††institutetext: Department of Physics, Boston University
590 Commonwealth Avenue, Boston, MA 02215, USA

# CFT2 in the Bulk

Cameron V. Cogburn<EMAIL_ADDRESS>
###### Abstract

We study non-Gaussian bulk 2d CFTs in AdS2 using boundary CFT techniques and
recent results in JT/Schwarzian gravity. We highlight the constraints on the
operator content of a theory imposed by the boundary conditions by examining
the relation between correlator coefficients and OPE coefficients in the
presence of a boundary. We then calculate bulk and boundary correlators for
various boundary conditions. Schwarzian techniques are used to calculate
gravitational correlators perturbatively in $1/c$.

##  1 Introduction

Studying strongly coupled QFTs is difficult as perturbative calculational
tools break down in the absence of a small expansion parameter. By “strongly
coupled” we mean the correlators of generic operators are non-Gaussian, i.e.,
they do not obey an expansion in a $``1/N"$ small parameter.

AdS is a convenient background in which to study strongly coupled systems as
it is a maximally symmetric space that inherently acts as a bulk IR regulator
Callan:1989em . One way to study strongly coupled QFTs in AdS non-
perturbatively is to make use of a numerical scheme such as a lattice
discretization Brower:2019kyh  or a Hamiltonian truncation Hogervorst:2021spa
. Another way is to use the property that a CFT in AdSd is related to its
boundary CFT (BCFT) in the $d$-dimensional upper half-plane
$\mathcal{H}^{d}\equiv\\{\vec{x}+iy\,|\,y>0;\vec{x}\in\mathbbm{R}^{d-1}\,,y\in\mathbbm{R}\\}$
via a Weyl transformation:

$\displaystyle
ds^{2}_{\text{AdS}_{d}}=y^{-2}(dy^{2}+d\vec{x}^{2})=\Omega^{2}ds^{2}_{\mathcal{H}^{d}}\;,$
(1.1)

with $\Omega=y^{-1}$. This has been utilized several times before Doyon:2004fv
; Aharony:2010ay ; Carmi:2018qzm ; Paulos:2016fap ; Herzog:2019bom ;
Aharony:2015hix ; Giombi:2020rmc ; therefore we can use tools from the
sizeable literature on BCFTs to study rational CFTs in AdS.

Although the QFT is strongly coupled, gravity can be weakly coupled by
inserting factors of the stress tensor into correlators, e.g.,

$\displaystyle\langle\phi\dots\phi T\dots T\rangle\;,$ (1.2)

where $\phi$ is a scalar and $T$ a stress tensor, the latter of which
gravitons are coupled to. For a general non-Gaussian CFT we do not know these;
similarly, in dimensions higher than two the $TT$ OPE structure is not unique
Osborn:1993cr . However, in CFT2, adding factors of $T$ is controlled by
conformal invariance, making such calculations tractable. Moreover, in 2d it
is possible to use the dual 1d Schwarzian theory to calculate the
gravitational correlators of a boundary quantity $\tilde{{\cal O}}$,

$\displaystyle\langle\tilde{{\cal O}}\rangle_{t}=\int[Dt]\tilde{{\cal
O}}e^{-S[t]}\;,\quad S[t]\propto\int dt\,\\{t,u\\}\;,$ (1.3)

that would otherwise be calculated by inserting multiples of the stress tensor
into a bulk correlator before taking it to the boundary.

The purpose of this work is to show that in AdS2 all the machinery exists to
do perturbative gravitational calculations, and some types of non-perturbative
ones, on strongly coupled conformal theories weakly coupled to gravity, using
the unitary minimal models as an explicit example. Section 2 provides an
overview of CFT2 in AdS2. We use the Ising model to explicitly show how
boundary correlators can be obtained solely from the bulk-boundary OPE, and
how boundary conditions constrain the bulk and boundary data. In Section 3, by
mapping to the infinite strip we can study the theory in the presence of two
boundaries, such as in RS1 models. In Section 4 we add gravity and calculate
the gravitational correction for a generic minimal model $\tilde{\phi}_{2,1}$
four-point function using Schwarzian techniques. Section 5 discusses the case
of higher dimensions and how to calculate boundary gravitational corrections
therein. Finally, in Section 6 we conclude with a discussion.

##  2 CFT2 in AdS2

We begin with a general discussion of putting a 2d CFT in AdS2. Since AdS has
a conformal boundary we need to understand how boundary conditions affect the
bulk theory. There is an abundance of literature on CFTs in the presence of a
boundary in flat space, from which we review the salient points and apply them
to AdS. The main idea is that we can obtain boundary correlators solely from
the bulk-boundary OPE, without needing to solve for the bulk correlators and
then taking them to the boundary. We also review the constraints imposed from
boundary conditions on the bulk correlators and the dual CFT1 boundary
operators. Boundary correlators are the main ingredient needed for the
gravitational calculations of Section 4.

###  2.1 Overview: boundary conditions

Many physical systems have some sort of boundary, so the study of conformal
theories in the presence of a boundary has been well developed. The upper
half-plane (UHP) is the prototypical space in which to study the consequences
of demanding conformal invariance for a two-dimensional system in the presence
of a boundary. A key point in the analysis is to use the “doubling trick”, in
which a mirror image of the system is introduced in the lower half-plane and
the now doubled system is analyzed in the full plane, $\mathbbm{R}^{2}$. Doing
this in AdS2 only introduces a Weyl factor. Upon taking the bulk correlators
to the boundary we encounter a small puzzle, the clarification of which is a
theme of Section 2.

We start by putting a CFT2 in the UHP with a flat metric and a conformal
boundary condition at the boundary $y=0$ by using the aforementioned doubling
trick CARDY1984514 :

$\displaystyle\langle{\cal O}_{1}(z_{1},\bar{z}_{1})\dots\cal
O_{n}(z_{n},\bar{z}_{n})\rangle_{\rm UHP}=\langle{\cal O}_{1}(z_{1}){\cal
O}_{1}(z^{*}_{1})\dots\cal O_{n}(z_{n}){\cal O}_{n}(z_{n}^{*})\rangle\;.$
(2.1)

Here ${\cal O}_{i}(z)$ is the holomorphic half of the operator ${\cal
O}_{i}(z,\bar{z})$ and $z=x+iy$. This construction explicitly preserves all of
the symmetries (conformal transformations that leave the $x$-axis invariant as
well as any holomorphic equations of motion for the ${\cal O}$s). We will see
later it also makes it convenient to set different boundary conditions by
making different choices about what terms in the OPE of ${\cal
O}_{i}(z_{i}){\cal O}_{i}(z^{*}_{i})$ are kept.

To map this correlator to AdS2, perform a Weyl transformation,

$\displaystyle g_{\mu\nu}(x,y)\rightarrow\Omega^{2}(x,y)g_{\mu\nu}(x,y)\;,$
(2.2)

with $\Omega(x,y)=y^{-1}$. Under this transformation the CFT operators
transform like $\Omega^{-\Delta}$, so we have

$G^{(n)}_{{\cal O}}(x_{i},y_{i})\equiv\langle{\cal
O}_{1}(x_{1},y_{1})\dots\cal O_{n}(x_{n},y_{n})\rangle_{{\rm
AdS}_{2}}=y_{1}^{\Delta_{1}}\dots y_{n}^{\Delta_{n}}\langle{\cal
O}_{1}(z_{1}){\cal O}_{1}(z^{*}_{1})\dots\cal O_{n}(z_{n}){\cal
O}_{n}(z_{n}^{*})\rangle\;.$ (2.3)

To obtain the boundary correlator, take the limit as the bulk operators
approach the boundary:

$\displaystyle G^{(n)}_{{\cal O},\partial}(x_{i})\equiv\langle\tilde{{\cal
O}}_{1}(x_{1})\dots\tilde{{\cal
O}}_{n}(x_{n})\rangle_{\mathbb{R}}=\lim_{y_{i}\rightarrow
0}y_{1}^{-\tilde{\Delta}_{1}}\dots y_{n}^{-\tilde{\Delta}_{n}}G^{(n)}_{{\cal
O}}(x_{i},y_{i})\;.$ (2.4)

Here, $\tilde{\Delta}_{i}$ are the dimensions of the dual CFT1 operators. By
“dual CFT1 operator” we mean the leading operator in the boundary limit of the
bulk operator coming from the bulk-boundary operator expansion

$\displaystyle{\cal
O}_{i}(x,y)=\sum_{j}(2y)^{\Delta_{j}-\Delta_{i}}C^{a}_{ij}\tilde{{\cal
O}}_{j}(x)+\cdots$ (2.5)

The $C^{a}_{ij}$ are the boundary OPE coefficients and the index $a$ signifies
that the precise operators in the expansion are boundary condition-dependent.
At the same time, because of the doubling trick, we can compare the bulk-
boundary OPE (2.5) with the boundary limit of the ${\cal O}_{i}\times{\cal
O}_{i}$ bulk OPE. Taking the bulk OPE

$\displaystyle{\cal O}_{i}(z){\cal O}_{i}(w)$ $\displaystyle=$
$\displaystyle\sum_{{\cal O}^{\prime}}\frac{c_{ii{\cal
O}^{\prime}}}{(z-w)^{2h_{i}-h_{{\cal O}^{\prime}}}}{\cal O}^{\prime}(w)$ (2.6)

in the boundary limit yields

$\displaystyle\lim_{y\rightarrow 0}{\cal O}_{i}(z){\cal
O}_{i}(z^{*})=\lim_{y\rightarrow 0}{\cal O}_{i}(x+iy){\cal
O}_{i}(x-iy)=\sum_{{\cal O}^{\prime}}c_{ii{\cal
O}^{\prime}}(2y)^{-\Delta_{i}+h_{{\cal O}^{\prime}}}{\cal
O}^{\prime}(x)+\dots$ (2.7)

We see that $\tilde{{\cal O}}$ is just the lowest-dimension operator ${\cal
O}^{\prime}$ in the ${\cal O}_{i}\times{\cal O}_{i}$ OPE, and the dimension
$\tilde{\Delta}$ of $\tilde{{\cal O}}$ is $h_{{\cal O}^{\prime}}$.

Usually the theory is defined so that the lowest-dimension operator in the
${\cal O}_{i}\times{\cal O}_{i}$ bulk OPE is the identity operator. However,
we may have or want boundary conditions that necessitate having a vanishing
identity contribution. By inspection of the above OPEs, we see that this means
setting the OPE coefficients for all operators with $h_{{\cal
O}^{\prime}}<2h_{i}$ to zero, seemingly eliminating the identity as the dual
CFT1 operator.

We can identify when this is the case by looking at the bulk-boundary OPE
(2.5). The identity is not the dual CFT1 operator when the bulk-boundary
coupling coefficient to the identity $A^{a}_{i}\equiv C^{a}_{i\mathbbm{1}}$
vanishes. This can easily be read off from the modular matrix, as shown in the
following subsection. The point is that we can obtain the boundary correlators
solely from the bulk-boundary OPE, without needing to solve for bulk
correlators and then taking them to the boundary.

As an example, let us compute the boundary $2n$-point functions from bulk
$\langle\sigma\sigma\dots\sigma\rangle$ correlators of the Ising model. The
$\sigma\times\sigma$ OPE is

$\displaystyle[\sigma]\times[\sigma]=[1]+[\epsilon]\;.$ (2.8)

where $[\dots]$ indicates the full Virasoro irrep of operators. Pulling off
the identity operator in order to impose Dirichlet boundary conditions, we
find that the boundary correlators are

$\displaystyle
G^{(2n)}_{\sigma,\partial}(x_{i})=\langle\epsilon(x_{1})\dots\epsilon(x_{2n})\rangle=\bigg{|}\text{Pf}\,\bigg{[}\frac{1}{x_{i}-x_{j}}\bigg{]}_{1\leq
i,j\leq 2n}\bigg{|}^{2}\;,$ (2.9)

where Pf is the Pfaffian. For example, with $n=2$ we have

$\displaystyle
G^{(4)}_{\sigma,\partial}(x_{i})=\langle\epsilon(x_{1})\epsilon(x_{2})\epsilon(x_{3})\epsilon(x_{4})\rangle=\frac{1}{x_{12}x_{34}}+\frac{1}{x_{13}x_{24}}+\frac{1}{x_{14}x_{23}}\;.$
(2.10)

Although the Ising model is particularly simple, the above logic applies to
any boundary correlator from a bulk 2d CFT.

Since the bulk-boundary identity coupling $A^{a}_{i}$ is dependent on the
boundary condition $a$, so is the dual CFT1 operator. Using Cardy boundary
states we can understand precisely what boundary conditions correlate to what
bulk-boundary coefficients, and hence to what dual CFT1 operators.

###  2.2 Boundary states

As we saw above, the bulk-boundary OPE coefficients are dependent on the
boundary condition, that is the physical boundary state. Since Cardy
classified the allowed physical boundary states they are also known as Cardy
boundary states.

A Cardy boundary state $|\tilde{l}\rangle$ can be expanded in the basis of
allowed states (Ishibashi states) $|j\rrangle$ corresponding to the bulk
primary fields as Cardy:1989ir

$\displaystyle|\tilde{l}\rangle=\sum_{j}\frac{S_{ij}}{\sqrt{S_{0j}}}|j\rrangle\;,$
(2.11)

where $S$ is the modular matrix.111A more detailed discussion about the
modular matrix, Cardy boundary states, and Ishibashi states can be found in
Appendix B. The matrix $S$ also encodes the coupling between bulk fields and
the identity boundary operator through CARDY1991274

$\displaystyle A^{a}_{j}\equiv C^{a}_{j\mathbbm{1}}=\frac{\llangle
j|a\rangle}{\llangle
0|a\rangle}=\frac{S_{ja}}{S_{0a}}\left(\frac{S_{00}}{S_{j0}}\right)^{1/2}\;,$
(2.12)

where $j$ is an Ishibashi state and $a$ is a Cardy boundary state defined in
(2.11). This means that when $C^{a}_{j\mathbbm{1}}=0$ the identity is not the
leading operator in the expansion of the associated bulk operator $\phi_{j}$
in terms of its dual boundary operators in the short-distance expansion (2.5).
For $C^{a}_{j\mathbbm{1}}=0$ we need the matrix element $S_{ja}=0$, which can
be calculated from (B.1). Since Dirichlet (free) boundary conditions
correspond to requiring there being no identity on the boundary, we can
identify the boundary conditions corresponding to the free states by
identifying the vanishing $S$-matrix elements.

###  2.3 Boundary data (dual CFT1 operators)

Because the boundary conditions determine the bulk-boundary coefficients
$C^{a}_{ij}$ they also specify the dual CFT1 operators. For the Ising case,
expanding $\sigma$ gives

$\displaystyle\sigma(z)$
$\displaystyle=(2y)^{\tilde{\Delta}_{1}-\Delta_{\sigma}}C^{a}_{\sigma\mathbbm{1}}\tilde{\mathbbm{1}}+(2y)^{\tilde{\Delta}_{\epsilon}-\Delta_{\sigma}}C^{a}_{\sigma\epsilon}\tilde{\epsilon}(x)+\cdots$
$\displaystyle=\frac{1}{(2y)^{1/8}}C^{a}_{\sigma\mathbbm{1}}\tilde{\mathbbm{1}}+(2y)^{3/8}C^{a}_{\sigma\epsilon}\tilde{\epsilon}(x)+\cdots$
(2.13)

Doing the same for $\epsilon$ gives

$\displaystyle\epsilon(z)$
$\displaystyle=(2y)^{\tilde{\Delta}_{1}-\Delta_{\epsilon}}C^{a}_{\epsilon\mathbbm{1}}\tilde{\mathbbm{1}}+(2y)^{\tilde{\Delta}_{\sigma}-\Delta_{\epsilon}}C^{a}_{\epsilon\sigma}\tilde{\sigma}(x)+\cdots$
$\displaystyle=\frac{1}{(2y)^{1/2}}C^{a}_{\epsilon\mathbbm{1}}\tilde{\mathbbm{1}}+\frac{1}{(2y)^{3/8}}C^{a}_{\epsilon\sigma}\tilde{\sigma}(x)+\cdots$
(2.14)

The bulk-boundary structure constants for the different boundary conditions in
the Ising model are calculated in (C.4) - (C.6). Applying the results here,
for the fixed $(\pm)$ case we get:

$\displaystyle\sigma(z)=\frac{\pm
2^{1/4}}{(2y)^{1/8}}\tilde{\mathbbm{1}}+\cdots\;,\quad\epsilon(z)=\frac{1}{(2y)^{1/2}}\tilde{\mathbbm{1}}+\cdots\;.$
(2.15)

Whereas for the free $(f)$ case we get:

$\displaystyle\sigma(z)=\frac{1}{(2y)^{3/8}}\left(\sqrt{2}\alpha^{ff}_{\tilde{\epsilon}}\right)^{-1/2}\tilde{\epsilon}(x)+\cdots\;,\quad\epsilon(z)=-\frac{1}{(2y)^{1/2}}\tilde{\mathbbm{1}}+\cdots\;.$
(2.16)

This shows that for $(\pm)$ boundary conditions, the dual CFT1 operator for
both $\sigma$ and $\epsilon$ is simply the boundary identity
$\tilde{\mathbbm{1}}$. For $(f)$ boundary conditions, the dual CFT1 operator
for $\epsilon$ is again $\tilde{\mathbbm{1}}$ but for $\sigma$ it is the dual
energy density operator $\tilde{\epsilon}$.

We saw that for the Ising model the form of the dual CFT1 operators were
determined by the bulk-boundary structure constants $C^{a}_{ji}$, which in
turn were dependent on the modular matrix through (2.12). The generalization
to other minimal models is straightforward: given a bulk minimal model and
boundary conditions we can apply (2.12) to determine the bulk-boundary
structure constants, and hence the allowed dual operator spectrum.

###  2.4 Bulk data from boundary conditions

We can see how the boundary conditions also set the form of the bulk
correlators. Let us restrict the bulk CFT2 theory in AdS2 to the unitary
minimal models. Our first goal is to calculate a bulk two-point function
$G^{(2)}_{\phi_{1}\phi_{2}}(z_{1},\bar{z}_{1},z_{2},\bar{z}_{2})=\langle\phi_{h_{1}}(z_{1},\bar{z}_{1})\phi_{h_{2}}(z_{2},\bar{z}_{2})\rangle$
in this geometry. Here, $\phi_{h_{i}}$ is a primary scalar field with
dimension $h_{i}$. Because of the doubling trick (2.1), the two-point function
will be a linear combination of the same conformal blocks as the associated
four-point function in the full space produced by the doubling:

$\displaystyle
G^{(4)}_{\phi_{1}\phi_{2}\phi_{1}\phi_{2}}(z_{1},z_{2},z_{3},z_{4})=\langle\phi_{h_{1}}(z_{1})\phi_{h_{2}}(z_{2})\phi_{h_{1}}(z_{3})\phi_{h_{2}}(z_{4})\rangle\;,$
(2.17)

with $z_{3}=z^{*}_{1}$ and $z_{4}=z^{*}_{2}$. The process of projecting out
null states to obtain correlator differential equations is famously well
known, as solving the appropriate differential equation amounts to a complete
determination of the correlators in a minimal theory DiFrancesco:639405 .

As a specific example, let us study the Ising spin two-point function
$\langle\sigma\sigma\rangle$ in AdS2 by putting it in the Poincaré UHP. From
(2.3) this is equal to a Weyl transformed full-plane four-point function222We
have intentionally absorbed a factor of $\sqrt{-1}$ into the second term so
that $c_{2}^{\sigma\sigma}$ is real. This is because the cross-ratio $x$ can
be written as  $\displaystyle
x=\frac{z_{12}z_{34}}{z_{13}z_{24}}=-\frac{\rho^{2}+(y_{1}-y_{2})^{2}}{4y_{1}y_{2}}\;,$
(2.18) where $\rho\equiv x_{2}-x_{1}$, meaning the physical region is given by
$x<0$. We have also picked a convenient overall constant $2^{-1/4}$ to keep
the relation between correlator and OPE coefficients simple.

$\displaystyle
G^{(2)}_{{}^{\sigma}}=\frac{1}{2^{1/4}(y_{1}y_{2})^{1/8}}\frac{1}{[x(x-1)]^{1/8}}\left(c^{\sigma\sigma}_{1}\frac{\sqrt{\sqrt{1-x}+1}}{\sqrt{2}}+c^{\sigma\sigma}_{2}\sqrt{2}\sqrt{\sqrt{1-x}-1}\right)\;.$
(2.19)

The coefficients $c^{\sigma\sigma}_{1}$, $c^{\sigma\sigma}_{2}$ are determined
from the boundary conditions, and could be found at this point from the
asymptotic behavior of the correlator near the real-axis boundary.

At the same time, the bulk OPE gives the form

$\displaystyle
G^{(2)}_{{}^{\sigma}}=C_{\sigma\sigma\mathbbm{1}}|z_{12}|^{-1/4}+C_{\sigma\sigma\epsilon}A_{\epsilon}|z_{12}|^{1/4}\;,$
(2.20)

with the bulk coefficients given by $C_{\sigma\sigma\mathbbm{1}}=1$,
$A_{\epsilon}=1$, and $C_{\sigma\sigma\epsilon}=\pm\frac{1}{2}$, the sign
being determined by the boundary condition.333 The “extraordinary transition”
gives the plus sign whereas the “ordinary transition” gives the minus sign.
Taking the limits $x\rightarrow 0$ and $x\rightarrow-\infty$ in (2.19) and
equating with (2.20) sets

$\displaystyle c_{1}^{\sigma\sigma}=C_{\sigma\sigma\mathbbm{1}}=1\;,\quad\quad
c_{2}^{\sigma\sigma}=C_{\sigma\sigma\epsilon}A_{\epsilon}=\pm\frac{1}{2}\;.$
(2.21)

What we have reviewed is that the physical boundary conditions change the OPE
coefficients and the conformal block coefficients; or equivalently, an
appropriate change in the OPE coefficients induces a change in the boundary
conditions.

##  3 Two Boundaries: The RS1 Setup

Suppose we are interested in analyzing the case of a CFT2 in the presence of
two boundaries in AdS2. Such a situation arises, for example, if we have a
Randall-Sundrum 1 (RS1) model with two branes Randall:1999ee . In this model
there is a Planck brane at the conformal boundary $y=0$ in the UHP and a TeV
brane at a distance $y=L$. Multiple boundaries are also useful, for example,
in describing the RG flow between theories using a RG brane Gaiotto:2012np .

A convenient geometry to analyze AdS2 with two boundaries is the conformally
equivalent infinite strip with finite width $L$. The strip geometry is
obtained from the UHP through the conformal map

$\displaystyle w=\frac{L}{\pi}\ln z\;,$ (3.1)

where $w=u+iv$ and $z=x+iy$ are the respective holomorphic coordinates for the
strip and for the UHP. Note how the UHP positive real axis $(x>0)$ is mapped
to the lower edge of the strip $(v=0)$ and the negative real axis $(x<0)$ to
the upper edge $(v=L)$. Therefore, we can obtain a strip with different
boundary conditions on its two edges by looking at the UHP with differing
boundary conditions along its positive and negative real axis, mediated by a
boundary condition-changing operator at the origin. We again using the Ising
model to illustrate calculations.

###  3.1 Boundary correlators

The analysis for the bulk spin two-point function in the flat half-plane
metric is done in DiFrancesco:639405 . For correlators in AdS there is a Weyl
factor to go from the flat metric to the Poincaré one. Boundary correlators on
the strip are obtained by taking $v\rightarrow 0$.

First consider the case where the two boundaries have the same boundary
condition. A two-point function on the strip is given by

$\displaystyle\langle{\cal O}_{1}(w_{1},\bar{w}_{1}){\cal
O}_{2}(w_{2},\bar{w}_{2})\rangle_{\text{strip}}=\prod_{i=1}^{2}\left(\frac{dz_{i}}{dw_{i}}\right)^{h_{i}}\left(\frac{d\bar{z}_{i}}{d\bar{w}_{i}}\right)^{\bar{h}_{i}}\Omega^{-\Delta_{i}}_{i}\langle{\cal
O}_{1}(z_{1}){\cal O}_{2}(z_{2}){\cal O}_{1}(z^{*}_{1}){\cal
O}_{2}(z^{*}_{2})\rangle\;,$ (3.2)

where the doubling trick has been applied and the bulk four-point function
pre-factor

$\displaystyle\prod_{i=1}^{2}\left(\frac{dz_{i}}{dw_{i}}\right)^{h_{i}}\left(\frac{d\bar{z}_{i}}{d\bar{w}_{i}}\right)^{\bar{h}_{i}}\Omega^{-\Delta_{i}}_{i}=\bigg{[}\Big{(}\frac{\pi}{L}\Big{)}^{2}e^{\frac{\pi
u_{1}}{L}}v_{1}^{2}\bigg{]}^{h_{1}}\bigg{[}\Big{(}\frac{\pi}{L}\Big{)}^{2}e^{\frac{\pi
u_{2}}{L}}v_{2}^{2}\bigg{]}^{h_{2}}\;,$ (3.3)

comes from the Weyl transformation from the Poincaré UHP to the flat UHP and
then to the strip. The boundary correlator can be obtained by going to the
boundary via

$\displaystyle\langle{\cal O}_{1}(w_{1},\bar{w}_{1}){\cal
O}_{2}(w_{2},\bar{w}_{2})\rangle_{\text{strip},\,\partial}^{(a)}=\lim_{v_{i}\rightarrow
0}v^{-\Delta_{\tilde{{\cal O}}_{1}^{(a)}}}_{1}v^{-\Delta_{\tilde{{\cal
O}}_{2}^{(a)}}}_{2}\langle{\cal O}_{1}(w_{1},\bar{w}_{1}){\cal
O}_{2}(w_{2},\bar{w}_{2})\rangle_{\text{strip}}\;,$ (3.4)

where the superscript notation $(a)$ indicates that the dual CFT1 operator is
boundary condition-dependent.

Let us once again use the Ising spin correlator as an example. For free $(f)$
boundary conditions (ordinary case), the dual boundary operator from (2.16) is
$\tilde{\epsilon}$ with scaling dimension $\Delta_{\tilde{\epsilon}}=1/2$.
This gives for the strip boundary spin-spin correlator

$\displaystyle\langle\sigma(w_{1},\bar{w}_{1})\sigma(w_{2},\bar{w}_{2})\rangle_{\text{strip},\,\partial}^{(ff)}=\frac{\pi}{2^{1/4}L}\operatorname{csch}\frac{\pi
u}{2L}\;,$ (3.5)

where $u\equiv u_{1}-u_{2}$. For fixed $(\pm)$ boundary conditions
(extraordinary case) the dual CFT1 operator from (2.15) is the identity,
$\Delta_{\tilde{\mathbbm{1}}}=0$. Taking the correlator to the edge yields

$\displaystyle\lim_{v_{i}\rightarrow
0}v_{1}^{0}v_{2}^{0}\langle\sigma(w_{1},\bar{w}_{1})\sigma(w_{2},\bar{w}_{2})\rangle_{\text{strip}}^{(\pm\pm)}=$
$\displaystyle
2^{3/4}+\frac{\pi^{2}}{24\,2^{1/4}L^{2}}(v_{1}^{2}+v_{2}^{2})+\frac{7\pi^{4}}{3840\,2^{1/4}L^{4}}(v_{1}^{4}+v_{2}^{4})+v_{1}^{2}v_{2}^{2}\left(\frac{\pi^{2}(1+36\operatorname{csch}^{4}\frac{\pi
u}{2L})}{1152\times 2^{1/4}L^{4}}\right)$
$\displaystyle=C_{0}+v_{1}^{2}v_{2}^{2}C_{2}+\cdots$ (3.6)

As expected, the first term is the boundary operator $\tilde{\mathbbm{1}}$ and
has a constant coefficient $C_{0}=2^{3/4}$. The next lowest dimension operator
is the dimension-2 “stress tensor” with coefficient $C_{2}$.

We can repeat the preceding steps to calculate the other Ising two-point
correlators on the strip, giving the results

$\displaystyle\langle\epsilon(w_{1},\bar{w}_{1})\epsilon(w_{2},\bar{w}_{2})\rangle_{\mathrm{strip},\,\partial}^{(ff)\,\mathrm{or}\,(\pm\pm)}$
$\displaystyle=\frac{1}{4}+\frac{\pi^{2}}{24L^{2}}(v_{1}^{2}+v_{2}^{2})+\frac{7\pi^{4}}{1440L^{4}}(v_{1}^{4}+v_{2}^{4})+v_{1}^{2}v_{2}^{2}\left(\frac{\pi^{4}(1+36\operatorname{csch}^{4}\frac{\pi
u}{2L})}{144\times L^{4}}\right)$
$\displaystyle=D_{0}+v_{1}^{2}v_{2}^{2}D_{2}+\cdots$ (3.7)

and

$\displaystyle\langle\epsilon(w_{1},\bar{w}_{1})\sigma(w_{2},\bar{w}_{2})\rangle_{\text{strip}}^{(\pm\pm)}$
$\displaystyle=\left(\frac{\pi}{128\,L}\right)^{1/8}+\cdots+v_{1}^{2}v_{2}^{2}\left(\frac{\pi^{33/8}(1+36\operatorname{csch}^{4}\frac{\pi
u}{2L})}{288\times 2^{7/8}L^{33/8}}\right)$
$\displaystyle=E_{0}+v_{1}^{2}v_{2}^{2}E_{2}+\cdots$ (3.8)

Similar to before, the first term is the boundary identity operator with
constant coefficient and the next term a dimension-2 operator.

###  3.2 Energy spectrum

Through the boundary correlators we can write down the spectrum of the
boundary theory. This is particularly useful if our setup is a RS1 model where
we would like to know the spectrum of particles on the boundary. The boundary
spectrum can easily be read off by expanding the boundary correlator

$\displaystyle\langle\tilde{{\cal O}}(u_{1})\tilde{{\cal
O}}(u_{2})\rangle=\sum_{n}|\langle 0|{\cal O}|n\rangle|^{2}e^{-\Delta
E_{n}u}\;,$ (3.9)

where $\Delta E_{n}=E_{n}-E_{0}$ gives the energy of the particle above the
ground state and $|c_{n}|^{2}=|\langle 0|{\cal O}|n\rangle|^{2}$ is the
expansion coefficient. Doing this for the Ising spin-spin correlator in the
ordinary transition gives

$\displaystyle\langle\sigma\sigma\rangle_{\text{strip}}^{\mathrm{(ff)}}=\frac{\pi}{2^{1/4}L}\operatorname{csch}\frac{\pi
u}{2L}$ $\displaystyle=\frac{\pi}{2^{1/4}L}\frac{2}{e^{\frac{\pi
u}{2L}}(1+e^{-\frac{\pi u}{L}})}$
$\displaystyle=\frac{2^{3/4}\pi}{L}\left(e^{-\frac{\pi u}{2L}}+e^{-\frac{3\pi
u}{2L}}+e^{-\frac{5\pi u}{2L}}+\cdots\right)\;.$ (3.10)

For fixed $(\pm)$ boundary conditions we have to look at the term $C_{2}$ to
expand. After normalizing by the short-distance two-point function we get

$\displaystyle 2\times 2^{1/4}\times
C_{2}=\frac{\pi^{4}}{L^{4}}\left(\frac{1}{576}+e^{-\frac{2\pi
u}{L}}+4e^{-\frac{3\pi u}{L}}+10e^{-\frac{4\pi u}{L}}+\cdots\right)\;.$ (3.11)

For the energy-energy correlator, we do the same for the term $D_{2}$ to get

$\displaystyle D_{2}=\frac{\pi^{4}}{L^{4}}\left(\frac{1}{576}+e^{-\frac{2\pi
u}{L}}+4e^{-\frac{3\pi u}{L}}+10e^{-\frac{4\pi u}{L}}+\cdots\right)\;,$ (3.12)

the same as the extraordinary spin-spin case. Finally, for the energy-spin
correlator, looking at the term $E_{2}$ gives

$\displaystyle\left(\frac{L}{2\pi}\right)^{1/8}E_{2}=\frac{\pi^{4}}{L^{4}}\left(\frac{1}{576}+e^{-\frac{2\pi
u}{L}}+4e^{-\frac{3\pi u}{L}}+10e^{-\frac{4\pi u}{L}}+\cdots\right)\;.$ (3.13)

It is interesting to note that
$\langle\sigma\sigma\rangle^{(ff)}_{\mathrm{strip}}$ is the least suppressed
by the system size $L$, compared to
$\langle\sigma\sigma\rangle^{(\pm)}_{\mathrm{strip}}$ or the
$\langle\epsilon\epsilon\rangle_{\mathrm{strip}}$ and
$\langle\sigma\epsilon\rangle_{\mathrm{strip}}$. Also note that the amplitude
for the mixed correlator $\langle\sigma\epsilon\rangle$ is suppressed by an
extra factor of $L^{-1/8}$ compared to the fixed $\langle\sigma\sigma\rangle$
and $\langle\epsilon\epsilon\rangle$ ones.

####  3.2.1 Two-point functions with differing boundary conditions

Generically, the two boundary conditions can be different. To induce a change
of boundary conditions, a boundary condition-changing operator can be inserted
at the origin on the UHP Cardy:1989ir .

How to calculate Ising correlators with differing boundary conditions on the
infinite strip in flat space can be found in BURKHARDT1991653 . A key relation
is that the conformal Ward identity in the presence of mixed boundary
conditions $ab$ on the UHP links the $n$-point correlator in this mixed
boundary, half-plane geometry with the bulk $(2n+2)$-point function:

$\displaystyle\langle{\cal O}_{1}(z_{1},\bar{z}_{1})\dots\cal
O_{n}(z_{n},\bar{z}_{n})\rangle_{ab}=\lim_{x_{1}\rightarrow
0\atop{x_{2}\rightarrow-\infty}}(x_{1}-x_{2})^{2h_{ab}}\langle{\cal
O}_{1}(z_{1}){\cal O}_{1}(z^{*}_{1})\dots\cal O_{n}(z_{n}){\cal
O}_{n}(z^{*}_{n})\,{\cal O}_{ab}(x_{1}){\cal O}_{ab}(x_{2})\rangle\;,$ (3.14)

where the $x_{i}$ are real coordinates on the boundary, and the $z_{i}$ are
complex coordinates in the bulk. Therefore, the same sequence of manipulations
are done as in the case where the boundary conditions are the same, just now
we need to calculate six-point functions instead of four-point functions.
Again we use the Ising model as an explicit example.

The spin-spin boundary correlator between fixed boundary conditions $(+-)$,
where the boundary condition-changing operator is given by the boundary
operator $\tilde{\epsilon}$, is found to be

$\displaystyle\langle\sigma(1)\sigma(2)\rangle_{\mathrm{strip},\,\partial}^{(+-)}$
$\displaystyle=1-\frac{23\pi^{2}}{48\,L^{2}}(v_{1}^{2}+v_{2}^{2})+\frac{247\pi^{4}}{7680\,L^{4}}(v_{1}^{4}+v_{2}^{4})$
$\displaystyle+v_{1}^{2}v_{2}^{2}\frac{\pi^{4}}{18432\,L^{4}}\left((-429+188\cosh\frac{\pi
u}{L}+529\cosh\frac{2\pi u}{L})\operatorname{csch}^{4}\frac{\pi u}{2L}\right)$
$\displaystyle=F_{0}+v_{1}^{2}v_{2}^{2}F_{2}+\cdots$ (3.15)

The first term is the boundary identity operator and has a constant
coefficient $F_{0}=1$ indicating the normalization. The next lowest dimension
operator is again the dimension-2 stress tensor term. Continuing with the
other two-point functions give

$\displaystyle\langle\sigma(1)\sigma(2)\rangle_{\mathrm{strip},\,\partial}^{(+f)}=$
$\displaystyle
1-\frac{\pi^{2}}{24\,L^{2}}(v_{1}^{2}+v_{2}^{2})-\frac{\pi^{4}}{960\,L^{4}}(v_{1}^{4}+v_{2}^{4})$
$\displaystyle+v_{1}^{2}v_{2}^{2}\frac{\pi^{4}}{4608\,L^{4}}\left((3+68\cosh\frac{\pi
u}{L}+\cosh\frac{2\pi u}{L})\operatorname{csch}^{4}\frac{\pi u}{2L}\right)$
$\displaystyle=G_{0}+v_{1}^{2}v_{2}^{2}G_{2}+\cdots$ (3.16)
$\displaystyle\langle\epsilon(1)\epsilon(2)\rangle_{\mathrm{strip},\,\partial}^{(+-)}=$
$\displaystyle\frac{1}{4}-\frac{23\pi^{2}}{24\,L^{2}}(v_{1}^{2}+v_{2}^{2})+\frac{247\pi^{4}}{1440\,L^{4}}(v_{1}^{4}+v_{2}^{4})$
$\displaystyle+v_{1}^{2}v_{2}^{2}\frac{\pi^{4}}{1152\,L^{4}}\left((-429+188\cosh\frac{\pi
u}{L}+529\cosh\frac{2\pi u}{L})\operatorname{csch}^{4}\frac{\pi u}{2L}\right)$
$\displaystyle=H_{0}+v_{1}^{2}v_{2}^{2}H_{2}+\cdots$ (3.17)
$\displaystyle\langle\epsilon(1)\epsilon(2)\rangle_{\mathrm{strip},\,\partial}^{(+f)}=$
$\displaystyle\frac{1}{4}-\frac{\pi^{2}}{12\,L^{2}}(v_{1}^{2}+v_{2}^{2})+\frac{\pi^{4}}{180\,L^{4}}(v_{1}^{4}+v_{2}^{4})$
$\displaystyle+v_{1}^{2}v_{2}^{2}\frac{\pi^{4}}{288\,L^{4}}\left((3+68\cosh\frac{\pi
u}{L}+\cosh\frac{2\pi u}{L})\operatorname{csch}^{4}\frac{\pi u}{2L}\right)$
$\displaystyle=I_{0}+v_{1}^{2}v_{2}^{2}I_{2}+\cdots$ (3.18)
$\displaystyle\langle\sigma(1)\epsilon(2)\rangle_{\mathrm{strip},\,\partial}^{(+-)}$
$\displaystyle=\frac{\pi^{1/8}}{2\,L^{1/8}}\bigg{[}1-\frac{23\pi^{2}}{48\,L^{2}}(v_{1}^{2}+184v_{2}^{2})+\frac{247\pi^{4}}{7680\,L^{4}}(v_{1}^{4}+\frac{64}{3}v_{2}^{4})$
$\displaystyle+v_{1}^{2}v_{2}^{2}\frac{\pi^{4}}{2304\,L^{4}}\left((-429+188\cosh\frac{\pi
u}{L}+529\cosh\frac{2\pi u}{L})\operatorname{csch}^{4}\frac{\pi
u}{2L}\right)\bigg{]}$ $\displaystyle=J_{0}+v_{1}^{2}v_{2}^{2}J_{2}+\cdots$
(3.19)
$\displaystyle\langle\sigma(1)\epsilon(2)\rangle_{\mathrm{strip},\,\partial}^{(+f)}$
$\displaystyle=\frac{\pi^{1/8}}{L^{1/8}}\bigg{[}1-\frac{\pi^{2}}{24\,L^{2}}(v_{1}^{2}+8v_{2}^{2})+\frac{\pi^{4}}{960\,L^{4}}(v_{1}^{4}+\frac{64}{3}v_{2}^{4})$
$\displaystyle+v_{1}^{2}v_{2}^{2}\frac{\pi^{4}}{576\,L^{4}}\left((3+68\cosh\frac{\pi
u}{L}+\cosh\frac{2\pi u}{L})\operatorname{csch}^{4}\frac{\pi
u}{2L}\right)\bigg{]}$ $\displaystyle=K_{0}+v_{1}^{2}v_{2}^{2}K_{2}+\cdots$
(3.20)

By expanding the stress tensor term, we can find the energy spectrums as well:

$\displaystyle F_{2}=\frac{\pi^{4}}{L^{4}}\left(\frac{529}{2304}+e^{-\frac{\pi
u}{L}}+\frac{9}{4}e^{-\frac{2\pi u}{L}}+4e^{-\frac{3\pi u}{L}}+\cdots\right)$
(3.21) $\displaystyle
G_{2}=\frac{\pi^{4}}{L^{4}}\left(\frac{1}{576}+\frac{1}{8}e^{-\frac{\pi
u}{L}}+\frac{1}{2}e^{-\frac{2\pi u}{L}}+\frac{11}{8}e^{-\frac{3\pi
u}{L}}+\cdots\right)$ (3.22) $\displaystyle
H_{2}=\frac{\pi^{4}}{L^{4}}\left(\frac{529}{144}+16e^{-\frac{\pi
u}{L}}+36e^{-\frac{2\pi u}{L}}+64e^{-\frac{3\pi u}{L}}+\cdots\right)$ (3.23)
$\displaystyle I_{2}=\frac{\pi^{4}}{L^{4}}\left(\frac{1}{36}+2e^{-\frac{\pi
u}{L}}+8e^{-\frac{2\pi u}{L}}+22e^{-\frac{3\pi u}{L}}+\cdots\right)$ (3.24)
$\displaystyle J_{2}=\frac{\pi^{4}}{L^{4}}\left(\frac{529}{288}+8e^{-\frac{\pi
u}{L}}+18e^{-\frac{2\pi u}{L}}+32e^{-\frac{3\pi u}{L}}+\cdots\right)$ (3.25)
$\displaystyle K_{2}=\frac{\pi^{4}}{L^{4}}\left(\frac{1}{72}+e^{-\frac{\pi
u}{L}}+4e^{-\frac{2\pi u}{L}}+11e^{-\frac{3\pi u}{L}}+\cdots\right)$ (3.26)

Although the above calculations have been done explicitly with the Ising
model, generalization to other minimal models is straightforward. Given any
minimal mode we can deduce the dual CFT1 operators using the logic of Sec.
2.3. Knowing these, we can calculate the boundary correlators on the infinite
strip and read off the spectrum.

##  4 Gravitational Corrections

###  4.1 Gravitational coupling to a non-Gaussian theory

Now we weakly couple gravity to the strongly coupled CFT2 and calculate
gravitational corrections. For a Gaussian bulk theory, perturbation theory
allows gravitational contributions to be calculated from the Feynman rules
(cf. Fig. 1). However, in a strongly coupled theory, normal perturbative
techniques are not available.

Instead, gravitational contributions are found by inserting factors of the
stress tensor into matter correlation functions. This is because the graviton
$h_{\mu\nu}$ couples to the stress tensor, as seen by expanding the matter
action around the classical AdS2 solution,
$g_{\mu\nu}=g^{(c)}_{\mu\nu}+h_{\mu\nu}$, giving

$\displaystyle S_{M}[g]=S_{M}[g^{(c)}]+\int\frac{\delta S}{\delta
g_{\mu\nu}}h_{\mu\nu}+\frac{1}{2}\int h_{\alpha\beta}\frac{\delta^{2}S}{\delta
g_{\alpha\beta}\delta g_{\mu\nu}}h_{\mu\nu}+O(h^{3})\;,$ (4.1)

where the stress tensor $T^{\mu\nu}(x)=(2/\sqrt{g})\delta S_{M}/\delta
g_{\mu\nu}(x)$. Gravitational correlators are then found by computing the path
integral

$\displaystyle\langle{\cal O}\rangle_{\text{grav}}=\int[Dh]\,\langle{\cal
O}\rangle\,e^{-S_{M}[h]}\;,$ (4.2)

where $\langle{\cal O}\rangle$ is the matter correlator. Expanding the action
(4.1) and the exponential in (4.2) generates $1/c$ gravitational corrections.

For example, suppose we want to calculate the leading $1/c$ gravitational
correction to a boundary scalar two-point function
$\langle\tilde{\phi}\tilde{\phi}\rangle$ where $\tilde{\phi}$ is the CFT1 dual
to $\phi$. The correction comes from expanding (4.1) in the path integral,
resulting in a term

$\displaystyle\int d^{2}x\,d^{2}y\sqrt{g_{x}}\sqrt{g_{y}}\,\langle\phi\phi
T(x)T(y)\rangle\,H(x,y)\;,$ (4.3)

where $H(x,y)=\langle h(x)h(y)\rangle$ is the graviton propagator. Doing this
to completion requires calculating $\langle\phi\phi T(x)T(y)\rangle$, working
out the $hT$ coupling in the bulk as well as the form of $H(x,y)$, and then
integrating the above directly before taking the result to the boundary.444We
are being schematic because we will not calculate gravitational corrections
this way. A similar calculation is detailed in Anand:2017dav .

More generally, being able to compute gravitational corrections like (4.3)
requires knowing the $TT$ OPE. For a general non-Gaussian CFT we do not know
this as for $d>2$ the $TT$ OPE does not have a universal structure
Osborn:1993cr . Conformal invariance via the Ward identities Belavin:1984vu
shows how to insert factors of $T$ into $n$-point matter correlators:

$\displaystyle\langle T(\xi)T(x_{1})$ $\displaystyle\dots
T(x_{M})\phi_{1}(z_{1})\dots\phi_{N}(z_{N})\rangle$
$\displaystyle=\left\\{\sum_{i=1}^{N}\left[\frac{h_{i}}{(\xi-
z_{i})^{2}}+\frac{1}{\xi-z_{i}}\frac{\partial}{\partial
z_{i}}\right]+\sum_{j=1}^{M}\left[\frac{2}{(\xi-x_{j})^{2}}+\frac{1}{\xi-
x_{j}}\frac{\partial}{\partial x_{j}}\right]\right\\}$
$\displaystyle+\sum_{j=1}^{M}\left[\frac{c/2}{(\xi-x_{j})^{4}}\right]\langle
T(x_{1})\dots T(x_{j-1})T(x_{j+1})\dots
T(x_{M})\phi_{1}(z_{1})\dots\phi_{N}(z_{N})\rangle\;.$ (4.4)

Therefore the power of conformal invariance in two dimensions allows
contributions from weakly-coupled gravity to be calculated, even if the matter
sector is strongly coupled.

However we are not going to calculate gravitational corrections for
correlators in the 1d boundary theory this way; the same corrections can be
streamlined using the 1d Schwarzian theory, as we will see below.

$+$ $\quad\longrightarrow$ $+$

$\quad\quad\longrightarrow$

Figure 1: Top: In a weakly coupled theory, gravitational soft modes become
loop corrections that can be calculated using perturbative techniques such as
propagators and vertices. Bottom: In a strongly coupled theory perturbative
techniques break down. To calculate gravitational loop corrections we have to
integrate matter correlators with insertions of the stress tensor. This can
equivalently be done using Schwarzian techniques.

###  4.2 Schwarzian effective action

To calculate gravitational corrections using the Schwarzian theory we use
Jackiw-Teitelboim (JT) gravity Jackiw:1984je ; Teitelboim:1983ux  of the type
studied in Refs. Almheiri:2014cka ; Maldacena:2016upp ; Jensen:2016pah ;
Engelsoy:2016xyb . In Appendix E we review how the Schwarzian theory comes
from JT gravity and how to calculate correlators in the Schwarzian theory, but
it useful to summarize the salient ideas below.

JT gravity is a 2d dilaton gravity whose boundary description is given by a 1d
Schwarzian action,

$\displaystyle S[t]=-C\int
du\,\\{t,u\\}\;,\quad\quad\\{t,u\\}\equiv\frac{t^{\prime\prime\prime}}{t^{\prime}}-\frac{3}{2}\left(\frac{t^{\prime\prime}}{t^{\prime}}\right)^{2}\;,$
(4.5)

where $C$ is a bulk-valued coupling constant. We are using the notation that
the AdS2 metric is given by $ds^{2}_{\text{AdS}_{2}}=z^{-2}(dt^{2}+dz^{2})$
and $u$ is the proper boundary time. Hence, $t(u)$ is the dynamical field
variable and primes denote derivatives with respect to $u$. Because the
Schwarzian action is just the 1d description of JT gravity, gravitational
contributions are calculated via Schwarzian correlators,

$\displaystyle\langle\tilde{{\cal O}}\rangle_{t}=\int[dt]\,\langle\tilde{{\cal
O}}\rangle\,e^{-S[t]}\;,$ (4.6)

where $\langle\tilde{{\cal O}}\rangle$ is the matter boundary correlator.

The 1d Schwarzian theory is a theory of time reparametrizations whose action
has a global SL(2,$\mathbbm{R}$) invariance. The fields $t(u)$ are Nambu-
Goldstone bosons generated from the spontaneous breaking of
SL(2,$\mathbbm{R}$) symmetry by the AdS2 solution in the JT gravity action
(E.1) and represent the zero modes involving large diffeomorphisms 555Although
technically different, for curves (1-manifolds) reparametrizations and
diffeomorphisms are the same. As discussed in Appendix E, it is necessary to
explicitly break the SL(2,$\mathbbm{R}$) symmetry by fixing the boundary
length to be large but finite in order to lift these zero modes to non-zero
modes needed for a dynamic gravity theory.

By varying the Schwarzian action it can be shown that one of the solutions to
the field equations of motion is a constant Schwarzian derivative (cf. eq.
(E.7)). Indeed, for $t(u)=u$ the Schwarzian derivative vanishes, corresponding
to a saddle point in the path integral language. Considering small
reparametrizations, $t(u)\rightarrow t(u)=\tan\frac{u+\epsilon k(t)}{2}$, an
effective local fluctuation action can be derived (E.12),

$\displaystyle
S_{\epsilon}[t]=\frac{C}{2}\int_{0}^{2\pi}du\,\left[(k^{\prime\prime})^{2}-(k^{\prime})^{2}\right]\;,$
(4.7)

from which the fluctuation propagator (E.3) can be found.

In 1d reparametrizations and conformal transformations are the same. Therefore
to calculate the leading perturbative gravitational correction we simply apply
a conformal transformation to the matter correlator,

$\displaystyle\tilde{{\cal
O}}(u_{1},\dots,u_{n})=(t^{\prime}_{1})^{\Delta_{1}/2}\dots(t^{\prime}_{n})^{\Delta_{n}/2}\tilde{{\cal
O}}(t_{1}(u_{1}),\dots,t_{n}(u_{n}))\;,$ (4.8)

where $t^{\prime}_{i}\equiv dt_{i}(u_{i})/du_{i}$, and then use the effective
Schwarzian action (4.7) to calculate the gravitational contribution.

###  4.3 Non-Gaussian $\tilde{\phi}_{2,1}$ boundary four-point functions

Aside from the Ising model, minimal model $\tilde{\phi}_{2,1}$ four-point
functions are non-Gaussian. A proof of this statement can be found in Appendix
D. The Ising model has the factorizable spin boundary four-point function

$\displaystyle\langle\sigma\sigma\sigma\sigma\rangle_{\partial}=\langle\epsilon_{1}\epsilon_{2}\epsilon_{3}\epsilon_{4}\rangle=\frac{1}{t_{12}t_{34}}+\frac{1}{t_{13}t_{24}}+\frac{1}{t_{14}t_{23}}\;,$
(4.9)

a reflection of the fact that the Ising model contains the theory of a free
fermion. Factorized four-point functions are a trivial extension of the two-
point function and can be found in Maldacena:2016upp . Mixed four-point
functions of the type
$\langle\phi_{h_{1}}\phi_{h_{1}}\phi_{h_{2}}\phi_{h_{2}}\rangle$ also fall
into this category since taking them to the boundary produces a factorizable
four-point function,

$\displaystyle\langle\phi_{h_{1}}\phi_{h_{1}}\phi_{h_{2}}\phi_{h_{2}}\rangle_{\partial}=\langle\tilde{{\cal
O}}_{h_{1}}\tilde{{\cal O}}_{h_{1}}\rangle\langle\tilde{{\cal
O}}_{h_{2}}\tilde{{\cal
O}}_{h_{2}}\rangle=\frac{1}{t^{2\tilde{\Delta}_{\tilde{{\cal
O}}_{1}}}_{12}}\frac{1}{t^{2\tilde{\Delta}_{\tilde{{\cal O}}_{2}}}_{34}}\;,$
(4.10)

of the type discussed before.

However, a boundary unitary minimal model $\mathcal{M}(m)$ four-point function
$\langle\tilde{\phi}_{2,1}\tilde{\phi}_{2,1}\tilde{\phi}_{2,1}\tilde{\phi}_{2,1}\rangle$
is given by a linear combination of two hypergeometric functions, each
multiplied by factors of the cross-ratio,

$\displaystyle
G_{4}(x)=(x(1-x))^{-\Delta}\bigg{(}{}_{2}F_{1}\big{(}\frac{1-2\Delta}{3},-2\Delta,\frac{2-4\Delta}{3};x\big{)}+A_{m}\,x^{\frac{1+4\Delta}{3}}{}_{2}F_{1}\big{(}\frac{1-2\Delta}{3},\frac{2+2\Delta}{3},\frac{4+4\Delta}{3};x\big{)}\bigg{)}\;,$
(4.11)

where $A_{m}=e^{\frac{2\pi
i}{m+1}}\frac{\Gamma(\frac{2}{m+1})\Gamma(2-\frac{3}{m+1})}{2\Gamma(\frac{1}{m+1})\Gamma(\frac{2m}{m+1})}$,
$\tilde{\Delta}\equiv\tilde{\Delta}_{\tilde{\phi}_{2,1}}$, and
$x=\frac{t_{12}t_{34}}{t_{13}t_{24}}$. Under a conformal transformation this
becomes

$\displaystyle
G_{4}(t_{1},t_{2},t_{3},t_{4})\longrightarrow(f^{\prime}_{1}f^{\prime}_{2}f^{\prime}_{3}f^{\prime}_{4})^{\tilde{\Delta}}G_{4}(f_{1},f_{2},f_{3},f_{4})\;.$
(4.12)

The gravitational correlator is found by plugging $f(t)=\tan\frac{t+\epsilon
k(t)}{2}$ and (4.11) into (4.12), expanding to quadratic order in $\epsilon$,
and then contracting using the $\epsilon$-propagator. In practice, the easiest
way to perform the calculation is to break it up into pieces. Start with the
conformal transformation factor

$\displaystyle(f^{\prime}_{1}f^{\prime}_{2}f^{\prime}_{3}f^{\prime}_{4})^{\tilde{\Delta}}\rightarrow
F_{0}(1+\epsilon F_{1}+\epsilon^{2}F_{2})+O(\epsilon^{3})\;,$ (4.13)

where

$\displaystyle F_{0}$ $\displaystyle=1$ $\displaystyle F_{1}$
$\displaystyle=\tilde{\Delta}\sum_{i=1}^{4}(k_{i}\tan\frac{t_{i}}{2}+k^{\prime}_{i})$
$\displaystyle F_{2}$
$\displaystyle=\frac{\tilde{\Delta}}{4}\sum_{i=1}^{4}(k^{2}_{i}\sec^{2}\frac{t_{i}}{2}-2k^{\prime
2}_{i})+\frac{\tilde{\Delta}^{2}}{2}\big{(}\sum_{i=1}^{4}(k_{i}\tan\frac{t_{i}}{2}+k^{\prime}_{i})\big{)}^{2}\;.$
(4.14)

Next note that the transformation for the cross-ratio goes as

$\displaystyle x=\frac{f_{12}f_{34}}{f_{13}f_{24}}\rightarrow x_{0}+\epsilon
x_{1}+\epsilon^{2}x_{2}+O(\epsilon^{3})$ (4.15)

where

$\displaystyle x_{0}$
$\displaystyle=\frac{\sin\frac{t_{12}}{2}\sin\frac{t_{34}}{2}}{\sin\frac{t_{13}}{2}\sin\frac{t_{24}}{2}}$
$\displaystyle x_{1}$
$\displaystyle=\frac{1}{8}\csc^{2}\frac{t_{13}}{2}\csc^{2}\frac{t_{24}}{2}\left(k_{12}\sin
t_{34}+k_{31}\sin t_{24}+k_{14}\sin t_{23}+k_{23}\sin t_{14}+k_{42}\sin
t_{13}+k_{34}\sin t_{12}\right)$ $\displaystyle x_{2}$
$\displaystyle=\left[-\frac{k^{2}_{1}}{8}\csc^{4}\frac{t_{13}}{2}\csc\frac{t_{24}}{2}\sin\frac{t_{23}}{2}\sin\frac{t_{34}}{2}\sin
t_{13}+\left(\begin{matrix}1\leftrightarrow 2\\\ 3\leftrightarrow
4\end{matrix}\right)+\left(\begin{matrix}1\leftrightarrow 3\\\
2\leftrightarrow 4\end{matrix}\right)+\left(\begin{matrix}1\leftrightarrow
4\\\ 2\leftrightarrow 3\end{matrix}\right)\right]$
$\displaystyle+\left[\frac{k_{1}k_{2}}{4}\csc^{2}\frac{t_{13}}{2}\csc^{2}\frac{t_{24}}{2}\csc^{2}\frac{t_{34}}{2}+\left(\begin{matrix}1\leftrightarrow
3\\\ 2\leftrightarrow 4\end{matrix}\right)\right]$
$\displaystyle-\bigg{[}\bigg{(}\frac{k_{1}k_{4}}{4}\csc^{2}\frac{t_{13}}{2}\csc^{2}\frac{t_{23}}{2}\csc^{2}\frac{t_{24}}{2}$
$\displaystyle+\frac{k_{1}k_{3}}{8}\left(\cos\frac{t_{13}+t_{24}}{2}-2\cos\frac{t_{12}+t_{34}}{2}+\cos\frac{t_{13}+t_{42}}{2}\right)\csc^{3}\frac{t_{13}}{2}\csc\frac{t_{24}}{2}\bigg{)}+\left(\begin{matrix}1\leftrightarrow
2\\\ 3\leftrightarrow 4\end{matrix}\right)\bigg{]}\;.$ (4.16)

Expanding (4.11) results in

$\displaystyle G_{4}(x)=\mathcal{G}_{0}+\epsilon^{2}\mathcal{G}_{2}$ (4.17)

with

$\displaystyle\mathcal{G}_{0}$
$\displaystyle=F_{0}(x_{0}(1-x_{0}))^{-\tilde{\Delta}}\left[{}_{2}F_{1}\left(\frac{1-2\tilde{\Delta}}{3},-2\tilde{\Delta},\frac{2-4\tilde{\Delta}}{3};x_{0}\right)+A_{m}x^{\frac{1+4\tilde{\Delta}}{3}}_{0}{}_{2}F_{1}\left(\frac{1-2\tilde{\Delta}}{3},\frac{2+2\tilde{\Delta}}{3},\frac{4+4\tilde{\Delta}}{3};x_{0}\right)\right]$
$\displaystyle\mathcal{G}_{2}$
$\displaystyle=\frac{1}{6}(x_{0}(1-x_{0}))^{-\tilde{\Delta}}\bigg{[}\frac{1}{(x_{0}(x_{0}-1))^{2}}3(2F_{2}(x_{0}(x_{0}-1))^{2}+\tilde{\Delta}(-2F_{1}x_{0}x_{1}(1-3x_{0}+2x^{2}_{0})$
$\displaystyle-2F_{0}x_{0}x_{2}(1-3x_{0}+2x^{2}_{0})+F_{0}x^{2}_{1}(1+\tilde{\Delta}-2x_{0}(1+2\tilde{\Delta})+x^{2}_{0}(2+4\tilde{\Delta}))))$
$\displaystyle(A_{m}x^{\frac{1+4\tilde{\Delta}}{3}}_{0}{}_{2}F_{1}\left(\frac{1-2\tilde{\Delta}}{3},\frac{2+2\tilde{\Delta}}{3},\frac{4+4\tilde{\Delta}}{3};x_{0}\right)+{}_{2}F_{1}\left(\frac{1-2\tilde{\Delta}}{3},-2\tilde{\Delta},\frac{2-4\tilde{\Delta}}{3};x_{0}\right)$
$\displaystyle+\frac{x_{1}(F_{1}x_{0}(1-x_{0})+F_{0}x_{1}\tilde{\Delta}(2x_{0}-1))}{(x_{0}-1)x^{5/3}_{0}}\bigg{(}-2A_{m}x_{0}^{4\tilde{\Delta}/3}(1+4\tilde{\Delta}){}_{2}F_{1}\left(\frac{1-2\tilde{\Delta}}{3},\frac{2+2\tilde{\Delta}}{3},\frac{4+4\tilde{\Delta}}{3};x_{0}\right)$
$\displaystyle+6\tilde{\Delta}x_{0}^{2/3}{}_{2}F_{1}\left(1-2\tilde{\Delta},\frac{4-2\tilde{\Delta}}{3},\frac{5-4\tilde{\Delta}}{3};x_{0}\right)+A_{m}x_{0}^{\frac{1+4\tilde{\Delta}}{3}}(-1+2\tilde{\Delta}){}_{2}F_{1}\left(\frac{4-2\tilde{\Delta}}{3},\frac{5+2\tilde{\Delta}}{3},\frac{7+4\tilde{\Delta}}{3};x_{0}\right)\bigg{)}$
$\displaystyle+6F_{0}\bigg{(}-\tilde{\Delta}x_{2}\,{}_{2}F_{1}\left(1-2\tilde{\Delta},\frac{4-2\tilde{\Delta}}{3},\frac{5-4\tilde{\Delta}}{3};x_{0}\right)+\frac{x^{2}_{1}\tilde{\Delta}(\tilde{\Delta}-2)(1-2\tilde{\Delta})}{5-4\tilde{\Delta}}{}_{2}F_{1}\left(2-2\tilde{\Delta},\frac{7-2\tilde{\Delta}}{3},\frac{8-4\tilde{\Delta}}{3};x_{0}\right)$
$\displaystyle+\frac{A_{m}}{18}x^{\frac{-5+4\tilde{\Delta}}{3}}\bigg{(}2(1+4\tilde{\Delta})(3x_{0}x_{2}+x^{2}_{1}(-1+2\tilde{\Delta})){}_{2}F_{1}\left(\frac{1-2\tilde{\Delta}}{3},\frac{2+2\tilde{\Delta}}{3},\frac{4+4\tilde{\Delta}}{3};x_{0}\right)$
$\displaystyle+\frac{x^{2}_{0}x^{2}_{1}(\tilde{\Delta}-2)(-1+2\tilde{\Delta})(5+2\tilde{\Delta})}{7+4\tilde{\Delta}}{}_{2}F_{1}\left(\frac{7-2\tilde{\Delta}}{3},\frac{8+2\tilde{\Delta}}{3},\frac{10+4\tilde{\Delta}}{3};x_{0}\right)$
$\displaystyle+x_{0}(1-2\tilde{\Delta})(3x_{0}x_{2}+x^{2}_{1}(1+4\tilde{\Delta})){}_{2}F_{1}\left(\frac{4-2\tilde{\Delta}}{3},\frac{5+2\tilde{\Delta}}{3},\frac{4+4\tilde{\Delta}}{3};x_{0}\right)\bigg{)}\bigg{)}\bigg{]}$
(4.18)

after expanding the hypergeometrics and $x(1-x)$ prefactor. The final
expression is found by inserting (4.3) into the above and contracting using
the fluctuation propagators (E.3). The linear term vanishes due to
$\langle\epsilon\rangle$ terms, which can also serve as a calculational check.

The point of the above is to explicitly show that we can in principle fully
calculate to arbitrary order the gravitational contribution to unitary minimal
model $\tilde{\phi}_{2,1}$ four-point functions, which aside from the Ising
model are non-Gaussian. The generalization to higher $n$-point functions on
the boundary is straight forward, if tedious.

##  5 Higher Dimensions

So far we have focused on two dimensions in the bulk, however, we can extend
some of the previous discussion to the higher dimensional case of CFTd+1 in
AdSd+1 with $d>1$. We aim to outline how to calculate gravitational
corrections to the CFTd boundary correlators like for the $d=1$ case of Sec.
4.1.

First, let us review a bulk scalar two-point function $\langle\phi\phi\rangle$
in AdSd+1 without gravity. Similar to before, after a coordinate
transformation and Weyl transformation this can be mapped to the flat UHP
metric $ds^{2}=y^{-2}(d\vec{x}^{2}+dy^{2})$ with $x^{\mu}=(y,\vec{x})$ and
$\vec{x}\in\mathbbm{R}^{d}$. A two-point function in the presence of the $y=0$
boundary will obey a crossing symmetry equating the bulk and boundary
decompositions. A thorough analysis of this bootstrap for boundary CFTd
(BCFTd) was done in Liendo:2012hy . The statement of crossing symmetry for the
two-point scalar function in BCFTd is given by

$\displaystyle 1+\sum_{{\cal O}^{\prime}}C_{\phi\phi{\cal O}^{\prime}}A_{{\cal
O}^{\prime}}f_{\text{bulk}}(\Delta_{{\cal
O}^{\prime}};\xi)=\xi^{\Delta}\left(A_{\phi}^{2}+\sum_{\tilde{{\cal
O}}}C^{2}_{\phi\tilde{{\cal O}}}f_{\text{bdry}}(\Delta_{\tilde{{\cal
O}}};\xi)\right)\;,$ (5.1)

where $\xi=(x_{1}-x_{2})^{2}/(4y_{1}y_{2})$ and $f_{\text{bulk}}$,
$f_{\text{bdry}}$ are the respective bulk and boundary conformal blocks given
by

$\displaystyle f_{\text{bulk}}(\Delta;\xi)$
$\displaystyle=\xi^{\Delta/2}{}_{2}F_{1}\left(\frac{\Delta}{2},\frac{\Delta}{2},\Delta+1-\frac{d}{2};-\xi\right)$
$\displaystyle f_{\text{bdry}}(\Delta;\xi)$
$\displaystyle=\xi^{-\Delta}{}_{2}F_{1}\left(\Delta,\Delta+1-\frac{d}{2},2\Delta+2-d;-\frac{1}{\xi}\right)\;.$
(5.2)

Crossing (5.1) is pictorially encapsulated in Fig. 2.

$\displaystyle\sum_{i}$=$\displaystyle\sum_{j}$$i$$j$

Figure 2: Two-point function crossing symmetry in BCFTd.

The power of this bootstrap approach is that given the bulk data we can solve
for the boundary data, which is necessary for computing boundary gravitational
correlators.

Therefore, suppose we are given the bulk data and thus can calculate the
boundary data. Recall that an expansion in the AdSd+1 gravitational constant
$G$ corresponds to a $1/c$ expansion on the CFTd side. Therefore the leading
correction to the two-point gravitational correlator is given by the
gravitational loop diagram in the bottom-right of Fig. 1. This corresponds to
calculating the higher-dimensional counterpart to (4.3),

$\displaystyle\int
d^{d+1}x_{1}d^{d+1}x_{3}\sqrt{g_{x_{1}}}\sqrt{g_{x_{3}}}\langle
T_{\mu\nu}(x_{1})\tilde{\phi}(x_{2})T_{\rho\sigma}(x_{3})\tilde{\phi}(x_{4})\rangle
G^{\mu\nu,\rho\sigma}_{BB}(x_{1},x_{3})\;,$ (5.3)

where $G^{\mu\nu,\rho\sigma}_{BB}$ is the $(d+1)$-dimensional graviton bulk
propagator DHoker:1999bve . On the CFTd side, this correponds to our object of
interest, a $1/c$ expansion in the two-point boundary correlator

$\displaystyle\langle\tilde{\phi}(\vec{x})\tilde{\phi}(0)\rangle=\frac{1+\frac{\tilde{N}^{(1)}_{\phi}}{c}+\cdots}{\vec{x}^{2\tilde{\Delta}_{\phi}}}\left(1-\frac{2\tilde{\Delta}^{(1)}_{\phi}}{c}\ln\vec{x}-\frac{2\tilde{\Delta}^{(2)}_{\phi}}{c^{2}}\ln\vec{x}+\frac{2(\tilde{\Delta}^{(1)}_{\phi})^{2}}{c^{2}}\ln^{2}\vec{x}+\cdots\right)$
(5.4)

after expanding the scaling dimension and normalization in $1/c$

$\displaystyle\tilde{\Delta}=\sum_{n=0}^{\infty}\frac{\tilde{\Delta}^{(n)}_{\phi}}{c^{n}}\;,\quad\quad\tilde{N}_{\phi}=\sum_{n=0}^{\infty}\frac{\tilde{N}^{(n)}_{\phi}}{c^{n}}\;,$
(5.5)

due to the presence of a finite gravitational constant.

To calculate (5.3) it is convenient to first expand $T$ in terms of boundary
quantities to be able to then work in a CFTd without a boundary. The bulk
stress tensor can be expanded in boundary operators $\tilde{{\cal O}}$ via

$\displaystyle
T^{\mu\nu}(y,\vec{x})=\sum_{i}(2y)^{\tilde{\Delta}_{i}-d}b^{\mu\nu}_{i,\alpha_{1}\dots\alpha_{l}}\tilde{{\cal
O}}^{\alpha_{1}\dots\alpha_{l}}_{i}(\vec{x})\;,$ (5.6)

corresponding to the configuration

which only involves boundary operators. Since all operators are on the
boundary, a CFTd, a conformal block decomposition for
$\langle\tilde{\phi}\tilde{{\cal O}}\tilde{\phi}\tilde{{\cal O}}\rangle$
applies:

Putting it all together gives

$\displaystyle\langle
T(x_{1})\tilde{\phi}(x_{2})T(x_{3})\tilde{\phi}(x_{4})\rangle=\sum_{ijk}(4y_{2}y_{4})^{\tilde{\Delta}_{\phi}-\Delta_{\phi}}(2y_{1})^{\tilde{\Delta}_{i}-d}(2y_{3})^{\tilde{\Delta}_{j}-d}C^{2}_{\phi\tilde{\phi}}b^{\mu\nu}_{i}b^{\rho\sigma}_{j}C_{\tilde{{\cal
O}}_{i}\tilde{\phi}}C_{\tilde{{\cal
O}}_{j}\tilde{\phi}}\mathcal{F}^{(k)}_{\tilde{{\cal
O}}_{i}\tilde{\phi}\tilde{{\cal
O}}_{j}\tilde{\phi}}(\\{\vec{x}_{\alpha}\\})\;.$ (5.7)

for the bulk stress tensor-scalar four-point function. After inserting (5.7)
into (5.3) and calculating the integrals, the result is the leading
gravitational correction to the CFTd boundary scalar two-point function: the
$O(1/c)$ term in (5.4).

##  6 Discussion

Analyzing minimal models in AdS2 allows us to take a strongly coupled CFT and
use the power of boundary CFT and JT/Schwarzian gravity. The result is a
family of “solved” bulk theories that we can calculate perturbative
gravitational corrections for to arbitrary order. We illustrated examples of
the types of calculations that can be done in this framework using the Ising
model, which can be generalized to any minimal model. In particular we saw how
the physical boundary conditions affect the operator content of the theory
through the relation between the boundary conditions, correlator coefficients,
and the OPE coefficients. For the case of AdS with two boundaries – as found,
e.g., in RS1 models – we calculated the energy spectrum from the boundary
correlators. JT gravity was added and the leading gravitational correction to
the non-Gaussian $\tilde{\phi}_{2,1}$ minimal model correlators was found.

It is interesting to think about the scenario studied here to the arguments
put forth in Heemskerk:2009pn . In that work, it is conjectured that a CFT
with a large-$N$ expansion and a sufficiently large gap has a local bulk dual
description. In the case at hand, the bulk is strongly coupled and the
boundary operator dimensions and correlators do not have a large-$N$
factorization. Therefore, in a sense holography is more general than the
requirements of Heemskerk:2009pn .

## Acknowledgements

This work grew out of a collaboration with Liam Fitzpatrick, to whom I owe
many thanks for his insight and guidance. We also thank Hongbin Chen for
helpful discussions. This work was supported in part by the Simons
Collaboration on the Nonperturbative Bootstrap and Boston University.

##  Appendix A OPE Conventions

In this work we deal with bulk, boundary, and bulk-boundary operator product
expansions, with the additional complication of these being dependent on
potentially changing boundary conditions. Here we define the OPE notations and
conventions used throughout the text, based on the notation used in
Lewellen:1991tb .

For the two-dimensional bulk theory, conformal invariance fixes the form of
the one- and two-point functions. We normalize the vacuum and the primary
field ${\cal O}_{i}$ with scaling dimensions $(\Delta_{i},\bar{\Delta}_{i})$
as

$\displaystyle\langle\mathbbm{1}\rangle=1,\quad\langle{\cal O}_{i}\rangle$
$\displaystyle=0\;,\quad\langle{\cal O}_{i}(z,\bar{z}){\cal
O}_{j}(0)\rangle=\delta_{ij}\,z^{-2\Delta_{i}}\bar{z}^{-2\bar{\Delta}_{i}}\,.$
(A.1)

At short distances the bulk OPE is given by

$\displaystyle{\cal O}_{i}(z,\bar{z}){\cal
O}_{j}(0)=\sum_{k}C_{ijk}\,z^{\Delta_{k}-\Delta_{i}-\Delta_{j}}\bar{z}^{\bar{\Delta}_{k}-\bar{\Delta}_{i}-\bar{\Delta}_{j}}{\cal
O}_{k}(z,\bar{z})+\cdots$ (A.2)

Restricting ourselves to the one-dimensional boundary, we now have operators
${\tilde{\cal O}}_{i}^{ab}(x)$ that are primaries under a single Virasoro
algebra with conformal dimension $\tilde{\Delta}_{i}$. The superscript indices
$ab$ indicate a boundary operator that mediates a change in boundary
conditions from type $a$ to type $b$. The boundary correlators are thus

$\displaystyle\langle\mathbbm{1}\rangle_{a}=\alpha^{a},\quad\langle{\tilde{\cal
O}}_{i}^{ab}\rangle$ $\displaystyle=0\;,\quad\langle{\tilde{\cal
O}}_{i}^{ab}(0){\tilde{\cal
O}}_{j}^{cd}(x)\rangle=\delta_{ad}\delta_{bc}\delta_{ij}\,x^{-2\tilde{\Delta}_{i}}\tilde{\alpha}^{ad}_{i}\;,$
(A.3)

with a boundary OPE given by

$\displaystyle{\tilde{\cal O}}_{i}^{ab}(0){\tilde{\cal
O}}_{j}^{cd}(x)=\sum_{k}\tilde{C}^{abc}_{ijk}\,x^{\tilde{\Delta}_{k}-\tilde{\Delta}_{i}-\tilde{\Delta}_{j}}{\tilde{\cal
O}}_{k}^{ac}(x)+\cdots$ (A.4)

The normalization constants $\alpha^{a}$ and $\alpha^{ad}_{i}$ are boundary
condition-dependent and arise from that fact that having canonically fixed the
bulk normalizations, we can not do the same with the boundary ones.

When a bulk operator approaches a boundary with boundary condition $a$, we get
a bulk-boundary OPE

$\displaystyle{\cal
O}_{j}(z,\bar{z})=\sum_{i}(2y)^{\tilde{\Delta}_{i}-\Delta_{j}-\bar{\Delta}_{j}}C_{ji}^{a}\,{\tilde{\cal
O}}_{i}^{aa}(x)+\cdots$ (A.5)

Due to the single Virasoro algebra, the explicit factor of 2 comes from
thinking about (A.5) as one chiral half of the bulk OPE.

For the sake of notational easement we drop the boundary condition
superscripts when dealing with constant and uniform boundary conditions or
otherwise when it is unambiguously clear (e.g., ${\tilde{\cal
O}}_{i}^{aa}\rightarrow{\tilde{\cal O}}_{i}$, $C_{ji}^{a}\rightarrow C_{ji}$).

##  Appendix B The Modular Matrix $S$

The purpose of this appendix is to show the relationship between the modular
matrix $S$ and the fusion matrices and algebra explicitly for the Ising model.
The matrix elements of $S$ are given by

$\displaystyle
S_{rs;\rho\sigma}=2\sqrt{\frac{2}{pq}}(-1)^{1+s\rho+r\sigma}\sin(\pi\frac{p}{q}r\rho)\sin(\pi\frac{q}{p}s\sigma)\,.$
(B.1)

For the minimal models, the central charge can be written as

$\displaystyle c=1-6\frac{(p-q)^{2}}{pq}\;,$ (B.2)

which corresponds to the irreducible representation with Kac indices $(r,s)$
in the range

$\displaystyle 1\leq r$ $\displaystyle\leq q-1$ $\displaystyle 1\leq s$
$\displaystyle\leq p-1$ $\displaystyle qs$ $\displaystyle<pr\,.$ (B.3)

It is common to denote $E_{p,q}$ as the set of pairs $(r,s)$ in the above
range (B) and that $\rho,\sigma\in E_{p,q}$ in (B.1). This means that if
$(q,p)=(4,3)$ corresponding to the Ising model, the allowed pairs are
$(r,s):\\{(1,1),\,(2,1),\,(2,2)\\}$. Indeed, plugging the allowed pairs into
(B.1) we obtain the following matrix elements:

$\displaystyle S_{11,11}=\frac{1}{2}\;,\quad S_{11,21}$
$\displaystyle=\frac{1}{2}\;,\quad S_{11,22}=\frac{1}{\sqrt{2}}\;,$
$\displaystyle S_{21,11}=\frac{1}{2}\;,\quad S_{21,21}$
$\displaystyle=\frac{1}{2}\;,\quad S_{21,22}=-\frac{1}{\sqrt{2}}\;,$
$\displaystyle S_{22,11}=\sqrt{\frac{1}{2}}\;,\quad S_{22,21}$
$\displaystyle=-\sqrt{\frac{1}{2}}\;,\quad S_{22,22}=0\;,$ (B.4)

corresponding to the Ising modular matrix $S$:

$\displaystyle
S_{\text{Ising}}=\begin{pmatrix}\frac{1}{2}&\frac{1}{2}&\sqrt{\frac{1}{2}}\\\
\frac{1}{2}&\frac{1}{2}&-\sqrt{\frac{1}{2}}\\\
\sqrt{\frac{1}{2}}&-\sqrt{\frac{1}{2}}&0\end{pmatrix}\;.$ (B.5)

Recall that $S$ is a transformation on the basis of the minimal characters,

$\displaystyle\chi_{r,s}(-1/\tau)=\sum_{(\rho,\sigma)\in
E_{p,q}}S_{rs,\rho\sigma}\chi_{\rho,\sigma}(\tau)\;,$ (B.6)

and can be calculated from explicit values for the characters $\chi$. In the
Ising case: $\chi_{0}(\tau),\,\chi_{\frac{1}{2}}(\tau),\,$ and
$\chi_{\frac{1}{16}}(\tau),$ respectively. The characters involve theta
functions and Dedekind eta functions. The fact that under a modular
transformation $S:\tau\mapsto-1/\tau$

$\displaystyle\chi_{\frac{1}{16}}(\tau)\mapsto\chi_{\frac{1}{16}}(-1/\tau)=\frac{1}{\sqrt{2}}\chi_{0}(\tau)-\frac{1}{\sqrt{2}}\chi_{\frac{1}{2}}(\tau)+0\cdot\chi_{\frac{1}{16}}(\tau)$
(B.7)

is reflected in the vanishing matrix element in the Ising matrix $S$.

The fusion matrices can be calculated from the Verlinde formula,

$\displaystyle
N^{k}_{\;ij}=\sum_{l=0}^{2}\frac{S_{jl}S_{il}(S^{-1})_{lk}}{S_{0l}}\;,$ (B.8)

to be

$\displaystyle N_{\mathbbm{1}}=\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix}\;,\quad N_{\epsilon}=\begin{pmatrix}0&1&0\\\ 1&0&0\\\
0&0&1\end{pmatrix}\;,\quad N_{\sigma}=\begin{pmatrix}0&0&1\\\ 0&0&1\\\
1&1&0\end{pmatrix}\;,$ (B.9)

so that

$\displaystyle S\;N_{\mathbbm{1}}\;S=\text{diag}(1,1,1)\;,\quad
S\;N_{\epsilon}\;S=\text{diag}(1,1,-1)\;,\quad
S\;N_{\sigma}\;S=\text{diag}(\sqrt{2},-\sqrt{2},0)\;.$ (B.10)

From (B.9) and the fusion algebra
$[\Phi_{i}]\times[\Phi_{j}]=\sum_{k}N^{k}_{\;ij}[\Phi_{k}]$ we can read off
the fusion rules

$\displaystyle\sigma\times\sigma$ $\displaystyle=\mathbbm{1}+\epsilon$
$\displaystyle\sigma\times\epsilon$ $\displaystyle=\sigma$
$\displaystyle\epsilon\times\epsilon$ $\displaystyle=\mathbbm{1}\;.$ (B.11)

##  Appendix C Boundary States and Bulk-Boundary Constants

Cardy boundary states are the physical boundary states and can be represented
as a linear combination of boundary states that preserve local conformal
invariance, so-called Ishibashi states. Given $S$, a Cardy boundary state
$|\tilde{l}\rangle$ can be expanded in the basis of Ishibashi states
$|j\rrangle$ corresponding to the bulk primary fields as Cardy:1989ir

$\displaystyle|\tilde{l}\rangle=\sum_{j}\frac{S_{ij}}{\sqrt{S_{0j}}}|j\rrangle\;.$
(C.1)

The matrix $S$ also encodes the coupling between bulk fields and the identity
on the boundary through CARDY1991274

$\displaystyle C^{a}_{j\mathbbm{1}}=A^{a}_{j}=\frac{\llangle
j|a\rangle}{\llangle
0|a\rangle}=\frac{S_{ja}}{S_{0a}}\left(\frac{S_{00}}{S_{j0}}\right)^{1/2}\;,$
(C.2)

where $j$ is an Ishibashi state and $a$ is a Cardy boundary state defined in
(C.1). Applying the above to the Ising modular matrix (B.5), the Cardy
boundary states are given by

$\displaystyle|\tilde{0}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}|0\rrangle+\frac{1}{\sqrt{2}}|\epsilon\rrangle+\frac{1}{2^{1/4}}|\sigma\rrangle$
$\displaystyle|\tilde{\frac{1}{2}}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}|0\rrangle+\frac{1}{\sqrt{2}}|\epsilon\rrangle-\frac{1}{2^{1/4}}|\sigma\rrangle$
$\displaystyle|\tilde{\frac{1}{16}}\rangle$
$\displaystyle=|0\rrangle-|\epsilon\rrangle\;.$ (C.3)

corresponding to the all spins pointing up $(+)$, down $(-)$, or left free
$(f)$, respectively. The boundary operators corresponding to these changes in
boundary conditions are $\tilde{\epsilon}^{\pm\mp}(x)$,
$\tilde{\epsilon}^{ff}(x)$, $\tilde{\sigma}^{\pm f}$, and
$\tilde{\sigma}^{f\pm}(x)$, with $\Delta_{\tilde{\epsilon}}=1/2$ and
$\Delta_{\tilde{\sigma}}=1/16$. These can be calculated from the fusion
coefficients $n^{h}_{\alpha\beta}=N^{h}_{\alpha\beta}$ and are derived in
CARDY1986200 .

The modular matrix can also be used to calculate the bulk-boundary
coefficients. Directly applying (C.2) to (B.5) gives

$\displaystyle C^{\pm}_{\tilde{\sigma}\mathbbm{1}}=A^{\pm}_{\sigma}=\pm
2^{1/4}\;,\quad\quad C^{f}_{\tilde{\sigma}\mathbbm{1}}=A^{f}_{\sigma}=0\;,$
(C.4) $\displaystyle
C^{\pm}_{\tilde{\epsilon}\mathbbm{1}}=A^{\pm}_{\epsilon}=1\;,\quad\quad
C^{f}_{\tilde{\epsilon}\mathbbm{1}}=A^{f}_{\epsilon}=-1\;,$ (C.5)
$\displaystyle\left(C^{f}_{\tilde{\sigma}\tilde{\epsilon}}\right)^{2}=\frac{1}{\sqrt{2}\alpha^{ff}_{\tilde{\epsilon}}}\;,$
(C.6)

for the bulk-boundary (and half-plane bulk one-point) structure constants.
These are consistent with the symmetries of the Ising model under spin
reversal
$(\sigma\rightarrow-\sigma,\;\epsilon\rightarrow\epsilon,\;|+\rangle\leftrightarrow|-\rangle,\;|f\rangle\rightarrow|f\rangle)$
and duality
$(\epsilon\rightarrow-\epsilon,\;|\pm\rangle\leftrightarrow|f\rangle)$.

##  Appendix D Non-Gaussianity of Minimal Model $\phi_{2,1}$ Four-Point
Functions

Apart from the Ising model, unitary minimal model $\phi_{2,1}$ four-point
functions are non-Gaussian. A “Gaussian” correlator is one that is entirely
determined by its two-point functions. This means that the connected four-
point function of a Gaussian correlator tends to zero when operator pairs are
pulled apart from each other due to clustering. The factorization of the Ising
energy correlator (2.10) is an example of this, which can be seen equivalently
through

$\displaystyle\lim_{\stackrel{{\scriptstyle x_{2}\rightarrow
x_{1}}}{{x_{4}\rightarrow
x_{3}}}}\frac{\langle\epsilon_{1}\epsilon_{2}\epsilon_{3}\epsilon_{4}\rangle}{\langle\epsilon_{1}\epsilon_{2}\rangle\langle\epsilon_{3}\epsilon_{4}\rangle}\rightarrow
1\;.$ (D.1)

Therefore, a test that a generic four-point correlator of the operator ${\cal
O}$ is Gaussian is that

$\displaystyle\lim_{\stackrel{{\scriptstyle x_{2}\rightarrow
x_{1}}}{{x_{4}\rightarrow x_{3}}}}x_{12}^{2\Delta_{{\cal
O}}}x_{34}^{2\Delta_{{\cal O}}}G_{4}^{{\cal O}}(x_{i})\rightarrow 1\;,$ (D.2)

where $G_{4}^{{\cal O}}(x_{i})$ is given by

$\displaystyle G_{4}^{{\cal
O}}(x)=\frac{1}{\left|x(1-x)\right|^{2\Delta}}\left(\left|\mathcal{F}^{{\cal
O}}_{1}(x)\right|^{2}+A^{2}_{m}\left|x\right|^{\frac{2+8\Delta}{3}}|\mathcal{F}^{{\cal
O}}_{2}(x)|^{2}\right)\;.$ (D.3)

with

$\displaystyle\mathcal{F}^{{\cal
O}}_{1}(x)={}_{2}F_{1}\left(\frac{1-2\Delta_{{\cal O}}}{3},-2\Delta_{{\cal
O}},\frac{2-4\Delta_{{\cal O}}}{3};x\right)\;,\quad\mathcal{F}^{{\cal
O}}_{2}(x)={}_{2}F_{1}\left(\frac{1-2\Delta_{{\cal
O}}}{3},\frac{2+2\Delta_{{\cal O}}}{3},\frac{4+4\Delta_{{\cal
O}}}{3};x\right)\;.$

and
$|A_{m}|^{2}=\Gamma(\frac{2}{m+1})\Gamma(2-\frac{3}{m+1})/\Gamma(\frac{1}{m+1})\Gamma(\frac{2m}{m+1})$.

It can be checked using (D.2) that the first term in (D.3) is Gaussian for an
operator of any scaling dimension whereas the second term is not. Therefore,
to be fully Gaussian a four-point function needs the second term to vanish.
From the fusion algebra

$\displaystyle\left[\phi_{2,1}\right]\times\left[\phi_{2,1}\right]=\left[\phi_{1,1}\right]+\cdots$
(D.4)

this occurs when the higher operators denoted by the ellipses are absent in
the Kac table of the minimal model in question associated with the
$\phi_{2,1}$ operator. This only occurs for the Ising minimal model where
$\phi_{2,1}=\epsilon$ and

$\displaystyle\left[\epsilon\right]\times\left[\epsilon\right]=\left[\mathbbm{1}\right]\;.$
(D.5)

##  Appendix E JT Gravity and the Schwarzian Action

This section aims to concisely review how the Schwarzian action arises as the
one-dimensional boundary theory to JT gravity, and then develops the effective
Schwarzian theory. There are several good review articles Sarosi_2018 ;
Trunin_2021  that cover much of this material in greater depth.

###  E.1 JT gravity

JT gravity is a theory with a two-dimensional metric tensor $g_{\mu\nu}$ and
dilaton field $\Phi$ coupling given by the Euclidean action

$\displaystyle S_{\text{JT}}=-\frac{1}{16\pi
G}\left[\int_{\mathcal{M}}d^{2}x\sqrt{g}\,\Phi\left(R-\frac{\Lambda}{L^{2}}\right)+2\int_{\partial\mathcal{M}}du\sqrt{h}\,\Phi_{b}K\right]\;,$
(E.1)

where $\mathcal{M}$ is the manifold the theory is defined on and
$\partial\mathcal{M}$ is its boundary with induced metric $h$ and extrinsic
curvature $K$.666We have suppressed the purely topological piece
$\displaystyle S_{\text{JT}}\supset\Phi_{b}\left(\int d^{2}x\sqrt{g}R+2\int
du\sqrt{h}K\right)$ (E.2) responsible for the extremal entropy. The bulk term
uses the dilaton as a Lagrange multiplier to set the curvature to the
cosmological constant $R=\Lambda/L^{2}=-2$ in the equations of motion. The
boundary term is the Gibbons-Hawking-York (GHY) term, which ensures a well-
defined variational principle for the Dirichlet problem.

To understand the boundary degrees of freedom the boundary proper length needs
to be fixed. This is done by cutting off the exactly Euclidean AdS2 space to a
nearly AdS2 (NAdS2) space with a bulk IR/boundary UV regulator by setting

$\displaystyle\Phi_{b}=\Phi_{\text{bdy}}=\frac{\Phi_{r}(u)}{\epsilon}\;,\quad\frac{1}{\epsilon^{2}}=g_{uu}=\frac{t^{\prime
2}+z^{\prime 2}}{z^{2}}\;,\quad
ds|_{\text{bdy}}=\frac{du}{\epsilon}\implies\int_{0}^{\beta}ds=\frac{\beta}{\epsilon}$
(E.3)

so that as $\epsilon\rightarrow 0$ the boundary proper length becomes large.
The quantity $\Phi_{r}(u)$ is a dimension $-1$ coupling that is “renormalized”
in the sense it remains finite in the above limit. The primes are derivatives
with respect to the proper boundary time $u$.

The result of this is a trajectory $(t(u),z(u))$ near the boundary
corresponding to a UV cutoff in the dual boundary theory, producing a nearly
CFT1 (NCFT1). Notice the parametrization $z(u)=\epsilon t^{\prime}(u)$
satisfies the above equations at linear order in epsilon for arbitrary $t(u)$,
so there is a family of boundary curves related by a reparametrization of
$t(u)$.

With this parametrization the induced metric $\sqrt{h}=1/\epsilon$ and the
remaining boundary term becomes

$\displaystyle S_{\text{JT}}\rightarrow-\frac{1}{8\pi
G}\int\frac{du}{\epsilon}\frac{\Phi_{r}(u)}{\epsilon}K\;,$ (E.4)

where the extrinsic curvature is given by

$\displaystyle
K=1+\epsilon^{2}\\{t,u\\}+\cdots\quad\quad\\{t,u\\}\equiv\frac{t^{\prime\prime\prime}}{t^{\prime}}-\frac{3}{2}\left(\frac{t^{\prime\prime}}{t^{\prime}}\right)^{2}\;,$
(E.5)

with $\\{t,u\\}$ the Schwarzian derivative. The result of this is the
linearized boundary action of (E.1) reduces to an effective Schwarzian action

$\displaystyle S=-\,C\int dt\,\Phi_{r}(u)\\{t,u\\}\;,$ (E.6)

where $C=\frac{1}{8\pi G}$. The normalizable part of the dilaton,
$\Phi_{r}(u)$, is constant on the boundary and serves as an external
coupling.777On the boundary, $\Phi_{r}/\epsilon=\Phi_{b}$, which can be
absorbed into the definition of $C$.

In the Schwarzian action (E.6), the dynamical field variable $t(u)$ represents
the Nambu-Goldstone bosons generated from the spontaneous breaking of SL$(2)$
symmetry by the AdS2 solution, and are zero modes involving large
diffeomorphisms. In the conformal limit where $\beta=\infty\leftrightarrow
T=0$, these zero modes have zero energy. Moving to NCFT1 lifts these zero
modes to nearly-zero modes needed for a dynamic gravity theory.

###  E.2 Effective Schwarzian action

By varying the action we get the equation of motion

$\displaystyle\left[\frac{1}{t^{\prime}}\left(\frac{(t^{\prime}\Phi_{r})^{\prime}}{t^{\prime}}\right)\right]^{\prime}\longrightarrow\frac{\\{t,u\\}^{\prime}}{t^{\prime}}=0$
(E.7)

where in moving from left to right we used that $\Phi_{r}$ is constant on the
boundary. We see that when $t$ is a linear function of $u$, the equations of
motion are satisfied, and represent a saddle in the path integral. Therefore
the Schwarzian derivative is invariant under SL(2) symmetry
$t\rightarrow(at+b)/(ct+d)$ and vanishes for $t(u)=u$.

Since we are on the disk and have a finite boundary length, it is convenient
unwrap the circle by reparametrizing

$\displaystyle u_{\text{line}}=\tan\frac{\pi}{\beta}t(u_{\text{circle}})$
(E.8)

in terms of the diff($S^{1}$) reparametrization $f$ satisfying,

$\displaystyle t(u_{\text{circle}}+\beta)=t(u_{\text{circle}})+\beta\;,\quad
t^{\prime}(u_{\text{circle}})\geq 0\;.$ (E.9)

We use $u\equiv u_{\text{circle}}$ everywhere else unless stated otherwise.
Because the boundary length $\beta$ can be interpreted as the inverse
temperature, this makes manifest the finite temperature theory.

Quantum gravitational thermal correlators are found by computing the thermal
Schwarzian path integral

$\displaystyle\langle\mathcal{O}\rangle_{\beta}\equiv\int_{\mathcal{M}}[\mathcal{D}t]\,\mathcal{O}[t]e^{C\int_{0}^{\beta}du\,\left\\{\tan\frac{\pi}{\beta}t,\,u\right\\}}$
(E.10)

where the integration space
$\mathcal{M}=\text{diff}(S^{1})/\text{SL}(2,\mathbb{R})$. Using the
composition property of the Schwarzian,
$\\{t(g(u)),u\\}=(g^{\prime})^{2}\\{t,g\\}+\\{t,u\\}$, this becomes

$\displaystyle S[t]=-C\int
du\,\left\\{\tan\frac{\pi}{\beta}t(u),\,u\right\\}=\frac{C}{2}\int
du\,\left[\left(\frac{t^{\prime\prime}}{t^{\prime}}\right)^{2}-\left(\frac{2\pi}{\beta}\right)^{2}(t^{\prime})^{2}\right]\;.$
(E.11)

###  E.3 Schwarzian fluctuation theory

To obtain the linearized theory we proceed perturbatively around the saddle by
writing $t(u)=u+\epsilon k(u)$ and expanding the action in $\epsilon$. To
simplify things, let $\beta=2\pi$; to recover the original $u$ variable
rescale $u\rightarrow(2\pi/\beta)u$. After expanding and integrating by parts,
the linearized action is

$\displaystyle
S_{\epsilon}[t]=\frac{C}{2}\int_{0}^{2\pi}du\,\left[(k^{\prime\prime})^{2}-(k^{\prime})^{2}\right]\;.$
(E.12)

To quantize the fluctuation theory we invert the kernel in the above action by
discrete Fourier transforming via $\epsilon(u)=\sum_{n}\epsilon_{n}e^{inu}$
with $n\in\mathbbm{Z}$ and $u\in[0,2\pi]$. This gives

$\displaystyle
S_{\epsilon}[t]=\frac{C}{2}\sum_{n\in\mathbbm{Z}}\,(n^{4}-n^{2})\epsilon_{n}\epsilon_{-n}\;.$
(E.13)

Inverting, we obtain the fluctuation propagator888Recall the integration space
is modded out by gauge redundancy SL(2,$\mathbbm{R}$) so we can fix the
redundant SL(2,$\mathbbm{R}$) singular modes $n=0,\pm 1$ in the sum and ignore
when inverting.

$\displaystyle\langle\epsilon(u_{1})\epsilon(u_{2})\rangle$
$\displaystyle=\frac{1}{2\pi C}\sum_{n\neq 0,\pm
1}\frac{e^{in\mathfrak{u}}}{n^{2}(n^{2}-1)}$ $\displaystyle=\frac{1}{2\pi
C}\left(1+\frac{\pi^{2}}{6}-\frac{(|\mathfrak{u}|-\pi)^{2}}{2}+(|\mathfrak{u}|-\pi)\sin|\mathfrak{u}|+\frac{5}{2}\cos|\mathfrak{u}|\right)\;,$
$\displaystyle\langle\epsilon(\tau_{1})\epsilon^{\prime}(\tau_{2})\rangle$
$\displaystyle=\frac{\text{sgn}(\mathfrak{u})}{2\pi
C}\left((\pi-|\mathfrak{u}|)(1-\cos|\mathfrak{u}|-\frac{3}{2}\sin|\mathfrak{u}|\right)\;,$
$\displaystyle\langle\epsilon^{\prime}(\tau_{1})\epsilon^{\prime}(\tau_{2})\rangle$
$\displaystyle=\frac{1}{2\pi
C}\left(1+\frac{1}{2}\cos|\mathfrak{u}|-(\pi-|\mathfrak{u}|)\sin|\mathfrak{u}|\right)\;,$
(E.14)

where $\mathfrak{u}\equiv\tau_{1}-\tau_{2}$.

###  E.4 Schwarzian correlation functions

To calculate gravitational correlators we need to know how the Schwarzian
action (E.6) couples to matter correlators.

As discussed in Section 2, we know how to quantify any $n$-point boundary
correlator $\tilde{{\cal O}}(u_{1},\dots,u_{n})$ of a bulk CFT2. In 1d
reparametrization invariance and conformal invariance are the same. Under a
conformal transformation $\tilde{{\cal O}}$ transforms as

$\displaystyle\tilde{{\cal
O}}(u_{1},\dots,u_{n})=(t^{\prime}_{1})^{\Delta_{1}/2}\dots(t^{\prime}_{n})^{\Delta_{n}/2}\tilde{{\cal
O}}(t_{1}(u_{1}),\dots,t_{n}(u_{n}))\;,$ (E.15)

where $t^{\prime}_{i}\equiv dt_{i}(u_{i})/du_{i}$. Subsequently, from the
Schwarzian path integral we can compute gravitational correlators
Maldacena:2016upp ; Blommaert:2019hjr  by computing the quantity

$\displaystyle\langle\tilde{{\cal
O}}\rangle_{t}\equiv\int[\mathcal{D}t]\,\tilde{{\cal O}}(t_{i})\,e^{-S[t]}$
(E.16)

after performing a conformal transformation (E.15) on $\tilde{{\cal O}}$.

To perform the path integral we expand around the saddle by letting
$t(u)=u+\epsilon k(u)$ with $\epsilon$ small. Expanding the action sends
$S[t]\rightarrow S_{\text{eff}}[k]$ and expanding $\tilde{{\cal O}}(t_{i})$ to
second order in $\epsilon$ lets us use the fluctuation propagator (E.3) to
obtain the full expression for the gravitational correlator
$\langle\tilde{{\cal O}}\rangle_{t}$.

###  E.5 Two-point functions

Now that know how to calculate gravitational boundary correlators using the
Schwarzian, all that remains is doing it. Here we review simple examples from
the literature.

Taking a bulk minimal model two-point function of the operator ${\cal O}$ to
the boundary gives

$\displaystyle G_{2}(t_{1},t_{2})=\langle\tilde{{\cal
O}}_{\tilde{\Delta}}(t_{1})\tilde{{\cal
O}}_{\tilde{\Delta}}(t_{2})\rangle=\frac{1}{t_{12}^{2\tilde{\Delta}}}\;,$
(E.17)

where $\tilde{{\cal O}}$ is the dual CFT1 operator to ${\cal O}$ consistent
with the boundary conditions. Under a conformal transformation this becomes

$\displaystyle\frac{1}{t_{12}^{2\tilde{\Delta}}}\longrightarrow\left(\frac{(f^{\prime}(t_{1})f^{\prime}(t_{2}))}{(f(t_{1})-f(t_{2}))^{2}}\right)^{\tilde{\Delta}}\;.$
(E.18)

To calculate the gravitational contribution to the two-point function, expand
(E.18) around the saddle $f(t)=\tan\frac{t+\epsilon(t)}{2}$ to second order in
$\epsilon$, giving

$\displaystyle
G_{2}(t_{1},t_{2})=\frac{1+\epsilon\mathcal{B}_{1}(t_{12})+\epsilon^{2}\mathcal{B}_{2}(t_{12})+O(\epsilon^{3})}{\left(2\sin\frac{t_{12}}{2}\right)^{2\Delta}}\;,$
(E.19)

where

$\displaystyle\mathcal{B}_{1}(t_{12})$
$\displaystyle=\tilde{\Delta}\left(k^{\prime}_{1}+k^{\prime}_{2}-\frac{k_{1}-k_{2}}{\tan\frac{t_{12}}{2}}\right)\;,$
$\displaystyle\mathcal{B}_{2}(t_{12})$
$\displaystyle=\frac{\tilde{\Delta}}{4}\left(\frac{k^{2}_{12}}{(\sin\frac{t_{12}}{2})^{2}}-2(k^{\prime
2}_{1}+k^{\prime
2}_{2})\right)+\frac{\tilde{\Delta}^{2}}{2}\left(k^{\prime}_{1}+k^{\prime}_{2}-\frac{k_{12}}{\tan\frac{t_{12}}{2}}\right)^{2}\;.$
(E.20)

To compute the gravitational correlator substitute (E.19) into the effective
Schwarzian path integral with action (E.12),

$\displaystyle\langle
G_{2}(t_{1},t_{2})\rangle_{f}=\frac{1+\epsilon^{2}\langle\mathcal{B}_{2}(t_{12})\rangle}{t^{2\tilde{\Delta}}_{12}}\;,$
(E.21)

where the linear term in $\epsilon$ vanishes. The final expression involves
contracting factors of the propagator using (E.3) to give

$\displaystyle\langle G_{2}(t_{1},t_{2})\rangle_{f}$
$\displaystyle=\frac{1}{(2\sin\frac{t_{12}}{2})^{2\tilde{\Delta}}}\bigg{(}1+\tilde{\Delta}\frac{2(1-\pi
t_{12})+t^{2}_{12}-2\cos t_{12}+2\sin t_{12}(\pi-t_{12})}{8\pi
C\sin^{2}\frac{t_{12}}{2}}$
$\displaystyle-\tilde{\Delta}^{2}\frac{\big{(}2+(2\pi-
t_{12})\cot\frac{t_{12}}{2}\big{)}\big{(}-2+t_{12}\cot\frac{t_{12}}{2}\big{)}}{4\pi
C}\bigg{)}\;,$ (E.22)

reproducing the result of Maldacena:2016upp . Factorized four-point functions
are a straightforward extension of the two-point function and are also treated
in the aforementioned reference.

###  E.6 Three-point functions

Three-point functions follow a similar pattern to the two-point ones above.
First start with the dual CFT1 boundary three-point function

$\displaystyle G_{3}(t_{1},t_{2},t_{3})=\langle\tilde{{\cal
O}}_{\tilde{\Delta}}(t_{1})\tilde{{\cal
O}}_{\tilde{\Delta}}(t_{2})\tilde{{\cal
O}}_{\tilde{\Delta}}(t_{3})\rangle=\frac{N_{3}}{(t_{12}t_{23}t_{13})^{\tilde{\Delta}}}\;,$
(E.23)

where $N_{3}$ is a normalization constant. Under a conformal transformation
this becomes

$\displaystyle
G_{3}(t_{1},t_{2},t_{3})\longrightarrow\left(\frac{f^{\prime}_{1}f^{\prime}_{2}f^{\prime}_{3}}{f_{12}f_{23}f_{13}}\right)^{\Delta}\;.$
(E.24)

Reparametrizing and expanding around the saddle gives

$\displaystyle G_{3}(t_{1},t_{2},t_{3})=N_{3}\frac{1+\epsilon
C_{1}(t_{123})+\epsilon^{2}C_{2}(t_{123})+O(\epsilon^{3})}{(t_{12}t_{23}t_{13})^{\tilde{\Delta}}}$
(E.25)

where

$\displaystyle C_{1}(t_{123})$
$\displaystyle=\tilde{\Delta}\left(k^{\prime}_{1}+k^{\prime}_{2}+k^{\prime}_{3}-\frac{k_{12}}{2\tan\frac{t_{12}}{2}}-\frac{k_{3}}{2\tan\frac{t_{23}}{2}}-\frac{k_{13}}{2\tan\frac{t_{13}}{2}}\right)$
$\displaystyle C_{2}(t_{123})$
$\displaystyle=\frac{\tilde{\Delta}}{8}\left(\frac{k^{2}_{12}}{\sin^{2}\frac{t_{12}}{2}}+\frac{k^{2}_{23}}{\sin^{2}\frac{t_{23}}{2}}+\frac{k^{2}_{13}}{\sin^{2}\frac{t_{13}}{2}}-4(k^{\prime}_{1}+k^{\prime}_{2}+k^{\prime}_{3})\right)$
$\displaystyle+\frac{\tilde{\Delta}^{2}}{2}\left((k^{\prime}_{1}+k^{\prime}_{2}+k^{\prime}_{3})+\frac{k_{21}\cos
t_{12}}{2\sin\frac{t_{12}}{2}\sin\frac{t_{23}}{2}\sin\frac{t_{13}}{2}}+\frac{k_{13}\cos
t_{13}}{2\sin\frac{t_{12}}{2}\sin\frac{t_{23}}{2}\sin\frac{t_{13}}{2}}+\frac{k_{32}\cos
t_{23}}{2\sin\frac{t_{12}}{2}\sin\frac{t_{23}}{2}\sin\frac{t_{13}}{2}}\right)^{2}\;.$
(E.26)

Therefore the gravitational contribution to the three-point function is given
by

$\displaystyle\langle
G_{3}(t_{1},t_{2},t_{3})\rangle_{f}=N_{3}\frac{1+\epsilon^{2}\langle
C_{2}(t_{123})\rangle+O(\epsilon^{3})}{(t_{12}t_{23}t_{13})^{\tilde{\Delta}}}\;,$
(E.27)

where again the $\epsilon$-propagator terms are contracted using (E.3),
reproducing Qi:2018rqm .

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