Solvable models of quantum black holes a review on JackiwTeitelboim gravity Thomas G. Mertens1and Gustavo J. Turiaci23

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Solvable models of quantum black holes: a

review on Jackiw–Teitelboim gravity

Thomas G. Mertens1and Gustavo J. Turiaci2,3

1Department of Physics and Astronomy, Ghent University,

Krijgslaan, 281-S9, 9000 Gent, Belgium.

2Institute for Advanced Study, Princeton, NJ, USA.

3Physics Department, University of Washington, Seattle, WA,

USA.

thomas.mertens@ugent.be;turiaci@uw.edu;

Abstract

We review recent developments in Jackiw–Teitelboim (JT) gravity.

This is a simple solvable model of quantum gravity in two dimen-

sions (that arises e.g. from the s-wave sector of higher dimensional

gravity systems with spherical symmetry). Due to its solvability, it

has proven to be a fruitful toy model to analyze important ques-

tions such as the relation between black holes and chaos, the role

of wormholes in black hole physics and holography, and the way

in which information that falls into a black hole can be recovered.

Contents

1 Introduction 3

2 Classical Jackiw–Teitelboim gravity 5

2.1 Dilaton gravity models ....................... 5

2.1.1 First-order formulation .................. 7

2.1.2 Motivation: near-extremal black holes .......... 9

2.2 Classical solutions ......................... 13

2.2.1 Metric solution ....................... 13

2.2.2 Dilaton solution ...................... 17

arXiv:2210.10846v2 [hep-th] 21 Jun 2023

2Solvable models of quantum black holes: a review on JT gravity

2.3 Boundary conditions and Schwarzian dynamics ......... 20

2.3.1 Real-time derivation .................... 20

2.3.2 Aside: general dilaton gravity ............... 25

2.3.3 The Schwarzian action ................... 26

2.3.4 Geometric derivation from the action ........... 29

2.4 Quantum matter in classical gravity ............... 32

2.4.1 Application: Hawking-Unruh eﬀect and information loss 33

3 Quantum Jackiw–Teitelboim gravity 36

3.1 Spectrum of quantum black holes ................. 36

3.1.1 Perturbative calculation .................. 37

3.2 Quantum Jackiw–Teitelboim gravity coupled to matter ..... 39

3.3 Correlators ............................. 41

3.3.1 Path integral representation ................ 41

3.3.2 A ﬁrst approach: Schwarzian perturbation theory . . . . 43

3.3.3 Exact two-point function ................. 44

3.3.4 Exact four-point function ................. 48

3.3.5 Application: quantum chaos ................ 51

3.4 Diagrammatic rules for exact correlators ............. 55

3.5 Pure states and end-of-the-world (EOW) branes ........ 57

3.6 Other operator insertions in JT gravity ............. 59

3.7 Outline of derivations in the literature .............. 62

3.7.1 Free particle approach ................... 63

3.7.2 Limit of Liouville theory .................. 63

3.7.3 Boundary particle approach ................ 65

3.7.4 Two-dimensional gauge theory .............. 66

4 Spacetime wormholes and random matrices 68

4.1 Motivation: information loss and late time decay ........ 68

4.2 Multiboundary higher genus amplitudes ............. 70

4.2.1 The disk Z0,1(β)...................... 73

4.2.2 The cylinder Z0,2(β1, β2).................. 73

4.2.3 The general case Zg,n(β1, . . . , βn)............. 75

4.3 JT and random matrices ...................... 80

4.3.1 SSS duality ......................... 80

4.3.2 Double-scaling limit of matrix integrals ......... 82

4.3.3 Derivation of the duality .................. 84

4.4 Non-perturbative eﬀects in topological expansion ........ 86

4.4.1 Density of states ...................... 86

4.4.2 Late time decay of spectral form factor ......... 88

4.5 Generalization ........................... 91

4.5.1 Other ensembles ...................... 92

4.5.2 2d dilaton gravity with general potential ......... 93

4.5.3 JT gravity coupled to gauge ﬁelds ............ 94

4.5.4 JT gravity coupled to matter ............... 96

T. G. Mertens and G. J. Turiaci 3

5 Applications and future directions 97

5.1 The entropy of Hawking radiation ................ 97

5.1.1 Entanglement islands in JT gravity ............ 98

5.1.2 Replica wormholes ..................... 101

5.1.3 Replica wormholes and EOW branes ........... 103

5.2 Factorization, discreteness, and ensemble averaging in gravity . 106

5.3 Near-extremal black holes ..................... 107

5.4 Supersymmetric JT ......................... 107

5.5 Two-dimensional cosmology .................... 108

5.6 Traversable wormholes ....................... 109

5.7 Finite cut-oﬀ and T¯

T....................... 111

5.8 Volume of black hole interior and complexity .......... 112

5.9 Bulk correlators and observables ................. 113

5.10 Liouville gravity and minimal string ............... 114

5.11 Universe ﬁeld theory and quantum chaos ............. 116

References 117

1 Introduction

Understanding quantum gravity and quantum black hole physics are some

of the most pressing open problems in contemporary theoretical physics. In

particular, deep questions arise in the problem of black hole formation and

subsequent evaporation, starting with Hawking’s work in the 1970s [1,2].

The road towards quantum gravity, starting with the problem of non-

renormalizability of pure Einstein–Hilbert gravity in 3+1d, has in the years

led us through higher dimensions, string theory, compactiﬁcation, branes ....

And some signiﬁcant successes have been made using this approach [3]. How-

ever, it is safe to say that we do not have a fully satisfying understanding, and

any alternative approaches that could shed new light on the problem would

be most welcome.

With this goal in mind, an alternative strategy toward quantum gravity

would be to work in lower dimensions (2d or 3d), where gravitational models

can make sense at the level of the Euclidean path integral. If we furthermore

do this in the framework of holography, then we have a preferred anchoring

point, and are guided by the major breakthroughs in that ﬁeld throughout the

years [4].

Even within this strategy, ﬁnding interesting lower-dimensional solvable

models of gravity is an art on its own. Few such models exist, but the degree

of solvability that we can obtain in 2d Jackiw–Teitelboim gravity is unprece-

dented as we aim to explain with this review. If we work in two spacetime

dimensions, the simplest candidate model would be Einstein–Hilbert gravity

S=1

16πGNZd2x√−gR +. . . . (1.1)

4Solvable models of quantum black holes: a review on JT gravity

However, this model is topological since its Euclidean action is just the Euler

characteristic, and the Einstein tensor vanishes identically. This model does

play an important role as the 2d gravity on the string worldsheet, weighing

diﬀerent topologies. Additionally coupling this model to a matter action Sm,

the gravitational equations lead to Tm

µν = 0 and no energy ﬂows exist. Hence

when using it as a classical toy model for black hole formation and evaporation,

this model has little value.1

To get something more interesting, the required adjustment in two dimen-

sions is to introduce a direct coupling of the Ricci scalar to a new scalar ﬁeld

Φ, called the dilaton ﬁeld for historical (string-theoretic) reasons:

S=1

16πGNZd2x√−gΦR+. . . (1.2)

Models of this kind have been introduced in 3+1d as scalar-tensor (Brans–

Dicke type) models, and provide deformations of general relativity with

spacetime-varying gravitational coupling. Here we view these dilaton grav-

ity models as quantum mechanical solvable toy models of quantum gravity

and black hole physics. Jackiw–Teitelboim (JT) gravity is one such particular

model that we will specify below. Interestingly, this particular model also cap-

tures the dynamics close to the horizon of near-extremal black holes in higher

dimensions, see the upcoming review by [8].

Gravitational models of this kind have attracted widespread attention over

the years (see e.g. the reviews by [9,10]), but it has been only recently, sparked

by developments in 2015 by A. Kitaev in solvable many-body models (the

Sachdev–Ye–Kitaev or SYK models; [11–14]), that we have reached a far deeper

understanding of their quantum mechanical aspects. This work reﬁned the

original proposal relating SYK and AdS2gravity put forward by [15].

Our aim here is to provide an in-depth review of these developments in JT

gravity. This review is organized into four main sections. In Sect. 2we introduce

the model, and fully solve its classical equations of motion, crucially incorpo-

rating boundary conditions at the holographic boundary that allow us to map

the dynamics to its boundary Schwarzian description. Sect. 3proceeds with

the exact quantization of the model, by computing the Euclidean gravitational

path integral in the Schwarzian language. In Sect. 4we furthermore include

non-trivial topological corrections (or wormholes) to the quantum amplitudes,

that modify the heavy quantum regime even further. Finally, Sect. 5contains

several applications of the exact solvability of JT gravity. We do not treat these

in technical detail, but refer to the literature for more information. In partic-

ular, in the past few years signiﬁcant progress has been made on Hawking’s

information loss paradox, made possible to a large extent due to the solvability

of JT gravity. We discuss some of these developments, but will not do justice

1Quantum mechanically, gauge ﬁxing this model with conformal matter, leads to Liouville

gravity which is an interesting model appearing in the context of the non-critical string [5–7]. We

do not follow this route.

T. G. Mertens and G. J. Turiaci 5

to this topic in this work. We refer e.g. to the excellent recent review by [16]

for details.

A few reviews have been written before on the connection between JT

gravity and SYK. We refer the interested reader to the excellent review on this

topic by [17] that focuses mostly on the SYK side. Other earlier reviews are [18]

and [19]. However, no comprehensive review exists that combines these earlier

developments with the current state-of-the-art of JT gravity speciﬁcally and

lower-dimensional gravitational models more generally, something we hope to

address with this review.

Finally, a few comments on conventions: in Lorentzian signature our metric

signature convention is (−,+, . . . +). We denote Lorentzian actions as Sand

Euclidean actions as I.

2 Classical Jackiw–Teitelboim gravity

We begin by introducing and motivating the JT model and its coupling to

matter. In this section, we study the classical solution of this model including

gravitational backreaction. Our endeavors will ultimately lead us to a descrip-

tion in terms of a boundary Schwarzian model that will be the starting point

for a quantum mechanical solution in the next Sect. 3.

2.1 Dilaton gravity models

To start, it is instructive to consider what is the most general theory of dilaton

gravity in two dimensions with a two-derivative action [20–22,9]. Working in

Euclidean signature, any such theory can be written as

I=−1

16πGNZM

d2x√gU1(˜

Φ)R+U2(˜

Φ)gµν ∂µ˜

Φ∂ν˜

Φ + U3(˜

Φ),(2.1)

where gµν is the two-dimensional metric and ˜

Φ(x) is the dilaton ﬁeld. The 2d

Newton’s constant GNis dimensionless, and hence unlike in higher dimensions

does not set the scale of physics.2This theory seems to be parametrized by

three functions U1(˜

Φ), U2(˜

Φ) and U3(˜

Φ) but two of them are redundant. To

see this, ﬁrst perform a ﬁeld redeﬁnition on the dilaton ˜

Φ→Φ = U1(˜

Φ). We

assume there is no value of ˜

Φ such that U′

1(˜

Φ) = 0 so the ﬁeld redeﬁnition is

invertible, otherwise the resulting kinetic term in Φ will be ill-deﬁned.

We can further set U2(Φ) = 0 by making a Weyl transformation on the

metric as follows. Under a local rescaling, the 2d Ricci scalar transforms as

g′

µν =e2ωgµν ,→g′1/2R′=g1/2(R−2∇2ω).(2.2)

2Later on, around Eq. (2.57) in JT gravity, we will see an eﬀective scale emerge nonetheless.

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Solvablemodelsofquantumblackholes:areviewonJackiw–TeitelboimgravityThomasG.Mertens1andGustavoJ.Turiaci2,31DepartmentofPhysicsandAstronomy,GhentUniversity,Krijgslaan,281-S9,9000Gent,Belgium.2InstituteforAdvancedStudy,Princeton,NJ,USA.3PhysicsDepartment,UniversityofWashington,Seattle,WA,USA.thomas.mer...

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Solvable models of quantum black holes a review on JackiwTeitelboim gravity Thomas G. Mertens1and Gustavo J. Turiaci23

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