A convergent genus expansion for the plateau Phil Saad1 Douglas Stanford2 Zhenbin Yang2 and Shunyu Yao2 1School of Natural Sciences

2025-04-24 0 0 663.52KB 36 页 10玖币

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A convergent genus expansion for the plateau

Phil Saad1, Douglas Stanford2, Zhenbin Yang2, and Shunyu Yao2

1School of Natural Sciences,

Institute for Advanced Study, Princeton, NJ 08540

2Stanford Institute for Theoretical Physics,

Stanford University, Stanford, CA 94305

Abstract

We conjecture a formula for the spectral form factor of a double-scaled matrix

integral in the limit of large time, large density of states, and ﬁxed temperature. The

formula has a genus expansion with a nonzero radius of convergence. To understand

the origin of this series, we compare to the semiclassical theory of “encounters” in

periodic orbits. In Jackiw-Teitelboim (JT) gravity, encounters correspond to por-

tions of the moduli space integral that mutually cancel (in the orientable case) but

individually grow at low energies. At genus one we show how the full moduli space

integral resolves the low energy region and gives a ﬁnite nonzero answer.

arXiv:2210.11565v1 [hep-th] 20 Oct 2022

Contents

1 Introduction 3

2 Tau scaling of the spectral form factor 4

3 Encounters in orbits and in JT 7

3.1 Review of periodic-orbit theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 Sieber-RichterpairinJT................................ 10

4 Beyond encounters 13

4.1 Kontsevich’s decomposition of moduli space . . . . . . . . . . . . . . . . . . . . . 13

4.2 Genusone-half ..................................... 15

4.3 Genusone........................................ 18

5 Discussion 21

A Airy sigma model 22

B Action correlation in the Airy model 27

C Soft mode action in orbits and gravity 29

C.1 Gauge ﬁxing in periodic orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

C.2 Gravitycalculation ................................... 31

D Formula for P(ρ)

g(β)32

1 Introduction

A longstanding challenge is to explain the discrete spectrum of black hole microstates using

spacetime geometry. In recent years, some statistical aspects of these microstates have been

explained using spacetime wormholes.1Examples include: aspects of the spectral form factor

[7, 8, 9] and late-time correlation functions [10, 11, 12, 13], the Page curve [14, 15] and matrix

elements [16] of an evaporating black hole, and the ETH behavior of matrix elements [17, 18, 19].

A statistical theory of microstates is far from a complete description, but it is enough to

probe discreteness of the energy spectrum. One tool to discuss this is the spectral form factor

Kβ(t) = hZ(β+ it)Z(β−it)i(1.1)

where Z(x) = Tr e−xH is the thermal partition function, and the brackets represent some form of

averaging for which the statistical description is suﬃcient. The discrete nature of chaotic energy

levels is reﬂected in the “plateau” to a late time value Kβ(∞) = Z(2β).

For systems in the unitary symmetry class (no time reversal symmetry), the random matrix

theory (RMT) prediction for the spectral form factor is simple. The microcanonical version

KE(t) = Zdβ

2πie2βEKβ(t) (1.2)

should have the form of a linear ramp connected to a plateau: min{t/2π, eS(E)}with S(E)

the microcanonical entropy at energy E. This sharp transition at tp= 2πeS(E)is a signature

of discretness of the spectrum; it arises from oscillations in the density pair correlator with

wavelength e−S(E), representing the mean spacing between discrete energy levels.

The sharpness of the transition from the ramp to the plateau is an apparent obstruction to

an explanation in terms of geometry. In particular, in two-dimensional dilaton gravity models

such as JT gravity, the genus expansion should roughly be thought of as an expansion in e−S(E).

But the transition from the ramp to the plateau comes from contributions have go as ei#eS(E),

nonperturbative in the genus counting parameter, suggesting that it is not captured by the

conventional sum over geometries.2

However, the spectral form factor Kβ(t) is an integral of KE(t) over energy, and this integral

has the potential to smooth out the transition to the plateau. As ﬁrst shown by [22, 23] for the

Airy matrix integral, the resulting function can have a convergent genus expansion, smoothly

transitioning from the ramp to the plateau. We conjecture a generalization of this result below,

in a limit that will be referred to as “τ-scaling.” This convergent series makes it possible to

explain the plateau in terms of a conventional sum over geometries, rather than from a radical

nonperturbative eﬀect.

In this paper, we will explain some features of this genus expansion for the spectral form factor,

primarily working in the low-energy limit of JT gravity: the Airy model. Our explanations will

connect with the encounter computations in semiclassical periodic orbit theory, used to explain

1The role of spacetime wormholes in quantum gravity has also been a longstanding puzzle, [1, 2, 3, 4, 5, 6].

2Some previous approaches to explaining the plateau through a sum over geometries have involved “spacetime

D-branes” [8, 20, 21], which generalize the sum over geometries to include contributions from an inﬁnite number

of asyptotic boundaries.

the RMT corrections to the ramp [24, 25, 26, 27, 28]. The sum over encounters is closely

analogous to a genus expansion, so it is natural to try interpret the genus expansion for the

plateau in terms of a gravitational analog of encounters. Encounters alone cannot be suﬃcient

to explain the genus expansion for Kβ(t) because without time-reversal symmetry, the encounters

cancel genus by genus.

The models that we study, in particular the Airy model, allow us to generalize the theory

of encounters beyond their usual regime of validity in the high-energy, semiclassical limit. At

very low energies, of order 1/t, the encounters receive large quantum corrections that disturb the

cancellation between encounters, reproducing the expected τ-scaled Kβ(t).

In Section Two, we introduce a formula for Kβ(t) in a double-scaled matrix integral in the

“τ-scaled” limit, generalizing [22, 29]. We reconcile the existence of a convergent genus expansion

for Kβ(t) with the absence of such an expansion for KE(t). In particular, one can think of the

genus expansion for Kβ(t) as coming entirely from very low energies.

In Section Three we review an analog of the genus expansion for KE(t) in periodic orbit

theory: the sum over encounters. The sum over encounters gives an expansion in e−S(E), valid

at high energies. For periodic orbit systems in the GUE symmetry class (no time-reversal),

corrections to the ramp coming from encounters cancel order by order [27, 28]. In JT gravity,

we discuss a direct analog of the simplest type of encounter contribution in a theory with time-

reversal symmetry, contributing to the SFF at genus one-half.

In Section Four we study the Airy model, the low-energy limit of JT gravity. The wormhole

geometries in this model are very simple, and in one-to-one correspondence with ribbon graphs

in the Feynman diagram expansion of Kontsevich’s matrix model. These graphs allow us to

generalize the encounter computations beyond the semiclassical, high-energy regime. At genus

one and high energies, the encounter contributions mutually cancel in the GUE symmetry class.

At low energies, quantum corrections to the encounters spoil this cancellation, leading to the

nonzero contribution to Kβ(t). The full answer at this genus comes from a large region of moduli

space, far from the semiclassical encounter regime.

Note: Two recent papers [30, 31] are closely related to our work. A preliminary version of

section two of this paper was shared with the authors of [30, 31] in October 2021.

2 Tau scaling of the spectral form factor

In this section we discuss the “τ-scaling” limit of matrix integrals in which we conjecture that

the spectral form factor has a simple form with a convergent genus expansion. Consider a

double-scaled matrix integral with unitary symmetry class and classical density of states

ρ(E) = eS0ρ0(E).(2.1)

The spectral form factor is deﬁned as

Kβ(t)≡ hZ(β+ it)Z(β−it)i, Z(x)≡Tr e−xH .(2.2)

Here the angle brackets represent the average in the matrix integral. We would like to analyze

this in a limit where tgoes to inﬁnity and eS0also goes to inﬁnity, holding ﬁxed β, and also

holding ﬁxed the ratio

τ=te−S0.(2.3)

This will be referred to as the “τ-scaled” limit.

In the τ-scaled limit, the time t=eS0τis large, so the SFF will be dominated by correlations

of nearby energy levels. Pair correlations of nearby levels are described by the universal sine-

kernel formula, which translates to a ramp-plateau structure min{t/2π, ρ(E)}as a function of

the center of mass energy E. By integrating this contribution over E, one gets the following

candidate expression for the spectral form factor

Kβ(t)?

≈Z∞

dE e−2βEmin t

2π, ρ(E).(2.4)

This was previously discussed as an uncontrolled approximation to the SFF [32]. Here we would

like to propose that it is exact in the τ-scaled limit,

lim

S0→∞ e−S0Kβ(τeS0) = Z∞

dE e−2βEmin nτ

2π, ρ0(E)o.(2.5)

Let’s try an example by taking ρ0(E) = √E

2π, which is sometimes called the Airy model, or

the Kontsevich-Witten model. Then (2.5) becomes

e−S0Kβ(τeS0) = 1

2πZ∞

dEe−2βE min nτ, √Eo(2.6)

2π

π1/2

25/2β3/2Erf(p2βτ) (2.7)

=τ

4πβ −τ3

6π+β

10πτ5−β2

21πτ7+. . . . (2.8)

We can compare this to the exact answer for the spectral form factor of the Airy model [23, 29]

Kβ(t) = hZ(2β)iErf(e−S0p2β(β2+t2)) (2.9)

=exp S0+1

3e−2S0β3

4√πβ3/2Erf(e−S0p2β(β2+t2)).(2.10)

This agrees with (2.7) in the τ-scaled limit.

As a second example, we can take ρ0(E) = 1

4π2sinh(2π√E), which corresponds to JT gravity:

e−S0Kβ(τeS0) = 1

2πZ∞

dEe−2βE min τ, 1

2πsinh(2π√E)(2.11)

=eπ2

2β

16√2πβ3/2"Erf β

πarcsinh(2πτ) + π

√2β!+ Erf β

πarcsinh(2πτ)−π

√2β!# (2.12)

=τ

4πβ −τ3

6π+β

10π+2π

15 τ5−β2

21π+4πβ

21 +64π3

315 τ7+. . . (2.13)

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AconvergentgenusexpansionfortheplateauPhilSaad1,DouglasStanford2,ZhenbinYang2,andShunyuYao21SchoolofNaturalSciences,InstituteforAdvancedStudy,Princeton,NJ085402StanfordInstituteforTheoreticalPhysics,StanfordUniversity,Stanford,CA94305AbstractWeconjectureaformulaforthespectralformfactorofadouble-scal...

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A convergent genus expansion for the plateau Phil Saad1 Douglas Stanford2 Zhenbin Yang2 and Shunyu Yao2 1School of Natural Sciences

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