∇ ††institutetext: Department of Physics, University of California at Santa
Barbara
Santa Barbara, CA 93106, U.S.A.

# Heavy States in 3d Gravity and 2d CFT

David Grabovsky<EMAIL_ADDRESS>
###### Abstract

We discuss correlators of light fields in heavy states in $\mathrm{AdS}_{3}$
gravity and holographic 2d CFTs. In the bulk, the propagator of free fields in
AdS backgrounds containing a conical defect or a BTZ black hole can be
obtained by solving the wave equation, as well as by the method of images. On
the boundary, these geometries are sourced by heavy operator insertions, and
the propagator is dual to a heavy-light (HHLL) correlator. By matching its
expansion in Virasoro blocks to our bulk results, we determine the OPE
coefficients of all contributing states in both the $s$ and $t$ channels. In
the $s$ channel, these states are excitations of the light field on top of the
heavy state, and their OPE coefficients are the amplitudes to create them. The
$t$-channel OPE is dominated by the Virasoro vacuum block, but there is also
an infinite family of light two-particle states that contribute to the
correlator. The OPE coefficients that couple these states to heavy operators
represent their expectation values in heavy backgrounds. We determine them
exactly, derive their asymptotic form at large twist, and discuss their
behavior near and above the BTZ threshold, where they become thermal one-point
functions.

##  1 Introduction and Overview

How can we understand high-energy states in quantum gravity? Our intuition
from general relativity suggests that the answer is universal: at high
energies, essentially any localized object will form a black hole. At the
semiclassical level, black holes are characterized by a universal Hawking
temperature, and in AdS/CFT Witten:1998qj ; Gubser:1998bc ; Banks:1998dd  they
are dual to thermal states of the boundary theory. One way to understand their
properties is to add light probe fields to the system and ask how they behave
in the presence of a heavy object. We can often answer such questions
precisely by studying the correlation functions of light fields in heavy
states, especially in holographic settings.

There are essentially two ways to understand how heavy objects affect light
ones in the bulk. From one perspective, the light state and the heavy state
come together, bind gravitationally, and form two-particle composite states
that can be thought of as orbiting configurations of the light field around
the heavy object. From another perspective, the light and heavy states each
propagate independently but exchange gravitons and other virtual particles,
like light-light or heavy-heavy composites, to mediate the interactions
between them. Both of these viewpoints are illustrated schematically below:

In the dual conformal field theory (CFT), the “binding” and “exchange”
philosophies correspond to the $s$\- and $t$-channel expansions of a
holographic correlator, respectively, and are related by crossing symmetry.
The requirement of crossing symmetry is highly constraining because the data
that enters a calculation in one channel can often be used to bootstrap what
must happen in the other channel Belavin:1984vu . In this way, one can gain
access to detailed information about CFT observables and their bulk duals. In
particular, (1) can teach us about the amplitudes to create the composite
states described above, as well as their expectation values in heavy states.

In this paper, we will make this precise within the framework of
$\mathrm{AdS}_{3}/\mathrm{CFT}_{2}$, a model where many calculations that are
essentially intractible in higher dimensions simplify and often yield exact
results. Below, we review how the relative simplicity of 3d gravity
Deser:1983tn ; Deser:1983nh  and the powerful tools available in 2d CFT
Belavin:1984vu  afford analytical control.

###  1.1 3d Gravity with Defects

In $(2+1)$-dimensional gravity, there are no local propagating bulk degrees of
freedom Deser:1983tn . As a consequence, every solution to pure 3d gravity
with a negative cosmological constant is locally isometric to the empty anti-
de Sitter (AdS) spacetime. The only solutions are $\mathrm{AdS}_{3}$ itself
and certain quotients of $\mathrm{AdS}_{3}$, the simplest of which is the BTZ
black hole Banados:1992wn . BTZ black holes of any size can exist, but their
masses must lie above a critical threshold mass $M_{*}=\frac{1}{8G}$
separating them from the pure $\mathrm{AdS}_{3}$ vacuum.111We work throughout
in natural units $c=\hbar=1$, and we set the AdS radius
$\ell_{\mathrm{AdS}}=1$. We also take the energy of the pure
$\mathrm{AdS}_{3}$ vacuum to be zero, instead of $-1$ as is more commonly
done. What happens to the geometry if the mass goes below the BTZ threshold?

In pure gravity, there are no such geometries. But if the theory is coupled to
pointlike sources, then it admits conical defect solutions describing a
massive particle in $\mathrm{AdS}_{3}$ with mass below the BTZ threshold
Deser:1983nh . An important class of defects consists of the
$\mathrm{AdS}_{3}/\mathbb{Z}_{N}$ orbifolds, which have been important in
string theory Lunin:2001jy ; Balasubramanian:2005qu ; Giusto:2004ip ;
Giusto:2020mup  and in connection with gravitational path integrals
Lewkowycz:2013nqa ; Benjamin:2020mfz . Generically, however, the parameter $N$
characterizing the defect’s strength does not need to be an integer, and most
conical defect solutions are not quotients of $\mathrm{AdS}_{3}$ in an obvious
way. In fact, the metric of conical AdS is identical to that of BTZ, but with
the horizon radius analytically continued so that the mass falls below the BTZ
threshold. This suggests that we should be able to probe the physics of black
holes from below the threshold by giving a unified treatment of heavy and
thermal states.

We pursue this treatment by studying the propagation of free fields in heavy
AdS backgrounds. We allow the fields to be massive, but we require them to be
light enough that they do not backreact on the geometry. Both below and above
the BTZ threshold, we construct the propagator for free fields by solving the
Klein–Gordon equation and summing over the spectrum of normal or quasinormal
modes, which play the role of the orbiting bound states on the LHS of (1). We
also obtain the propagator in a different way: using the method of images for
quotients of $\mathrm{AdS}_{3}$. A sum over images is always available in BTZ,
but only makes sense below the threshold in $\mathrm{AdS}_{3}/\mathbb{Z}_{N}$.
In both cases, it may be viewed as a saddle-point sum over the lengths of all
geodesics between two points in the quotient geometry. Although it is not
immediately obvious, this representation of the propagator corresponds to the
“exchange” calculation on the RHS of (1).

###  1.2 HHLL Correlators in 2d CFT

Semiclassical $\mathrm{AdS}_{3}$ gravity has a dual description in terms of a
holographic 2d CFT living at the asymptotic AdS boundary.222The existence of
such a CFT has not been shown, and the appropriate holographic dual may
instead involve an ensemble of theories Chandra:2022bqq . However, since every
member of this ensemble must share the same semiclassical properties, we may
refer to “the” boundary theory when discussing its large-$c$ behavior.
Holographic CFTs have a very large central charge $c=\frac{3}{2G}\gg 1$
corresponding to large-$N$ limit of gauge theories. We work at leading order
in the $1/c$ expansion, where many aspects of the CFT become universal
Hartman:2014oaa ; Fitzpatrick:2014vua ; Fitzpatrick:2015zha ; Collier:2019weq
.

In the CFT, the bulk field is dual to a light primary operator
$\mathcal{O}_{\mathrm{L}}$ with conformal dimension
$\Delta_{\mathrm{L}}=\Delta$, where “light” means that
$\frac{\Delta}{c}\longrightarrow 0$ as $c\longrightarrow\infty$. Since the
bulk field is free, its CFT dual is a generalized free field: this encodes the
notion of an effective field theory where fluctuations on top of the heavy
state are essentially free. Meanwhile, a conical defect is created by a heavy
CFT operator $\mathcal{O}_{\mathrm{H}}$ whose conformal dimension
$\Delta_{\mathrm{H}}=O(c)$ lies below the BTZ threshold.333Some authors use
“heavy” to refer to conformal dimensions that lie strictly above the BTZ
threshold. We relax this condition slightly and refer to both conical defects
and BTZ black holes as heavy. The defects we consider are static and non-
rotating, so in the CFT they are represented by scalar primaries. By the
operator-state correspondence, they can also be thought of as highly excited
pure states $\ket{\mathcal{O}_{\mathrm{H}}}$. By contrast, BTZ black holes are
dual to thermal states in the CFT. But the Eigenstate Thermalization
Hypothesis (ETH) Srednicki:1994mfb  tells us that up to exponentially small
corrections, it does not matter whether we consider a thermal state or a pure
state of the same average energy. Therefore both above and below the
threshold, we may describe heavy states by CFT primaries.

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{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{gathered}$ Figure
1: The configuration of fields defining the HHLL correlator is shown on the
left in AdS (where we have a nontrivial background geometry) and on the right
in the CFT (where, in radial quantization, the background is created by two
heavy insertions).

With heavy AdS backgrounds viewed in this way, our bulk propagator becomes a
heavy-heavy-light-light (HHLL) CFT correlator of the form
$\langle\mathcal{O}_{\mathrm{H}}\mathcal{O}_{\mathrm{L}}\mathcal{O}_{\mathrm{L}}\mathcal{O}_{\mathrm{H}}\rangle$.
This situation is illustrated in Figure 1, both in AdS (left) and in radial
quantization (right). CFT correlators are computed by using the operator
product expansion (OPE) to sum over intermediate states exchanged by the
external operators, as in (1). The organization of CFT states into irreducible
representations of the Virasoro algebra turns the OPE into an expansion in
functions called Virasoro conformal blocks, which have figured prominently in
the conformal bootstrap program Heemskerk:2009pn ; El-Showk:2011yvt ;
Fitzpatrick:2012yx ; Komargodski:2012ek ; Jackson:2014nla  and the study of
holographic entanglement Ryu:2006ef ; Headrick:2010zt ; Hartman:2013mia ;
Asplund:2014coa . Fortunately, the semiclassical HHLL Virasoro blocks are
known Fitzpatrick:2015zha , leaving only the questions of which states are
exchanged and how much they each contribute.

The main contributions to the OPE come from double-trace, or double-twist,
operators built from $\mathcal{O}_{\mathrm{L}}$ and
$\mathcal{O}_{\mathrm{H}}$. In the $s$ channel, these are heavy-light
composite states that describe excitations of the light field in the bulk and
correspond to the Fourier modes one obtains by solving the wave equation in
AdS. The associated OPE coefficients, which we deduce by matching the OPE to
our bulk result, are interpreted as the amplitudes of these excitations. In
the $t$ channel, the dominant contribution comes from the Virasoro vacuum
block, which encodes gravitational interactions between
$\mathcal{O}_{\mathrm{L}}$ and $\mathcal{O}_{\mathrm{H}}$. The vacuum block
gives the same contribution to the correlator as the minimal geodesic in the
sum over images Hartman:2013mia  and fully captures the singular structure of
the OPE limit where the two light operators are brought together
Berenstein:2022ico . The non-minimal geodesics making up the rest of the
correlator are then responsible for the regular terms in the OPE, and should
have a CFT interpretation in terms of non-vacuum exchanges. Indeed, there is
an infinite family of light-light double-twist states that provide exactly
these contributions.

The OPE coefficients of these primaries are identified as their expectation
values in heavy states, and much of the technical work in this paper is
devoted to computing them explicitly by matching the $t$-channel OPE to the
sum over images. Our main results, which comprise Sections 5–8, include
expressions for the expectation values of low-lying operators, asymptotic
formulæ for the HHLL Virasoro blocks and OPE coefficients at large twist, and
a discussion of their behavior near and above the BTZ threshold.

###  1.3 Main Results and Outline

By carrying out the analysis described above, we aim to achieve four
interrelated goals:

  1. 1.

Understand the emergence of thermal behavior. By studying conical defects of
any mass (not just $\mathrm{AdS}_{3}/\mathbb{Z}_{N}$), we fill in the gaps
between the integers and illustrate how physics below the BTZ threshold
smoothly connects to the physics in the black hole regime. In particular, we
describe how to analytically continue the sum over images for non-integer $N$.
The expectation values we compute analytically continue, above the threshold,
to thermal one-point functions in BTZ or thermal AdS.

  2. 2.

Give a unified treatment from a Lorentzian perspective. By treating both
defects and black holes as heavy operator insertions on the Lorentzian
cylinder, we avoid the need to switch to Euclidean signature above the BTZ
threshold. Thus we do not need a thermal circle to see thermal behavior.

  3. 3.

Move beyond vacuum dominance. By focusing on non-vacuum $t$-channel exchanges,
we point out that there is more to learn about CFT correlators beyond their
perturbative expansions in singular powers. These effects are inaccessible to
large-spin asymptotics in the $s$ channel, but are required by crossing
symmetry.

  4. 4.

Understand double-trace expectation values. In higher dimensions, it is of
great interest—but even greater difficulty—to understand correlators of
double-trace operators in black hole backgrounds. Now, at least in a
simplified toy model, we have the answers. The expectation values we compute
do not factorize at large $N$, in the sense that
$\langle\mathcal{O}_{\mathrm{L}}^{2}\rangle\neq\langle\mathcal{O}_{\mathrm{L}}\rangle\langle\mathcal{O}_{\mathrm{L}}\rangle$.
Instead, the effective classical background created by
$\mathcal{O}_{\mathrm{H}}$ modifies the expectation values of
$\langle\mathcal{O}_{\mathrm{L}}^{2}\rangle$ directly.

This paper is organized into two broad parts. In the first part, we compute
the HHLL correlator in four different ways: twice in the bulk, first by
solving the wave equation (Section 2 and Appendix A) and then using the method
of images (Section 3); and twice on the boundary, using the $s$-channel and
$t$-channel OPEs (Section 4). These computations are the bulk and boundary
avatars of the LHS and RHS of (1), respectively. In the second part, we
calculate the expectation values of light double-trace operators (Section 5)
and analyze their properties. We derive an asymptotic formula for highly
excited expectation values (Section 6), study their behavior as a function of
the mass of the defect or black hole (Section 7 and Appendix B), and discuss
their interpretation as thermal one-point functions (Section 8). We conclude
and indicate future directions in Section 9.

##  2 The Wave Equation

The background geometry created by a static, heavy object in
$\mathrm{AdS}_{3}$ can be written in global coordinates $(t,\theta,r)$, where
$\theta\sim\theta+2\pi$, as follows:

$\displaystyle\differential s^{2}$ $\displaystyle=-f(r)\differential
t^{2}+\frac{\differential r^{2}}{f(r)}+r^{2}\differential\theta^{2},\qquad
f(r)=\begin{cases}\displaystyle r^{2}+\frac{1}{N^{2}}&\text{below the
threshold},\\\\[11.0pt] r^{2}-r_{0}^{2}&\text{above the
threshold}.\end{cases}$ (4)

Below the BTZ threshold, the parameter $N\geq 1$ describes the strength of a
conical defect. Integer values of $N$ correspond to the cyclic orbifolds
$\mathrm{AdS}_{3}/\mathbb{Z}_{N}$, and the limit $N\longrightarrow\infty$
corresponds to the BTZ threshold. Above the threshold, a horizon of radius
$r_{0}\in\mathbb{R}_{+}$ forms, and the state becomes thermal with a
temperature given by $r_{0}=\frac{2\pi}{\beta}=2\pi T_{\mathrm{H}}$. The
geometries above and below the threshold are related by the analytic
continuation $N\longleftrightarrow-\frac{i}{r_{0}}$, so conical defects can be
viewed as black holes with imaginary temperature, and conversely BTZ can be
thought of as a conical geometry with an imaginary angular deficit. To unify
our descriptions of the two regimes, we write $f(r)=r^{2}+\alpha^{2}$, where

$\displaystyle\alpha=\frac{1}{N}\longleftrightarrow ir_{0}=\frac{2\pi
i}{\beta},\qquad\mathrm{i.e.}\qquad\alpha\equiv\begin{cases}\frac{1}{N}\in[0,1]&\text{below
the threshold}\\\ ir_{0}\in i\mathbb{R}_{+}&\text{above the
threshold}.\end{cases}$ (5)

Consider a free massive scalar field $\phi$ propagating in this background. We
decompose the solutions of its equation of motion, the Klein–Gordon equation,
into Fourier modes:

$\displaystyle\MT@delim@Auto\p@star{\Box-m^{2}}\phi_{\omega,\ell}=0,\qquad\phi_{\omega,\ell}(t,\theta,r)=e^{-i\omega
t}e^{i\ell\theta}R_{\omega,\ell}(r),\qquad\ell\in\mathbb{Z}$

On this ansatz, (2) reduces to a differential equation for the radial modes:

$\displaystyle\quantity[\frac{1}{r}\derivative{r}\MT@delim@Auto\p@star{rf(r)\derivative{r}}+\frac{\omega^{2}}{f(r)}-\frac{\ell^{2}}{r^{2}}-m^{2}]R_{\omega,\ell}(r)=0.$
(6)

The solutions to the radial equation can be expressed in terms of
hypergeometric functions. With appropriate boundary conditions, these mode
solutions can be used to construct the propagator. We review how this is done
in Appendix A, which includes a detailed treatment of the wave equation, its
solutions in various backgrounds, the relevant boundary conditions, and the
resulting propagators. Below, we summarize the essential results.

###  2.1 Correlators Below the Threshold

In conical AdS, we require the radial modes to be regular at the origin and
normalizable, i.e. decaying at infinity. Those solutions $R_{\omega,\ell}(r)$
which are regular at $r=0$ behave near the AdS boundary as linear combinations
of normalizable and non-normalizable modes:

$\displaystyle
R_{\omega,\ell}(r)\sim\mathcal{N}_{\omega,\ell}\MT@delim@Auto\p@star{A(\omega,\ell)r^{\Delta-2}+B(\omega,\ell)r^{-\Delta}}\qquad\mathrm{as}\quad
r\longrightarrow\infty.$ (7)

Here $\mathcal{N}_{\omega,\ell}$ is a normalization constant, $\Delta$ is the
conformal dimension of the CFT operator dual to $\phi$, and $A(\omega,\ell)$
and $B(\omega,\ell)$ are certain ratios of Gamma functions. To ensure the
normalizability of the modes, we must have $A(\omega,\ell)=0$. This condition
is satisfied on a discrete set of normal frequencies $\omega_{n\ell}$ derived
in Appendix A.3 and given by

$\displaystyle\omega_{n\ell}=|\ell|+\frac{1}{N}\big{(}\Delta+2n\big{)}\in\mathbb{R}_{+},\qquad\ell\in\mathbb{Z},\;\,n\in\mathbb{N}.$
(8)

The resulting normal modes can be made orthonormal with respect to the
Klein–Gordon inner product, and any solution to (2) can be expanded in normal
modes:

$\displaystyle\phi(x)=\sum_{n=0}\sum_{\ell\in\mathbb{Z}}\quantity[\phi_{n\ell}(x)a_{n\ell}+\phi_{n\ell}^{*}(x)a_{n\ell}^{\dagger}],\qquad\phi_{n\ell}(t,\theta,r)=e^{-i\omega_{n\ell}t}e^{i\ell\theta}R_{\omega_{n\ell},\ell}(r).$
(9)

We quantize the field by promoting $a_{n\ell}$ and $a_{n\ell}^{\dagger}$ to
operators obeying the canonical commutation relations
$[a_{n\ell},a_{n^{\prime}\ell^{\prime}}^{\dagger}]=\delta_{nn^{\prime}}\delta_{\ell\ell^{\prime}}$.
We choose the Hartle–Hawking vacuum $\ket{0}$ as the one annihilated by all of
the lowering operators $a_{n\ell}$, and finally we construct the bulk-to-bulk
propagator $G(x,x^{\prime})$ as a sum over the normal modes:

$\displaystyle
G(x,x^{\prime})=\matrixelement{0}{\phi(x)\phi(x^{\prime})}{0}=\sum_{n\ell}e^{-i\omega_{n\ell}(t-t^{\prime})}e^{i\ell(\theta-\theta^{\prime})}R_{\omega_{n\ell},\ell}(r)R_{\omega_{n\ell},\ell}^{*}(r^{\prime}).$
(10)

As we send $r\longrightarrow\infty$, The boundary values of the field modes
are rescaled by $r^{\Delta}$ Gubser:1998bc :

$\displaystyle\phi_{n\ell}(t,\theta)$
$\displaystyle=\lim_{r\to\infty}r^{\Delta}\phi_{n\ell}(t,\theta,r)=\mathcal{C}_{n\ell}^{\mathrm{HL}}e^{-i\omega_{n\ell}t}e^{i\ell\theta},\qquad\mathcal{C}_{n\ell}^{\mathrm{HL}}\equiv\lim_{r\to\infty}r^{\Delta}R_{\omega_{n\ell},\ell}(r).$
(11)

The coefficients
$\mathcal{C}_{n\ell}^{\mathrm{HL}}=\mathcal{N}_{\omega_{n\ell},\ell}B(\omega_{n\ell},\ell)$,
given by (20a), are computed in Appendix A.3 by normalizing the modes
$\phi_{n\ell}(x)$. These coefficients represent the amplitudes to produce an
excitation of the scalar on the boundary with energy $\omega_{n\ell}$ and
angular momentum $\ell$; we will see later that they are also OPE coefficients
for the CFT operators dual to $\phi_{n\ell}$.

We obtain the boundary-to-boundary propagator by sending
$r,r^{\prime}\longrightarrow\infty$ in (10) with
$t^{\prime}=\theta^{\prime}=0$ and using (11). The result is a sum over
Fourier modes on the boundary:

$\displaystyle
G(t,\theta)=\lim_{r,r^{\prime}\to\infty}\Big{(}r^{\Delta}r^{\prime\Delta}G(x,x^{\prime})\Big{)}=\sum_{n\ell}\MT@delim@Auto\p@star{\mathcal{C}_{n\ell}^{\mathrm{HL}}}^{2}e^{-i\omega_{n\ell}t}e^{i\ell\theta}=\sum_{n\ell}\mathcal{G}(\omega_{n\ell},\ell)e^{-i\omega_{n\ell}t}e^{i\ell\theta}.$
(12)

Here we have introduced the momentum-space boundary propagator
$\mathcal{G}(\omega,\ell)$ as the Fourier inverse of $G(t,\theta)$. In conical
AdS, it is defined only at the normal frequencies.

###  2.2 Correlators Above the Threshold

The story in BTZ is similar, but there are some important differences. For
instance, normal modes are not well defined because waves in black hole
backgrounds can fall through the horizon. In order to probe the black hole’s
thermal response to perturbations, we need to impose boundary conditions
suitable for the retarded Green’s function $G_{\mathrm{R}}(x,x^{\prime})$. The
modes should be purely ingoing at the horizon and decaying at infinity:

$\displaystyle
R_{\omega,\ell}(r)\sim\begin{cases}(r-r_{0})^{-\frac{i\omega}{2r_{0}}},&r\longrightarrow
r_{0},\\\ r^{-\Delta},&r\longrightarrow\infty\end{cases}$ (13)

The ingoing modes behave like (7) near the boundary, and the condition
$A(\omega,\ell)=0$ (which ensures that $R_{\omega,\ell}(r)$ decays at
infinity) is satisfied on a discrete set of complex frequencies
$\widetilde{\omega}_{n\ell}$ in the lower half plane associated to the
quasinormal modes (QNMs):

$\displaystyle\widetilde{\omega}_{n\ell}=-\ell-
ir_{0}\big{(}\Delta+2n\big{)}\in\mathbb{H}_{-},\qquad\ell\in\mathbb{Z},\;\;n\in\mathbb{N}.$
(14)

The real and imaginary parts of $\widetilde{\omega}_{n\ell}$ represent the
energy and decay rate of the QNMs, respectively;
$\mathrm{Im}(\widetilde{\omega}_{n\ell})<0$ ensures that the black hole is
stable to small perturbations. The QNMs have a characteristic lifetime
$\tau_{n}=\frac{1}{r_{0}\MT@delim@Auto\p@star{\Delta+2n}}$, independent of
their angular momentum. The longest-lived modes ($n=0$) dominate the retarded
propagator at late times, leading to an exponential decay
$G_{\mathrm{R}}(t,\theta)\sim e^{-r_{0}\Delta t}$ in the semiclassical limit.

Following the usual holographic prescription Klebanov:1999tb ;
Skenderis:2008dg , we identify $A(\omega,\ell)$ and $B(\omega,\ell)$ in (7) as
the source $J$ and linear response $\langle\mathcal{O}\rangle_{J}$,
respectively, for small perturbations of the boundary theory by an operator
dual to the bulk field $\phi$. In momentum space, the linear response is the
retarded Green’s function multiplied by the source, so

$\displaystyle\langle\mathcal{O}(\omega,\ell)\rangle_{J}=\mathcal{G}_{\mathrm{R}}(\omega,\ell)J(\omega,\ell)\implies\mathcal{G}_{\mathrm{R}}(\omega,\ell)=\frac{B(\omega,\ell)}{A(\omega,\ell)}.$
(15)

To obtain the position-space propagator from (15), we perform a Fourier
transform. The spectrum of normal modes in conical AdS becomes dense near the
BTZ threshold, so an integral over all frequencies is understood to replace
the discrete sum in (12):

$\displaystyle
G_{\mathrm{R}}(t,\theta)=\sum_{\ell\in\mathbb{Z}}\int_{-\infty}^{\infty}\frac{\differential\omega}{2\pi}\,\mathcal{G}_{\mathrm{R}}(\omega,\ell)e^{-i\omega
t}e^{i\ell\theta}.$ (16)

The momentum-space retarded Green’s function
$\mathcal{G}_{\mathrm{R}}(\omega,\ell)$ is a meromorphic function of $\omega$
whose only singularities are simple poles at the QNM frequencies
$\widetilde{\omega}_{n\ell}$.444There is one important exception. At $\ell=0$,
the QNM singularities with $\ell>0$ and $\ell<0$ coalesce into double poles on
the negative imaginary axis. These poles still have nontrivial residues; see
below. The integral in (16) may then be performed by closing the contour at
infinity in the lower half-plane and picking up all of the residues of
$\mathcal{G}_{\mathrm{R}}(\omega,\ell)$, as shown in Figure 2:

$\displaystyle
G_{\mathrm{R}}(t,\theta)=\sum_{n\ell}\big{(}\widetilde{\mathcal{C}}_{n\ell}^{\mathrm{HL}}\big{)}^{2}e^{-i\widetilde{\omega}_{n\ell}t}e^{i\ell\theta},\qquad\textbf{}\big{(}\widetilde{\mathcal{C}}_{n\ell}^{\mathrm{HL}}\big{)}^{2}\equiv
i\,\mathrm{Res}\big{[}\mathcal{G}_{\mathrm{R}}(\omega,\ell),\,\widetilde{\omega}_{n\ell}\big{]}.$
(17)

Both $\mathcal{G}_{\mathrm{R}}(\omega,\ell)$ and its residues are computed in
Appendix A.4; we discuss them below.

$r_{0}\Delta$$\omega$$\mathfrak{R}(\omega)$$\mathfrak{I}(\omega)$ Figure 2:
The poles of the momentum-space propagator in BTZ lie in the lower half-plane
and are located at the QNM frequencies
$\widetilde{\omega}_{\ell}=-\ell+ir_{0}\MT@delim@Auto\p@star{\Delta+2n}$,
where $\ell\in\mathbb{Z}$ and $n\in\mathbb{N}$. They are all simple poles
(red), except for the double poles at $\ell=0$ (orange). The contour used to
perform the integral (16) encloses all of these poles and is shown in blue.

###  2.3 Analytic Continuation to BTZ

The formal similarity between the conical and BTZ propagators (12) and (17) is
striking, especially given that their derivations are somewhat different. It
is natural to expect that the two should be related by an analytic
continuation in $\alpha$, perhaps in the sense that taking
$N\longrightarrow-\frac{i}{r_{0}}$ will convert (12) into (17) by sending

$\displaystyle\omega_{n\ell}\stackrel{{\scriptstyle?}}{{\longrightarrow}}\widetilde{\omega}_{n\ell}\qquad\mathrm{and}\qquad\MT@delim@Auto\p@star{\mathcal{C}_{n\ell}^{\mathrm{HL}}}^{2}\stackrel{{\scriptstyle?}}{{\longrightarrow}}\big{(}\widetilde{\mathcal{C}}_{n\ell}^{\mathrm{HL}}\big{)}^{2}.$
(18)

This naïve expectation turns out to be almost, but not quite, correct. As we
show in Appendix A.4, it is actually the momentum-space Green’s function,
_not_ the coefficients $\mathcal{C}_{n\ell}^{\mathrm{HL}}$, which analytically
continues across the BTZ threshold via $N\longrightarrow-\frac{i}{r_{0}}$. In
terms of the parameter $\alpha$, the Green’s function both above and below the
threshold is Son:2002sd

$\displaystyle\mathcal{G}(\omega,\ell)\propto\frac{\Gamma\Big{(}\frac{1}{2}\big{(}\Delta+\frac{\omega+\ell}{\alpha}\big{)}{}_{\\!}\Big{)}\,\Gamma\Big{(}\frac{1}{2}\big{(}\Delta+\frac{\omega-\ell}{\alpha}\big{)}{}_{\\!}\Big{)}}{\Gamma\Big{(}\frac{1}{2}\big{(}2-\Delta+\frac{\omega+\ell}{\alpha}\big{)}{}_{\\!}\Big{)}\Gamma\Big{(}\frac{1}{2}\big{(}2-\Delta+\frac{\omega-\ell}{\alpha}\big{)}{}_{\\!}\Big{)}}.$
(19)

What about the amplitudes $\mathcal{C}_{n\ell}^{\mathrm{HL}}$ and the residues
$\widetilde{\mathcal{C}}_{n\ell}^{\mathrm{HL}}$? They end up being nearly
identical, but they differ by an important factor that appears only in the BTZ
regime:

$\displaystyle\MT@delim@Auto\p@star{\mathcal{C}_{n\ell}^{\mathrm{HL}}}^{2}$
$\displaystyle=\frac{N^{1-2\Delta}}{\Gamma(\Delta)^{2}}\frac{\Gamma(\Delta+n+N|\ell|)\Gamma(\Delta+n)}{\Gamma(1+n+N|\ell|)\,\Gamma(1+n)},$
(20a)
$\displaystyle\big{(}\widetilde{\mathcal{C}}_{n\ell}^{\mathrm{HL}}\big{)}^{2}$
$\displaystyle=\frac{r_{0}^{2\Delta-1}}{\Gamma(\Delta)^{2}}\frac{\Gamma(\Delta+n-i|\ell|/r_{0})\Gamma(\Delta+n)}{\Gamma(1+n-i|\ell|/r_{0})\Gamma(1+n)}f_{n\ell}.$
(20b)

The BTZ factor $f_{n\ell}$ is given by

$\displaystyle f_{n\ell}$
$\displaystyle=\begin{cases}\displaystyle\frac{\sinh\MT@delim@Auto\p@star{\pi|\ell|/r_{0}+i\pi\Delta}}{\sinh\MT@delim@Auto\p@star{\pi|\ell|/r_{0}}},&\ell\neq
0,\\\ \displaystyle
2\cos\MT@delim@Auto\p@star{\pi\Delta}-\frac{2}{\pi}\sin\MT@delim@Auto\p@star{\pi\Delta}\big{(}\psi(1+n)-\psi(\Delta+n)\big{)},&\ell=0,\end{cases}$
(21)

where $\psi(z)=\frac{\Gamma^{\prime}(z)}{\Gamma(z)}$ is the digamma function.
This factor arises from the use of Euler’s reflection formula
$\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi z)}$ in evaluating the QNM residues
(17). If we had naïvely taken $N\longrightarrow-\frac{i}{r_{0}}$ in (20a), we
would have gotten (20b) wrong.

One way to understand the apparent asymmetry between these coefficients is to
view $f_{n\ell}$ as a thermal factor that becomes trivial at zero temperature
and influences the high-temperature behavior of the correlator. As a function
of the inverse temperature,

$\displaystyle
f_{n\ell}(\beta)=\frac{\sinh\MT@delim@Auto\p@star{\beta|\ell|/2+i\pi\Delta}}{\sinh\MT@delim@Auto\p@star{\beta|\ell|/2}},\qquad\big{|}f_{n\ell}(\beta)\big{|}^{2}=\cos^{2}\MT@delim@Auto\p@star{\pi\Delta}+\coth^{2}\MT@delim@Auto\p@star{\frac{\beta|\ell|}{2}}\sin^{2}\MT@delim@Auto\p@star{\pi\Delta}$
(22)

for $\ell\neq 0$. Up to an overall phase $e^{-i\pi\Delta}$ that can be
absorbed into the normalization of the propagator, $f_{n\ell}(\beta)$ quickly
approaches 1 at low temperatures and stays frozen at 1 below the BTZ
threshold: this explains its apparent absence in conical AdS.

###  2.4 Asymptotics at Large Spin

With this understanding, we can write down the propagator both above and below
the threshold in a unified manner. We adopt complex coordinates
$z=e^{\tau+i\theta}$ and $\bar{z}=e^{\tau-i\theta}$ with $\tau=it$ on the AdS
boundary cylinder, so that (12) and (17) become555To see this, observe that
the modes $\phi_{n\ell}(t,\theta)=e^{-i\omega_{n\ell}t}e^{i\ell\theta}$ become
$\phi_{n\ell}(z,\bar{z})=|z|^{-\Delta}|z|^{\alpha(\Delta+2n)}z^{\ell}$ in
complex coordinates. The leading factor $|z|^{-\Delta}$ is the cost of the
conformal transformation from the plane to the cylinder. It is understood in
the definition of $\mathcal{C}_{n\ell}^{\mathrm{HL}}$ that $|f_{n\ell}|\equiv
1$ below the BTZ threshold.

$\displaystyle G(z,\bar{z})$
$\displaystyle=\absolutevalue{z}^{(\alpha-1)\Delta}\sum_{n\ell}\MT@delim@Auto\p@star{\mathcal{C}_{n\ell}^{\mathrm{HL}}}^{2}\absolutevalue{z}^{2\alpha
n}z^{\ell},$ (23)
$\displaystyle\MT@delim@Auto\p@star{\mathcal{C}_{n\ell}^{\mathrm{HL}}}^{2}$
$\displaystyle=\frac{\alpha^{2\Delta-1}}{\Gamma(\Delta)^{2}}\frac{\Gamma(\Delta+n+|\ell|/\alpha)\Gamma(\Delta+n)}{\Gamma(1+n+|\ell|/\alpha)\,\Gamma(1+n)}f_{n\ell}(\beta).$

Figure 3: The equal-time propagator $G(\theta)$ is shown for $\Delta=2$ and
for several values of the mass, both below and above the BTZ threshold. The UV
divergence at $\theta=0$ is universal, and the propagator is periodic in
$\theta$. Long-range correlations are suppressed as the mass increases, and
for larger black holes $G(\theta)$ decays exponentially in $\theta$ away from
$\theta=0$.

The power series (23) is analytic in $|z|<1$, but it diverges at
$z,\bar{z}=1$. In Lorentzian signature, these singularities occur when
$e^{i(t\pm\theta)}=1$, i.e. on the boundary light cones $t\pm\theta=2\pi m$,
where $m\in\mathbb{Z}$ is the number of times the light cones wrap the
cylinder. To see this more clearly, it is useful to study the correlator at
equal times. Setting $t=0$ puts $|z|=1$ and makes $G(z,\bar{z})$ a function
only of the angular separation $\theta$ between the field insertions on the
boundary. In this case the sum over $n$ can be performed, and we find

$\displaystyle
G(\theta)=\frac{\alpha^{2\Delta-1}}{2\Gamma(2\Delta)\cos\MT@delim@Auto\p@star{\pi\Delta}}\sum_{\ell\in\mathbb{Z}}\MT@delim@Auto\p@star{\frac{\Gamma(\Delta+|\ell|/\alpha)}{\Gamma(1+|\ell|/\alpha-\Delta)}}f_{0\ell}(\beta)e^{i\ell\theta}.$
(24)

Figure 3 illustrates the behavior of $G(\theta)$ as $\alpha$ varies across the
BTZ threshold. The $z,\bar{z}=1$ singularity occurs at vanishing separation
$\theta=0$; this is the standard UV divergence that arises when two CFT
operators are brought close together. To diagnose the strength of the
singularity, we examine the behavior of the Fourier coefficients of (24) at
large $|\ell|$. The ratio of Gamma functions grows like
$\big{(}|\ell|/\alpha\big{)}^{2\Delta-1}$; meanwhile, $|f_{n\ell}|^{2}$
approaches 1 nonperturbatively quickly as $|\ell|\longrightarrow\infty$.666By
this we mean that the Taylor series of $|f_{n\ell}|^{2}$ at $|\ell|=\infty$,
i.e. in powers of $\frac{1}{|\ell|/\alpha}$, is identically 1. We substitute
$f_{n\ell}\sim 1$ and the asymptotic series for the ratio of Gamma functions
Erdelyi:1951aa  into (24), resum the resulting Fourier series term by term,
and expand in powers of $\theta$ to extract the divergence structure:

$\displaystyle G(\theta)$
$\displaystyle\sim\frac{\alpha^{2\Delta-1}}{2\Gamma(2\Delta)\cos\MT@delim@Auto\p@star{\pi\Delta}}\sum_{\ell\in\mathbb{Z}}\MT@delim@Auto\p@star{\big{(}|\ell|/\alpha\big{)}^{2\Delta-1}+O\MT@delim@Auto\p@star{\big{(}|\ell|/\alpha\big{)}^{2\Delta-3}}}e^{i\ell\theta}$
(25)
$\displaystyle=\theta^{-2\Delta}\MT@delim@Auto\p@star{1+\frac{\Delta\alpha^{2}}{12}\theta^{2}+\frac{\Delta(5\Delta+1)\alpha^{4}}{1440}\theta^{4}+O\MT@delim<EMAIL_ADDRESS>
This analysis shows that $f_{n\ell}$ is not felt at large $|\ell|$ and
therefore cannot contribute to the singular part of the correlator. It does,
however, strongly affect the first few terms of the series, which contribute
mainly to the part of $G(z,\bar{z})$ that is regular at $z,\bar{z}=1$. We will
show in Section 5 that the regular part of $G(z,\bar{z})$ contains important
information about the expectation values of certain two-particle states in
heavy backgrounds, and that it encodes certain universal OPE data in the dual
boundary theory.

##  3 The Method of Images

In spacetimes that can be realized as quotients of $\mathrm{AdS}_{3}$ by the
action of some group, we can obtain the bulk and boundary propagators by
passing to the pure AdS covering space and using the method of images. In this
section, we review this construction, discuss the singular and regular parts
of the propagator, explain the connection between the method of images and the
mode sums of Section 2, and relate the images to bulk geodesics.

###  3.1 The Sum Over Images

The bulk-to-bulk propagator in pure $\mathrm{AdS}_{3}$ is known and can be
expressed as a function of the geodesic distance $\sigma(x,x^{\prime})$
between two bulk points. It takes the form Hijano:2015qja

$\displaystyle
G_{\mathrm{AdS}}(x,x^{\prime})=\frac{e^{-\Delta\sigma(x,x^{\prime})}}{e^{-2\sigma(x,x^{\prime})}-1}=\MT@delim@Auto\p@star{\frac{\xi}{2}}^{\Delta}{}_{2}F_{1}\MT@delim@Auto\p@star{\frac{\Delta}{2},\frac{\Delta+1}{2},\Delta;\xi^{2}}=\frac{\xi^{\Delta}\MT@delim@Auto\p@star{1+\sqrt{1-\xi^{2}}}}{2\sqrt{1-\xi^{2}}},$
(26)

where $\sigma(x,x^{\prime})=\cosh^{-1}(1/\xi)$, and $\xi$ is given in global
coordinates by

$\displaystyle\xi\equiv\frac{1}{\cosh\sigma(x,x^{\prime})}=\frac{1}{\sqrt{\MT@delim@Auto\p@star{r^{2}+1}\MT@delim@Auto\p@star{r^{\prime
2}+1}}\cos(t-t^{\prime})-rr^{\prime}\cos(\theta-\theta^{\prime})}.$ (27)

On the boundary, (26) reduces to the standard result

$\displaystyle G_{\mathrm{AdS}}(t,\theta)=\frac{1}{\big{(}2\cos
t-2\cos\theta\big{)}^{\Delta}}.$ (28)

Below the BTZ threshold, the method of images applies only to the orbifolds
$\mathrm{AdS}_{3}/\mathbb{Z}_{N}$, i.e. for $N\in\mathbb{Z}_{+}$. Above the
threshold, it applies to black holes of any mass, since every BTZ spacetime is
a quotient of $\mathrm{AdS}_{3}$ by an action of $\mathbb{Z}$. The quotient
structure is depicted in Figure 4. In both cases, we start from the pure AdS
propagator (26), pass to the quotient by rescaling the coordinates
$(t,\theta,r)\mapsto(\alpha t,\alpha\theta,\frac{r}{\alpha})$, and sum over
images that rotate one of the insertion points by an angle $2\pi k\alpha$,
where $k\in\\{0,...,N-1\\}$ in conical AdS and $k\in\mathbb{Z}$ in BTZ. Each
field insertion also picks up an factor of $\alpha^{\Delta}$ in response to
the coordinate transformation. As a result, the bulk-to-bulk propagator in an
$\mathrm{AdS}_{3}$ quotient is

$\displaystyle G(x,x^{\prime})$
$\displaystyle=\alpha^{2\Delta}\sum_{k}\frac{e^{-\Delta\sigma_{k}(x,x^{\prime})}}{e^{-2\sigma_{k}(x,x^{\prime})}-1},\quad\sigma_{k}(x,x^{\prime})=\sigma\MT@delim@Auto\p@star{\alpha
t,\alpha t^{\prime};\,\alpha(\theta+2\pi
k),\alpha\theta^{\prime};\,\frac{r}{\alpha},\frac{r^{\prime}}{\alpha}}.$ (29)

It will also be useful, for future reference, to record the boundary-to-
boundary propagator below and above the threshold. Sending
$r\longrightarrow\infty$ in (29), we recover

$\displaystyle G_{\mathrm{con}}(t,\theta)$
$\displaystyle=\sum_{k=0}^{N-1}\frac{N^{-2\Delta}}{\Big{[}2\cos\MT@delim@Auto\p@star{\frac{t}{N}}-2\cos\MT@delim@Auto\p@star{\frac{\theta+2\pi
k}{N}}\Big{]}^{\Delta}},$ (30a) $\displaystyle G_{\mathrm{BTZ}}(t,\theta)$
$\displaystyle=\sum_{k\in\mathbb{Z}}\frac{r_{0}^{2\Delta}}{\Big{[}2\cosh\\!\big{(}r_{0}(\theta+2\pi
k)\big{)}-2\cosh\MT@delim<EMAIL_ADDRESS>(30b)

$\mathcal{H}$

Figure 4: The quotient structure of $\mathrm{AdS}_{3}/\mathbb{Z}_{N}$ (left)
and BTZ (middle) is shown on constant-time slices of $\mathrm{AdS}_{3}$. The
fundamental domain (gray) is embedded in pure AdS, and images of this domain
(dashed lines) are identified in the quotient. Two field insertions (black
points) are separated by some opening angle (red), and the action of the
quotient group rotates one of them to image points (brown) in the AdS cover.
As the fields are brought together, the renormalized length of the minimal
geodesic (cyan) diverges, while the other geodesics have finite lengths
because their endpoints remain well separated. The winding number of each
geodesic is equal to the number of dashed lines it crosses. In BTZ, we also
require that the geodesics do not cross the horizon (dotted line). The view
from the quotient geometry (right) shows that in addition to $\gamma_{0}$,
there is always another non-winding geodesic (green) which competes for
dominance with $\gamma_{0}$ for opening angles near $\theta=\pi$. In BTZ and
$\mathrm{AdS}_{3}/\mathbb{Z}_{N}$ with $N\geq 3$, there are also winding
geodesics (blue).

The results (30) can be unified using the holomorphic coordinates
$(z,\bar{z})$, which become $(z^{\alpha},\bar{z}^{\alpha})$ in the quotient
geometry. The images are generated by the action
$z^{\alpha}\mapsto\zeta^{k}z^{\alpha}$, where $\zeta=e^{2\pi i\alpha}$
represents a $2\pi\alpha$ rotation.777Below the BTZ threshold, $\zeta=e^{2\pi
i/N}$ is a root of unity. Above the threshold, $\zeta=e^{-2\pi r_{0}}$ is
real. In holomorphically factorized expressions like (31), we take the
“conjugate” of $\zeta$ to be $e^{+2\pi r_{0}}$. The boundary propagator in
$\mathrm{AdS}_{3}$ is the CFT two-point function
$G_{\mathrm{AdS}}(z,\bar{z})=\absolutevalue{1-z}^{-2\Delta}$, so we have

$\displaystyle
G(z,\bar{z})=\absolutevalue{z}^{(\alpha-1)\Delta}\sum_{k}\absolutevalue{\frac{1-\zeta^{k}z^{\alpha}}{\alpha}}^{-2\Delta}\qquad\mathrm{with}\quad\zeta=e^{2\pi
i\alpha}$ (31)

for any value of $\alpha$. The only remaining difference between the conical
and BTZ cases is the range of the sum ($\mathbb{Z}_{N}$ vs. $\mathbb{Z}$),
reflecting the structure of the quotient.

###  3.2 Singular Structure

As the two field insertions are brought close together, the propagator
diverges. The nature of this singularity is controlled by the direct ($k=0$)
image in (29–31), which inherits its UV behavior from the pure AdS propagator.
All of the other images are regular in the coincident limit, since they
correspond to configurations where the field insertions remain well separated
(see Figure 4). This gives us a natural decomposition of the propagator into
its regular and singular parts: we have
$G(x,x^{\prime})=G_{0}(x,x^{\prime})+G_{\mathrm{reg}}(x,x^{\prime})$, with

$\displaystyle
G_{0}(x,x^{\prime})=\alpha^{2\Delta}\frac{e^{-\Delta\sigma_{0}(x,x^{\prime})}}{e^{-2\sigma_{0}(x,x^{\prime})}-1},\qquad
G_{\mathrm{reg}}(x,x^{\prime})=\alpha^{2\Delta}\sum_{k\neq
0}\frac{e^{-\Delta\sigma_{k}(x,x^{\prime})}}{e^{-2\sigma_{k}(x,x^{\prime})}-1}.$
(32)

At short distances, the bulk-to-bulk propagator exhibits a divergence that
goes like the inverse of the shortest geodesic length
$\sigma_{0}(x,x^{\prime})$ between nearby points:

$\displaystyle G(x,x^{\prime})\approx
G_{0}(x,x^{\prime})=\frac{\alpha^{2\Delta}e^{-\Delta\sigma_{0}}}{e^{-2\sigma_{0}}-1}=-\frac{\alpha^{2\Delta}}{2\sigma_{0}}+(\mathrm{regular})\qquad\mathrm{as}\quad
x\longrightarrow x^{\prime}.$ (33)

For short radial geodesics, the geodesic length is approximately
$\sigma_{0}(x,x^{\prime})\sim\frac{r}{\alpha}$; therefore $G(x,x^{\prime})$
diverges like $1/r$. We may understand this behavior physically by zooming in
on two nearby points with $|r-r^{\prime}|\ll R_{\mathrm{AdS}}=1$. At these
scales the curvature of AdS becomes negligible, so $G(x,x^{\prime})$
approaches the flat-space propagator in 3 dimensions—but this is nothing more
than the Green’s function $\frac{1}{|r-r^{\prime}|}$ for the flat-space
Laplacian.

The short-distance behavior of the boundary propagator is markedly different,
since points on the conformal boundary with very small $t-t^{\prime}$ and
$\theta-\theta^{\prime}$ are still formally infinitely far apart. The
“coincident” limit is still controlled by the $k=0$ image, but this time
$G(z,\bar{z})$ exhibits a much stronger power-law divergence. From (31), near
$z,\bar{z}=1$ we have

$\displaystyle
G_{0}(z,\bar{z})=|z|^{(\alpha-1)\Delta}\absolutevalue{\frac{1-z^{\alpha}}{\alpha}}^{-2\Delta}=\absolutevalue{1-z}^{-2\Delta}\Big{(}1+O\big{(}|1-z|\big{)}{}_{\\!}\Big{)}.$
(34)

Meanwhile, the regular part of $G(z,\bar{z})$ remains finite at $z,\bar{z}=1$:

$\displaystyle
G_{\mathrm{reg}}(z,\bar{z})=\absolutevalue{z}^{(\alpha-1)\Delta}\sum_{k\neq
0}\absolutevalue{\frac{1-\zeta^{k}z^{\alpha}}{\alpha}}^{-2\Delta}=\sum_{k\neq
0}\absolutevalue{\frac{2}{\alpha}\sin(\pi
k\alpha)}^{-2\Delta}+O\big{(}|1-z|\big{)}.$ (35)

It can be checked that $G_{0}(z,\bar{z})$ has the same singular structure as
the full mode sum (23). This is easiest to see explicitly by comparison with
the equal-time propagator. Taking $z=e^{i\theta}$ and expanding (34) in powers
of $\theta\ll 1$, we get

$\displaystyle
G_{0}(\theta)=\MT@delim@Auto\p@star{\frac{2}{\alpha}\sin\MT@delim@Auto\p@star{\frac{\alpha\theta}{2}}}^{-2\Delta}=\theta^{-2\Delta}\MT@delim@Auto\p@star{1+\frac{\Delta\alpha^{2}}{12}\theta^{2}+\frac{\Delta(5\Delta+1)\alpha^{4}}{1440}\theta^{4}+O\big{(}\theta^{6}\big{)}{}_{\\!}},$
(36)

in agreement with (25). We see that $G_{0}(z,\bar{z})$ characterizes the
asymptotic behavior of the mode sum at large $|\ell|$, while
$G_{\mathrm{reg}}(z,\bar{z})$ is influenced by the small-$|\ell|$ terms in the
series, and is sensitive to the thermal factor $f_{n\ell}(\beta)$ above the
threshold.

###  3.3 Analytic Continuation

While the BTZ propagator is manifestly analytic in $r_{0}$, in conical AdS the
sum over images is finite and thus only defined for integer $N$. However, most
conical geometries have $N\notin\mathbb{Z}$ and are not quotients of
$\mathrm{AdS}_{3}$, so it is desirable to analytically continue (30a) to
arbitrary $N$. A partial solution to this problem is to approximate
$G(z,\bar{z})$ by its singular term $G_{0}(z,\bar{z})$, which is analytic in
$N$ and dominates near $z,\bar{z}=1$. But this is a poor approximation away
from $z,\bar{z}=1$, and it does not explain how one might also analytically
continue $G_{\mathrm{reg}}(z,\bar{z})$ to non-integer $N$. Below, we explain
how to proceed more carefully.

#### The full correlator.

We begin by applying Newton’s binomial formula

$\displaystyle\MT@delim@Auto\p@star{1-x}^{-\Delta}=\frac{1}{\Gamma(\Delta)}\sum_{m=0}^{\infty}\frac{\Gamma(\Delta+m)}{\Gamma(1+m)}x^{m},\qquad|x|<1,$
(37)

in (31), taking $x=\zeta^{k}z^{\alpha}$. Expanding in $z$ and $\bar{z}$
separately, we obtain

$\displaystyle G(z,\bar{z})$
$\displaystyle=\frac{\alpha^{2\Delta}}{\Gamma(\Delta)^{2}}\absolutevalue{z}^{(\alpha-1)\Delta}\sum_{k=0}^{N-1}\sum_{m,m^{\prime}=0}^{\infty}\frac{\Gamma(\Delta+m)\Gamma(\Delta+m^{\prime})}{\Gamma(1+m)\Gamma(1+m^{\prime})}z^{\alpha
m}\bar{z}^{\alpha m^{\prime}}\zeta^{k(m-m^{\prime})}.$ (38)

Now we can sum over the roots of unity $\zeta^{k(m-m^{\prime})}$ to get rid of
the sum over $k$. When this is done carefully,888The sum of the roots of unity
is a Kronecker-delta comb spaced by multiples of $N$,
$\displaystyle\sum_{k=0}^{N-1}e^{2\pi
ik(m-m^{\prime})/N}=N\sum_{\ell\in\mathbb{Z}}\delta_{m,m^{\prime}+N\ell}.$
(39) Substituting this into (38) sets $m^{\prime}=m+N|\ell|$ and replaces the
sum over $m^{\prime}$ with a sum over $\ell$. The absolute value is necessary
because we may have either $\ell\geq 0$ or $\ell<0$, but $m^{\prime}$ must be
nonnegative. one obtains precisely the sum over normal modes (23). We conclude
that the method of images knows about the wave equation; and vice versa, the
sum over modes analytically continues the sum over images to cases where it
does not apply. As we will discuss in Section 4.4, this calculation also
implements the crossing symmetry of CFT four-point functions and is related to
the conformal bootstrap program.

#### Regular and singular parts.

This calculation shows that $G_{\mathrm{reg}}(z,\bar{z})$ may be analytically
continued to non-integer $N$ by writing it as the difference between the mode
sum (23) and the singular image (34). This prescription can be somewhat
unweildy, however, because it involves the direct subtraction of two divergent
quantities.

We can do better by taking advantage of a symmetry: in
$\mathrm{AdS}_{3}/\mathbb{Z}_{N}$, the $k$th image in (31) is equal to the
$(N-k)$th image; and in BTZ, the $k$th image is equal to the $-k$th image.
Moreover, near $z,\bar{z}=1$, the dominant contribution to
$G_{\mathrm{reg}}(z,\bar{z})$ comes from (twice) the $k=1$ image. All of the
others are suppressed because they correspond to geodesics with nontrivial
winding around the singularity—see the discussion below.999Another way to
explain the dominance is to observe that the $k$th image in $G(z,\bar{z})$ has
a singularity at $\zeta^{k}z^{\alpha}=1$. The closest of these points to $z=1$
occurs at $k=1$ (as well as at $k=N-1$ in conical AdS or $k=-1$ in BTZ), so
the corresponding images contribute the most to $G_{\mathrm{reg}}$ near $z=1$.
By keeping only one contribution, we get an approximate analytic continuation
of $G_{\mathrm{reg}}(z,\bar{z})$:

$\displaystyle G_{\mathrm{reg}}(z,\bar{z})\approx
2|z|^{(\alpha-1)\Delta}\absolutevalue{\frac{1-\zeta
z^{\alpha}}{\alpha}}^{-2\Delta},\qquad\alpha\in
i\mathbb{R}_{+}\quad\mathrm{or}\quad\alpha\leq\frac{1}{3}.$ (40)

In conical AdS, the condition $N\geq 3$ is necessary because the $k=1$ and
$k=N-1$ images do not really exist for defect geometries with fewer than 3
distinct images. That said, (40) is exact for $N=3$, and is also exact for
$N=2$ without the leading factor of 2.

###  3.4 Images and Geodesics

The form of the propagator obtained by the method of images has a nice
geometric interpretation in terms of the geodesic method for heavy
correlators. For sufficiently massive fields, $G(z,\bar{z})$ is well
approximated by a saddle-point sum over all bulk geodesics between two
boundary points Balasubramanian:1999zv . There are $N$ such geodesics in
$\mathrm{AdS}_{3}/\mathbb{Z}_{N}$, and infinitely many in BTZ. Each geodesic
$\gamma_{k}$ contributes $e^{-m\mathcal{L}_{\mathrm{ren}}[\gamma_{k}]}$ to the
propagator, where $\mathcal{L}_{\mathrm{ren}}[\gamma_{k}]$ is the renormalized
geodesic length of $\gamma_{k}$. Direct calculations of the geodesic length
Berenstein:2022ico ; Martinez:2019nor  show that each of the images in (30)
have precisely this form.

In general, the geodesic approximation is valid only for very heavy fields; in
particular, one requires that $\Delta=1+\sqrt{1+m^{2}}\approx m\gg 1$.
However, in the present case the method of images is exact for any $\Delta$
and matches the geodesic method, so in fact the assumption $\Delta\gg 1$ is
unnecessary. What we do need to assume is that the field is light enough that
it does not backreact on the geometry. This means that the CFT operator dual
to $\phi$ has conformal dimension independent of the central charge, which is
taken to be very large. Probe operators with dimensions $1\ll\Delta\ll c$ are
sometimes called “hefty,” to distinguish them from truly heavy operators whose
dimensions scale linearly with $c$.

For nearby boundary points, the singular part dominates the correlator because
it corresponds to $\gamma_{0}$, which is always the shortest geodesic between
nearby two points. The other geodesics either pass around the other side of
the conical singularity or horizon, or else have a nontrivial winding number
around it. As a result, they are all longer than $\gamma_{0}$, and their
contributions to the propagator are exponentially suppressed at large
$\Delta$. But for larger opening angles, $\gamma_{0}$ starts to compete for
dominance with another geodesic on the other side of the boundary cylinder. In
conical AdS, this is $\gamma_{N-1}$, while in BTZ, it is $\gamma_{-1}$: see
Figure 4. This Ryu–Takayanagi phase transition Hartman:2013mia  occurs at
$\theta=\pi$ (i.e. $z,\bar{z}=-1$) and cannot be seen solely by looking at the
singular part of the correlator.

##  4 HHLL Correlators in 2d CFT

We now turn to the boundary description of heavy states in $\mathrm{AdS}_{3}$
gravity. We consider a holographic 2d CFT with large central charge $c\gg 1$,
working at leading order in the $1/c$ expansion. In this setting, the
propagator $G(z,\bar{z})$ becomes a heavy-heavy-light-light (HHLL) four-point
function, and to compute it we need to understand the spectrum of the theory
and the Virasoro blocks that contribute to the OPE. The computation of
$G(z,\bar{z})$ is based on techniques developed by Fitzpatrick, Kaplan, and
Walters Fitzpatrick:2014vua ; Fitzpatrick:2015zha , and in this section we use
their methods to interpret the results of Sections 2 and 3 holographically.

###  4.1 The Spectrum of Primaries

At the semiclassical level, the spectrum of a holographic 2d CFT dual to pure
$\mathrm{AdS}_{3}$ gravity is universal: besides the identity operator, it
contains a continuum of heavy primaries $\mathcal{O}_{\mathrm{H}}$ whose
conformal dimensions lie above the BTZ threshold $\Delta_{*}=\frac{c}{12}$,
with a Cardy density of states Hartman:2014oaa . These states are dual to
black holes,101010This is true in a coarse-grained sense. The spectrum of a
holographic CFT is conjectured to be discrete and chaotic, and one should
really average over the heavy states in a small energy window or over an
ensemble of 2d CFTs Collier:2019weq ; Chandra:2022bqq . However, all of these
subtleties concern effects that are subleading or exponentially suppressed in
$c$, and are therefore undetectable in the semiclassical limit. and their
dimensions

$\displaystyle\Delta_{\mathrm{H}}=M=\frac{c}{12}\MT@delim@Auto\p@star{1-\alpha^{2}},\qquad\alpha\equiv\sqrt{1-\frac{12\Delta_{\mathrm{H}}}{c}}$
(41)

correspond to the ADM masses of the BTZ spacetimes if $\alpha\in
i\mathbb{R}_{+}$ is identified with the parameter introduced in (5). To this
universal part of the spectrum, we can add matter in two ways. Firstly, we
will add conical defects by allowing $\alpha$ to take on real values in
$[0,1]$, so that $\Delta_{\mathrm{H}}$ falls below the threshold and causes
$\mathcal{O}_{\mathrm{H}}$ to create a static particle in the bulk with mass
(41). And secondly, we will add by hand a light scalar primary
$\mathcal{O}_{\mathrm{L}}$ of conformal dimension $\Delta_{\mathrm{L}}=\Delta$
independent of $c$. We take $\mathcal{O}_{\mathrm{L}}$ to be a generalized
free field, so that it is dual in the bulk to the free scalar $\phi$ studied
in Section 2. By definition, all correlators of $\mathcal{O}_{\mathrm{L}}$ are
determined from its two-point function via Wick’s theorem, and its OPE data is
universal and governed by mean-field theory (MFT).

We may use both $\mathcal{O}_{\mathrm{L}}$ and $\mathcal{O}_{\mathrm{H}}$ to
build composite states. Given any two primaries $\mathcal{O}_{1}$ and
$\mathcal{O}_{2}$, one can construct an infinite family of double-trace or
_double-twist_ primaries that must appear in their OPE and have the schematic
form
$[\mathcal{O}_{1}\mathcal{O}_{2}]_{n\ell}\sim\,:\\!\mathcal{O}_{1}\Box^{n}\overset{{}_{\leftrightarrow}}{\partial}{}^{\ell}\mathcal{O}_{2}\\!:-\,(\mathrm{traces})$.
The trace subtraction is necessary to ensure that
$[\mathcal{O}_{1}\mathcal{O}_{2}]_{n\ell}$ is a primary; moreover, if
$\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ are identical, then the spin $\ell$
must be even. These operators should be thought of as two-particle states
whose bulk description involves an orbiting configuration of $\mathcal{O}_{1}$
and $\mathcal{O}_{2}$, with angular momentum $\ell$ and excitation energy
labeled by $n$. Depending on $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$, there
are three classes of double-twist states to consider, described below. The
operator content that will be important to us is summarized in Table 1.

#### LL double-twists.

If both $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ are light, then at large
$|\ell|$ the spectrum of $[\mathcal{O}_{1}\mathcal{O}_{2}]_{n\ell}$ approaches
the universal MFT spectrum of double-trace operators Fitzpatrick:2014vua :

$\displaystyle
h_{[\mathrm{LL}]_{n\ell}}=\begin{cases}h_{1}+h_{2}+n+\ell,&\ell\geq 0,\\\
h_{1}+h_{2}+n,&\ell<0;\end{cases}\qquad\bar{h}_{[\mathrm{LL}]_{n\ell}}=\begin{cases}\bar{h}_{1}+\bar{h}_{2}+n,&\ell\geq
0,\\\ \bar{h}_{1}+\bar{h}_{2}+n-\ell,&\ell<0.\end{cases}$ (42)

In dealing with OPEs that sum over the light-light (LL) double-twist spectrum,
we will occasionally find it useful to change summation variables from
$(n,\ell)$ to $(n,n+|\ell|)$. To accomplish this, we can take advantage of
holomorphic factorization to decompose
$[\mathcal{O}_{1}\mathcal{O}_{2}]_{n\ell}(z,\bar{z})$ into a product of chiral
and anti-chiral primaries labeled by nonnegative integers
$m,m^{\prime}\in\mathbb{N}$ that satisfy $m+m^{\prime}=2n+|\ell|$ and
$m-m^{\prime}=\ell$. We write

$\displaystyle[\mathcal{O}_{1}\mathcal{O}_{2}]_{n\ell}(z,\bar{z})=[\mathcal{O}_{1}\mathcal{O}_{2}]_{m}(z)[\mathcal{O}_{1}\mathcal{O}_{2}]_{m^{\prime}}(\bar{z}),\qquad
n=\min(m,m^{\prime}),\quad\ell=m-m^{\prime}.$ (43)

These primaries have weights $h_{m}=h_{1}+h_{2}+m$ and
$\bar{h}_{m^{\prime}}=\bar{h}_{1}+\bar{h}_{2}+m^{\prime}$, respectively.

We will focus on LL states with
$\mathcal{O}_{1}=\mathcal{O}_{2}=\mathcal{O}_{\mathrm{L}}$. The simplest such
primary is the composite field
$[\mathcal{O}_{\mathrm{L}}\mathcal{O}_{\mathrm{L}}]_{00}=\,:\\!\mathcal{O}_{\mathrm{L}}^{2}\\!:$,
but in fact the whole family of LL double-twists will play an important role
throughout the rest of our discussion.111111From here on out we will drop the
normal-ordering symbol on $:\\!\mathcal{O}_{\mathrm{L}}^{2}\\!:$ for
notational simplicity.

#### HL double-twists.

If $\mathcal{O}_{1}$ is heavy and $\mathcal{O}_{2}$ is light, then the twist
$\tau=\min(h,\bar{h})$ of the states
$[\mathcal{O}_{\mathrm{H}}\mathcal{O}_{\mathrm{L}}]_{n\ell}$ must be rescaled
by a factor of $\alpha$ relative to (42):

$\displaystyle\Delta_{[\mathrm{LL}]_{n\ell}}\equiv\Delta_{n\ell}=\Delta_{1}+\Delta_{2}+2n+|\ell|,\qquad\Delta_{[\mathrm{HL}]_{n\ell}}=\Delta_{1}+\alpha\MT@delim<EMAIL_ADDRESS>(44)

The HL double-twists
$[\mathcal{O}_{\mathrm{H}}\mathcal{O}_{\mathrm{L}}]_{n\ell}$ are dual to the
bulk field modes $\phi_{n\ell}$. This can be seen from their spectrum, which
reproduces the energies (8) of the modes on top of the heavy state:
$\Delta_{[\mathrm{HL}]_{n\ell}}=\Delta_{\mathrm{H}}+\omega_{n\ell}$. (In pure
AdS, these modes are just descendants of $\mathcal{O}_{\mathrm{L}}$.)

#### HH double-twists.

Heavy-heavy (HH) double-twist operators are dual to bulk states containing
multiple defects. One needs to be careful about their construction: if the
defects are too heavy, it is impossible to combine them without creating a BTZ
black hole that swallows them both up Gott:1990zr . Such high-energy states
make for very short-lived, unstable virtual particles, and processes involving
them are exponentially suppressed in the central charge. Therefore in the
semiclassical limit it is safe to essentially ignore them.

Primary |  Dimension $\Delta$ |  Spin $\ell$ | AdS interpretation
---|---|---|---
$\mathcal{O}$ | $h+\bar{h}$ | $h-\bar{h}$ | (bulk excitations)
$\mathbbm{1}$ | $0$ | $0$ | pure AdS vacuum
$\mathcal{O}_{\mathrm{L}}$ | $\Delta_{\mathrm{L}}=\Delta$ | $0$ | free scalar in AdS
$\mathcal{O}_{\mathrm{H}}$ | $\Delta_{\mathrm{H}}=\frac{c}{12}\MT@delim@Auto\p@star{1-\alpha^{2}}$ | $0$ | heavy background
$[\mathcal{O}_{\mathrm{L}}\mathcal{O}_{\mathrm{L}}]_{n\ell}$ | $\Delta_{n\ell}=2\Delta+2n+|\ell|$ | $\ell\in 2\mathbb{Z}$ | light 2-particle state
$[\mathcal{O}_{\mathrm{H}}\mathcal{O}_{\mathrm{L}}]_{n\ell}$ | $\Delta_{\mathrm{H}}+\alpha\MT@delim@Auto\p@star{\Delta+2n}+|\ell|$ | $\ell\in\mathbb{Z}$ | orbiting light mode
Table 1: The spectrum $(h,\bar{h})$ of the primary operators important for our
calculations.

###  4.2 The Structure of the OPE

We work in radial quantization, using the coordinates $z=e^{\tau+i\theta}$ and
$\bar{z}=e^{\tau-i\theta}$ (with $\tau=it$ the Euclidean time) to map the
cylinder at the boundary of $\mathrm{AdS}_{3}$ to the complex plane. The
boundary correlator $G(z,\bar{z})$ is the two-point function of
$\mathcal{O}_{\mathrm{L}}$ in the heavy state
$\ket{\mathcal{O}_{\mathrm{H}}}=\mathcal{O}_{\mathrm{H}}(0)\ket{0}$, or in
other words the normalized HHLL four-point function

$\displaystyle
G(z,\bar{z})=\langle\mathcal{O}_{\mathrm{H}}(\infty)\mathcal{O}_{\mathrm{L}}(1)\mathcal{O}_{\mathrm{L}}(z,\bar{z})\mathcal{O}_{\mathrm{H}}(0)\rangle.$
(52)

(Here the normalization
$\mathcal{O}(\infty)=\lim_{y\to\infty}|y|^{2\Delta_{\mathcal{O}}}\mathcal{O}(y,\bar{y})$
is understood implicitly.)

Let us review how to compute correlation functions like $G(z,\bar{z})$ by
using the OPE to sum over intermediate states. We start by inserting into (52)
a complete set of states $\mathbbm{1}=\sum_{\psi}\ket{\psi}\\!\bra{\psi}$,
where the basis runs over all of the primaries $\mathcal{O}_{h,\bar{h}}$ in
the spectrum and their descendants. This decomposes $G(z,\bar{z})$ into a sum
of primary and descendant three-point functions; by convention, we keep the
sum over descendants implicit. The OPE becomes a sum over the weights
$(h,\bar{h})$ of all primaries in the theory, and assumes one of two forms
depending on whether distinct or identical operators are paired up in (52):

$\displaystyle G(z,\bar{z})$
$\displaystyle=\langle\wick[sep=0pt,offset=1.3em]{\c{1}\mathcal{O}_{\mathrm{H}}\c{1}\mathcal{O}_{\mathrm{L}}\c{2}\mathcal{O}_{\mathrm{L}}\c{2}\mathcal{O}_{\mathrm{H}}}\rangle=\sum_{\psi}\bra{\mathcal{O}_{\mathrm{H}}\mathcal{O}_{\mathrm{L}}}\ket{\psi}\\!\bra{\psi}\ket{\mathcal{O}_{\mathrm{L}}\mathcal{O}_{\mathrm{H}}}=\sum_{h,\bar{h}}\big{(}\mathcal{C}_{h,\bar{h}}^{\mathrm{HL}}\big{)}^{2}\mathcal{F}^{(\mathrm{s})}_{h}\overline{\mathcal{F}}^{(\mathrm{s})}_{\bar{h}}$
(53a)
$\displaystyle=\langle\wick{\c{1}\mathcal{O}_{\mathrm{H}}\c{2}\mathcal{O}_{\mathrm{L}}\c{2}\mathcal{O}_{\mathrm{L}}\c{1}\mathcal{O}_{\mathrm{H}}}\rangle=\sum_{\psi}\bra{\mathcal{O}_{\mathrm{L}}\mathcal{O}_{\mathrm{L}}}\ket{\psi}\\!\bra{\psi}\ket{\mathcal{O}_{\mathrm{H}}\mathcal{O}_{\mathrm{H}}}=\sum_{h,\bar{h}}\mathcal{C}^{\mathrm{LL}}_{h,\bar{h}}\mathcal{C}^{\mathrm{HH}}_{h,\bar{h}}\mathcal{F}^{(\mathrm{t})}_{h}\overline{\mathcal{F}}^{(\mathrm{t})}_{\bar{h}}.$
(53b)

The holomorphically factorizing functions
$\mathcal{F}_{h}(z)\overline{\mathcal{F}}_{\bar{h}}(\bar{z})$ are Virasoro
blocks: they encode the contributions to $G(z,\bar{z})$ of a primary and all
of its descendants by summing up the descendant three-point functions implicit
in (53). Because CFT three-point functions are fixed by conformal symmetry up
to normalization, the Virasoro blocks are (in principle) also fixed by
symmetry, while the corresponding OPE coefficients $\mathcal{C}_{h,\bar{h}}$
are left unfixed.

What are the intermediate states that propagate in each OPE channel? Because
$\mathcal{O}_{\mathrm{L}}$ is a generalized free field, only the identity and
the double-twist primaries can contribute. Their contributions in both OPE
channels can be summarized pictorially:

$\displaystyle\displaystyle\sum_{n\ell}\scalebox{0.7}{$\absolutevalue{\begin{gathered}\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \large\vertex(a);
\vertex[above left = 0.1cm and 1.5cm of a] (a1) {$\mathcal{O}_{\mathrm{L}}$};
\vertex[above right = 0.1cm and 1.5cm of a] (a2) {$\mathcal{O}_{\mathrm{H}}$};
\vertex[above = 0.1cm of a] (c1) {$\mathcal{C}_{n\ell}^{\mathrm{HL}}$};
\vertex[below = 3cm of a] (b); \vertex[below left = 0.1cm and 1.5cm of b] (b1)
{$\mathcal{O}_{\mathrm{L}}$}; \vertex[below right = 0.1cm and 1.5cm of b] (b2)
{$\mathcal{O}_{\mathrm{H}}$}; \vertex[below = 0.1cm of b]
(c2){$\mathcal{C}_{n\ell}^{\mathrm{HL}}$}; \diagram* { (a1) -- [thick] (a) --
[thick] (a2), (a) -- [scalar, edge label =
$\;{[\mathcal{O}_{\mathrm{H}}\mathcal{O}_{\mathrm{L}}]}_{n\ell}$, thick] (b),
(b1) -- [thick] (b) -- [thick] (b2), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{gathered}}$}^{\,2}$
$\displaystyle=\scalebox{0.7}{$\absolutevalue{\begin{gathered}\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \large\vertex(a);
\vertex[left = 0.1cm of a] (a1); \vertex[above left = 1.5cm and 0.1cm of a]
(b) {$\mathcal{O}_{\mathrm{L}}$}; \vertex[below left = 1.5cm and 0.1cm of a]
(d) {$\mathcal{O}_{\mathrm{L}}$}; \vertex[right = 3cm of a] (f); \vertex[right
= 0.1cm of f] (g); \vertex[above right = 1.5cm and 0.1cm of f] (c)
{$\mathcal{O}_{\mathrm{H}}$}; \vertex[below right = 1.5cm and 0.1cm of f] (e)
{$\mathcal{O}_{\mathrm{H}}$}; \diagram* { (b) -- [thick] (a) -- [thick] (d),
(a) -- [scalar, edge label = {$\mathbbm{1},\;T(z),\;...$}, thick] (f), (c) --
[thick] (f) -- [thick] (e), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{gathered}}$}^{\,2}+\displaystyle\sum_{\begin{subarray}{c}n\ell\\\
\ell\;\mathrm{even}\end{subarray}}\scalebox{0.7}{$\absolutevalue{\begin{gathered}\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \large\vertex(a);
\vertex[left = 0.1cm of a] (a1); \vertex[above left = 1.5cm and 0.1cm of a]
(b) {$\mathcal{O}_{\mathrm{L}}$}; \vertex[below left = 1.5cm and 0.1cm of a]
(d) {$\mathcal{O}_{\mathrm{L}}$}; \vertex[right = 3cm of a] (f); \vertex[right
= 0.1cm of f] (g); \vertex[above right = 1.5cm and 0.1cm of f] (c)
{$\mathcal{O}_{\mathrm{H}}$}; \vertex[below right = 1.5cm and 0.1cm of f] (e)
{$\mathcal{O}_{\mathrm{H}}$}; \vertex[below right = 0.0cm and 0.0cm of a]
(a2){$\mathcal{C}_{n\ell}^{\mathrm{LL}}$}; \vertex[below left = 0.0cm and
0.0cm of f] (a2){$\mathcal{C}_{n\ell}^{\mathrm{HH}}$}; \diagram* { (b) --
[thick] (a) -- [thick] (d), (a) -- [scalar, edge label =
{$[\mathcal{O}_{\mathrm{L}}\mathcal{O}_{\mathrm{L}}]_{n\ell}$}, thick] (f),
(c) -- [thick] (f) -- [thick] (e), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{gathered}}$}^{\,2}$
(57) $\displaystyle G(z,\bar{z})\qquad\quad$ $\displaystyle=\qquad\quad
G_{0}(z,\bar{z})\qquad\quad\;+\qquad\qquad G_{\mathrm{reg}}(z,\bar{z}).$

In the $s$ channel, $\mathcal{O}_{\mathrm{H}}$ and $\mathcal{O}_{\mathrm{L}}$
are brought together and combine to form the HL double-twists
$[\mathcal{O}_{\mathrm{H}}\mathcal{O}_{\mathrm{L}}]_{n\ell}$. The bulk picture
is that of a light field mode
$\phi_{n\ell}(t,\theta)=\mathcal{C}_{n\ell}^{\mathrm{HL}}e^{-i\omega_{n\ell}t}e^{i\ell\theta}$
orbiting a heavy object. We will show soon that the $s$-channel sum over HL
double-twists agrees with the sum over these bulk field modes in (23), and
this will allow us to identify the amplitudes
$\mathcal{C}_{n\ell}^{\mathrm{HL}}$ with the OPE coefficients of the same name
in (57).

In the $t$ channel, pairwise identical operators are brought together. The
dominant OPE contribution comes from the Virasoro vacuum block, which encodes
gravitational interactions between $\mathcal{O}_{\mathrm{H}}$ and
$\mathcal{O}_{\mathrm{L}}$. As we will see, the vacuum block reproduces the
singular image $G_{0}(z,\bar{z})$. But there is more to $G(z,\bar{z})$ than
its singular part, so there must be other states besides the vacuum running in
the $t$ channel. In other words, gravitons cannot be the only virtual
particles that mediate long-range interactions between
$\mathcal{O}_{\mathrm{H}}$ and $\mathcal{O}_{\mathrm{L}}$. There are two
natural candidates for additional states that could be exchanged: two
$\mathcal{O}_{\mathrm{L}}$ states could fuse to form
$[\mathcal{O}_{\mathrm{L}}\mathcal{O}_{\mathrm{L}}]_{n\ell}$, or two
$\mathcal{O}_{\mathrm{H}}$ states could fuse to form
$[\mathcal{O}_{\mathrm{H}}\mathcal{O}_{\mathrm{H}}]_{n\ell}$. But as discussed
above, the latter process is strongly suppressed, so we focus on the former.

The fact that the LL double-twists
$[\mathcal{O}_{\mathrm{L}}\mathcal{O}_{\mathrm{L}}]_{n\ell}$ contribute
nontrivially to $G(z,\bar{z})$ implies that the corresponding OPE coefficients
$\mathcal{C}_{n\ell}^{\mathrm{LL}}$ and $\mathcal{C}_{n\ell}^{\mathrm{HH}}$
are nonzero. We will study these coefficients extensively in the remainder of
the paper. For now, the bulk interpretation is that whenever heavy states in
3d gravity couple to matter, they interact at long range by exchanging an
infinite family of light two-particle states. Another way to say this is that
conical defects and BTZ black holes admit infinitely many species of quantum
hair.

###  4.3 Heavy-Light Virasoro Blocks

Now we shall assemble the complete $s$\- and $t$-channel OPEs for
$G(z,\bar{z})$, and then match them to our bulk results to find the OPE
coefficients in both channels.

#### The $s$ channel.

The $s$-channel Virasoro blocks $\mathcal{F}_{h}^{(\mathrm{s})}(z)$ are sums
of three-point functions of the form
$\langle\mathcal{O}_{\mathrm{H}}\mathcal{O}_{\mathrm{L}}\mathcal{O}_{h}\rangle$,
where the internal operator $\mathcal{O}_{h}$ is either a double-twist primary
$[\mathcal{O}_{\mathrm{H}}\mathcal{O}_{\mathrm{L}}]_{n\ell}$ or one of its
descendants. In the case of primary exchange, we get

$\displaystyle\langle\mathcal{O}_{h}(\infty)\mathcal{O}_{\mathrm{L}}(z)\mathcal{O}_{\mathrm{H}}(0)\rangle=\mathcal{C}^{\mathrm{HL}}_{h}z^{h-h_{\mathrm{H}}-h_{\mathrm{L}}}\implies\mathcal{F}_{h}^{(\mathrm{s})}(z)\supset
z^{h-h_{\mathrm{H}}-h_{\mathrm{L}}},$ (58)

where $\mathcal{C}^{\mathrm{HL}}_{h}$ is an OPE coefficient and $\supset$
means “contains as a term.” In fact, this is all we need, since all descendant
OPE contributions are suppressed at large $c$. To see this, note that
descendant three-point functions can be obtained from (58) by applying a
differential operator to $\mathcal{O}_{h}$ and then dividing by the norm of
the descendant state. The differential operator generates a function of $z$
with a prefactor polynomial in $h-h_{\mathrm{H}}-h_{\mathrm{L}}$, while the
norm of the descendant is a polynomial in $c$ and $h$ of the same degree as
the level of the descendant. (For instance,
$L_{-1}\mathcal{O}_{h}=\partial\mathcal{O}_{h}$ has norm
$\matrixelement{h}{L_{1}L_{-1}}{h}=2h$.) The HL double-twists have $h=O(c)$
but $h-h_{\mathrm{H}}-h_{\mathrm{L}}=O(1)$, so the contribution of a level-$k$
descendant is $O(c^{-k})$. Thus the $s$-channel blocks of the
$[\mathcal{O}_{\mathrm{H}}\mathcal{O}_{\mathrm{L}}]_{n\ell}$ reduce to scaling
blocks,

$\displaystyle\mathcal{F}_{h}^{(\mathrm{s})}(z)=z^{h-h_{\mathrm{H}}-h_{\mathrm{L}}}\big{(}1+O(1/c)\big{)},\qquad
h=h_{[\mathrm{HL}]_{n\ell}}.$ (59)

To put together the full correlator, we multiply in the anti-holomorphic part
of the block and sum over the double-twist spectrum, as in (57):

$\displaystyle
G(z,\bar{z})=\sum_{n\ell}\MT@delim@Auto\p@star{\mathcal{C}_{n\ell}^{\mathrm{HL}}}^{2}\mathcal{F}_{[\mathrm{HL}]_{n\ell}}^{(\mathrm{s})}(z)\overline{\mathcal{F}}_{[\mathrm{HL}]_{n\ell}}^{(\mathrm{s})}(\bar{z})=\absolutevalue{z}^{(\alpha-1)\Delta}\sum_{n\ell}\MT@delim@Auto\p@star{\mathcal{C}_{n\ell}^{\mathrm{HL}}}^{2}\absolutevalue{z}^{2\alpha
n}z^{\ell}.$ (60)

As promised, this OPE takes the same form as the mode sum (23). Matching the
two expressions allows us to identify the OPE coefficients
$\mathcal{C}_{n\ell}^{\mathrm{HL}}$ with the normal mode amplitudes given in
(20).121212Above the BTZ threshold, the frequencies
$\widetilde{\omega}_{n\ell}$ and coefficients
$\widetilde{\mathcal{C}}_{n\ell}^{\mathrm{HL}}$ are associated to quasinormal
modes in the bulk. Since they are complex-valued, they should not be
interpreted as the conformal dimensions or OPE data of physical states in a
unitary CFT. See Dodelson:2022eiz ; Giusto:2023awo  for a discussion of
similar issues. We conclude that the bulk theory knows how to compute
$s$-channel OPEs, and moreover it also knows the HL double-twist OPE data.

#### The $t$ channel.

The semiclassical HHLL $t$-channel Virasoro blocks were worked out by
Fitzpatrick, Kaplan, and Walters Fitzpatrick:2015zha . Their key insight was
that the presence of heavy operators in the correlator modifies the background
geometry in which the light fields propagate. The result is that the HHLL
blocks reduce to the all-light (i.e. global) conformal blocks, with the light
fields evaluated in a new set of coordinates:

$\displaystyle\mathcal{F}_{h}^{(\mathrm{t})}(z)$
$\displaystyle=z^{(\alpha-1)h_{\mathrm{L}}}\MT@delim@Auto\p@star{\frac{w}{\alpha}}^{h-2h_{\mathrm{L}}}{}_{2}F_{1}\MT@delim@Auto\p@star{h,h;2h;w},\qquad
w(z)=1-z^{\alpha}.$ (61)

The replacement $z\mapsto z^{\alpha}$ above is the same as the change of
boundary coordinates from pure $\mathrm{AdS}_{3}$ to a heavy background that
shows up in the method of images. From (61), we obtain the vacuum block by
setting $(h,\bar{h})=(0,0)$, and we get the LL double-twist blocks by
substituting the MFT spectrum (42). The vacuum contribution is

$\displaystyle
G_{0}(z,\bar{z})=\absolutevalue{\mathcal{F}_{0}^{(\mathrm{t})}(z)}^{2}=\absolutevalue{z}^{(\alpha-1)\Delta}\absolutevalue{\frac{1-z^{\alpha}}{\alpha}}^{-2\Delta},$
(62)

in agreement with the singular image (34), while the contributions of the LL
double-twist blocks must sum together to reproduce the regular piece of the
correlator:

$\displaystyle G_{\mathrm{reg}}(z,\bar{z})$
$\displaystyle=\absolutevalue{z}^{(\alpha-1)\Delta}\sum_{\begin{subarray}{c}n\ell\\\
\ell\;\mathrm{even}\end{subarray}}\mathcal{C}_{n\ell}^{\mathrm{LL}}\mathcal{C}_{n\ell}^{\mathrm{HH}}\absolutevalue{\frac{w}{\alpha}}^{2n}\MT@delim@Auto\p@star{\frac{w}{\alpha}}^{\ell}\Big{|}{}_{2}F_{1}(h_{n\ell},h_{n\ell};2h_{n\ell};w)\Big{|}^{2}.$
(63)

Here we set $h_{n\ell}=h_{[\mathrm{LL}]_{n\ell}}$ and recall that absolute
values are shorthand for holomorphically factorized products. The products
$\mathcal{C}_{n\ell}^{\mathrm{LL}}\mathcal{C}_{n\ell}^{\mathrm{HH}}$ must be
even under $\ell\longrightarrow-\ell$, because the primaries
$[\mathcal{O}_{\mathrm{L}}\mathcal{O}_{\mathrm{L}}]_{n,\pm\ell}$ should
contribute equally to the OPE
$\mathcal{O}_{\mathrm{L}}\times\mathcal{O}_{\mathrm{L}}$ of two scalars.

###  4.4 Crossing and Lorentzian Inversion

Let us make several bootstrap-adjacent comments complementary to our
perspective.

#### The vacuum block.

The Virasoro vacuum block in 2d CFT is nontrivial: it contains the identity,
the stress tensor, and an infinite tower of multi-stress tensor descendants.
In the bulk, these states excite topological degrees of freedom called
boundary gravitons, and their $t$-channel exchange between
$\mathcal{O}_{\mathrm{H}}$ and $\mathcal{O}_{\mathrm{L}}$ creates the
effective classical background defined by the $w$ coordinate. In the
$w$-plane, the vacuum block becomes the two-point function of
$\mathcal{O}_{\mathrm{L}}$, which is determined by a geodesic length. This is
one way in which the vacuum block knows about bulk geometry. But the vacuum
does not know everything, even at the semiclassical level: it is not single-
valued in $z$, and as we have discussed it fails to capture
$G_{\mathrm{reg}}(z,\bar{z})$. None of the blocks (61) are single-valued
either, but the fact that their sum must be analytic puts strong constraints
on the coefficients
$\mathcal{C}_{n\ell}^{\mathrm{LL}}\mathcal{C}_{n\ell}^{\mathrm{HH}}$.

#### Crossing symmetry.

Crossing symmetry is the statement that the $s$-channel and $t$-channel OPEs
compute the same four-point function $G(z,\bar{z})$. This is formally manifest
in (53), and our calculation (38) proving the equivalence of the mode sum and
the method of images provides an explicit demonstration of crossing. The fact
that $G(z,\bar{z})$ is a linear combination of Virasoro blocks in both
channels implies that each block in one channel can be expressed as a linear
combination of the blocks in the other. The matrix that inverts individual
$t$-channel blocks into combinations of $s$-channel blocks is is known exactly
and is called the Virasoro fusion kernel Collier:2018exn , and the use of this
kernel to bootstrap OPE data gives rise to the Euclidean and Lorentzian
inversion formulæ Simmons-Duffin:2017nub .

We are essentially working in the opposite direction: by independently
computing $G(z,\bar{z})$ using bulk methods, we start out with the answer for
$G(z,\bar{z})$ as well as all of its contributing Virasoro blocks. And by
matching the answer to its OPE, we can deduce all of the OPE coefficients and
thereby “derive” the fusion kernel.

#### Lorentzian inversion.

We have seen that the singular structure of the $t$-channel vacuum block (62)
controls the asymptotic behavior of the $s$-channel OPE (60). This is the
statement that applying an inversion formula to the vacuum block produces the
dominant contribution to the $s$-channel OPE data. As emphasized above, the
vacuum cannot provide the only contribution; indeed, the LL double-twist
blocks were inverted in Collier:2018exn  and shown to give nontrivial
subleading contributions to the $s$-channel OPE data. Our “inversion” of the
full sum over images in (38) is another way to see that the OPE coefficients
$\mathcal{C}_{n\ell}^{\mathrm{HL}}$ receive important contributions from both
vacuum and non-vacuum exchange.

#### A conical Farey tail?

It was proposed in Maloney:2016kee  that the full correlator can be built from
the vacuum block by summing over its $\mathrm{PSL}(2,\mathbb{Z})$ images. In
the case of HHLL correlators above the BTZ threshold, this prescription
coincides with the method of images as we have discussed it. But for conical
defects, the situation is less clear: in $\mathrm{AdS}_{3}/\mathbb{Z}_{N}$ we
have only finitely many images, and for defects that are not AdS quotients the
sum over images is not well defined in an obvious way. (See Arefeva:2016nic ;
li2024spinning  for proposals, however.) In any case, there should be a
definite relation between the $\mathrm{PSL}(2,\mathbb{Z})$ proposal and the
$t$-channel sum over LL double-twist blocks discussed here. Below we develop a
partial understanding of this correspondence by matching the image
representation of $G_{\mathrm{reg}}(z,\bar{z})$ to its $t$-channel OPE, but it
would be desirable to understand the situation in greater generality.

##  5 Expectation Values in Heavy States

In this section, we initiate our study of the OPE coefficients
$\mathcal{C}_{n\ell}^{\mathrm{LL}}$ and $\mathcal{C}_{n\ell}^{\mathrm{HH}}$.
Both sets of coefficients are normalization constants for three-point
functions involving
$[\mathcal{O}_{\mathrm{L}}\mathcal{O}_{\mathrm{L}}]_{n\ell}$:

$\displaystyle\mathcal{C}_{n\ell}^{\mathrm{LL}}=\langle\mathcal{O}_{\mathrm{L}}[\mathcal{O}_{\mathrm{L}}\mathcal{O}_{\mathrm{L}}]_{n\ell}\mathcal{O}_{\mathrm{L}}\rangle,\qquad\mathcal{C}_{n\ell}^{\mathrm{HH}}=\langle\mathcal{O}_{\mathrm{H}}[\mathcal{O}_{\mathrm{L}}\mathcal{O}_{\mathrm{L}}]_{n\ell}\mathcal{O}_{\mathrm{H}}\rangle.$
(64)

The coefficients $\mathcal{C}_{n\ell}^{\mathrm{LL}}$ represent the amplitudes
to fuse two copies of $\mathcal{O}_{\mathrm{L}}$ to create
$[\mathcal{O}_{\mathrm{L}}\mathcal{O}_{\mathrm{L}}]_{n\ell}$; they are known
in MFT. The coefficients $\mathcal{C}_{n\ell}^{\mathrm{HH}}$ are more
interesting: they are the expectation values of the LL double-twist states in
heavy backgrounds. Above the BTZ threshold, they become thermal one-point
functions—see Section 8 for a discussion.

In order to find the OPE coefficients for a correlator whose form is already
known, we can Taylor-expand both the correlator and each of the blocks in its
OPE, and then match the Taylor coefficients on both sides order by
order.131313We could also use an orthogonality relation for Virasoro blocks or
conformal partial waves Simmons-Duffin:2017nub , which would enable us to
obtain the OPE data by integrating each block against the correlator. The
results should be the same, and we leave a more detailed exploration of this
approach to future work. We begin by demonstrating how this strategy works in
the case of the light double-twist amplitudes
$\mathcal{C}_{n\ell}^{\mathrm{LL}}$ before adapting it to determine the
products $\mathcal{C}_{n\ell}^{\mathrm{LL}}\mathcal{C}_{n\ell}^{\mathrm{HH}}$,
first in general and then at large twist.

###  5.1 Universal OPE Data in Mean-Field Theory

Consider the four-point function of $\mathcal{O}_{\mathrm{L}}$ in MFT. By
Wick’s theorem, it is given by

$G_{\mathrm{L}}(z,\bar{z})\equiv\langle\mathcal{O}_{\mathrm{L}}(\infty)\mathcal{O}_{\mathrm{L}}(1)\mathcal{O}_{\mathrm{L}}(z,\bar{z})\mathcal{O}_{\mathrm{L}}(0)\rangle=\frac{1}{|z|^{2\Delta}}+\frac{1}{|1-z|^{2\Delta}}+1.$
(65)

But we can also compute $G_{\mathrm{L}}(z,\bar{z})$ using an OPE. The only
primaries that appear in the OPE of $\mathcal{O}_{\mathrm{L}}$ with itself are
the identity and the LL double-twists. The Virasoro blocks of these primaries
reduce to global conformal blocks in the semiclassical limit, so

$\displaystyle G_{\mathrm{L}}(z,\bar{z})$
$\displaystyle=\frac{1}{|z|^{2\Delta}}\MT@delim@Auto\p@star{1+\sum_{n\ell}\MT@delim@Auto\p@star{\mathcal{C}_{n\ell}^{\mathrm{LL}}}^{2}\big{|}k_{h_{n\ell}}(z)\big{|}^{2}},\qquad
k_{h}(z)=z^{h}{}_{2}F_{1}\MT@delim@Auto\p@star{h,h;2h;z},$ (66)

where as before $h_{n\ell}=h_{[\mathrm{LL}]_{n\ell}}$ with $\ell\in
2\mathbb{Z}$. Now we set (65) equal to (66) and strip off the vacuum
contribution $1/|z|^{2\Delta}$ from both sides: what remains is the part of
$G_{\mathrm{L}}(z,\bar{z})$ that is regular at $z,\bar{z}=0$. We change
summation variables from $(n,\ell)$ to $(m,m^{\prime})$ as in (43) and define
$p^{n}_{\ell}=\MT@delim@Auto\p@star{\mathcal{C}_{n\ell}^{\mathrm{LL}}}^{2}$.
The vacuum-subtracted correlator becomes

$\displaystyle
1+\frac{1}{|1-z|^{2\Delta}}=\frac{1}{|z|^{2\Delta}}\sum_{m,m^{\prime}=0}^{\infty}\frac{1}{2}\quantity[1+(-1)^{m-m^{\prime}}]p^{\mathrm{min}(m,m^{\prime})}_{m-m^{\prime}}k_{\Delta+m}(z)k_{\Delta+m^{\prime}}(\bar{z}).$
(67)

Next, we Taylor-expand both sides of (67) in powers of $z,\bar{z}$ around the
origin:

$\displaystyle
1+\sum_{r,r^{\prime}=0}^{\infty}a^{(r,r^{\prime})}z^{r}\bar{z}^{r^{\prime}}=\sum_{m,m^{\prime}=0}^{\infty}\frac{1}{2}\quantity[1+(-1)^{m-m^{\prime}}]p^{\mathrm{min}(m,m^{\prime})}_{m-m^{\prime}}\sum_{r,r^{\prime}=0}^{\infty}b_{m,m^{\prime}}^{(r,r^{\prime})}z^{r}\bar{z}^{r^{\prime}},$
(68)

where the Taylor coefficients
$a^{(r,r^{\prime})}=\alpha^{(r)}\alpha^{(r^{\prime})}$ and
$b_{m,m^{\prime}}^{(r,r^{\prime})}=\beta_{m}^{(r)}\beta_{m^{\prime}}^{(r^{\prime})}$
are given by

$\displaystyle\alpha^{(r)}=\frac{\Gamma(\Delta+r)}{\Gamma(\Delta)\Gamma(1+r)},\qquad\beta_{m}^{(r)}=\frac{\Gamma(\Delta+r)^{2}\Gamma(2\Delta+2m)}{\Gamma(\Delta+m)^{2}\Gamma(1+r-m)\Gamma(2\Delta+r+m)}.$
(69)

Finally, we match Taylor coefficients at each order
$z^{r}\bar{z}^{r^{\prime}}$ on both sides of (68):

$\displaystyle\delta_{r,0}\delta_{r^{\prime},0}+a^{(r,r^{\prime})}=\sum_{m,m^{\prime}=0}^{\infty}\frac{1}{2}\quantity[1+(-1)^{m-m^{\prime}}]b_{m,m^{\prime}}^{(r,r^{\prime})}\,p^{\mathrm{min}(m,m^{\prime})}_{m-m^{\prime}}.$
(70)

This power-matching condition reflects the fact that multiple LL double-twist
conformal blocks contribute to the regular part of the correlator at each
order in $z,\bar{z}$. The nontrivial Taylor expansion of $k_{h}(z)$ is what
causes the products $p^{n}_{\ell}$ to appear in complicated linear
combinations at each order; to disentangle them, we must view (70) as an
infinite linear system, with one equation at each order $(r,r^{\prime})$, to
be solved for the $p^{n}_{\ell}$.

In the simplest case $(r,r^{\prime})=(0,0)$, we are looking at the constant
term in the Taylor series. From (67), the value of the vacuum-subtracted
correlator at $z,\bar{z}=0$ is 2, and the only block that contributes to the
constant term is the one with $m=m^{\prime}=0$, corresponding to the composite
operator
$[\mathcal{O}_{\mathrm{L}}\mathcal{O}_{\mathrm{L}}]_{00}=\mathcal{O}_{\mathrm{L}}^{2}$.
Thus we find

$\displaystyle
2=p^{0}_{0}=\MT@delim@Auto\p@star{\mathcal{C}_{00}^{\mathrm{LL}}}^{2}\implies\mathcal{C}_{00}^{\mathrm{LL}}=\langle\mathcal{O}_{\mathrm{L}}\,\mathcal{O}_{\mathrm{L}}^{2}\,\mathcal{O}_{\mathrm{L}}\rangle=\sqrt{2}.$
(71)

This OPE coefficient reflects the correct Boltzmann counting one has to do to
account for the bosonic exchange statistics that arise when combining two
$\mathcal{O}_{\mathrm{L}}$ states to form $\mathcal{O}_{\mathrm{L}}^{2}$. The
higher coefficients $\mathcal{C}_{n\ell}^{\mathrm{LL}}$ should be understood
as combinatorial generalizations of the Boltzmann factor which serve to
properly normalize the states
$[\mathcal{O}_{\mathrm{L}}\mathcal{O}_{\mathrm{L}}]_{n\ell}$.

The linear system (70) is upper-triangular, which means it can be solved
recursively by back-substitution. To see how this works, observe that the sums
over $m,m^{\prime}$ are all finite: they truncate when $m$ reaches $r$ or when
$m^{\prime}$ reaches $r^{\prime}$, due to the Gamma function poles in the
denominator of $b^{(r,r^{\prime})}_{m,m^{\prime}}$. Therefore only finitely
many of the OPE coefficients enter at each order. In fact, the set of
equations one gets by truncating the expansion at order $(r,r^{\prime})$ is
sufficient to determine all of the OPE coefficients
$\mathcal{C}_{n\ell}^{\mathrm{LL}}$ with $n\leq\min(r,r^{\prime})$ and
$|\ell|\leq|r-r^{\prime}|$. One can then increase $r$ and $r^{\prime}$ until
the next “batch” of equations closes, solve for the remaining undetermined OPE
coefficients, and repeat the process.

Remarkably, this system has a closed-form solution, first written down in
Heemskerk:2009pn :

$\displaystyle\MT@delim@Auto\p@star{\mathcal{C}_{n\ell}^{\mathrm{LL}}}^{2}=\quantity[1+(-1)^{\ell}]c_{n}c_{n+\ell},\qquad
c_{n}=\frac{\Gamma(\Delta+n)^{2}\Gamma(2\Delta+n-1)}{\Gamma(\Delta)^{2}\Gamma(1+n)\Gamma(2\Delta+2n-1)}.$
(72)

This can be deduced either by guessing a pattern from the first few OPE
coefficients and checking that (72) solves (70), or by applying the Lorentzian
inversion formula.141414We thank João Penedones for discussion on this point.

As an aside, another way to formulate the problem is to ask for the
differential operator
$\mathcal{D}_{n\ell}\sim\Box^{n}\overset{{}_{\leftrightarrow}}{\partial}{}^{\ell}$
for which
$[\mathcal{O}_{\mathrm{L}}\mathcal{O}_{\mathrm{L}}]_{n\ell}=\mathcal{O}_{\mathrm{L}}\mathcal{D}_{n\ell}\mathcal{O}_{\mathrm{L}}$
is a properly normalized primary state. Then one may obtain the desired OPE
coefficients by taking derivatives of (65):

$\displaystyle\mathcal{C}_{n\ell}^{\mathrm{LL}}=\langle\mathcal{O}_{\mathrm{L}}(\infty)[\mathcal{O}_{\mathrm{L}}\mathcal{O}_{\mathrm{L}}]_{n\ell}(1)\mathcal{O}_{\mathrm{L}}(0)\rangle=\mathcal{D}_{n\ell}\langle\mathcal{O}_{\mathrm{L}}(\infty)\mathcal{O}_{\mathrm{L}}(1)\mathcal{O}_{\mathrm{L}}(z,\bar{z})\mathcal{O}_{\mathrm{L}}(0)\rangle\Big{|}_{z,\bar{z}=1}$
(73)

This was the perspective of Berenstein:2021pxv , where a recursion relation
for $\mathcal{D}_{n\ell}$ was found and solved, in the case of SL(2), for the
normalization coefficients of quasiprimaries.

###  5.2 Heavy-Heavy-Light OPE Coefficients

Let us return to the problem of matching the HHLL correlator $G(z,\bar{z})$ to
its $t$-channel OPE. This problem is similar to the one treated above: in both
cases, the vacuum-subtracted correlator $G_{\mathrm{reg}}(z,\bar{z})$ is set
equal to an infinite sum of LL double-twist Virasoro blocks and then Taylor-
expanded to extract the OPE coefficients. In the present case, we have

$\displaystyle\widetilde{G}_{\mathrm{reg}}(w,\overline{w})=\sum_{k\neq
0}\absolutevalue{\frac{1-\zeta^{k}(1-w)}{\alpha}}^{-2\Delta}=\frac{1}{|w|^{2\Delta}}\sum_{n\ell}\mathcal{C}_{n\ell}^{\mathrm{LL}}\mathcal{C}_{n\ell}^{\mathrm{HH}}|\alpha|^{-(2n+|\ell|)}\big{|}k_{h_{n\ell}}(w)\big{|}^{2}.$
(74)

Here we have stripped off an unimportant prefactor from (35) and (63) by
defining
$\widetilde{G}_{\mathrm{reg}}(w,\overline{w})\equiv|z|^{-(\alpha-1)\Delta}G_{\mathrm{reg}}(z,\bar{z})$.
Taking the same approach as in Section 5.1, we change summation variables from
$(n,\ell)$ to $(m,m^{\prime})$, define
$\widetilde{p}^{n}_{\ell}=\mathcal{C}_{n\ell}^{\mathrm{LL}}\mathcal{C}_{n\ell}^{\mathrm{HL}}$,
expand in powers of $w,\overline{w}$, and equate the Taylor coefficients on
both sides order by order:

$\displaystyle\widetilde{G}_{\mathrm{reg}}(w,\overline{w})=\sum_{r,r^{\prime}=0}^{\infty}\widetilde{a}^{(r,r^{\prime})}w^{r}\overline{w}^{r^{\prime}}=\sum_{m,m^{\prime}=0}^{\infty}|\alpha|^{-(m+m^{\prime})}\widetilde{p}^{\mathrm{min}(m,m^{\prime})}_{m-m^{\prime}}\sum_{r,r^{\prime}=0}^{\infty}b_{m,m^{\prime}}^{(r,r^{\prime})}w^{r}\overline{w}^{r^{\prime}}.$
(75)

The coefficients $b_{m,m^{\prime}}^{(r,r^{\prime})}$ are the same as the ones
in (69), while the new coefficients $\widetilde{a}^{(r,r^{\prime})}$ can be
obtained by expanding the images
$\MT@delim@Auto\p@star{1-\zeta^{k}(1-w)}^{-\Delta}$ in powers of $w$:

$\displaystyle\widetilde{a}^{(r,r^{\prime})}$
$\displaystyle=\frac{1}{r!\,r^{\prime}!}\partial^{r}\overline{\partial}{}^{r^{\prime}}\widetilde{G}_{\mathrm{reg}}(w,\overline{w})\evaluated{}_{w,\overline{w}=0}=\sum_{k\neq
0}\widetilde{\alpha}_{k}^{(r)}\widetilde{\alpha}_{-k}^{(r^{\prime})},$ (76)
$\displaystyle\widetilde{\alpha}^{(r)}_{k}$
$\displaystyle=\frac{|\alpha|^{\Delta}\MT@delim@Auto\p@star{-\zeta^{k}}^{r}}{\MT@delim@Auto\p@star{1-\zeta^{k}}^{\Delta+r}}\MT@delim@Auto\p@star{\frac{\Gamma(\Delta+r)}{\Gamma(\Delta)\Gamma(1+r)}}.$

From (75), the linear system that determines the products
$\widetilde{p}^{n}_{\ell}$ is

$\displaystyle\widetilde{a}^{(r,r^{\prime})}=\sum_{m,m^{\prime}=0}^{\infty}|\alpha|^{-(m+m^{\prime})}b_{m,m^{\prime}}^{(r,r^{\prime})}\widetilde{p}^{\mathrm{min}(m,m^{\prime})}_{m-m^{\prime}}.$
(77)

This matrix can, at least in principle, be inverted to yield OPE coefficients
schematically of the form
$\mathcal{C}_{n\ell}^{\mathrm{HH}}\sim(b_{n\ell}^{-1}\cdot\widetilde{a})/\mathcal{C}_{n\ell}^{\mathrm{LL}}$.
However, it is unlikely that the system can be solved in closed form in the
same way as (70), due to the complicated $r$-dependence of
$\widetilde{a}^{(r,r^{\prime})}$.

Nevertheless, the first few OPE coefficients can be found by truncating the
expansion at low orders in $w,\overline{w}$. For instance, consider
$[\mathcal{O}_{\mathrm{L}}\mathcal{O}_{\mathrm{L}}]_{00}=\mathcal{O}_{\mathrm{L}}^{2}$,
the simplest operator with a nontrivial expectation value in a heavy state.
Its expectation value can be obtained by looking at the constant term in (74),
or from the $(r,r^{\prime})=(0,0)$ equation of (77):

$\displaystyle\mathcal{C}_{00}^{\mathrm{HH}}=\langle\mathcal{O}_{\mathrm{L}}^{2}\rangle_{\mathrm{H}}=\frac{1}{\sqrt{2}}\widetilde{G}_{\mathrm{reg}}(0,0)=\frac{1}{\sqrt{2}}\sum_{k\neq
0}\absolutevalue{\frac{2}{\alpha}\sin(\pi k\alpha)}^{-2\Delta}.$ (78)

The next few OPE coefficients are readily obtained explicitly in terms of
$\widetilde{G}_{\mathrm{reg}}(w,\overline{w})$:

$\displaystyle\mathcal{C}_{10}^{\mathrm{HH}}$
$\displaystyle=\frac{\alpha^{2}}{\mathcal{C}_{10}^{\mathrm{LL}}}\MT@delim@Auto\p@star{\widetilde{G}_{\mathrm{reg}}^{(1,1)}(0,0)-\frac{\Delta^{2}}{4}\widetilde{G}_{\mathrm{reg}}(0,0)},$
$\displaystyle\mathcal{C}_{02}^{\mathrm{HH}}$
$\displaystyle=\frac{\alpha^{2}}{\mathcal{C}_{02}^{\mathrm{LL}}}\MT@delim@Auto\p@star{\frac{1}{2}\widetilde{G}_{\mathrm{reg}}^{(2,0)}(0,0)-\frac{\Delta(1+\Delta)^{2}}{4(1+2\Delta)}\widetilde{G}_{\mathrm{reg}}(0,0)},$
(79) $\displaystyle\mathcal{C}_{20}^{\mathrm{HH}}$
$\displaystyle=\frac{\alpha^{4}}{\mathcal{C}_{20}^{\mathrm{LL}}}\bigg{(}\frac{1}{4}\widetilde{G}_{\mathrm{reg}}^{(2,2)}(0,0)-\frac{\Delta(1+\Delta)^{2}}{8(1+2\Delta)}\widetilde{G}_{\mathrm{reg}}^{(2,0)}(0,0)-\frac{(1+\Delta)^{2}}{4}\widetilde{G}_{\mathrm{reg}}^{(1,1)}(0,0)+\frac{\Delta^{2}(1+\Delta)^{2}}{16}\widetilde{G}_{\mathrm{reg}}(0,0)\bigg{)},$

and so on. The main point is that (77) provides an algorithm for computing the
OPE coefficients $\mathcal{C}_{n\ell}^{\mathrm{HH}}$ whose only input data are
the universal coefficients (69) and (72), both derived from the semiclassical
Virasoro blocks, and the local behavior of $G_{\mathrm{reg}}(z,\bar{z})$ near
$z,\bar{z}=1$. This data can be obtained either by summing over images, as in
(76), or by direct numerical evaluation of the difference
$G_{\mathrm{reg}}=G-G_{0}$, i.e. using

$\displaystyle\widetilde{G}_{\mathrm{reg}}(w,\overline{w})\equiv\frac{G(z,\bar{z})-G_{0}(z,\bar{z})}{|z|^{(\alpha-1)\Delta}}=\sum_{n\ell}\MT@delim@Auto\p@star{\mathcal{C}_{n\ell}^{\mathrm{HL}}}^{2}|1-w|^{2n}(1-w)^{\ell/\alpha}-\absolutevalue{\frac{w}{\alpha}}^{-2\Delta}.$
(80)

Both terms in (80) are formally divergent at $w=0$, but their difference can
be computed numerically at very small $|w|\ll 1$. The results of this
subtraction are shown in Figure 5.2, which plots the expectation values
(78–5.2) in conical AdS as a function of $N$.

Figure 5: The first few low-lying LL double-twist expectation values in
conical AdS are plotted numerically as a function of the defect strength. We
take $\Delta=2$ and focus on primarily on the regime of light conical defects
($1<N<3$), where the method of images is not reliable. The expectation values
vanish in pure AdS ($N=1$), exhibit nontrivial behavior at finite $N$, and
tend to small but nonzero limits as $N\longrightarrow\infty$.

##  6 Asymptotic Analysis at Large Twist

For larger values of $n$, it becomes cumbersome to evaluate the
$\mathcal{C}_{n\ell}^{\mathrm{HH}}$ analytically. However, the Virasoro blocks
of LL double-twist operators with _very_ large $n$ simplify dramatically, and
we regain analytical control over the OPE. This will lead us to an asymptotic
formula for the expectation values of highly excited double-twists.

###  6.1 Virasoro Blocks with Heavy Exchange

For preparation, consider the semiclassical $t$-channel blocks (61) at
asymptotically large internal dimension, $h\gg 1$.151515Here the large-$c$
limit is taken first, at fixed $h$, before letting $h$ become large. In other
words we take the limits $h\longrightarrow\infty$ and
$c\longrightarrow\infty$, but with $h/c\longrightarrow 0$ so that the internal
operator is still “light.” An old result of Watson on the asymptotic behavior
of hypergeometric functions at large parameters Luke1969  states that

$\displaystyle
k_{h}(w)=w^{h}{}_{2}F_{1}(h,h;2h;w)\sim\frac{(4\rho)^{h}}{\sqrt{1-\rho^{2}}}\qquad\mathrm{as}\;\,h\longrightarrow\infty,$
(81)

where $\rho$ is the “radial coordinate” defined by

$\displaystyle
w=\frac{4\rho}{\MT@delim@Auto\p@star{1+\rho}^{2}}\iff\rho=\frac{1-\sqrt{1-w}}{1+\sqrt{1-w}},\qquad|\rho|<1.$
(82)

The transformation $w\mapsto\rho(w)$ maps the cut plane
$\mathbb{C}\mathbin{\vbox{\hbox{$\scriptstyle\setminus{}_{{}_{\\!\\!\\!\\!}}\scriptstyle\setminus$}}}[1,\infty)$
to the unit disk. It was first studied in Pappadopulo:2012jk ;
Hogervorst:2013sma , where its use improved the convergence rate of certain
OPEs.

It follows from (81) that the blocks (61) have the following asymptotics:

$\displaystyle\mathcal{F}_{h}^{(\mathrm{t})}(z)\sim
z^{(\alpha-1)h_{\mathrm{L}}}\frac{\MT@delim@Auto\p@star{1+\rho}^{4h_{\mathrm{L}}}}{\sqrt{1-\rho^{2}}}\MT@delim@Auto\p@star{\frac{4\rho}{\alpha}}^{h-2h_{\mathrm{L}}}\qquad\mathrm{as}\;\,h\longrightarrow\infty.$
(83)

Evidently at large $h$, the semiclassical HHLL blocks reorganize into scaling
blocks in the radial variable $\rho$. In this new coordinate, no descendants
are excited at all, and the OPE (74) turns into an ordinary power series in
$\rho$. This fact will allow us to carry out our power-matching procedure much
more directly than in the case of arbitrary $h$.

Since we want both the holomorphic and anti-holomorphic blocks to behave like
(83), it is natural to take the twist $\tau=\min(h,\bar{h})$ to be large,
rather than either $h$ or $\bar{h}$. In the case of LL double-twist exchange,
we have $\tau=\Delta+n$, so (83) is an asymptotic formula either in the hefty
regime $1\ll\Delta\ll c$ or in the highly excited regime $1\ll n\ll c$.

###  6.2 Darboux’s Method: Preliminaries

Our next step is to reformulate the matching problem (74) in terms of
$\rho,\bar{\rho}$ at large twist. We substitute the asymptotic Virasoro blocks
(83) into the LL double-twist OPE, change variables
$(n,\ell)\longrightarrow(m,m^{\prime})$ as before, and write the OPE in the
following form:

$\displaystyle F(\rho,\bar{\rho})$
$\displaystyle\equiv\absolutevalue{1-\rho^{2}}\big{|}1+\rho\big{|}^{-4\Delta}\widetilde{G}_{\mathrm{reg}}(w,\overline{w})\sim\sum_{m,m^{\prime}=0}^{\infty}\widetilde{p}^{\mathrm{min}(m,m^{\prime})}_{m-m^{\prime}}\MT@delim@Auto\p@star{\frac{4\rho}{\alpha}}^{m}\MT@delim<EMAIL_ADDRESS>(84)

Here $\sim$ indicates that the blocks in the OPE asymptotically approach (83)
at large $m,m^{\prime}$. From (74), $F(\rho,\bar{\rho})$ also has a
representation as a sum over images:

$\displaystyle F(\rho,\bar{\rho})=\sum_{k\neq
0}f_{k}(\rho)f_{-k}(\bar{\rho}),\qquad
f_{k}(\rho)=\alpha^{\Delta}\frac{\sqrt{1-\rho^{2}}}{\MT@delim@Auto\p@star{1+\rho}^{2\Delta}}\quantity[1-\zeta^{k}\MT@delim<EMAIL_ADDRESS>(85)

If we can find a formula for the Taylor coefficients of $f_{k}(\rho)$, then by
comparing to the expansion (84) we can determine the products
$\widetilde{p}^{n}_{\ell}$ at large twist. More precisely, if we expand
$f_{k}(\rho)$ in a Taylor series around $\rho,\bar{\rho}=0$ with coefficients
$a_{m}^{(k)}$, then

$\displaystyle\widetilde{p}^{\mathrm{min}(m,m^{\prime})}_{m-m^{\prime}}\sim\absolutevalue{\frac{\alpha}{4}}^{m+m^{\prime}}\sum_{k\neq
0}a_{m}^{(k)}a_{m^{\prime}}^{(-k)}\implies\mathcal{C}_{n\ell}^{\mathrm{HH}}\sim\frac{1}{\mathcal{C}_{n\ell}^{\mathrm{LL}}}\absolutevalue{\frac{\alpha}{4}}^{2n+|\ell|}\sum_{k\neq
0}a_{n+\ell}^{(k)}a_{n}^{(-k)}.$ (86)

Thus to determine the expectation values of operators with large twist, it
suffices to understand the asymptotic behavior of the Taylor coefficients
$a_{m}^{(k)}$ at large $m$. This can be done using the method of Darboux,
which says that the asymptotic Taylor coefficients of an analytic function are
controlled by its behavior near the closest of its singularities to the
origin. If the function diverges like $f(z)\sim(z-z_{0})^{\beta}$, then the
Taylor coefficients of $f$ are asymptotic to those of $(z-z_{0})^{\beta}$
itself. The precise statement is as follows wilf2005 :

######  Theorem 1 (Darboux).

Let $f(z)$ be analytic in the disk $|z|<R$, and suppose that $f$ has a branch-
point singularity at $z=z_{0}$ on the circle $|z|=R$. Without loss of
generality, write $f(z)=(z-z_{0})^{\beta}g(z)$, where $\beta$ is not a
positive integer and $g$ is analytic in a neighborhood of $z_{0}$. Then the
Taylor coefficients of $f$ around the origin are asymptotic to

$\displaystyle a_{n}\sim
g(z_{0})\frac{z_{0}^{\beta-n}\Gamma(n-\beta)}{\Gamma(-\beta)\Gamma(n+1)}=O(z_{0}^{-n}n^{-\beta-1})\qquad\mathrm{as}\;\,n\longrightarrow\infty.$
(87)

###  6.3 Darboux’s Method: Results

$1$$-1$$\rho_{k}^{+}$$\rho_{k}^{-}$$\rho$${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>Figure 6: The singular structure of $f_{k}(\rho)$, given in (85), is shown.
One of the two singular points $\rho_{k}^{+}$ or $\rho_{k}^{-}$ (blue) always
lies inside the unit disk. We have chosen the corresponding branch cuts (red)
so that $f_{k}(\rho)$ is analytic in the disk $|\rho|<|\rho_{k}^{+}|$.

As shown in Figure 6, each image $f_{k}(\rho)$ has four singular points. Two
are at $\rho=\pm 1$, and near these points $f_{k}(\rho)\sim\sqrt{1\pm\rho}$.
The other two singularities are located at

$\displaystyle\rho_{k}^{+}=i\tan\MT@delim@Auto\p@star{\frac{k\pi\alpha}{2}},\qquad\rho_{k}^{-}=-i\cot\MT@delim<EMAIL_ADDRESS>(88)

Note that in conical AdS, $\rho_{k}^{+}$ is the same point as
$\rho_{N-k}^{-}$, and moreover $\rho_{k}^{+}$ lies inside the unit circle for
$k<\frac{N}{2}$. Meanwhile, in BTZ we have $\rho_{k}^{+}=\rho_{-k}^{-}$, and
$|\rho_{k}^{+}|<1$ for all $k$. Thus by Theorem 1 we do not need to worry
about the singularities at $\pm 1$, and without loss of generality we may
restrict our attention to the behavior of $f_{k}(\rho)$ near
$\rho_{k}^{+}$:161616The $\mathrm{AdS}_{3}/\mathbb{Z}_{2}$ orbifold is
special: it admits only one nontrivial image, and all four of its singular
points lie on the unit circle. Its Taylor coefficients are asymptotic to a sum
of four terms of the form (87); of these, the two coming from the power-law
divergences at $\pm i$ dominate over the two coming from the square-root
singularities at $\pm 1$. The Taylor coefficients of $F(\rho,\bar{\rho})$ can
be computed exactly in this case, and their asymptotic behavior agrees with
Darboux’s method and is in accordance with the results below.

$\displaystyle
f_{k}(\rho)=\MT@delim@Auto\p@star{\frac{\alpha}{4}}^{\Delta}\frac{e^{-i\pi
k\alpha\Delta}}{\cos\MT@delim@Auto\p@star{\frac{\pi
k\alpha}{2}}}\MT@delim@Auto\p@star{\rho-\rho_{k}^{+}}^{-\Delta}\Big{[}1+O\MT@delim<EMAIL_ADDRESS>(89)

Figure 7: The exact scalar expectation values $\mathcal{C}_{n0}^{\mathrm{HH}}$
(points), as well as the asymptotic predictions of (91) (lines), are plotted
in conical AdS as a function of $n$ for $\Delta=2$. The asymptotics are
already reasonably good in the $\mathrm{AdS}_{3}/\mathbb{Z}_{2}$ orbifold, and
are essentially indistinguishable from the exact answers for heavier defects.

By Theorem 1, the Taylor coefficients of $f_{k}(\rho)$ are asymptotic to

$\displaystyle
a_{m}^{(k)}\sim\MT@delim@Auto\p@star{\frac{\alpha}{4}}^{\Delta}\frac{e^{-i\pi
k\alpha\Delta}}{\cos\MT@delim@Auto\p@star{\frac{\pi
k\alpha}{2}}}\frac{\Gamma(\Delta+m)}{\Gamma(\Delta)\Gamma(1+m)}\quantity[i\tan\MT@delim@Auto\p@star{\frac{\pi
k\alpha}{2}}]^{-\Delta-m}$ (90)

for $1<k\leq\frac{N}{2}$ in conical AdS and for $k\geq 1$ in BTZ. Substituting
this into (86) and using the invariance of $f_{k}(\rho)f_{-k}(\rho)$ under
$k\longleftrightarrow N-k$ or $k\longleftrightarrow-k$, we obtain

$\displaystyle\mathcal{C}_{n\ell}^{\mathrm{HH}}\sim\frac{i^{\ell}}{\mathcal{C}_{n\ell}^{\mathrm{LL}}}\MT@delim@Auto\p@star{\frac{\Gamma(\Delta+n+\ell)\Gamma(\Delta+n)}{\Gamma(\Delta)^{2}\Gamma(1+n+\ell)\Gamma(1+n)}}\sum_{k\neq
0}\frac{1}{\cos^{2}\MT@delim@Auto\p@star{\frac{\pi
k\alpha}{2}}}\absolutevalue{\frac{4}{\alpha}\tan\MT@delim@Auto\p@star{\frac{\pi
k\alpha}{2}}}^{-\Delta_{n\ell}},$ (91)

where $\Delta_{n\ell}=2\Delta+2n+|\ell|$. This is one of our main results: it
gives an explicit formula for the expectation values of highly excited
composite operators in the presence of a conical defect or a BTZ black hole.
It approximates the exact results of (5.2) surprisingly well, even for modest
values of $n$ and $\ell$: see Figure 7. For example, it predicts that

$\displaystyle\mathcal{C}_{00}^{\mathrm{HH}}\;\text{``}=\\!\text{''}\;\frac{1}{\sqrt{2}}\sum_{k\neq
0}\frac{1}{\cos^{2}\MT@delim@Auto\p@star{\frac{\pi
k\alpha}{2}}}\absolutevalue{\frac{4}{\alpha}\tan\MT@delim@Auto\p@star{\frac{\pi
k\alpha}{2}}}^{-2\Delta},$ (92)

which agrees with the exact result (78) near the BTZ threshold, where
$|\alpha|\ll 1$.

The large-$n$ and large-$\Delta$ limits of our results are also worth
recording: see Table 2. There we have kept only the $k=1$ contribution to
$\mathcal{C}_{n\ell}^{\mathrm{HH}}$: this term dominates over the others,
because $\rho_{1}^{+}$ is the closest singularity of $F(\rho,\bar{\rho})$ to
the origin, and as before this furnishes an approximate analytic continuation
of the expectation values to arbitrary $\alpha$. In both limits, the OPE
coefficients are exponentially suppressed in $n$ and $\Delta$, indicating that
highly energetic states leave a vanishing imprint on the background geometry.

| $n\longrightarrow\infty$ | $\Delta\longrightarrow\infty$
---|---|---
$\mathcal{C}_{n\ell}^{\mathrm{LL}}$ | $\displaystyle\frac{4\sqrt{2\pi}}{\Gamma(\Delta)^{2}}2^{-\Delta_{n\ell}}n^{2\Delta-\frac{3}{2}}$ | $\displaystyle\sqrt{\frac{2\MT@delim@Auto\p@star{\Delta/2}^{2n+|\ell|}}{n!(n+\ell)!}}^{{}^{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{-}}}$
$\mathcal{C}_{n\ell}^{\mathrm{HH}}$ | $\displaystyle\frac{i^{\ell}}{\cos^{2}\MT@delim@Auto\p@star{\frac{\pi\alpha}{2}}}\absolutevalue{\frac{2}{\alpha}\tan\MT@delim@Auto\p@star{\frac{\pi\alpha}{2}}}^{-\Delta_{n\ell}}\frac{1}{2\sqrt{2\pi n}}$ | $\displaystyle\frac{\sqrt{2}i^{\ell}}{\cos^{2}\MT@delim@Auto\p@star{\frac{\pi\alpha}{2}}}\absolutevalue{\frac{4}{\alpha}\tan\MT@delim@Auto\p@star{\frac{\pi\alpha}{2}}}^{-\Delta_{n\ell}}\sqrt{\frac{\MT@delim@Auto\p@star{2\Delta}^{2n+|\ell|}}{n!(n+\ell)!}}$
Table 2: Asymptotic behavior of the OPE data
$\mathcal{C}_{n\ell}^{\mathrm{LL}}$ and $\mathcal{C}_{n\ell}^{\mathrm{HH}}$ in