Explorations of Non–Perturbative JT Gravity and Supergravity

Clifford V. Johnson∗

Department of Physics and AstronomyUniversity of Southern California

Los Angeles, CA 90089-0484, U.S.A.

Some recently proposed definitions of Jackiw–Teitelboim gravity and supergravities in terms ofcombinations of minimal string models are explored, with a focus on physics beyond the perturbativeexpansion in spacetime topology. While this formally involves solving infinite order non–lineardifferential equations, it is shown that the physics can be extracted to arbitrarily high accuracy ina simple controlled truncation scheme, using a combination of analytical and numerical methods.The non–perturbative spectral densities are explicitly computed and exhibited. The full spectralform factors, involving crucial non–perturbative contributions from wormhole geometries, are alsocomputed and displayed, showing the non–perturbative details of the characteristic “slope”, “dip”,“ramp” and “plateau” features. It is emphasized that results of this kind can most likely be readilyextracted for other types of JT gravity using the same methods. Preliminary results also suggestthat a new well–defined non–perturbative completion of ordinary JT gravity using the Hermitianmatrix models may exist.

I. INTRODUCTION

There are many reasons to study Jackiw–Teitelboim(JT) gravity [1, 2]. One of them is the fact that it is atheory of a two dimensional quantum gravity, where thespacetime is allowed to split and join, changing its topol-ogy (characterized by Euler characteristic χ=2−2g−b−cwhere g counts handles, b boundaries, and c crosscaps).In the full theory the partition function Z(β) is a sumover the contributions from all topologies as well as anon–perturbative part that is not captured by the per-turbative expansion in topology:

Z(β) =∑χ

Zχ(β) + non–perturb. (1)

Here, Zχ(β) stands for the contribution to the partitionfunction from surfaces of Euler characteristic χ. It comeswith a factor eχS0 , as S0 is a coupling which multipliesthe Einstein–Hilbert action in the model. (Although χ=1for the (leading) disc order quantities, the subscript 0 willbe widely used at leading order henceforth. So the disclevel partition function is Z0, spectral density is ρ0, etc.)

The focus of this paper will be on characterizing thefull partition function of the theory, including the fullnon–perturbative physics, by making explicit aspects ofthe double–scaled matrix model definitions suggested inrefs. [3, 4], which should be considered companion pa-pers to this one. The beautiful work of refs. [5, 6] indefining double–scaled matrix models of (various kindsof) JT gravity is intrinsically perturbative in spirit, since

∗ johnson1@usc.edu

they use recursion relations connecting different topolo-gies, and the work in refs. [3, 4] is intended as a comple-mentary construction (using minimal strings) that allowsmore direct access to non–perturbative quantities. Theoutput of this paper will be the first explicit computa-tion of the full spectral densities (and hence the parti-tion functions, by Laplace transform), and explorationsof several important phenomena that depend crucially onbeing able to compute non–perturbative physics.

0 0.5 1 1.5 2 2.5 3-1.1

-1.09

-1.08

-1.07

-1.06

-1.05

-1.04

-1.03

Figure 1: The full spectral form factor, showing the classic(saxophone) shape made up of a slope, dip, ramp, and plateau.This is computed using the methods of this paper for the (2,2)model of JT supergravity. Here, β=35, ~=1/5. (See text.)

An example of the latter is the 2–point “spectralform factor” shown in figure 1, a quantity that helpsin diagnosing universal aspects of quantum chaotic be-

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haviour [7, 8]. It was computed using the methods of thispaper. This is the first time this quantity (and others likeit to be presented later) has been computed fully in JTgravity or supergravity for generic values of β and S0, andso some time will be spent unpacking the techniques andresults1. The late time “plateau” feature of the curve,and the transition to it from the “ramp” behaviour, areintrinsically non–perturbative features of wide interest.There are important non–perturbative effects that showup in the slope part too, in some cases, as will be demon-strated. They can sometimes be dramatic, as will be seenin the supergravity examples presented.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

5

10

15

20

25

30

35

40

45

50

Figure 2: The full spectral density, computed using the methodsof this paper, for the (2,2) model of JT supergravity. The dashedblue line is the disc level result of equation (5). Here ~=1.

Another example of this paper’s results is given in fig-ure 2. It is the full spectral density (the thicker line, actu-ally made out of data dots) of the model ρSJT(E) with theclassical result (see equation (5)) plotted as a dashed linefor comparison. By Laplace transform, this function de-fines the full non–perturbative partition function for thesupergravity theory, and is computed here explicitly forthe first time. This JT supergravity model is in fact the(α,β)=(2, 2) matrix model in the Altland–Zirnbauer[10]classification scheme, or the case Γ= 1

2 in the notation of

ref. [4]. The result for the companion (0, 2) case (Γ=− 12 )

will be displayed later (see figure 8 on page 8). As canbe seen in figure 2, for the (2,2) case non–perturbativeeffects entirely erase the characteristic classical peak inthe spectrum at low energy, which dramatically altersthe “slope” part of the spectral form factor as comparedto the analogous result for the (0,2) case where a peakpersists in the full spectrum.

1 An interesting recent paper [9] presented an expression for thespectral form factor of JT gravity, but in a very special ultra–low temperature scaling limit that allowed a closed form to bewritten. In this paper, no special limits on the parameters aretaken, and while no closed forms are presented, answers can besystematically extracted at a wide range of β and S0.

While ordinary JT gravity is important and interesting(and results will be presented for it), a good deal of at-tention will be given to these two particular models of JTsupergravity. They are of particular interest because thenon–perturbative physics is more dramatic, in a sense. Itwas observed in ref. [6] (and confirmed to be manifest inthe minimal model construction of ref. [4]) that beyondthe first one or two leading orders of perturbation the-ory (depending upon the quantity being computed) theentire topological perturbative series vanishes. Thereforethe non–perturbative effects uncovered in these models(as will be done here) are placed more in stark relief thanother JT gravity systems.

Having shown examples of the key results, the job ofthe rest of the paper is to explain how to get them, andthen to interpret them. The results follow from the non–perturbative construction, proposed in refs. [3, 4], of JTgravity and supergravity in terms of minimal string mod-els (of a special type). The basic idea, building on sug-gestions in refs. [5, 11], is to reinterpret the JT system asan infinite set of minimal models (non–linearly) coupledtogether in a particular way, or equivalently (as explainedin ref. [4]) by turning on an infinite set of operators inthe minimal string model obtained by taking the k→∞limit2. Since the full information about the kth mini-mal string model in question (see section III for a quickreview of the essentials) involves solving an order 2k+1highly non–linear ordinary differential equation (ODE),this way of defining JT gravity or supergravity involvessolving an infinite order differential equation. This mightseem rather daunting, or even formal, but from a prag-matic point of view it is rather straightforward to im-plement an approximation scheme that allows compu-tation of an answer to a specific concrete question, towhatever accuracy is desired. The point is that the con-tribution to the model of successively higher orders ofderivatives in the ODE grows smaller with increasing k,and so there is a point at which truncating the ODE andsolving a finite order equation will give access to the fullspectrum all the way up to a given desired energy, tosome required accuracy. In other words, this is hardlyany different from computing Feynman diagrams up tosome sufficiently high order for some field theory prob-lem (except that here the formalism is computing non–perturbative physics, and moreover the series is conver-gent, not asymptotic.)An outline of the paper is as follows: Section II

is a brief summary of some of the (now standard) keyideas in the study of JT gravity that will be used inthis paper. It is entirely optional for those who knowthe subject well, but serves to set context, notation, and(perhaps) some motivation. The deconstruction in termsof minimal models will be lightly explained in Section III.

2 Other recent work exploring connections between the formalismof Liouville theory and minimal strings on the one hand, and JTgravity on the other, includes refs. [9, 12, 13].

3

Refs. [3, 4] should be consulted for further details, andthe non–perturbative explorations of key toy models pre-sented there. The main task of this paper is to show howto extract non–perturbative results for the full JT (su-per)gravities. In particular, this section will explain how(using the supergravity examples) the truncation schemeof the previous paragraph works. Section IV will solvethe full quantum mechanical system to yield the non–perturbative spectral density (and hence the partitionfunction), for the supergravity cases. Then Section Vturns to the non–perturbative spectral form factor forthe supergravities, explaining how it is computed andthen displaying several results.

Section VI then discusses the analogous constructionand results for a non–perturbative definition of ordinaryJT gravity obtained (as first presented in ref. [3]) by em-bedding it into a larger framework that it matches pertur-batively (at high energy) but which supplies it with non–perturbatively well–behaved low energy physics. Sec-tion VII contains a proposal and brief preliminary dis-cussion of an alternative non–perturbative definition ofordinary JT gravity that may be more natural than thatin the previous section. For it to work, a particular kindof solution to the string equations should exist that hasseems to have not been considered in this context before,and sample truncated solutions are displayed.

Since most of the results of this paper come fromnumerically unpacking the highly non–linear system ofequations (and also using computer algebra to help un-pack them), some Appendices are included with some (itis hoped) helpful technical notes and suggestions aboutthe methods employed, for the reader interested in com-puting these or other results using this formalism. Ap-pendix A presents a numerical study of the spectral formfactor of the Airy model (the double–scaled GaussianHermitian matrix model) and compares the results tothe known exact expressions, showing how the effects ofthe truncation to a numerical system are extremely wellcontrolled. This serves as a demonstration of the trust-worthiness of the numerical results obtained for the JTgravity and supergravity models in the main body of thepaper. Appendix B 1 describes aspects of solving high or-der differential equations numerically, and Appendix B 2describes how to solve for the energies and eigenfunctionsneeded to build the spectrum and spectral form factor.Appendix C lists some important quantities needed in thebody of the paper (the Gel’fand–Dikii differential poly-nomials) and a recursion relation for getting the higherorder expressions.

There are some brief closing remarks in the final sec-tion, VIII, with thoughts about the potential applicationof these methods to other systems.

II. JT GRAVITY LIGHTNING TOUR

Although it is a 2D theory of quantum gravity, byvirtue of a coupling to a scalar, the dynamics of JT

gravity is all on the 1D spacetime boundary. (A goodreview of much of this is ref. [14].) The boundary canchange its shape while keeping its total length fixed to bethe inverse temperature β=1/T , the period of Euclideantime. Meanwhile, the bulk spacetime has constant neg-ative curvature (the Ricci scalar R=−2). So the theoryis locally AdS2, and the leading spacetime (disc topol-ogy, i.e., no handles or crosscaps, one boundary) is oftencalled “nearly–AdS2” [15–18], in the sense that, e.g. inPoincare coordinates, the boundary is not a fixed cir-cle an infinite distance away, but instead a finite loop oflength β that is allowed to change its shape. See figure 3.

Figure 3: The “nearly AdS2” geometry, presented in twoequivalent ways.

At this order the dynamics of the loop is controlled bya Schwarzian action[16], and the result is:

ZJT0 (β) =

eS0eπ2

β

4π1/2β3/2=

∫ ∞0

ρJT0 (E)e−βEdE , (2)

related to the disc order spectral density ρJT0 (E) by a

Laplace transform. There is a JT supergravity general-ization of this result [6, 19]:

ZSJT0 (β) =

√2eS0e

π2

β

π1/2β1/2=

∫ ∞0

ρSJT0 (E)e−βEdE , (3)

defining a disc order spectral density ρSJT0 (E). In each

case, the densities are given by:

ρJT0 (E) = eS0

sinh(2π√E)

4π2, and (4)

ρSJT0 (E) =

√2eS0

cosh(2π√E)

π√E

. (5)

(Henceforth the redefinition√

2ρSJT0 →ρSJT

0 will bedone, to adapt JT conventions of ref. [6] to the matrixmodel normalization to be used here.) The coupling e−S0

will be denoted ~ in what follows, and indeed it will bethe ~ of a key quantum–mechanical system to appearshortly. One interpretation of S0 is that it is simply theleading (T=0, disc topology) contribution to the entropy.For the ordinary JT case:

S =

(1− β ∂

∂β

)lnZ0(β) = S0+

2π2

β− 3

2lnβ+· · · , (6)

4

This leads to a second reason (beyond the one mentionedin the introduction) to study JT gravity. It is a modelof the low–temperature (near–extremal) dynamics of cer-tain higher dimensional black holes and branes (see e.g.refs. [20–24]). For example, the metric of a charged blackhole in d=4 is well known to become AdS2×S2 at T=0,and the area A of the two–sphere, S2 sets the T=0 en-tropy: A=4S0. Turning on a small temperature replacesAdS2 by “nearly-AdS2”, and the horizon area and hencethe entropy gets corrections. The JT gravity model cap-tures the dynamics of these corrections. (The dynamicalscalar represents the deviation of the area away from ex-tremality.) The 2D dynamics can be thought of as con-taining black holes in its own right as well, worth study-ing in their own terms. These are, at leading order, thedisc geometries already described.

A third reason for studying JT gravity is that it is alow energy holographic dual, in a certain sense [15, 17,18, 25] of a class of 1D quantum systems that exhibitquantum chaos, such as the Sachdev–Ye–Kitaev (SYK)model [16, 26, 27]. A key diagnostic of the quantumchaotic behaviour of the system is the 2–point “spec-tral form factor” 〈Z(β−it)Z(β+it)〉, which exhibits cer-tain key universal features [7, 8, 28, 29]. Starting out at〈Z(β)2〉, it decays down a “slope” to a “dip” at duringthe first epoch of time t, rises along a “ramp” at inter-mediate times, before levelling off to a “plateau” at latetimes at a value given by 〈Z(2β)〉. (See all these featuresin figure 1, but recall that it is not an SYK spectral formfactor, but a gravity one; See below).

The timescales over which these features manifest areimportant, especially the time to when the plateau setsin, as it gives a measure of how long correlations taketo wash away. No single SYK dual cleanly exhibits theuniversal behaviour individually. There are wild oscil-lations in the spectral form factor at intermediate andlate times 3. Instead, these features emerge as the time–averaged behaviour, as can be seen by averaging over anensemble of models [30]. An important idea in quan-tum chaos is the notion that random matrix ensemblesshould capture the universal features seen in the aver-aged behaviour of a chaotic system (for a review seeref. [7]). This led to the suggestion of refs. [30, 31] thata random matrix description of averaged SYK could beavailable. On the other hand, random matrix models areknown to describe, in a “double–scaling” limit [32–35],the sum over surfaces of a 2D quantum gravity, so this isanother way of seeing that there ought to be a dual gravi-tational description of SYK–like models. This was shownto be more than a coincidence of ideas in ref. [5], whereJT gravity was demonstrated to be explicitly consistentwith —order by order in the topological expansion— theproperties of a double–scaled matrix model. Ref. [6] fur-nished several more examples and a classification of the

3 In the phraseology of the moment, these later eras are “difficulttimes” for an SYK model.

possibilities in terms of the ten standard random matrixensembles.

So the JT gravity dual (or supergravity dual, for theappropriate generalization of SYK [6, 19, 36–41]) per-forms the ensemble average directly. The early time be-haviour is controlled by the disconnected diagram con-structed of two discs (a pair of AdS2 black holes), pluscorrections, while the later ramp and plateau featurescome from the cylinder diagram (an AdS2 wormhole) [42]plus corrections. See figure 4. These amplitudes do not

Figure 4: Black holes vs. wormholes.

fluctuate chaotically in time, but have smooth behaviourto be expected from geometric objects in a theory ofgravity. This can be seen already in the leading com-putation for the cylinder diagram [5, 43], which yields a

simple t dependence: 〈Z(β−it)Z(β+it)〉∼β−1√β2 + t2

gives a rise for the initial part of the ramp behaviour ina regime that would already be beset by fluctuations inany given SYK model. For t�β, assuming the transi-tion to the plateau has not yet occurred, this yields alinear rise. In this paper it will be observed that non–perturbative effects can, depending upon the value of ~,take over rapidly to generate the ramp, and so the linearpart is hardly visible at moderate β.

As already mentioned, the plateau in the spectral formfactor (and the transition to it from the ramp) is a resultof perturbative and, especially, non–perturbative correc-tions to the leading cylinder contribution. The purpose ofthis paper is to focus on unpacking the non–perturbativedefinitions of refs. [3, 4] in order to explicitly uncover sucheffects. Figure 1, already shown above, is a sample of thework reported on in this paper. It is the full spectralform factor for a particular model of JT supergravity. Itwill be discussed more fully in section V. Now, on to thecomputations.

III. CONSTRUCTING JT (SUPER)GRAVITYFROM MINIMAL STRINGS

The key ingredients are certain double scaled matrixmodels that have been used in the past to study certainkinds of “minimal” string theories. (See e.g. refs. [43, 44]for reviews.) The details of the string theory construc-tions do not matter here. The most important fact toknow is that some of the models (a subset of the “one–cut” matrix models) can be described in terms of an as-sociated 1D quantum mechanics problem [45, 46], with

5

Hamiltonian:

H = −~2 ∂2

∂x2+ u(x) , (7)

where the potential u(x) satisfies a non–linear ordinarydifferential equation (ODE) called a “string equation”.The key task is to build the correct u(x) for the problemin hand. Once it is known, the full spectral density canbe extracted by simply solving the spectrum of H andevaluating the fully non–perturbative ρ(E), using an ex-pression given in the next section. It is useful to note thatin the limit where just the disc–level physics is kept, thespectral density at this order can be written as a sim-ple integral involving the leading part of the potential,u0(x)= lim~→0 u(x):

ρ0(E) =1

π~

∫ E

0

f(u0)√E − u0

du0 , (8)

where f(u0)=−∂x/∂u0(x).Turning back to the ingredients, the minimal models

will be labelled by an integer index, k. As mentionedbefore, the models will be combined together to yield theJT (super)gravity. There is a parameter, tk, that will beused to turn on the kth model in the mix. The modelis turned on if tk is non–zero. The minimal models inquestion can be obtained [47–51]4 by taking the double–scaling limit of models of a complex N×N matrix M ,with a potential V (M†M) (see also footnote 5). Thestring equation that needs to be solved is:

uR2 − ~2

2RR′′ + ~2

4(R′)2 = ~2Γ2 , (9)

where the constant Γ will be discussed shortly and

R ≡∞∑k=1

tkRk[u] + x . (10)

Here, Rk[u] is the kth order polynomial in u(x) and itsx–derivatives defined by Gel’fand and Dikii [53]. Theyhave a purely polynomial in u(x) piece, which is u(x)k, apurely derivative linear piece, u(x) x–differentiated 2k−2times, and then non–linear mixed terms involving u(x)and its x–derivatives. Here, they are normalized so thatthe coefficient of uk is unity. The first three are:

R1[u] = u , R2[u] = u2 − 1

3u

′′, and

R3[u] = u3 − 1

2(u

′)2 − uu

′′+

1

10u

′′′′, (11)

where a prime denotes an x–derivative times a factor of ~.It will transpire that R4, R5, R6 and R7 will be used in

4 These minimal models were later identified by ref. [52] as the(4k, 2) superconformal minimal models coupled to gravity witha type 0A projection.

this paper too, but since they are rather lengthy, some arelisted in Appendix C, along with methods for generatingothers if needed.

The boundary condition that ensures good non–perturbative behaviour is, for each model,

u(x)→ 0 as x→ +∞ ,

u(x)→ (−x)1k as x→ −∞ . (12)

Note the presence of ~=e−S0=1 in the string equation(and the various quantities that make it up). It is veryuseful for separating the classical parts from the rest, bysending ~→0, or equivalently, dropping derivatives. Forthe study of non–perturbative physics, solutions u(x) ofthe equation will be extracted for ~=1. Several resultspresented in figures to come will be for this value, be-cause it allows the non–perturbative effects to be writlarge in the results (for spectra, etc.), and therefore seeneasily. When instructive to do so, comparison to resultswith ~ dialled down will be discussed. It is interestingthat it is in fact more difficult to solve the string equa-tions for smaller ~. This is because when derivatives havesmaller coefficients, they are allowed to fluctuate more,contributing to the sensitivity when solving these highlynon–linear equations, as will be discussed later. (Thisincreased difficulty to get smaller ~ results has the char-acter of a sort of strong/weak coupling duality, in fact.)

Turn now to the constant Γ in the string equation (9).With it present, the matrix model is in the (2Γ+1, 2) classin the (α,β) Altland–Zirnbauer classification of matrixensembles 5. The two choices Γ=± 1

2 will mostly be con-sidered in this paper, and the two JT supergravity modelsdiscussed here are will be labelled (2, 2) and (0, 2).

A particular (say, themth) minimal model can be stud-ied by setting all the tk to zero except for k=m. As dis-cussed in the previous two papers [3, 4], the m=1 case inparticular was important as it models the low energy tailof the eigenvalue distribution very well. To get the fullbehaviour, all of the tk must be turned on in a particularcombination. For example, in the case of JT supergrav-ity, the combination (derived in ref. [4]) is:

tk =π2k

(k!)2. (13)

5 Double scaling means that in the matrix model[47–51] of thecomplex matrix M , the size N is taken to infinity while couplingsin the potential V (M†M) are tuned to certain critical [54] values.Diagonalizing M turns this into a problem involving its eigen-values λi (i=1 · · ·N) at a cost of introducing a van der Mondedeterminant J=

∏i<j(λ2

i − λ2j )2 for the Jacobian. The effec-

tive Dyson gas problem for the λ2i can be thought of as existing

on the positive real line, with a wall at zero. The constant Γ inequation (9) can be thought of as arising from the coefficient of alogarithmic term in the potential of the model (see e.g. ref. [55]),and as such, results in an extra factor λ2Γ

i to the effective integra-

tion measure over the ith eigenvalue, giving∏

i λ2Γ+1i dλi. With

the factor J included, the model is seen to be in the (2Γ+1, 2)class in the (α,β) Altland–Zirnbauer classification of matrix en-sembles, defined for α=0, 1, 2. Actually Γ can be more generalintegers or half–integers that just these values.

6

So all the infinite models are turned on and the stringequation becomes a highly complicated object. But thepurpose of this paper is to show that physics can be read-ily extracted nonetheless.

Here is the reason why. First, look at the disc level.The string equation is expression (9) with the three partswith ~2 in front of them removed, and so the solutionsare either u0=0 or

R =

∞∑k=1

tkuk0 + x = 0 , (14)

corresponding to the asymptotics given in equation (12).For the second choice, the x<0 regime, the combina-tion (13) of tk’s amounts to f(u0)=πI1(2π

√u0)/√u0 in

equation (8), yielding the part of the spectral density ex-

panded in positive powers of E. The simple E−12 part

comes from the u0=0 behaviour. Integrating f(u0) withrespect to u0, or simply by looking at equation (14), theexplicit potential that gives JT supergravity on the discis given by the equation

x = 1− I0(2π√u0) , (15)

where I0(s) is the zeroth modified Bessel function of s.This is a remarkably simple form.

The issue of tractability becomes the simple issue ofhow well this potential can be approximated by truncat-ing to a finite number of tks. The answer boils down towhat maximum energy scale E one wants to know thespectrum up to, and to what accuracy. As an exam-ple, the full classical potential (15) is plotted in figure 5alongside two truncations.

0

Figure 5: The complete classical potential for JT supergravity(solid line). The uppermost dotted line is a truncation up to t2,the lower dotted is a truncation up to t4.

The first truncation contains just t1 and t2:

x = −π2u0 −1

4π4u2

0 , (16)

and it is clear that it is a good approximation for energiesup to approximately E∼0.1 after which it begins to de-viate considerably. The next example truncation adds t3and t4:

x = −π2u0 −1

4π4u2

0 −1

36π6u3

0 −1

576π8u4

0 , (17)

and for energies up to order E∼0.5 it serves rather well.Further improvements come by adding higher orders.

The next issue to appreciate is how much the solu-tion changes when all the non–perturbative correctionsare included. For all k a shallow well can develop in thecentral region (slightly to the right of x=0). Crucially,moving away from that region, the deviation of the so-lution from the disc level behaviour rapidly dwindles, asit matches on to the asymptotic behaviour. The same istrue for the coupled solution. Moreover as can be seenfrom equation (13), the form of the tk as k grows is suchthat good approximations at the disc level can be foundby adding only a small number of minimal models, fora given needed accuracy. For larger Es, the solution be-comes hard to distinguish from the classical result, andin that case the exact classical potential can be used, toa good approximation.

The highest truncation levels chosen for the purposesof this paper was to keep all the minimal models up tok=6 (this section) and k=7 (for section VI), althoughvery good results were obtained for lower order trunca-tions too. Since R7[k] has the twelfth order derivativeof u(x) in it, the string equation (9) is a 14th order dif-ferential equation. (It is 12th order for the k=6 trunca-tion.) In general, it is easier to take a derivative of thestring equation, whereupon an overall factor of R canbe divided out, reducing some of the non–linearity some-what, at the expense of an increase in the order. For theboundary conditions in question, a 15th (or 13th) orderdifferential equation is not too hard to solve numerically,with care. Some suggestions and notes are given in ap-pendix B 1, for those who wish to carry out their owncomputations using this framework. Part of the full non–perturbative potential u(x) for the truncation to k=6 isdisplayed in figure 6.

This is for the case Γ=+ 12 , i.e., the (2,2) JT super-

gravity. (It was solved between x=−200 and x=+200.)Notice that it approaches the classical solution and agreeswith it rather well up (and beyond, it turns out) toE∼1.3, and so the full solution, out to beyond thex'−100 shown, can be used to capture the spectrumof (2,2) JT supergravity with good accuracy (see more inappendix B 2 on how to do this). It is (relatively) easy todo better, if desired, but little visible change was noticedin going to higher orders, in exchange for accessing onlya slightly larger maximum energy.

Figure 7 shows the solution for Γ=− 12 , which will be

used to study the properties of the (0,2) JT supergravity.

This solution has a well in the interior (as is quite typi-cal of these solutions), and so slightly more rapid changes

7

Figure 6: The Γ=+ 12

solution (solid line) of the string equationfor truncation up to t6. The inset shows a closeup of the smoothtransitional region in the interior. For comparison, the fullclassical solution line is shown too (dotted).

Figure 7: The Γ=− 12

solution (solid line) of the string equationfor truncation up to t6. The inset shows the smooth well thatdeveloped in the interior. For comparison, the full classicalsolution is shown too (dotted).

take place there. Rather than use numerical methods tosolve for this directly (which are inevitably more sensi-tive to error in this case), a handy solution–generatingtechnique derived in ref. [56] was used6, that allowed itto be generated from the Γ=+ 1

2 solution already found.See Appendix B 1 for more on this.

6 It is inherited from the rather rich underlying Korteweg–de Vries(KdV) hierarchy integrable structure that underpins this entireformalism.

IV. THE SPECTRAL DENSITY

A. General Remarks

The next step is to solve the full spectral problem forthe Hamiltonian H, given the potential u(x) found in theprevious section by solving the (truncated) string equa-tion. The relation between the spectrum ofH and the JTpartition function is as follows. From the minimal stringperspective, the JT (super)gravity partition function issimply [5, 11] the expectation value of a “macroscopicloop” of length β. The technology for working this outwas derived long ago in ref. [45]. (Ref. [43] unpacks theformalism in a useful review.) It is the trace of the ex-ponentiated H, with a projection P inserted:

Z(β) = Tr(e−βHP) (18)

where the operator P≡∫ µ−∞ dx |x〉〈x|, and the upper

limit µ will be discussed shortly. Inserting a completeset of states: ∫

dψ|ψ〉〈ψ| = 1 , (19)

yields

Z(β) =

∫ µ

−∞dx〈x|e−βH|x〉

=

∫ µ

−∞dx

∫dψ〈x|e−βH|ψ〉〈ψ|x〉

=

∫ µ

−∞dx

∫dψE〈x|ψE〉〈ψE |x〉e−βE

=

∫dEe−βEρ(E) , (20)

where

ρ(E) =

∫ µ

−∞ψ(x,E)ψ∗(x,E)dx , (21)

is the spectral density.To understand the upper limit µ, return to the leading

order expression for the density given in equation (8), andchange variables to x. The solid black line in figure 5 isa reminder of the behaviour of u0(x). The part of theintegral where u0(x)=0 begins at x=0 and extends topositive x to some value denoted µ, while (at a given E)the lowest value for x is set by the turning point whereE=u(x), at a position denoted −|x0| and so:

ρ0(E) =

1

π~

∫ µ

−|x0|

dx√E − u0(x)

. (22)

In the full quantum expression (21), there are contri-butions to the spectral density from the whole integra-tion range down to x=−∞. Intuitively, this is becausewavefunctions penetrate to the left beyond the classicalturning point E=u(x). (Often, in the classical expres-sion (22), the lower limit is also written as x=−∞ with

8

the understanding that only the real part of the integralcontributes to ρ

0)7. The role of µ becomes clear from fo-

cussing on just the contribution from the u0(x)=0 part in

equation 22. It adds µ/(π~√E) to the classical spectral

density. Physically, µ controls how much the classicalspectral density is “piled up” against the natural wall atE=0 that stops the energies in the complex matrix mod-els from going negative (see footnote 5). Vanishing µcorresponds to just touching the wall. In the normaliza-tion of equation (5) (without the

√2), the value of µ is 1,

and this will be used for much of the rest of this paper.However, it is useful for later to write a more generalclassical spectral density with a different µ:

ρ0

SJT(E,µ) =1

π~

[cosh(2π

√E)√

E+µ− 1√E

]. (23)

B. Computation

Now to the matter of computing the full spectral den-sity (21). Just as in refs. [3, 4], a matrix Numerovmethod [57] was used to solve for the spectrum of H.Conceptually it is a simple problem in finding eigenval-ues and eigenfunctions for a Schrodinger operator in 1D.The wavefunctions are free and oscillatory to the far right(x>0), begin to feel the presence of the potential as theymove further into the interior (it starts as a 1/x2 de-pendence), and then once they hit the potential there isan exponential decay to the left (x<0). As a guide toextracting them with good accuracy, some extra sugges-tions and notes for interested readers are given in ap-pendix B 2. The same normalization method as the oneused in ref. [3] was used for the resulting eigenfunctions.The key point explained there is that in the far x>0 re-gion, wavefunctions are known to asymptote to a simpleform involving the Bessel function of order Γ, where thenormalization can be analytically chosen to yield the cor-rect contribution to the disc level spectral density.

The outcome of the numerical spectrum solving wasapproximately 1000 accurate normalized wavefunctionsand their energies, for use in constructing the density inthis section, and the spectral form factor in the next.Using them, the spectral density can be constructed us-ing a simple trapezoidal integration to implement equa-tion (21) and the result (for µ=1) is shown in figure 2 for

7 Of course, the integrand of (21) is not written in terms of Eand the full u(x), but in terms of the wavefunctions. The pre-cise connection between the two expressions is via the diago-nal of the resolvent, (H−E)−1, of H, denoted R(x). It canbe built out of ψ(x,E)ψ∗(x,E), using standard Green func-tion methods. Gel’fand and Dikii [53] derived an equation

for R(x) which is 4(u(x)−E)R2−2RR′′+(R′)2=1. The lead-ing order solution comes again from dropping derivatives, andis R(x)=±(2

√u(x)−E)−1. In the normalization of this paper

ρ0=(2/π~)Im∫R(x)dx, yielding the form (22).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

5

10

15

20

25

30

35

40

45

50

Figure 8: The full spectral density (made out of red dots),computed using the methods of this paper, for the (0,2) model ofJT supergravity. The dashed blue line is the disc level result ofequation (5). Here, ~=1. The companion result for the (2,2)model is in figure 2.

Γ= 12 (shown on page 2) and figure 8 for Γ=− 1

2 . Plottedalongside the (dense) line of dots (the computed spec-tral data) is a dashed line showing the disc level spectral

density. Strikingly, the 1/√E divergence present at disc

level is erased entirely by non–perturbative effects in theΓ= 1

2 (2,2) case, but not in the Γ=− 12 (0,2) case. As dis-

cussed in refs. [4, 6], for Γ=± 12 , there are no perturba-

tive corrections to the spectral density beyond the disc,so all the differences here are due to non–perturbativephysics, which makes these JT supergravity cases partic-ularly interesting to study when investigating the resultsof non–perturbative physics. This stark difference willbe reflected in comparisons of the spectral form factor,to be studied in the next section.

Satisfyingly, what has appeared here for the full (2,2)and (0,2) cases are actually grown–up versions of whatwas discussed for two baby (Bessel) models in ref. [6],and so this constitutes a nice consistency check of themethods of this paper. The Bessel special cases are mod-els of the very low energy tip of the spectral density—thepart that (classically) pushes up against the E=0 wall,as mentioned earlier. Generalizing to include arbitrarypositive µ (see also refs. [3, 4]), the full non–perturbativeBessel spectral density is (after a change of conventionsfrom those papers):

ρ(E) =µ

π~√E

+sin(−π2α+ 2µ

√E

~

)2πE

, (24)

where (as a reminder) α=2Γ+1 takes the values 0 and 2.The leading part is the disc contribution and all otherperturbative contributions vanish. The oscillating term isthe full non–perturbative contribution. At E=0 for α=2,(Γ= 1

2 ) there is the aforementioned cancellation betweenthe divergent disc contribution and the E→0 piece of

9

the oscillating non–perturbative part. This is what isnow seen to be present (as it ought to be) in the full JTsupergravity spectral density computed explicitly above.

C. A Special Formula, and a Tale of Instantons

Actually, it is possible to go considerably further thanthe Bessel tail comparison. The structure of the Besselmodels above (along with various other technical fea-tures of the matrix model recursion relations) led Stan-ford and Witten to suggest (see appendix E of ref. [6])that more generally, the JT supergravity spectral densityfor the (0,2) and (2,2) cases should closely approximatethe form8:

ρ(E) ≈ ρ0(E) +

sin(−π2α+ π

∫ E0ρ

0(E′)dE′

)2πE

. (25)

(Note that their conventions were adapted to match thoseof the current paper.) Again, the perturbative terms be-yond the disc vanish. The argument of the sinusoidalnon–perturbative piece is determined in terms of the disccontribution9. Crucially, they note that this does notinclude possible instanton corrections, non–perturbativephysics of a form not accounted for by the sinusoidalmodulation (hence the use of the ≈ sign in the above).

While the expression above was written for the specificvalue µ=1 (µ is not a variable parameter in ref. [6]), othervalues of µ (see below equation (21) for its explanation)are quite readily incorporated by this form. This is al-ready obvious for the simple Bessel prototype (24), butfurther evidence comes from inserting into it the expres-sion (23) for ρ

0(E,µ) written earlier for the disc level JT

supergravity density for other µ. Plotting this analyti-cal result against the numerical results for the spectraldensity obtained using this paper’s methods yields a re-markable agreement, as shown in figures 9 and 10 forΓ= 1

2 and Γ=− 12 , for the values µ=1, 5, and 10. As a

reminder, the red lines are each made of ∼1800 pointsindividually computed using the truncation and numeri-cal methods outlined, while the blue dots are samples ofthe analytical formula (25). It is very striking how wellthey agree.

The areas of disagreement suggest interesting physics,in fact. For smaller µ (e.g., see the case of µ=1 in eachfigure) it is easy to see disagreement. It is especiallyquite visible at smaller energies (although at E=0 theyagree as they must, since this is covered by the Besselcases above). The clear pattern in the discrepancy issuggestive of physics and not numerical inaccuracies.

8 The author thanks Douglas Stanford for asking a helpful questionabout this issue after the 1st version of this manuscript appeared.

9 In fact, it is easy to guess generalizations of this formula forother half–integer Γ, which seem to give sensible physics usingthe present methods. It would be interesting to test them.

For example, the fact that the disagreement rapidlydisappears as µ increases is striking. Nothing about thenumerical methods should show this sort of systematicdependence. As a further example, the form of the dis-agreement suggests that there is a change in the ampli-tude of the sinusoidal modulation, again a pattern thatis hard to reproduce with mere numerical systematics. Anatural guess is that there are further multiplicative fac-tors in the non–perturbative parts that are not presentin the analytic formula. They must be small, and ofan instanton form (as already anticipated in remarks inref. [6]).

0 0.1 0.2 0.3 0.4 0.5 0.6

E

0

5

10

15

20

25

30

(E)

Figure 9: Comparison of the full spectral density for the (2,2)model of JT supergravity, with equation (25) for the leading form(dark blue dots). The dashed line is the disc level result (23). Thecases µ=1, 5 and 10 are shown (lower to upper). Here ~=1.

0 0.1 0.2 0.3 0.4 0.5 0.6

E

0

5

10

15

20

25

30

(E)

Figure 10: Comparison of the full spectral density for the (0,2)model of JT supergravity, with equation (25) for the leading form(dark blue dots). The dashed line is the disc level result (23). Thecases µ=1, 5 and 10 are shown (lower to upper). Here ~=1.

10

In the kth minimal model the height of the effectivepotential barrier for one eigenvalue in the background ofall the others yields the action of the instanton effects [58](they are “ZZ–branes” in the modern parlance [59]). Al-ternatively, it can be computed using a WKB analysisof the string equation itself. This was all worked outlong ago for the complex matrix models in ref. [50] (seeref. [60] for a review and more contemporary presen-tation), and the dependence of the action on µ and ~is ( 4k

2k+1 )µ1+ 12k /~. Therefore a dependence exp(−2µ/~)

should be expected for instanton corrections to JT super-gravity constructed out of minimal models in the mannerof this paper.

To test this, a rough estimate of the dependence wasdone by measuring the size of the deviations of the typeseen in the figures (9 and 10) for some sample cases, as µvaried from 0 to 5. The deviations fall rapidly in thatrange (as is evident from the figures), and a logarithmicplot of them against µ yielded a straight line of negativeunit slope to good (better than 1%) accuracy. Whilethis can be made more precise, it is already a strongindication that the instanton effects of the expected formare present and accounted for. It will be interesting toexplore this further, but this will be left for later work.

D. Going to Weak Coupling

Before moving on it is worth showing the effect of goingto weak coupling by reducing ~. The spectra shown so farare for ~=1, and working in this regime has the advantageof making much of the non–perturbative physics readilyvisible. Nevertheless, it is useful to be able to gain accessto weaker coupling, not the least because certain dualsystems of interest (such as SYK–type models, or largeblack holes (where S0 is large)) may be in that regime.

It is possible to reduce the value of ~, but it is at theexpense of not being able to access as high energies inthe spectrum for a given truncation, unless there is acompensation in terms of increased computational effort(working on a finer grid and producing more wavefunc-tions). Additionally, solving the string equation takesmore care, as already mentioned, because the derivativesare more in play. Nevertheless, progress is possible. Forexample, dialling down to e.g., ~= 1

5 , the string equa-tions can be coaxed to produce the solutions shown (justas close–ups, for brevity) in figure 11. These should becompared to what was shown in the insets of figures 6and 7. For the (2,2) case, the curve turns the cornermore sharply (showing the increased role of the deriva-tives), while for the (0,2) case, the well is far deeper andnarrower (these two features come together to continueto ensure there are no bound states).

It is easy to imagine how these developments continuein order to get to the classical limit ~→0 (the red dottedline in the figures). The (2,2) curve turns the corner evermore sharply, and for the (0,2) curve the well deepensand narrows and eventually is entirely squeezed away.

Figure 11: The solid lines show close–ups of the Γ= 12

or (2,2)

solution (left) and the Γ=− 12

or (0,2) solution (right) of the

string equation for truncation up to t6, for ~= 15

. For comparison,the full classical solution is shown too (dotted).

The techniques already described in the previous twosections can be carried out to study the spectral density,and the result in figure 12 displays the results (red dots)for the Γ= 1

2 or (2,2) case, for the cases µ=1 and 5. (Theµ=10 case, which rises to 508 vertically, was omitted forclarity.) Again, the blue dots are samples of the analyticformula (25) with (23) input for ρ

0(E,µ) (now ~= 1

5 isinserted), showing again how well this procedure works.Notice that, in accord with the fact that the coupling isweaker, the exp(−µ/~) instanton effects described in theprevious section, which would appear as deviations of thered curves from the blue dots, are too small to seen here.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

20

40

60

80

100

120

140

Figure 12: Two examples (µ=1, lower; µ=5, upper) of the fullspectral density for the (2,2) model of JT supergravity, for ~reduced to 1

5. The (blue) dots are points from the equation (25).

In each case the dashed line is the disc result of equation (23).c.f. the ~=1 cases of figure 9.

Comparing to the ~=1 curves in figure 9, the quantumundulations are smaller, for generic E, and the curvesstray less from the classical curve, beginning marked de-viations only at lower energies compared to the ~=1 sit-uation, until the curve falls quickly to zero. This is as itshould be.

11

V. THE SPECTRAL FORM FACTOR

A. General Remarks

As mentioned in the brief review of section II, thespectral form factor is derived from the two point func-tion of Z(β). This has two parts, a disconnected piece〈Z(β)〉〈Z(β′)〉 and a connected piece 〈Z(β)Z(β′)〉. Inthe old matrix model language, the connected piece isthe connected correlator of two “macroscopic loops”, andthis is readily written down as [43, 45]:

〈Z(β)Z(β′)〉 = Tr(e−βH(1− P)e−β′HP) (26)

= Tr(e−(β+β′)H)− Tr(e−βHPe−β′HP)

= Z(β+β′)−∫dE

∫dE′ρ(E,E′)ρ∗(E′, E) ,

where P is discussed below equation (18) and

ρ(E,E′) =

∫ µ

−∞dxψ∗(x,E)ψ(x,E′) . (27)

The quantity µ, discussed in the previous section, will beset to unity henceforth, corresponding to having the disclevel spectral density given in equation (5) (without the

overall√

2).

B. A Phase transtition

Set β=β′ for a while. Generically, the disconnectedpart (corresponding to two black holes) and the con-nected part (a wormhole) are worth studying in theirown right as distinct sectors of the quantum gravity thatcompete for dominance [61] as a function of β. The dis-connected part, being the square of the partition func-tion, rapidly decreases with increasing β while the con-nected part increases. At some point βcr there is a tran-sition, and the connected diagram becomes more dom-inant. This is also true in the complete (not just per-turbative) theory discussed here. This is all nicely un-der control in the current definitions of JT supergravity.The spectrum has been computed in the previous sectionand so all the elements in equation (26) are readily com-putable to the desired accuracy. Figure 13 shows a plotof the (log10 of the) connected and disconnected pieces —with all perturbative and non–perturbative contributionsincluded— as a function of β, for the (2,2) JT supergrav-ity case, showing the transition at βcr=13.56. A similarcomputation (yielding a similar graph, omitted) showsthat βcr=28.30 for the (0,2) JT supergravity case.

Notably, for the (0, 2) case the amplitudes are overan order of magnitude larger. This is a striking effectattributable entirely to non–perturbative effects. The(2,2) JT supergravity has, as mentioned in the previ-ous section IV, non–perturbative effects that cancel the1/√E behaviour at low energy, coming from the leading

disc amplitude. The (0,2) version does not cancel this

12 12.5 13 13.5 14 14.5 15-2.1

-2.05

-2

-1.95

-1.9

-1.85

-1.8

-1.75

-1.7

-1.65

Figure 13: The disconnected (starting higher at the left) vs. theconnected (lower) two–point function of the partition function asa function of β at ~=1, showing a phase transition at βcr'13.56.

away, and so while to larger E the two models’ spec-tra are roughly similar (see figures 2 and 8), there is anenhancement at low E that means much larger contri-butions to the partition function for any fixed β. Thismarked difference between the non–perturbative physicswill make a major appearance in the temporal behaviourof the spectral form factor too, studied next.

C. Time Dependence

The above computation served as a useful guide forwhat to expect for the spectral form factor, which tracksthe correlation function over time. This is done byputting β→β+it and β′→β−it, with β fixed, studyingthe dependence on t. The fixed β can be above or be-low βcr. The disconnected part will start out as thesquared partition function and then decrease with t. Thisis the “slope” behaviour of the spectral form factor. Onthe other hand, looking at equation (26), it can be seenthat the connected part has a t–independent part, Z(2β),from which is subtracted a positive piece which gets smallat large t, with only significant contributions from ener-gies that are close to each other. The value of Z(2β)therefore sets the height of the universal “plateau” fea-ture and the approach to it, the “ramp” has its size set byhow rapidly the energy correlations die away at large t.The “dip” region is formed by the process of handingover from the decreasing disconnected part to the increas-ing connected part. The formalism here allows, usingthe complete package of almost 1000 good wavefunctionsand energies, for this all to be computed to good accu-racy for JT supergravity (and for a non–perturbativelywell–behaved definition of JT gravity in section VI) forthe first time. Using again the JT supergravity exam-ple, figure 14 shows the disconnected contribution to the

12

spectral density function, for β=50>βcr, with log10 axes.

-0.5 0 0.5 1 1.5 2 2.5 3-8.5

-8

-7.5

-7

-6.5

-6

-5.5

-5

-4.5

-4

Figure 14: The disconnected part of the (2,2) JT supergravityspectral density function vs. t, at β=50 and ~=1, showing theclassic slope feature.

The slope behaviour is quite evident. Since the axes arelogarithmic, it is easy to see from the figure that the slopeof it is roughly −3, suggesting a fall–off of ∼t−3. Thismight seem surprising since it is similar to the pertur-bative fall off rate for ordinary JT gravity (see ref. [30]).The reason for this faster rate is clear, and again at-tributable to non–perturbative physics. The slope’s fall–off is controlled by the behaviour of the endpoint of thespectral density. While at the disc level for JT supergrav-ity ρ(E)∼1/

√E there, producing a t−1 fall–off [62], the

fact that in the (2,2) case non–perturbative corrections

remove this 1/√E behaviour results in the faster fall off

more usually associated with the ordinary JT case (andHermitian matrix models). There are far fewer states inthe vicinity of the endpoint. This reasoning predicts thatfor the (0,2) supergravity, the slope should be closer toa t−1 fall-off. Indeed, this is clear from the behaviourof the disconnected piece for (0,2) supergravity, shownin figure 15, for β=50. There, the slope of the linearpart shows t−1 fall off. Continuing in line with these ex-pectations is a computation of the same quantity for the~=1/5 case mentioned earlier. The spectrum was shownin figure 12, and from there it is natural to guess that thefall-off would be even faster since the non–perturbativeeffects have scooped away even more states near the end-point. A check showed that this is indeed correct, al-though another figure will not be presented to displaythe result, to save repetition.

Moving to the connected contribution’s time depen-dence, the (2,2) case is shown in figure 16 for the samevalue of β=50. The ramp and plateau structures, and thetransition between them, are visible. Strikingly, the riseto the plateau is very short–lived, the ramp regime ris-ing only a small (on the logarithmic scale) amount before

0.5 1 1.5 2 2.5 3

-2.6

-2.4

-2.2

-2

-1.8

-1.6

-1.4

-1.2

Figure 15: The disconnected part of the (0,2) JT supergravityspectral density function vs. t, at β=50 and ~=1.

-0.5 0 0.5 1 1.5 2 2.5 3-2.668

-2.666

-2.664

-2.662

-2.66

-2.658

-2.656

Figure 16: The connected part of the (2,2) JT supergravityspectral density function vs. t, at β=50 and ~=1, showing theclassic ramp and plateau features.

transitioning to the plateau.

The beginning shape of the ramp is already anticipated

in the perturbative answer, ∼β−1√β2 + t2, long known

for two–macroscopic–loop correlators [43] (appropriatelycontinued to yield the t–dependence [5]), but there arestrong non–perturbative corrections such that before thelong–time linear part can manifest, the other effects turnthe ramp over into the plateau.

A similar story is told, initially, by the ramp shape seenfor the (0,2) case, shown in figure 17 (again for β=50).However, there’s a new feature. The rise is indeed slow,but it is remarkably slow. After almost two orders ofmagnitude more time has elapsed, as compared to the(2,2) case, the saturation to the plateau has still not quitecompleted.

13

This is a rather novel feature of this case, and worthfurther investigation. The origin of this physics is most

0.5 1 1.5 2 2.5 3 3.5 4-0.98

-0.96

-0.94

-0.92

-0.9

-0.88

-0.86

-0.84

-0.82

-0.8

-0.78

Figure 17: The connected part of the (0,2) JT supergravityspectral density function vs. t, at β=50 and ~=1.

likely again to be attributed to the peculiar pileup ofstates in the vicinity of E=0 that this model has. There’san endless supply of closely spaced low–lying states con-tributing to the part of the form factor that subtractsfrom the saturation value Z(2β) (see equation (26)). Atlonger and longer times there are even more low lyingstates to contribute, and still closely spaced, maintainingtheir effect of slowing the saturation.

Turning back to the ramp itself, note that this was(for both (2,2) and (0,2)) for the case ~=1. At smallervalues of ~ there is more time for the ramp to develop,with an increased rate of rise before the turnover. Notehowever that it is only for extremely small ~ that thelinear behaviour of the ramp has a chance to appear.

There are two important lessons here. The first is thatassociating the ramp with linear behaviour (as is some-times done in the literature) is maybe not the most ac-curate descriptor. The second is that non–perturbativeeffects can enhance the appearance of the ramp in the JTsupergravity case (in the full spectral form factor madeby taking the sum of disconnected and connected parts,even though perturbative expectations might have sug-gested a reduction [62]. The potential reduction of theramp feature is based on the idea that a slow t−1 rate offall might not give the ramp time to develop before theplateau sets in. In fact, non–perturbative effects are seenhere to produce a rapid fall (sometimes faster even thanthe perturbative bosonic t−3) giving plenty of time to de-velop a sharp “dip”, a clear ramp, and a smart turnoverinto the plateau for the (2,2) case.

The sum of the connected and disconnected piecesgives, for (2,2) supergravity, the classic saxophone

shape10 known from studying spectral density functionsin a wide range of contexts. It is displayed in figure 18.This is for ~=1, and the case of ~=1/5 was already pre-

-0.5 0 0.5 1 1.5 2 2.5 3-2.65678

-2.65677

-2.65676

-2.65675

-2.65674

-2.65673

-2.65672

Figure 18: The full spectral form factor for the (2,2) model of JTsupergravity at β=50, ~=1. (The case of ~=1/5 is in figure 1).

sented in figure 1, at β=35. The latter, being at smaller βand ~, is larger overall and develops a wider variation.The linear part of the ramp would be even more visiblefor lower values of these parameters.

For (0, 2) JT supergravity, figure 19 shows the result-ing ~=1 full spectral form factor, again at β=50. It is, asto be anticipated, almost two orders of magnitude largerthan for the (2,2) case, because of non–perturbative ef-fects (already discussed). There is (on the logarithmic

0.5 1 1.5 2 2.5 3-0.79385

-0.7938

-0.79375

-0.7937

-0.79365

-0.7936

-0.79355

-0.7935

-0.79345

-0.7934

Figure 19: The full spectral form factor for (0,2) JT supergravity.Here, β=50 and ~=1. c.f. the (2,2) case in figure 18.

10 The shape deserves a name, and saxophone seems a good choice,to balance out the many uses of the name “trumpet” in otheraspects of JT gravity.

14

scale) a ramp–to–plateau transition (although it is, fromthe discussion above, much slower than for (2,2)). Alsovisible is the slower slope–to–dip time (due to its slowerdecay rate), as already discussed.

D. Temperature Dependence

It is of interest to see how the spectral form factorevolves as a function of temperature. The results fora series of increasing temperatures, β=50, 46, 42, 38, 34and 30 are presented for the disconnected part of thespectral density in figure 20.

-0.5 0 0.5 1 1.5 2 2.5 3-9

-8

-7

-6

-5

-4

-3

Figure 20: A series showing the evolution, with β, of thedisconnected part of the spectral form factor, for (2,2) JTsupergravity. Here ~=1.

The highest temperature curve is at the top. Strik-ingly, the curves soon merge into each other and followthe fall–off already discussed, regardless of the startingtemperature. In figure 21, there is a series of the fullspectral density, for the same set of temperatures. Cru-cially, for comparison purposes, the curves are all uni-formly scaled (on the vertical axis) to have the same ini-tial height as the highest temperature (β=30) case, whichis the lowermost curve. Therefore, in this figure, relativeslopes should not be taken literally. This scaling allowsfor ready access to some of the more meaningful compar-isons to be made, such as the relative size of the curves:Higher temperature (smaller β) gives a vertically largercurve: Higher temperature “shakes” the system up more,resulting in a wider amplitude of deviation from the ini-tial value before it settles down. Interestingly though,the dip time increases slightly with higher temperature,although not dramatically. The rapid ramp time alsochanges very slowly with β.

Heading toward smaller (but still moderate) values of β(in the region of β'βcrit) there are small modulations in

-0.5 0 0.5 1 1.5 2 2.5 3-2.658

-2.6578

-2.6576

-2.6574

-2.6572

-2.657

-2.6568

-2.6566

Figure 21: A series showing the evolution, with β, of the fullspectral form factor, for (2,2) JT supergravity. Here ~=1. Forcomparison, the curves have been rescaled to start out at thesame height as the highest temperature curve.

both the disconnected and connected parts of the formfactors, accumulating (in the latter) near the cross–overfrom ramp to plateau. The combined result of thesehigher temperature structures is that there is a dampedwobble as the ramp merges into the plateau. An exam-ple of the full spectral form factor showing this featureis given in figure 22 for the case β=14. Whether this is

-0.5 0 0.5 1 1.5 2 2.5 3-1.67

-1.66

-1.65

-1.64

-1.63

-1.62

-1.61

-1.6

-1.59

Figure 22: The full spectral form factor at β=14 ~=1, for (2,2)JT supergravity. A (relatively) high temperature feature appearsnear the cross–over from ramp to plateau. See text for discussion.

interesting physics or not is not clear 11. The value of the

11 It is reminiscent of features of an interesting exact expressionderived in ref. [63] in the context of an SYK model with sourceterms.

15

temperature at which this can be seen seems comfortablybelow the highest energy allowed by the truncation.

Some other other fascinating structures become appar-ent in the very high temperature regime. How useful theyare for the physics in question is debatable since thiswhole context (the Schwarzian, the connection to blackholes, SYK, etc.,) is in a low energy limit. Moreover,high temperature also begins to go beyond the energiesfor which the truncation of the string equation remainsreliable. However it is interesting to observe the featuresanyway, and could well be instructive for understandingJT models with a cutoff [64–67]. Looking at the dis-connected part of the spectral form factor for the (2,2)supergravity case, a series of dips evolve, becoming morepronounced toward higher temperature (smaller β). Seefigure 23.

-0.5 0 0.5 1 1.5 2 2.5 3-9

-8

-7

-6

-5

-4

-3

-2

-1

Figure 23: Disconnected part of the spectral form factor at β=9~=1, for (2,2) JT supergravity. The growing dips are consistentwith a pattern of zeros developing at infinite temperature.

They have a clear pattern and structure and are con-sistent with observations made for large N random ma-trix systems (even without double scaling). (They arealso analytically obtainable in the exact Airy examplereviewed in Appendix A.)

In fact, the disconnected function is beginning to re-semble the form J1(t)2/t2 that has been derived analyt-ically for the infinite temperature case. After taking thelogarithm, the zeroes of the Bessel function J1 becomethe dips in the logarithmic plot. It would be interestingto show that this (or a variation thereof) analytical formemerges in this JT supergravity context as well. Follow-ing the numerics to smaller β seems to confirm this (azero in the connected function also appears, in some ex-amples), although eventually numerical inaccuracies be-gin to overwhelm the results, presumably because thecorrect physics needs to include contributions from en-ergies that lie beyond the cutoff on the spectrum up towhich the truncated equations are valid.

VI. JT GRAVITY: NON–PERTURBATIVEDEFINITION A

This section presents results analogous to those shownin earlier sections, for the non–perturbative completionof ordinary JT gravity presented in ref. [3]. It mightseem odd to have studied the JT supergravity examplesfirst, leaving this case for last, but there is good reason.The non–perturbative physics of this case is more subtle.JT gravity was shown, in ref. [5], to be perturbatively (inthe topological expansion) equivalent to a double–scaledHermitian matrix model, i.e. classified in the GaussianUnitary ensemble (β=2 in the Dyson–Wigner series). Ongeneral grounds, such double–scaled Hermitian matrixmodels are known to sometimes have non–perturbative(in topology) instabilities, and so it is possible that theJT gravity definition inherits them. More specifically,thinking about the model in terms of constituent minimalmodels, as in section III, it is made up of an interpolatingfamily of minimal models that have the x→−∞ bound-ary condition in equation (12) in both directions, andinstead solve the string equation R=0. (Recall that R isgiven in equation (10)). These are the (2k−1, 2) bosonicminimal string models. For k even, these models arenon–perturbatively unstable, as has been known for sometime [45, 68–70]. From the point of view of the spec-trum, all the models, when non–perturbative effects aretaken into account, have contributions from arbitrarilynegative energy sectors. Even though exponentially sup-pressed, for even k the effective potential turns downwardfor states at these energies, signalling the system’s wishto tunnel to an entirely new solution that is quite differ-ent from the one around which perturbation theory wasdeveloped. From the perspective of this paper (solvingstring equations non–perturbatively), this means that foreach of those (k even) models, there simply are no realsmooth solutions of the kth equation with those condi-tions [45, 69, 71, 72]. Since JT gravity is made up of, inequal measure, even and odd k models, this strongly sug-gests that it inherits these problems, as already noted inref. [5]. (However, see section VII for a possible evasion.)

The route that ref. [3] took to supply a non–perturbative definition of JT gravity was to embed it intoa larger problem. The minimal models used in previoussections for JT supergravity, also indexed by k, have thesame x→−∞ boundary condition as the bosonic mini-mal models, and in fact when Γ=0 they have identicalperturbation theory as solutions to the differential equa-tion (9). Put differently they solve R=0 perturbativelyat large −x. This means that if used to construct a JTgravity model, they will yield the same physics at highenergies E, but yield different physics as lower energiesthat is untroubled by the stability issues. The combi-nation of models needed (that will give the Schwarzianspectral density (4)) at high E is as follows:

tk =1

2

π2k−2

k!(k − 1)!. (28)

16

(This relation was first derived in ref. [11], but with differ-ent normalization.) In fact it is possible to integrate thef(u0) that results from this combination to find the ex-plicit classical potential u0(x) that yields the Schwarziandensity, through:

x = −√u0

πI1(2π

√u0) , (29)

the analogue of the case (15) for the JT supergravityfound earlier. This non–perturbative taming of JT grav-ity can be explored extensively along the same lines asdone for the JT supergravity models. The truncationscheme works along the same lines, and so there is noneed to retread the ideas again. A solution to the stringequation (9) with the bosonic JT combination of minimalmodels (28) was found with the first 7 models turned on,constituting a very good truncation where energies up toE∼1.5 can be trusted. A combination of numerical andanalytic methods were used to find the solution to the15th order differential equation. See appendix B 1 forsome tips on how this was done. Figure 24 shows the so-lution, with the classical (disc level) potential that givesthe JT spectral density (4) displayed as a dashed line.

Figure 24: The solution (solid line) of the string equation fortruncation up to t7. The inset shows the well that developed inthe interior. The full classical solution is shown too (dotted).

Again there is a small well (not deep enough to sup-port bound states) in the interior, and then u(x) settlesto zero to the right. These are features shared by the JTsupergravity models, as should be expected since theircomponents are being used here as a non–perturbativelow energy “regulator” in a sense, removing the leakageto negative energy. Remarkably, the non–perturbativeeffects will (as happened for the (2,2) JT supergravity

case) remove all signs of the classical 1/√E contribution

to the low energy that would make for a striking andunsatisfactory departure from the physics of JT gravity.The methods of section IV then allowed for the spectrumto be computed (here, µ=0 is used for the upper limit in

equation (21)), and figure 25 displays, for the first time,the spectral density of a non–perturbative completion ofJT gravity with all perturbative and non–perturbativecorrections included (up to this energy). It is clear from

0 0.2 0.4 0.6 0.8 1 1.2

E

0

2

4

6

8

10

12

(E)

Figure 25: The full spectral density, for refs.[3]’snon–perturbative completion of JT gravity. The dashed blue lineis the disc level result of equation (4).

the figure that the non–perturbative ripples have alreadybegun to die away and merge into the classical smoothregion, showing that this truncation has captured thekey physics that is affected by non–perturbative contri-butions. Notice that the spectrum naturally is boundedbelow by E=0, as already shown explicitly in prelimi-nary studies in ref. [3]. Note that ref. [3] generalized theconstruction by turning on the parameter µ that was dis-cussed in the supergravity context in section IV. This isstraightforward to do, and there were no additional in-sights to be gained, and so results are not presented here.The tail of the resulting distribution, not surprisingly, re-sembles the tails already displayed in ref. [3].

Of course, with the spectrum in hand (approximately1800 normalized wavefunctions and their energies) thenext natural step is to compute the spectral form factor,using the methods of section V. The correlator of twoboundaries is readily computed, and the phase transi-tion where the disconnected part (two black holes) handsover to the connected part (wormhole) happens at βcr.(A figure similar to figure 13 is omitted here to avoidrepetition.) For β=50, the disconnected, connected, andcombined spectral form factor are shown in figures 26, 27,and 28, respectively.

The most striking feature overall is in the disconnectedportion of the form factor, controlling the initial slope.As might be expected from the absence of the 1/

√E low

energy behaviour of the regulating models (happily re-moved by non–perturbative effects), the time dependenceof the slope is not t−1 (as it is for the (0,2) JT supergrav-ity, but nor is it the t−3 expected from the classical lowenergy physics to be read off from the β−3/2 dependenceof the partition function (2). Rather, it interpolates be-

17

0 0.5 1 1.5 2 2.5 3

log10(t)

-8

-7.5

-7

-6.5

-6

-5.5

-5

-4.5lo

g 10(

Z(

+it)

Z(

-it)

)

Figure 26: The disconnected part of the JT gravity spectral formfactor vs. t, at β=50 and ~=1, for the non–perturbative scheme ofref. [3].

0 0.5 1 1.5 2 2.5 3

log10(t)

-2.9525

-2.952

-2.9515

-2.951

-2.9505

-2.95

-2.9495

-2.949

-2.9485

-2.948

log 10

( Z

(+

it)Z(

-it)

)

Figure 27: The connected part of the JT gravity spectral formfactor vs. t, at β=50 and ~=1, for the non–perturbative scheme ofref. [3].

tween them, and is ∼t−2, to the nearest integer. Fromwhat was learned from the supergravity cases of last sec-tion, the origin of this is clear. The endpoint of the distri-bution has a different structure (see figure 25), with somenon–zero ρ(0) at the end. There are far fewer states thanfor the (0,2) supergravity case but more than the (2,2)case, and hence the fall–off rate (at least for this valueof ~) is between that of those two cases.

Again, as seen in all the models studied in this paper(and also the special Airy model recalled in Appendix A)non–perturbative effects hasten the transition from dipto ramp to plateau such that the linear part of the classi-cal contribution to the ramp that emerges at long timessimply does not have time to develop, for moderate val-ues of ~.

0 0.5 1 1.5 2 2.5

log10(t)

-2.9482114

-2.9482112

-2.948211

-2.9482108

-2.9482106

-2.9482104

-2.9482102

log 10

( Z

(+

it)Z(

-it)

)

Figure 28: The full JT gravity spectral form factor vs. t, atβ=50 and ~=1, for the non–perturbative scheme of ref. [3].

VII. JT GRAVITY: NON–PERTURBATIVEDEFINITION B

It is worth revisiting the idea mentioned at the begin-ning of the previous section that the presence of the non–perturbatively unstable even k members of the (2k−1, 2)minimal model family renders unstable the JT gravityfrom which they are built. This is what motivated thenon–perturbative definition of ref. [3] in terms of other,more manifestly stable minimal models. There seems tobe another possibility, however. It arises naturally in thecontext of this paper, and so it will be briefly discussedhere, but details of further explorations will be presentedelsewhere, if successful.

Simply put, perhaps the presence of the odd k mini-mal models helps the system avoid the pathologies of thelone even k models. From the perspective presented inref. [4] that the construction of JT gravity from minimalmodels is equivalent to being in a particular model andthen turning on operators that mixes in the presence ofall the others, the view could be taken that the particularmodel is an odd k model (k can be taken to infinity atthe end), and then the (k−1) lower models are turnedon. The key here is that they are turned on a finiteamount. The Hermitian matrix model string equationR=0 should have a well–defined solution in such a case.This is because the pathologies representing instabilitiesof the mth (even) model appear in this language onlyif the coupling tm is sent to infinity, performing the RGflow to land entirely in that model. In that case the so-lution develops increasingly steep oscillatory components(as tm grows) that herald the appearance of problematicdouble poles, as noticed early on in ref. [69]. However,the JT prescription only requires the couplings to be cer-tain finite values (28). So, there could well be the rightkind of solution present after all, probably a unique one,and it would supply the u(x) needed to define the full

18

non–perturbative JT gravity model.From the perspective of this paper, all that is needed

is to find such a solution at a given truncation, and thetruncation should be at an odd k for a solution to befound. Remarkably, a search for such a truncated so-lution eventually (after some careful coaxing: R=0 is asimpler equation than equation (9), but for the boundaryconditions needed it is a skittish creature) yielded solu-tions of just the right sort. A truncation containing justt1, t2, and t3 is presented in figure 29. In both directions

Figure 29: The solution of the R=0 string equation fortruncation up to t3. Its asymptote (30) is shown (dotted, blue).The full classical solution is shown too (lower dotted to the left).

it asymptotes to the solution of the cubic equation

x = −1

2u0 −

1

4π2u2

0 −1

24π4u3

0 . (30)

Meanwhile a solution (but for ~=3.26; see below) for thetruncation up to t5 is presented in figure 30. It asymp-totes to the quintic equation:

x = −1

2u0 −

1

4π2u2

0 −1

24π4u3

0 −1

288π6u4

0 −1

5760π8u5

0 .

(31)Notice that in this latter case 12, the solution quicklyapproaches (for x<0) the exact disc level potential (29)(plotted as a (red) dotted line) that yields the Schwarzian

12 The solution for u(x) that was sought in this case was for ~=1, tomatch the other examples presented, and to be in the physicallysensible regime (recall that ~=e−S0≤1, since the leading entropyS0 is supposed to be positive). However, this solution was for~=3.26. Getting a solution with ~ all the way down to unity orbelow has so far proven rather difficult (the equations are easierto solve when the derivatives have larger coefficients), but theevolution of the shape as ~ was reduced toward one seemed tobe converging nicely, so it may be possible to complete the jobwith more care.

Figure 30: The solution of the R=0 string equation fortruncation up to t5, for ~=3.26. Its asymptote (31) is shown(dotted, blue). The full classical solution is shown too (lowerdotted to the left).

density (4). So this could be an excellent starting pointfor solving the non–perturbative spectrum of JT gravityto good accuracy, and computing the spectral form fac-tor. It will have a tail penetrating to negative energy (inthis way it is a grown–up version of the k=1 Airy modelreviewed in Appendix A), but this will be just fine if theeffective potential for one eigenvalue in this region pre-vents tunnelling, as happens in pure odd k models [70],and so may well be likely here.

It would be very attractive to find a non–perturbativesolution of the JT gravity Hermitian matrix model def-inition that uses the string equations of double scaledHermitian matrix models after all, and this seems a verypromising (and exciting) possibility. Further explorationof this proposal is left for future work.

VIII. CLOSING REMARKS

The purpose of this paper was to explicitly uncover andexamine the non–perturbative physics for JT gravity andsupergravity that is accessible if they are formulated us-ing minimal model building blocks. This constructionis not a simple large k limit of a minimal model, buta more refined affair involving coupling them togetherin a particular combination, as suggested perturbativelyin ref. [11], and extended to non–perturbative physics inrefs. [3, 4]. The principal applications that demonstratedthe facility of the technique was the explicit computation,for the first time, of the full non–perturbative spectraldensities of various JT gravity and supergravity mod-els, and the use of these spectral densities to computethe spectral form factor in each case, showing how the

19

non–perturbative effects affect the shape (sometimes dra-matically) of this important diagnostic quantity. Havingexplicit access to the non–perturbative features in thismanner turned out to be rather instructive, as exten-sively discussed in the body of the paper.

Techniques to allow such non–perturbative propertiesto be extracted, in a consistent and well–defined scheme(for generic values of β and ~), have not been presentedin the literature before, and it is hoped that these meth-ods and results will go some way to helping uncover moreof the fascinating web of interconnections between geom-etry, quantum mechanics, gravity, and chaos that seemsto be emerging from these studies.

It is likely that other models of JT gravity can be “de-constructed” in terms of minimal models in a way anal-ogous to what was done here, and thereby be given anon–perturbative definition, not just in principle, but (asshown here) in useful accessible terms. This includes thepossible alternative non–perturbative definition of ordi-nary JT gravity (using a smooth non–perturbative inter-polating solution of the original Hermitian matrix modelstring equation) suggested in section VII. It could pos-sibly also encompass some of the new kinds of matrixmodel descriptions of JT gravity black holes mentionedrecently in ref. [73]. Perhaps the results of explorationsalong these lines will be reported soon.

Appendix A: Numerical Testbed: The Airy Model

As a means of sharpening understanding of some ofthe key features of the spectral form factor, and for mod-elling what kinds of physics can be reliably captured bythe numerical approaches used in the main body of thepaper, this Appendix presents a numerical explorationof the exactly solvable Airy model, which is the double–scaled limit of the simple Hermitian matrix model withGaussian potential, obtained by magnifying the infinites-imal region at the edge of Wigner’s semi–circle [43]. Itis the k=1 model of the (2k−1, 2) minimal string series,and as such is also a model of the extreme low energytail of ref.[5]’s matrix model of JT gravity.

In the language of this paper, it comes from using thesimple linear potential u(x)=−x in the Hamiltonian (7),and the resulting equation to solve for the spectrum issimply (after a change of variables) Airy’s differentialequation. The wavefunctions for energy E are:

ψ(E, x) = ~−23 Ai(−~− 2

3 (E + x)) , (A1)

and the spectral density that results is:

ρ(E) =

∫ 0

−∞|ψ(E, x)|2dx = ~−

23 [Ai′(ζ)2 − ζAi(ζ)2] ,

(A2)

where ζ=−~− 23E. See figure 31 for a plot of the spec-

trum, showing the exponential tail running to negative E.At large E, the non–perturbative oscillations of the Airy

function die out, leaving the classical (disc) contribution

ρ0(E)=(π~)−1

√E. This is shown as a dashed line.

-2 0 2 4 6 8 10 12 14 16 18 20

E

0

0.5

1

1.5

(E)

Figure 31: The full spectral density for the exactly solvable Airymodel, at ~=1. The dashed line is the disc level result.

The correlator of two boundaries can be computedexactly using properties of the Airy functions out ofwhich the wavefunction is built13. The disconnectedpart is simply the square of the partition function, whichcan be evaluated by Laplace transform, remembering toinclude negative energies to incorporate the full non–perturbative spectrum:

ZAi(β) =

∫ +∞

−∞ρAi(E)e−βEdE =

e~2

12 β3

2π1/2~β3/2. (A3)

This gives

〈Z(β)〉〈Z(β′)〉 =e

~2

12 (β3+β′3)

4π~2(ββ′)3/2, (A4)

while implementing equation (26) yields the connectedpiece to be:

〈Z(β)Z(β′)〉 = (A5)

e~2

12 (β+β′)3

2π1/2~(β + β′)3/2Erf

(1

2~√ββ′(β + β′)

),

These are exact expressions, i.e., both perturbative andnon–perturbative parts are incorporated (see refs. [43, 74]and the review in the Appendix of ref. [11]). Neverthe-less, it is instructive to pretend that only a finite numberof wavefunctions are known, (only numerically), for a dis-crete set of energies up to some maximum energy. Thequestion is then how well the exact expressions can be re-produced. This is the situation of the body of the paper,

13 In fact, ref.[74] writes down expressions for correlators of multipleloops in this model.

20

resulting from the controlled truncation of the infiniteorder string equation.

The answer to the question, reassuringly, is that a greatdeal of the important physics is accessible. To show this,a set of 1000 of the wavefunctions (A1) were discretizedon the same size grid used in the body of the paper (xis broken up into 20,000 points), for a range of energies−20≥E≥20, and the same code that performed the nu-merical implementation of the expressions given in equa-tions (20) and (26) were carried out for this exact model.The result presented in figure 32 shows the crossover be-tween the two portions of the correlator as a functionof β, and figures 33, 34 and 35 show the disconnected,

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

log 10

( Z(

)2 ),

log 10

( Z(

)2)

Figure 32: The disconnected (starting higher at the left) vs. theconnected (lower) two–point function of the Airy model’spartition function as a function of β, showing a phase transitionat β=βcr'0.92. A dashed line shows the exact result, dots are fora numerical truncation.

connected, and combined parts of the spectral form fac-tor, for temperature β=1/2. The dashed lines are theplots of the exact functions (A4) and (A5) while the dotsshow the results of the numerical computations.

As might be expected, significant deviations from thedashed lines occur when β becomes too small, includingpatterns of zeros representing finite size effects. This sig-nals temperatures that excite higher energies that are notincluded in the numerical scheme (but are in the exactexpressions) and hence the results deviate. As long assuch extremes are avoided (depending upon the trunca-tion energy chosen), the numerical results are very reli-able. This is a good controlled model of the truncationscheme used in the body of the paper, showing that theresults obtained are robust.

-2 -1.5 -1 -0.5 0 0.5 1

log10(t)

-15

-10

-5

0

log 10

( Z

(+

it) Z

(-it

))

Figure 33: The disconnected part of the Airy model’s spectralform factor vs. t, at β=1/2, showing the classic slope feature. Adashed line shows the exact result, dots are for a numericaltruncation. (The undulations at the end are numerical errors atultra–small values and so should be ignored.)

-2 -1.5 -1 -0.5 0 0.5 1

log10(t)

-1.1

-1

-0.9

-0.8

-0.7

-0.6

-0.5lo

g 10(

Z(

+it)

Z(-it

))

Figure 34: The connected part of the Airy model’s spectral formfactor vs. t, at β=1/2. A dashed line shows the exact result, dotsare for a numerical truncation.

Appendix B: Numerical Recipes

1. Suggestions for solving the string equation

The string equation that supplies the potential u(x)for a particular problem is highly non–linear, and of highorder (2k) in derivatives if truncating to the kth model.Even the simplest solution (with the boundary conditionsof interest given in equation (12)), where all tk are set tozero except k=1, requires numerical techniques to extractits explicit form (it is related to a Painleve transcendent,and hence cannot be written in terms of other elementaryfunctions). Here are some suggestions for finding numer-

21

-2 -1.5 -1 -0.5 0 0.5 1

log10(t)

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1lo

g 10(

Z(

+it)

Z(-it

))

Figure 35: The full spectral form factor of the Airy model vs. t,at β=1/2.

ical solutions, to help the interested reader learn how toextract useful information for themselves. First, Maplewas used in this case (although MATLAB works well too,as probably would other programs). The dsolve routinewas used, with an error tolerance of 10−5−10−10, de-pending upon the equation being solved. (In fact whenthese equations were first solved [48, 75] it was for thecases k=1, 2, and 3, including cases where the modelswere non–trivially coupled [76]. Back then, their solu-tion was found by writing a program in FORTRAN thatcalled the routine D02RAF, part of the NAG libraries.)

As mentioned in the text, for various reasons, it makessense to take an additional derivative of the equation.This reduced the non–linearity somewhat, at the expenseof increasing the order, which is a small price to pay. Thisis because the first derivative results in an overall factorof R, which can be divided out since it will not van-ish for the solutions of interest. A derivative explicitlyremoves Γ from the equation, however. Now, the onlyknowledge the system has of the desired choice of Γ isthrough subleading (in the small ~ expansion) terms inthe boundary conditions, which need to be solved for withall tk present. This is nicely organized on x>0 boundarybecause the presence of the tk come in one by one at suc-cessively higher terms in the 1/x2 expansion (see ref. [4]).It is less nice to do analytically on the x<0 boundary.There, all the tk contribute at the next order and solvingfor the order ~Γ correction requires solving a kth orderpolynomial. By beyond the k=4 truncation this becomesunpleasant at best. However, if the system is solved on alarge enough region, with a small enough discretization,terms beyond the leading left boundary condition can besafely ignored, and a good approximation to the the solu-tion found anyway. (If needed, however, a recursive codefor solving for the subleading corrections to the boundaryconditions numerically can be employed.)

This works well for Γ= 12 , while for Γ=− 1

2 a different

technique was used, because the numerical approach wasless stable due to a more complicated well shape appear-ing in the interior, which is hard to control in a 13th or-der differential equation. Ref. [56] noticed that Γ can bechanged by an integer using a special “Backlund” trans-formation, and actually derived an analytic expressionexpression showing how to build the new u(x) at Γ±1from the old u(x) (and its derivatives) at Γ. Here it is:

uΓ±1 =3(R′)2 − 2RR′′ ± 8~ΓR′ + 4~2Γ2

4R2, (B1)

(where R≡R(uΓ) and a sign has been switched to matchthe current conventions). So once u(x) for the case ofΓ= 1

2 was found using Maple, the output of dsolve con-tains all the derivatives of u(x) needed to construct u(x)at Γ=− 1

2 , giving the well structure seen in figure 7.In fact, a similar story held for the case of Γ=0 used in

section VI. There is also a well structure in the interior,which is hard to solve for numerically when at high or-der and with boundary conditions far from the structureitself. Experience from studying positive integer Γ sug-gested that Γ=1 would be smoother to solve for and thatwas indeed the case. From there, transformation (B1)was used to construct the desired Γ=0 solution.

2. Suggestions for solving the spectrum

As mentioned in section IV, in order to solve for the so-lutions to the eigenvalue problem, {E,ψ(E, x)}. a matrixNumerov method [57] was used, as it was in refs.[3, 4].This simply puts the system into a box, and turns theproblem into a large matrix diagonalization problem,for a given input potential u(x). This was done usingMATLAB. A key point is that it is desirable to have a largenumber of eigenvalues in the energy range from zero tothe chosen highest energy (determined by the level of thetruncation of the string equation). So two choices weremade to ensure a good set of solutions. The first was touse a large grid, so a grid of 20000×20000 was used. Thesecond was to use a large box. As stated, the spectrumsolving method is essentially putting the system into abox, and a portion of the output eigenvalues and eigen-functions will be affected by the edges of the box. Thoseshould be discarded, and the larger the box, the moreuseable eigenstates will be available in the reliable energywindow. Since, as already observed in section III, the so-lution for u(x) becomes similar to the disc level behaviourfar away enough from the central region, the box can beeasily made larger by connecting the solution (solved nu-merically out to −200≤x≤+200) to a wider region (e.g.,−2645≤x≤+2645 for the (2,2) and (0,2) models) wherejust the exact disc solution u0(x) is used. (A smooth(enough) transition between the two solutions was doneat x<−100, corresponding to energies well above the cut-off determined by good matching for the truncation, sothis does not affect the physics.)

22

Appendix C: Gel’fand–Dikii Polynomials

In case they are needed, here are some of the higherorder Gel’Fand–Dikii polynomials. In the following equa-tion a prime denotes an x–derivative times a factor of ~.For high numbers of derivatives, instead, a notation u(n)

is used for n primes. The first five are listed here:

R1[u] = u , (C1)

R2[u] = u2 − 1

3u

′′,

R3[u] = u3 − 1

2(u

′)2 − uu

′′+

1

10u

′′′′,

R4[u] = u4 − 2u(u′)2 − 2u2u

′′+

4

5u

′u

′′′+

3

5(u

′′)2

+2

5uu

′′′′− 1

35u(6) ,

R5[u] = u5 − 5u2(u′)2 − 10

3u3u

′′+

11

3u

′′(u

′)2

+3u(u′′)2 + 4uu

′u

′′′+ u2u

′′′′− 23

42(u

′′′)2

−19

21u

′′′′u

′′− 3

7u(5)u′ − 1

7u(6)u+

1

126u(8) .

It will transpire that R6[u] and R7[u] will be needed aswell, in order to get the required level of accuracy forthe quantities computed in this paper. They are ratherlengthy quantities, so it is not clear if there is much value

in listing them here. Instead, they (and higher oder ones)can be easily computed using the recursion relation:

R′k+1 =~3

4R

′′′

k − u~R′k −~2u′Rk , (C2)

and the requirement that they vanish when u does.More useful therefore are the following three lines of

Maple code, which can be iterated, with obvious adjust-ments, to get to arbitrarily high order:

> R[1] := u(x); (C3)

> DR[2] := (1/4) ∗ h3 ∗ (diff(R[1], x$3))

−u(x) ∗ h ∗ (diff(R[1], x))

−(1/2) ∗ h ∗ (diff(u(x), x)) ∗ R[1];

> R[2] := simplify(−4 ∗ integrate(DR[2], x)/(3 ∗ h));

ACKNOWLEDGMENTS

CVJ thanks Felipe Rosso and for helpful questionsand comments, Douglas Stanford and Edward Witten forhelpful questions, the US Department of Energy for sup-port under grant DE-SC 0011687, and, especially duringthe pandemic, Amelia for her support and patience.

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