{h}_{p};\bar{\tau})\mathcal{F}_{1}^{g=1}(h_{q};\tau^{\prime})\overline{\mathcal{F}}_{1}^{g=1}(\bar{h}_{q};\bar{\tau}^{\prime})$
(B.8)
$\displaystyle=2\sum_{p}\overline{c_{1pp}^{2}}\mathcal{F}_{1}^{g=1}(h_{p};\tau)\overline{\mathcal{F}}_{1}^{g=1}(\bar{h}_{p};\bar{\tau})\mathcal{F}_{1}^{g=1}(h_{p};\tau^{\prime})\overline{\mathcal{F}}_{1}^{g=1}(\bar{h}_{p};\bar{\tau}^{\prime})$
(B.9)
$\displaystyle=2\sum_{p}\overline{c_{1pp}^{2}}\mathcal{F}_{1}^{g=1}(h_{p};\tau)\overline{\mathcal{F}}_{1}^{g=1}(\bar{h}_{p};\bar{\tau})\overline{\mathcal{F}}_{1}^{g=1}(h_{p};-\tau^{\prime})\mathcal{F}_{1}^{g=1}(\bar{h}_{p};-\bar{\tau}^{\prime})$
(B.10) $\displaystyle\approx 2\big{|}\int
dh_{p}\rho_{0}(h_{p})C_{0}(h_{1},h_{p},h_{p})\mathcal{F}_{1}^{g=1}(h_{p};\tau)\overline{\mathcal{F}}_{1}^{g=1}(h_{p};-\tau^{\prime})\big{|}^{2}$
(B.11) $\displaystyle\approx
2\braket{O_{1}}^{L}_{T^{2}(\tau,-\tau^{\prime})}\braket{O_{1}}^{L}_{T^{2}(-\bar{\tau}^{\prime},\bar{\tau})}$
(B.12)

###  B.2 Calculating product of two torus two-point functions

The two-point function can be written as

$\displaystyle\braket{O_{1}(v,\bar{v})O_{2}(0)}_{T^{2}(\tau,\bar{\tau})}$
$\displaystyle=(q\bar{q})^{-c/24}\mathrm{Tr}(q^{L_{0}}\bar{q}^{\bar{L}_{0}}O_{1}O_{2})$
(B.13)
$\displaystyle=(q\bar{q})^{-c/24}\sum_{h,\bar{h},h^{\prime},\bar{h}^{\prime}}\braket{h,\bar{h}}{e^{i(\tau-v)L_{0}}e^{-i(\bar{\tau}-\bar{v})\bar{L}_{0}}O_{2}e^{ivL_{0}}e^{-i\bar{v}\bar{L}_{0}}}{h^{\prime},\bar{h}^{\prime}}\braket{h^{\prime},\bar{h}^{\prime}}{O_{1}}{h,\bar{h}}$
(B.14)
$\displaystyle=\sum_{h,h^{\prime}}\braket{h,\bar{h}}{O_{2}}{h^{\prime},\bar{h}^{\prime}}\braket{h^{\prime},\bar{h}^{\prime}}{O_{1}}{h,\bar{h}}e^{i\tau(h-c/24)}e^{-i\bar{\tau}(\bar{h}-c/24)}e^{iv(h^{\prime}-h)}e^{-i\bar{v}(\bar{h}-\bar{h}^{\prime})}$
(B.15)
$\displaystyle=\sum_{h_{p},h_{q}}\braket{h_{p},\bar{h}_{p}}{O_{2}}{h_{q},\bar{h}_{q}}\braket{h_{q},\bar{h}_{q}}{O_{1}}{h_{p},\bar{h}_{p}}$
$\displaystyle\times\sum_{h\in\mathcal{V}_{h_{p}},h^{\prime}\in\mathcal{V}_{h_{q}}}\frac{\braket{h,\bar{h}}{O_{2}}{h^{\prime},\bar{h}^{\prime}}}{\braket{h_{p},\bar{h}_{p}}{O_{2}}{h_{q},\bar{h}_{q}}}\frac{\braket{h^{\prime},\bar{h}^{\prime}}{O_{1}}{h,\bar{h}}}{\braket{h_{q},\bar{h}_{q}}{O_{1}}{h_{p},\bar{h}_{p}}}e^{i\tau(h-c/24)}e^{-i\bar{\tau}(\bar{h}-c/24)}e^{iv(h^{\prime}-h)}e^{-i\bar{v}(\bar{h}-\bar{h}^{\prime})}$
(B.16)
$\displaystyle=\sum_{h_{p},h_{q}}c_{1pq}c_{2pq}\mathcal{F}_{12}^{g=1}(h_{p},h_{q};\tau,v)\overline{\mathcal{F}}_{12}^{g=1}(\bar{h}_{p},\bar{h}_{q};\bar{\tau},\bar{v})$
(B.17)

Comparing these two results, we get an expression for the conformal block

$\mathcal{F}_{12}^{g=1}(h_{p},h_{q};\tau,v)\overline{\mathcal{F}}_{12}^{g=1}(\bar{h}_{p},\bar{h}_{q};\bar{\tau},\bar{v})\\\
=\sum_{h\in\mathcal{V}_{h_{p}},h^{\prime}\in\mathcal{V}_{h_{q}}}\frac{\braket{h,\bar{h}}{O_{2}}{h^{\prime},\bar{h}^{\prime}}}{\braket{h_{p},\bar{h}_{p}}{O_{2}}{h_{q},\bar{h}_{q}}}\frac{\braket{h^{\prime},\bar{h}^{\prime}}{O_{1}}{h,\bar{h}}}{\braket{h_{q},\bar{h}_{q}}{O_{1}}{h_{p},\bar{h}_{p}}}e^{i\tau(h-c/24)}e^{-i\bar{\tau}(\bar{h}-c/24)}e^{iv(h^{\prime}-h)}e^{-i\bar{v}(\bar{h}-\bar{h}^{\prime})}$
(B.18)

In particular we have

$\mathcal{F}_{12}^{g=1}(h_{p},h_{q};\tau^{\prime},v^{\prime})\overline{\mathcal{F}}_{12}^{g=1}(\bar{h}_{p},\bar{h}_{q};\bar{\tau}^{\prime},\bar{v}^{\prime})=\overline{\mathcal{F}}_{12}^{g=1}(h_{p},h_{q};-\tau^{\prime},-v^{\prime})\mathcal{F}_{12}^{g=1}(\bar{h}_{p},\bar{h}_{q};-\bar{\tau}^{\prime},-\bar{v}^{\prime})$
(B.19)

Thus the averaged product of two torus two-point functions is given by

$\displaystyle\overline{\braket{O_{1}(v,\bar{v})O_{2}(0)}_{T^{2}(\tau,\bar{\tau})}\braket{O_{1}(v^{\prime},\bar{v}^{\prime})O_{2}(0)}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}}$
$\displaystyle=$
$\displaystyle\sum_{p,q,r,s}\overline{c_{1pq}c_{2pq}c_{1rs}c_{2rs}}\mathcal{F}_{12}^{g=1}(h_{p},h_{q};\tau,v)\overline{\mathcal{F}}_{12}^{g=1}(\bar{h}_{p},\bar{h}_{q};\bar{\tau},\bar{v})\mathcal{F}_{12}^{g=1}(h_{r},h_{s};\tau^{\prime},v^{\prime})\overline{\mathcal{F}}_{12}^{g=1}(\bar{h}_{r},\bar{h}_{s};\bar{\tau}^{\prime},\bar{v}^{\prime})$
(B.20) $\displaystyle=$ $\displaystyle
2\sum_{p,q}\overline{|c_{1pq}^{2}|}\overline{|c_{2pq}^{2}|}\mathcal{F}_{12}^{g=1}(h_{p},h_{q};\tau,v)\overline{\mathcal{F}}_{12}^{g=1}(\bar{h}_{p},\bar{h}_{q};\bar{\tau},\bar{v})\mathcal{F}_{12}^{g=1}(h_{p},h_{q};\tau^{\prime},v^{\prime})\overline{\mathcal{F}}_{12}^{g=1}(\bar{h}_{p},\bar{h}_{q};\bar{\tau}^{\prime},\bar{v}^{\prime})$
(B.21) $\displaystyle=$ $\displaystyle
2\sum_{p,q}\overline{|c_{1pq}^{2}|}\overline{|c_{2pq}^{2}|}\mathcal{F}_{12}^{g=1}(h_{p},h_{q};\tau,v)\overline{\mathcal{F}}_{12}^{g=1}(\bar{h}_{p},\bar{h}_{q};\bar{\tau},\bar{v})\overline{\mathcal{F}}_{12}^{g=1}(h_{p},h_{q};-\tau^{\prime},-v^{\prime})\mathcal{F}_{12}^{g=1}(\bar{h}_{p},\bar{h}_{q};-\bar{\tau}^{\prime},-\bar{v}^{\prime})$
(B.22) $\displaystyle\approx$ $\displaystyle 2\big{|}\int
dh_{p}dh_{q}\rho_{0}(h_{p})\rho_{0}(h_{q})C_{0}(h_{1},h_{p},h_{q})C_{0}(h_{2},h_{p},h_{q})\mathcal{F}_{12}^{g=1}(h_{p},h_{q};\tau,v)\overline{\mathcal{F}}_{12}^{g=1}(h_{p},h_{q};-\tau^{\prime},-v^{\prime})\big{|}^{2}$
(B.23) $\displaystyle\approx$ $\displaystyle
2\braket{O_{1}(v,-v^{\prime})O_{2}(0)}^{L}_{T^{2}(\tau,-\tau^{\prime})}\braket{O_{1}(-\bar{v}^{\prime},\bar{v})O_{2}(0)}^{L}_{T^{2}(-\bar{\tau}^{\prime},\bar{\tau})}$
(B.24)

###  B.3 Averaged product of two torus one-point function in
$\Delta\rightarrow 0$ limit

Let us begin with our formula of the averaged product of two one-point
functions

$\overline{\braket{O_{1}}_{T^{2}(\tau,\bar{\tau})}\braket{O_{1}}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}}\approx
2\big{|}\int
dh_{p}\rho_{0}(h_{p})C_{0}(h_{1},h_{p},h_{p})\mathcal{F}_{1}^{g=1}(h_{p};\tau)\overline{\mathcal{F}}_{1}^{g=1}(h_{p};-\tau^{\prime})\big{|}^{2}$
(B.25)

where $C_{0}$ is given by

$C_{0}(P_{1},P_{2},P_{3})=\frac{\Gamma_{b}(2Q)}{\sqrt{2}\Gamma_{b}(Q)^{3}}\frac{\Gamma_{b}(\frac{Q}{2}\pm
iP_{1}\pm iP_{2}\pm iP_{3})}{\prod_{k=1}^{3}\Gamma_{b}(Q\pm 2iP_{k})}$ (B.26)

Now we take the weight of $O_{1}$ to zero i.e.
$\Delta=2h=2\alpha(Q-\alpha)\rightarrow 0$. Without loss of generality we take
$\alpha=\epsilon\rightarrow 0$, so $\Delta=2\epsilon Q$. More specifically
$iP_{1}=-\frac{Q}{2}+\epsilon$ so

$C_{0}(h_{1},h_{p},h_{p})=C_{0}(i(\frac{Q}{2}-\epsilon),P,P)=\frac{\Gamma_{b}(\epsilon)^{2}\Gamma_{b}(\pm
2iP)}{\sqrt{2}\Gamma_{b}(Q)\Gamma_{b}(2\epsilon)\Gamma_{b}(Q\pm 2iP)}$ (B.27)

Remember the definition of $\Gamma_{b}$ in terms of double gamma function

$\Gamma_{b}(P)=\frac{\Gamma_{2}(P|b,b^{-1})}{\Gamma_{2}(\frac{Q}{2}|b,b^{-1})}$
(B.28)

where

$\Gamma_{2}(w|a_{1},a_{2})=\frac{e^{\lambda_{1}w+\lambda_{2}w^{2}}}{w}\prod_{(n_{1},n_{2})\in\mathbb{N}^{2},(n_{1},n_{2})\neq(0,0)}\frac{e^{\frac{w}{n_{1}a_{1}+n_{2}a_{2}}-\frac{1}{2}\frac{w^{2}}{(n_{1}a_{1}+n_{2}a_{2})^{2}}}}{1+\frac{w}{n_{1}a_{1}+n_{2}a_{2}}}$
(B.29)

so as we take $\epsilon\rightarrow 0$

$\Gamma_{2}(\epsilon|b,b^{-1})=\frac{1}{\epsilon}$ (B.30)

and

$C_{0}(h_{1},h_{p},h_{p})=\frac{\sqrt{2}}{\epsilon\Gamma_{2}(Q|b,b^{-1})}\frac{\Gamma_{b}(\pm
2iP)}{\Gamma_{b}(Q\pm 2iP)}$ (B.31)

Now observe that $\Gamma_{b}$ satisfies the following relations

$\Gamma_{b}(w+b)=\sqrt{2\pi}\frac{b^{bw-\frac{1}{2}}}{\Gamma(bw)}\Gamma_{b}(w)\quad\quad\Gamma_{b}(w+b^{-1})=\sqrt{2\pi}\frac{b^{-b^{-1}w+\frac{1}{2}}}{\Gamma(b^{-1}w)}\Gamma_{b}(w)$
(B.32)

so correspondingly

$\Gamma_{b}(Q+2iP)=\frac{2\pi
b^{1+i2(b-b^{-1})P}\Gamma_{b}(2iP)}{\Gamma(1+2ibP)\Gamma(2ib^{-1}P)}\quad\quad\Gamma_{b}(Q-2iP)=\frac{2\pi
b^{1-i2(b-b^{-1})P}\Gamma_{b}(-2iP)}{\Gamma(1-2ibP)\Gamma(-2ib^{-1}P)}$ (B.33)

so

$\frac{\Gamma_{b}(\pm 2iP)}{\Gamma_{b}(Q\pm 2iP)}=\frac{\Gamma(1\pm
2ibP)\Gamma(\pm 2ib^{-1}P)}{(2\pi)^{2}b^{2}}=\frac{1}{4\sinh(2\pi
bP)\sinh(2\pi b^{-1}P)}$ (B.34)

where we have used

$\displaystyle\Gamma(1+z)\Gamma(1-z)$
$\displaystyle=z\Gamma(z)\Gamma(1-z)=\frac{z\pi}{\sin\pi z}$ (B.35)
$\displaystyle\Gamma(z)\Gamma(-z)$
$\displaystyle=\frac{\Gamma(z)\Gamma(1-z)}{-z}=-\frac{\pi}{z\sin\pi z}$ (B.36)

so

$\displaystyle C_{0}(h_{1},h_{p},h_{p})$
$\displaystyle=\frac{1}{16\epsilon\Gamma_{2}(Q|b,b^{-1})\sinh^{2}(2\pi
bP)\sinh^{2}(2\pi b^{-1}P)}$ (B.37)
$\displaystyle=\frac{Q}{8\Delta\Gamma_{2}(Q|b,b^{-1})\sinh^{2}(2\pi
bP)\sinh^{2}(2\pi b^{-1}P)}$ (B.38)

From [3] equation (2.14) we know that

$\rho(P,\bar{P})=\rho_{0}(P)\rho_{0}(\bar{P})=|4\sqrt{2}\sinh(2\pi
bP)\sinh(2\pi b^{-1}P)|^{2}$ (B.39)

so now we can simplify the product of two one-point functions

$\overline{\braket{O_{1}}_{T^{2}(\tau,\bar{\tau})}\braket{O_{1}}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}}\approx\frac{32Q^{2}}{\Delta^{2}\Gamma_{2}(Q|b,b^{-1})^{2}}\left|\int
dP\,\mathcal{F}_{1}^{g=1}(h_{p};\tau)\overline{\mathcal{F}}_{1}^{g=1}(h_{p};-\tau^{\prime})\right|^{2}$
(B.40)

Now we evaluate $\Gamma_{2}(Q|b,b^{-1})$ using the formula

$\Gamma_{N}(w|a_{1},\ldots,a_{N})=\Gamma_{N-1}(w|a_{1},\ldots,a_{N-1})\Gamma_{N}(w+a_{N}|a_{1},\ldots,a_{N})$
(B.41)

For $N=2$

$\Gamma_{2}(w|a_{1},a_{2})=\Gamma_{1}(w|a_{1})\Gamma_{2}(w+a_{2}|a_{1},a_{2})$
(B.42)

Then

$\displaystyle\Gamma_{2}(\epsilon+b+\frac{1}{b})$
$\displaystyle=\frac{\Gamma_{2}(\epsilon+b|b,\frac{1}{b})}{\Gamma(\epsilon+b|b)}=\frac{\Gamma_{2}(\epsilon|b,\frac{1}{b})}{\Gamma(b|b)\Gamma(\epsilon|\frac{1}{b})}$
(B.43)

Using

$\Gamma(w|a)=\frac{a^{a^{-1}w-\frac{1}{2}}}{\sqrt{2\pi}}\Gamma(a^{-1}w)$
(B.44) $\displaystyle\Gamma(b|b)$
$\displaystyle=\sqrt{\frac{b}{2\pi}}\Gamma(1)=\sqrt{\frac{b}{2\pi}}$ (B.45)
$\displaystyle\Gamma(\epsilon|\frac{1}{b})$
$\displaystyle=\sqrt{\frac{b}{2\pi}}\Gamma(b\epsilon)$ (B.46)

Recall that as $w\rightarrow 0$

$\Gamma(w)=\frac{1}{w}+O(w^{0})\quad\quad\Gamma_{2}(w|b,b^{-1})=\frac{1}{w}+O(w^{0})$
(B.47)

so

$\Gamma_{2}(\epsilon+b+\frac{1}{b}|b,\frac{1}{b})=\frac{\frac{1}{\epsilon}}{\frac{b}{2\pi}\frac{1}{b\epsilon}}=2\pi$
(B.48)

Therefore,

$\overline{\braket{O_{1}}_{T^{2}(\tau,\bar{\tau})}\braket{O_{1}}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}}\approx\frac{8Q^{2}}{\Delta^{2}\pi^{2}}\left|\int
dP\,\mathcal{F}_{\mathbb{1}}^{g=1}(h_{p};\tau)\overline{\mathcal{F}}_{\mathbb{1}}^{g=1}(h_{p};-\tau^{\prime})\right|^{2}$
(B.49)

###  B.4 Saddle-point approximation in 3d

Saddle-point approximation was proven to work in an analogous situation in 2d
(for details see Appendix A.3). In 2d, we calculate the averaged product of
two two-point functions and take the saddle-point approximation. We compare
this result with contribution to two-point function from disk with crosscap
(see Appendix A.3.1) and results are the same. We then assume here that
saddle-point approximation can be generalized to 3d even though we do not have
a direct exact calculation in this case.

We already know from calculations in previous sections

$\overline{\braket{O_{1}(v,\bar{v})O_{2}(0)}_{T^{2}(\tau,\bar{\tau})}\braket{O_{1}(v^{\prime},\bar{v}^{\prime})O_{2}(0)}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}}=\\\
2\big{|}\int
dh_{2}dh_{3}\rho_{0}(h_{2})\rho_{0}(h_{3})C_{0}(h_{1},h_{2},h_{3})C_{0}(h_{1},h_{2},h_{3})\mathcal{F}_{11}^{g=1}(h_{2},h_{3};\tau,v)\mathcal{F}_{11}^{g=1}(h_{2},h_{3};\tau^{\prime},v^{\prime})\big{|}^{2}$
(B.50)

where

$\tau=i\beta\quad\tau^{\prime}=i\beta\quad v=i\frac{\beta}{2}-t\quad
v^{\prime}=i\frac{\beta}{2}+t$ (B.51)

Recall that

$\rho_{0}(h)\approx\exp\left(2\pi\sqrt{\frac{c}{6}(h-\frac{c}{24})}\right)$
(B.52)

and recall that the functions $C_{0}$ are given by

$C_{0}(P_{1},P_{2},P_{3})=\frac{\Gamma_{b}(2Q)}{\sqrt{2}\Gamma_{b}(Q)^{3}}\frac{\prod_{\pm_{1,2,3}}\Gamma_{b}(\frac{Q}{2}\pm_{1}iP_{1}\pm_{2}iP_{2}\pm_{3}iP_{3})}{\prod_{k=1}^{3}\Gamma_{b}(Q+2iP_{k})\Gamma_{b}(Q-2iP_{k})}$
(B.53)

so we can again define

$P_{+}=P_{2}+P_{3}\quad\quad P_{-}=P_{2}-P_{3}$ (B.54)

then the saddle points are at $P_{+}=0$ and $P_{-}=0$, i.e.
$h_{2}=h_{3}=h_{p}$. In particular, we should note that for the particular
values of $\tau,\tau^{\prime},v,v^{\prime}$ that we are considering

$\mathcal{F}_{11}^{g=1}(h_{p},h_{p};\tau,v)=\mathcal{F}_{11}^{g=1}(h_{p},h_{p};\tau^{\prime},v^{\prime})$
(B.55)

Therefore, we have

$\sqrt{\overline{\braket{O_{1}(v,\bar{v})O_{2}(0)}_{T^{2}(\tau,\bar{\tau})}\braket{O_{1}(v^{\prime},\bar{v}^{\prime})O_{2}(0)}_{T^{2}(\tau^{\prime},\bar{\tau}^{\prime})}}}\approx|\int
dh_{p}\rho_{0}(h_{p})C_{0}(h_{1},h_{p},h_{p})\mathcal{F}_{11}^{g=1}(h_{p},h_{p};\tau,v)|^{2}$
(B.56)

## References

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