A JT supergravity as a double-cut matrix model
AA JT supergravity as a double-cut matrix model
Clifford V. Johnson, ∗ Felipe Rosso, † and Andrew Svesko ‡ Department of Physics and Astronomy, University of Southern California,Los Angeles, California 90089-0484, USA Department of Physics and Astronomy, University College London,Gower Street, London, WC1E 6BT, UK
We study a Jackiw-Teitelboim (JT) supergravity theory, defined as an Euclidean path integral overorientable supermanifolds with constant negative curvature, that was argued by Stanford and Wittento be captured by a random matrix model in the β =2 Dyson-Wigner class. We show that the theoryis a double-cut matrix model tuned to a critical point where the two cuts coalesce. Our formulationis fully non-perturbative and manifestly stable, providing for explicit unambiguous computationof observables beyond the perturbative recursion relations derivable from loop equations. Ourconstruction shows that this JT supergravity theory may be regarded as a particular combinationof certain type 0B minimal string theories, and is hence a natural counterpart to another family ofJT supergravity theories recently shown to be built from type 0A minimal strings. We conjecturethat certain other JT supergravities can be similarly defined in terms of double-cut matrix models. I. INTRODUCTION
Jackiw-Teitelboim (JT) gravity [1, 2], a theory of 2Ddilaton gravity, has emerged as one of the simplest mod-els for studying non-trivial problems in quantum grav-ity. Describing the low-energy dynamics of a wide classof near-extremal black holes and branes [3–7], JT grav-ity also features in a precise realization of holographicAdS /CFT , arising as the low-energy gravitational dualto the Sachdev-Ye-Kitaev (SYK) 1D quantum mechani-cal system [8–13]. It has also been a crucial ingredient inrecent work that has yielded new insights into how theblack hole information puzzle may be resolved [14–17].The partition function of the theory Z ( β ) has two nat-ural parameters: The length, β , of a boundary, whichdefines the temperature via β =1 /T and the quantity S (which can be thought of as the extremal entropy of theparent black hole), which defines a coupling (cid:126) = e − S .Surfaces of Euler character χ =2(1 − g ) − n (there are g handles and n boundaries) contribute with a factor (cid:126) − χ to the path integral. It is vitally important to be able tofully formulate JT gravity as a complete theory of quan-tum gravity in its own right, not just perturbatively inthe parameters, but beyond. A landmark discovery inthis regard was the remarkable demonstration by Saad,Shenker and Stanford [18] that JT gravity can be formu-lated as certain random Hermitian matrix models, in adouble-scaling [19–22] limit: Z ( β , . . . , β n ) ←→ (cid:10) Tr e − β H . . . Tr e − β n H (cid:11) c . (1)On the left hand side Z ( β , ..., β n ) is the JT gravitypath integral, computed as an Euclidean path integralover (connected) Riemann surfaces with constant nega-tive curvature and n asymptotic boundaries of lengths β i . ∗ [email protected] † [email protected] ‡ [email protected] On the right hand side is the connected correlation func-tion of insertions of the operator Tr e − β i H , with (cid:104) . . . (cid:105) c implying an ensemble average of the Hermitian ma-trix H . The correspondence was established in ref. [18] via powerful recursion relations, matching the volumesof moduli spaces of Riemann surfaces [23] contributingto Z ( β , . . . , β n ) to a genus expansion of a Hermitianmatrix integral [24, 25]. While equation (1) is under-stood to be order by order in genus g and then summed,a matrix model definition can in principle supply non-perturbative completions that go beyond the sum overgenus. Non-perturbative physics is extremely important,especially for studying the low energy regime, and so itis of great interest to fully understand and characterizematrix model formulations, as a means of extracting it. The correspondence was broadened rather elegantly byStanford and Witten [27] to include wider classes of JTgravity, incorporating models with fermions and/or time-reversal symmetry. Such models were shown to be clas-sified according to one of the ten types of random ma-trix ensemble: The three Dyson-Wigner β ensembles andthe seven Altland-Zirnbauer ( α , β ) ensembles [28, 29].Many detailed gravity computations (now involving themoduli space of orientable and/or unorientable (super)Riemann surfaces) were shown to be captured by de-tailed matrix model integrals, establishing that expres-sion (1) still holds (for an appropriate choice of H ). Inthe case of N =1 JT supergravity, which will be the fo-cus of this paper, the Hermitian matrix is written H = Q ,where it is the supercharge Q that is (on general grounds)classified in the ten-fold way, depending upon the num-ber, N, of fermions the effective SYK-like “boundary”dual has, and the action of ( − F and T , where F and T While ref. [18]’s Hermitian matrix model definition has a prob-lematic non-perturbative definition, a complete non-perturbativecompletion of ordinary JT gravity that is naturally rooted in ma-trix models can be found, as shown in ref. [26]. a r X i v : . [ h e p - t h ] F e b are fermion number and the time reversal operator, re-spectively [27, 30–36].Symmetry under T results in the inclusion of unori-entable surfaces in the path integral, but this paper willfeature JT supergravity theories models that do not pre-serve T . So all that is left is to consider whether ( − F is a symmetry or not— i.e., whether N is even (let’s callthis Case A) or odd (Case B). In Case A, since the su-percharge anti-commutes with ( − F , it can be written(in a basis where ( − F is block diagonal) as: Q = (cid:18) MM † (cid:19) , (2)where M is a complex matrix, and M † its Hermitianconjugate (see Section 2.6.2 in ref. [27]), and the naturalcombination the matrix model (1) cares about is M † M .In Case B, there is no grading due to ( − F , and so Q isitself just an Hermitian matrix. This case should there-fore fall into the β =2 Dyson-Wigner class.Recently, it was shown in refs. [37–39] how to usedouble–scaled random complex matrix models [40–44]with a potential of the form V ( M † M ) to yield a com-plete perturbative and non-perturbative definition of theCase A supergravity above. The formulation was shownto be equivalent to combining together (in a precise sensereviewed later) an infinite family of models known [45] tobe equivalent to type 0A minimal string theories. Forother studies of JT gravity and supergravity using mini-mal models, see refs. [46–50].In retrospect, the involvement of type 0A minimalstrings is entirely natural. The worldsheet physics ofminimal string theories are themselves theories of two di-mensional quantum gravity. When fermions are present,there is a spin structure on the surfaces that enter thepath integral, and how these are treated defines differ-ent types of string theories [51]. Type 0A string theoriessum over such spin structures and keep track of them byincluding a weight factor of ( − ζ in the sum, where ζ (0or 1 mod 2) is the parity of the spin structure. On theother hand, in the JT classifications above, the bound-ary ( − F directly correlates with the bulk ( − ζ . Hencewith hindsight it is natural that, if minimal strings aregoing to be relevant at all (and that is thrust upon usby the presence of double-scaled matrix models in bothsettings) it is indeed type 0A minimal strings that re-late to Case A above. Indeed, many special propertiesof the non-perturbative “string equations” that describetype 0A strings, when deployed appropriately (as demon-strated in ref. [37]) yield many of the remarkable proper-ties of the supergravity models noticed in ref. [27].This paper answers the natural question as to whetherthe remaining T –breaking JT supergravity, Case B In fact, the same framework includes some T -invariant modelstoo. They all fall into the ( α , β ) = (2Γ+1 ,
2) Altland-Zirnbauerseries, for Γ = 0 , + , − . Here, integer Γ breaks T . above, can be given an analogous treatment, yielding afully computable stable non-perturbative definition of thetheory’s observables. In light of the previous paragraphthere is a natural guess: Since it does not respect ( − F and hence results in a sum over bulk geometries thatsimply ignores ( − ζ , it ought to have a connection totype 0B minimal strings, which themselves ignore thatweight factor by definition. The answer will turn out tobe yes, and the 0B connection (anticipated in ref. [27])will prove to be correct.Two more clues help lead to these answers, and theycan be found in the leading form of the spectral densityfor Q , denoted here ρ ( q ). Since H = Q , this is deter-mined by the Laplace transform of the super–Schwarzianpartition function, which yields [31] the density ρ SJT0 ( E )of H (with energies E ≥ ρ SJT0 ( E ) = √ π √ E ) π (cid:126) √ E = ⇒ ρ ( q ) = cosh(2 πq ) π (cid:126) , (3)since E = q , and after taking into account the Jacobian dE = 2 qdq in the Laplace transform integral, where the 2is absorbed by integrating q over R . (A translation factorof γ − =1 / √ Q therefore has (after double–scaling to match grav-ity) a spectral density that naturally spreads over theentire real line. This is our first clue: The most com-monly studied cases of double-scaled Hermitian matrixmodels usually have support on the half–line, resultingfrom the fact that the double-scaling limit “zooms in” tocapture the universal physics to be found at one criticalendpoint of the (unscaled) spectral density, as shown inthe upper part of Figure 1 (see e.g. ref. [52] for a review). qq FIG. 1. The double-scaling limit as a “zoom in” to the neigh-bourhood of the endpoints of spectral densities, resulting ineither a semi-infinite cut starting from a single cut case, or re-sulting in an infinite cut starting from a double-cut case thathas a merger.
The simplest way of arriving at a density that spreadsover the whole real line from double-scaling is to “zoomin” on two endpoints that are colliding with each other,as can happen with critical double-cut matrix models (seethe lower part of Figure 1). The second clue is that ρ ( q )is symmetric on the real line. As will be reviewed later,the string equations that arise from double-scaling sym-metric two-cut models (pioneered in refs. [53–58]) havealready been identified in ref. [45] as capturing type 0Bminimal string physics. Moreover, there is a curiousuniversal feature of the string equations that arises inthe symmetric sector (the Painlev´e II heirarchy) that,as we will show, reproduces certain characteristic fea-tures of observables in the JT supergravity. Combiningtogether an infinite family of 0B minimal string mod-els in a way precisely analogous to what was done inrefs. [37–39] for the other JT supergravities provides a fully non-perturbative definition of the theory. As an ex-ample of what can be computed from it, Figure 2 dis-plays (red dots) the full non-perturbative spectral den-sity for Q . The black dashed line is the perturbative re-sult of equation (3). Other important quantities can becomputed in this way, such as the spectral form factordisplayed later in Figure 10. -0.6 -0.4 -0.2 0 0.2 0.4 0.6024681012 FIG. 2. Numerical solution (red dots) for ρ ( q ) includingboth perturbative and non-perturbative contributions, ob-tained from eq. (16) after solving for the spectrum of Q ,with r ( x ) computed in the k max =6 truncation (see text). Thedashed black line corresponds to the leading solution ρ ( q ) ineq. (3) that receives no perturbative corrections. Here (cid:126) =1. The rest of this paper is organized as follows. Section IIreviews the construction of double-cut Hermitian matrixmodels and their double scaling limit. Section III definesthe particular model that describes the Dyson-Wigner β =2 JT supergravity, showing that it reproduces all ofthe characteristic perturbative results for this model de-rived in ref. [27]. Non-perturbative contributions of thematrix model are studied in Section IV, where the fullspectral density and spectral form factor are obtained bycombining analytic and numerical techniques. We finishin Section V with some discussion. There are a numberof appendices included that expand some computations,and help keep this work somewhat self-contained with-out diluting the narrative flow. Appendices A and Bare devoted to computing multi-correlators at leading genus and higher genus corrections to the 1-point func-tion (cid:104) Tr e − βQ (cid:105) , respectively. Appendix C studies theWKB approximation of the eigenfunctions of the Hamil-tonian operator appearing in the matrix model. Theseeigenfunctions are used to compute a formula for theleading non-perturbative corrections to the spectral den-sity that is tested in Section IV. The normalization ofeigenfunctions is discussed in Appendix D. II. DOUBLE-CUT MATRIX MODELS
Very soon after the discovery of the double-scalinglimit in the context of single-cut Hermitian matrix mod-els in the 90s [19–22], the study of the double–scaled limitof multi-cut Hermitian matrix models was initiated in aseries of works [55–59], with most of the focus on two-cut models. Closely allied to the latter are the double-scaled unitary matrix models, discovered slightly earlierin refs. [53, 54], which yield a subset of the same physics. We shall see that in the end, the physics that we need willbe accessible from either system, but Hermitian double-cut matrix model language is perhaps the most natu-ral here given the considerations of the previous Section.They are built from an N × N Hermitian matrix Q , sothat the expectation value of a matrix observable O is: (cid:104)O(cid:105) = 1 Z (cid:90) dQ O e − N Tr V ( Q ) , (4)where Z = (cid:82) dQ e − N Tr V ( Q ) is the matrix partition func-tion. The probability measure is determined by the po-tential V ( Q ), that is typically taken to be a polyno-mial. See refs. [52, 62, 63] for useful sources of infor-mation about random matrix theory and applications ofthe double-scaling limit.A central observable that determines the average dis-tribution of eigenvalues λ i ∈ R of the matrix Q is thespectral density ρ ( q ), defined as ρ ( q ) = (cid:68) N N (cid:88) i =1 δ ( q − λ i ) (cid:69) . (5)Depending on the particular number and location of theminima of V ( Q ), the spectral density in the large N limitbecomes a smooth function of q supported on a finitenumber of disjoint intervals in q ∈ R . These intervalsultimately become cuts in the complex q –plane when us-ing complex analysis to analyze the large N limit. When ρ ( q ) is supported on more than one interval, the matrixmodel is said to be “multi-cut”. Transitions of the kind that interest us, involving mergers ofcuts, have an older history from before double-scaling and gravityapplications (see e.g. refs. [60, 61]). Note that the symbol N , the dimension of the square matrix Q ,should not be confused with the N describing the number of SYKfermions, as used in Section I. FIG. 3. In blue we plot the singular spectral density (6) with k = 2, where the spectral density vanishes at the origin werethe two cuts meet lim N →∞ ρ k ( q ) ∝ q . The dashed curvecorresponds to the critical potential (7), with the two minimasymmetrically located around q =0. The focus here is on systems that transition betweendouble- and single-cut phases. Precisely at the transition,the corresponding potential V ( Q ) is said to be “critical”,with different critical systems characterized [64] by therate at which the spectral density vanishes where thetwo cuts meet at q =0, i.e. , lim N →∞ ρ m ( q ) ∼ q m with m ∈ N . For even potentials we have m =2 k with k ∈ N ,and the large N spectral density of a critical system canbe written aslim N →∞ ρ k ( q ) = b k π (cid:16) qa (cid:17) k (cid:114) a − q a , (6)where a > b k = 2 k +1 ( k + 1)!( k − /a (2 k − q ∈ ( − a, ∪ (0 , a ) and arises from a two-cutspectral density transitioning to a single-cut phase. Us-ing standard methods (see e.g., refs. [65, 66]), the appro-priate potential needed to produce this spectral densitycan be easily worked out to be: V (cid:48) k ( q ) = b k k (cid:88) n =0 (cid:18) / n (cid:19) ( − n (cid:16) qa (cid:17) k − n )+1 , (7)which defines the critical model for arbitrary N . Theeven potential V k ( q ) contains two minima symmetricallylocated in the interval | q | < a . Figure 3 is a plot of thepotential and spectral density for k = 2. Note that it ispossible to construct models whose spectral density be-haves like lim N →∞ ρ ( q ) ∼ q k +1 , but it is more subtle,as these models by themselves are not well defined sincethe resulting (unscaled) densities are not positive defi-nite. Instead, they can be introduced as perturbations ofthe even model potentials, as described in refs. [58, 67]. A. The double scaling limit
The double scaling limit involves approaching the crit-ical potential (7) while simultaneously taking the large N limit in a way that captures only certain universal physics associated to the merging of the two cuts. The leadingspectral density in the double scaled model correspondsto the behavior of ρ ( q ) at q ∼
0. For even potentials, thisis easily obtained from equation (6) by taking a →∞ ,which gives ρ ( q )= lim N →∞ ρ k ( q ) ∼ q k . There will be1 /N (topological) perturbative corrections to this lead-ing large N behaviour, and non-perturbative contribu-tions too. There is a powerful formulation (the orthogo-nal polynomial methods [68, 69]) that allows for them tobe efficiently extracted. These methods are reviewed in e.g., refs. [52, 62, 63, 65].A brief summary of the logic is as follows. After writingthe matrix model in terms of integrals over the N eigen-values λ n , a family of N polynomials P n ( λ ) = λ n + · · · that are orthogonal with respect to the matrix modelmeasure dλ e − NV ( λ ) are introduced. A simple argumentshows the different polynomials are related according to λP n ( λ ) = (cid:112) R n +1 P n +1 ( λ ) + S n P n ( λ ) + (cid:112) R n P n − ( λ ) , (8)where R n and S n depend on the particular theory. Thematrix model can be used to derive a family of recur-sion relations for R n and S n , and solving the model isequivalent to finding them. The P n ( λ ) themselves sup-ply a Hilbert space description of the system, on whichmatrix model observables become operators. This willbe very useful shortly. At large N , the R n and S n de-fine smooth functions R ( X ) and S ( X ) of the variable X = n/N ∈ (0 , b whichgoes to zero as N → ∞ , and the rate at which it doesso is set by: (cid:126) b /m =1 /N (with m =2 k ), determininga scaled topological expansion parameter (cid:126) . The rate atwhich various quantities in the model approach their crit-ical values at the X =1 endpoint is controlled by a powerof b e.g., R ( X )= R c + ( − n b /m f ( x ) + · · · where x ∈ R is the scaling part of X =1 − b ( x − µ ), and the parame-ter µ will be fixed later. A different scaling function g ( x )arises from S ( X ). Inserting all these relations into therecursion relations and taking the limit yields non-trivialcoupled ordinary differential equations for f ( x ) and g ( x ).These are the so-called “string equations” for the systemindexed by m .More generally, the critical potentials (7) can besummed together with coefficients t m , and, definingthe alternative set of functions r ( x ) and α ( x ) by f ( x )= r ( x ) cosh( α ( x )) and g ( x )= r ( x ) sinh( α ( x )) the dif-ferential equations can be conveniently written in theconventions of ref. [45] as: ∞ (cid:88) m =1 t m K m + xK = 0 , ∞ (cid:88) m =1 t m H m + xH + η = 0 , (9)where η is a parameter of the model, arising as an inte-gration constant (see ref. [45]) and { K m , H m } are poly-nomials of { r ( x ) , α ( x ) } and their derivatives determinedfrom the following recursion relations: K m +1 = (cid:126) α (cid:48) K m + rH m − (cid:126) ( H (cid:48) m /r ) (cid:48) ,H (cid:48) m +1 = (cid:126) α (cid:48) H (cid:48) m − rK (cid:48) m , (10)where ( K , H ) = ( r ( x ) ,
0) and primes are derivativeswith respect to x (see ref. [45] for explicit expressions forthe first few values of m ). The parameter t m controls the double scaling limitof the critical model with ρ ( q ) ∝ q m , e.g. the dou-ble scaled 2 k th model associated to the critical potentialin equation (7) is obtained by setting all t m =0 exceptfor t k =1. As differential equations, they encode notjust the corrections to the leading large N behavior, butnon-perturbative information too, as we shall see. Theycan be derived from the Zakharov-Shabat (ZS) integrablehierarchy [70], in an analogue of the manner in whichthe equations of type 0A minimal strings can be derivedfrom the Korteweg-de Vries (KdV) integrable hierarchy.These models were identified as type 0B minimal stringsin ref. [45].Strictly speaking, we have written here a large familyof string equations labeled by any integer m . For ourpurposes, it will be enough to restrict ourselves to the m even models with α (cid:48) ( x ) = 0. Using the recursion rela-tions (10) it is straightforward to show that H k = 0, sothat the second string equation (9) is automatically sat-isfied after fixing η = 0. The K k are polynomials in r ( x )and its derivatives, computed from the following closedrecursion relation obtained from equation (10)2 m + 12( m + 1) K m +2 = r ( x ) (cid:90) x d ¯ x r (¯ x ) K (cid:48) m − (cid:126) K (cid:48)(cid:48) m , (11)where we have rescaled them so that they are normalizedas K k = r ( x ) k +1 + O ( (cid:126) ). The first few are easily The string equations for the multi-cut matrix models were firstderived in full generality in refs. [56, 58]. We are followingthe conventions in ref. [45], slightly changing some of the no-tation in order to avoid confusion: α here = β there , K here m = R there m and η here = q there . As pointed out in ref. [45], q there arises as anintegration constant, and counts the amount of R-R flux in aminimal string interpretation. Note the actual constant value α ( x ) = α does not appear in thestring equations and is therefore irrelevant. The rescaling can be applied at the level of the string equationby shifting t k → t k /P k (0), where P k ( z ) is the Legendre poly-nomial. computed and given by: K = r ( x ) ,K = r ( x ) − (cid:126) r (cid:48)(cid:48) ( x ) ,K = r ( x ) + 203 (cid:126) r ( x ) (cid:18) r (cid:48) ( x ) r ( x ) (cid:19) (cid:48) + 83 (cid:126) r (4) ( x ) , ... K k = r ( x ) k +1 + · · · + ( − k k !( k − k − (cid:126) ) k r (2 k ) ( x ) , (12)where r (2 k ) ( x ) corresponds to the 2 k x -derivatives actingon r ( x ).Finally, the full string equation for this class of evendouble-cut models is given by: ∞ (cid:88) k =1 t k K k + r ( x ) x = 0 . (13)Restricting to the even m models and turning off α ( x ) re-duces the two string equations (9), to this simpler singleequation. It is in fact the Painlev´e II hierarchy of ODEs,related to the modified KdV (mKdV) integrable hierar-chy, and was first discovered for double–scaled unitarymatrix models [53, 54]. We shall see that equation (13)is enough for describing the JT supergravity theory inwhich we are interested. B. Observables
We shall need to compute certain observables from thefunction r ( x ) that solves the string equation (13). Theexpectation value of any single trace observable can becomputed from a certain “macroscopic loop” formula de-rived in ref. [58], which in our conventions is written as: (cid:10) Tr (cid:0) e − βQ (cid:1)(cid:11) = 1 √ (cid:90) µ −∞ dx (cid:104) x | (cid:0) e β Q + e − β Q (cid:1) | x (cid:105) , (14)where Q = (cid:112) ˆ p + r (ˆ x ) with ˆ p = − i (cid:126) ∂ x . This quan-tum mechanical system arose [21, 22, 71] from the con-tinuum limit of the orthogonal polynomial system de-scribed in the previous Subsection. The Hilbert space This is obtained by writing equation (5.1) of ref. [58] in our con-ventions ( x, t ) → − ( x, µ ) and f ± g = re ± α , see just above equa-tion (9) and footnote 5. We have also replaced (cid:96) → iβ and com-puted the matrix trace in equation (5.1). The formula holds alsofor arbitrary α ( x ). Crucially, the overall normalization is fixed byensuring that the universal cylinder term has the correct factors,as will be discussed in Subsection III B See refs. [52, 63] for detailed reviews on how this effective descrip-tion arises and refs. [26, 37, 39, 72] for more recent discussionsin the JT supergravity context. with P n ( λ ) ∼ | n (cid:105) was promoted to a full quantum me-chanics with position variable x . The integral arose as acontinuum limit of a sum over n of orthogonal polynomialquantities, becoming an X integral from 0 to 1. Whatremains is the part that survived in the double-scalinglimit, which zooms into the neighbourhood of the X =1endpoint defined by X =1 − b ( x − µ ), giving −∞ and µ asthe limits on the x integral. The value of µ will be fixedlater by comparison to the supergravity theory. The righthand side is computed in one-dimensional quantum me-chanics in the usual way with (cid:104) p | x (cid:105) = e ipx/ (cid:126) / √ π (cid:126) . Thisprovides an extremely concrete formalism that allows usto compute arbitrary single trace observables to all ordersin (cid:126) , and non-perturbatively.Our supergravity instruction from equation (1) (giventhat H = Q for an Hermitian matrix Q ) is to insteadwork with the expectation value of the matrix operatorTr e − βQ . It is straightforward to derive a new macro-scopic loop analogous to equation (14): (cid:10) Tr (cid:0) e − βQ (cid:1)(cid:11) = 1 √ ∞ (cid:88) n =0 ( − β ) n n ! (cid:10) Tr (cid:0) Q n (cid:1)(cid:11) = √ (cid:90) µ −∞ dx (cid:104) x | e − β Q | x (cid:105) , (15)where each term in the series is evaluated using equa-tion (14) and a factor of 2 arises from having twoterms there. Note that while the differential opera-tor Q plays the role of the supercharge, we now ar-rive at Q = ˆ p + r (ˆ x ) that has the natural structureof a Hamiltonian of a one-dimensional quantum mechan-ical system, i.e., no square root. As explained andshown in Appendix A, analogous formulae can be de-rived for higher number of insertions of the matrix op-erator Tr e − βQ . This will be useful when we compareresults of the multi-cut matrix model to the JT super-gravity predictions.This is all a beautiful counterpart to what occurredfor the other JT supergravity model (Case A in the In-troduction) and its cousins in refs. [37, 39, 72]. There,the natural system that arose from double-scaled com-plex matrix models produced a Schr¨odinger Hamiltonian H = ˆ p + u (ˆ x ), where u ( x ) solved a different string equa-tion. All of the techniques applied there can be broughtto this system in order to fully solve this JT supergrav-ity. For a start, we can use equation (15) to write a for-mula for the spectral density ρ ( q ) using the relation (cid:104) Tr e − βQ (cid:105) = (cid:82) + ∞−∞ dqρ ( q ) e − βq . Defining wavefunctions ψ q ( x ) as Q ψ q ( x ) = q ψ q ( x ), a simple calculation In the discussion of Section V, there are further comments aboutthe similarities, and crucial differences, between these two sys-tems. yields ρ ( q ) = √ | q | (cid:90) µ −∞ dx | ψ q ( x ) | . (16)Now we are ready to apply all of this technology to theJT supergravity of interest. III. PERTURBATIVE PHYSICS
An important test is to see if the double-cut double-scaled model reproduces the perturbative JT supergrav-ity results obtained in Section 5.2.1 of ref. [27]. Recallthat this is a supergravity theory with contributions fromorientable supermanifolds of constant negative curvaturewith both even and odd spin structures weighted equally.As a reminder, we denote as Z ( β , . . . , β n ) the partitionfunction with n asymptotic boundaries of renormalizedlength β i , that includes all possible (connected) genussurfaces weighted by (cid:126) g − n , where (cid:126) = e − S . Theseobservables were computed in ref. [27] to all orders inperturbation theory in (cid:126) . They were shown to vanish(see ref. [27]’s Appendices A and D), except for the cases n =1 ,
2, which are given by: Z ( β ) γ = e π /β (cid:126) √ πβ , Z ( β , β ) γ = √ β β π ( β + β ) . (17)Included with each supergravity observable is an inversefactor of γ = √ γ is needed to convertevery matrix model trace involved when comparing tosupergravity results.The spectral density of the supergravity theory, de-fined from Z ( β ) = (cid:82) ∞ dEρ SJT0 ( E ) e − βE , is given on theleft of equations (3). Meanwhile, the spectral density ρ ( q ) of the supercharge via the matrix model is given onthe right of equations (3). While all of these results aresurprisingly simple, we stress that they get corrected byimportant non-perturbative effects, which we will com-pute using the methods of this paper in Section IV. A. Matrix model matching
Section II reviewed a general class of double scaledmatrix models, specified by the coefficients t k that de-termine the string equation (13) satisfied by r ( x ), as wellas by µ appearing in the computation of observables (15).Fixing to a specific { t k , µ } determines a particular dou-ble scaled model. Matching the leading genus behavior of γ (cid:104) Tr e − βQ (cid:105) to the supergravity partition function Z ( β )in equation (17) uniquely fixes these parameters.We start by writing a perturbative expansion in (cid:126) forthe function r ( x ): r ( x ) = r ( x ) + ∞ (cid:88) n =1 r n ( x ) (cid:126) n . (18)Inserting this into the string equation (13), the leadingcontribution r ( x ) is determined from the following sim-ple algebraic constraint r ( x ) (cid:34) ∞ (cid:88) k =1 t k r ( x ) k + x (cid:35) = 0 , (19)which admits two possible solutions: Either r ( x ) = 0 orthe quantity in parentheses vanishes, giving r ( x ) (cid:54) = 0.To get a non-trivial function r ( x ) defined over the wholereal line x ∈ R , we use the piecewise and continuoussolution r ( x ) : ∞ (cid:88) k =1 t k r ( x ) k + x = 0 , x < ,r ( x ) = 0 , x > . (20)From this solution, let us compute the leading genus con-tribution to (cid:104) Tr e − βQ (cid:105) in equation (15), using the follow-ing identity (cid:104) x | e − β Q | x (cid:105) = e − βr ( x ) − β ( x − x (cid:126) ) (cid:126) √ πβ + O ( (cid:126) ) , (21)which can be easily proven by inserting a complete set ofeigenstates of the momentum operator ˆ p and solving theresulting p integral. Using this in equation (15) we find √ (cid:10) Tr e − βQ (cid:11) = 1 (cid:126) √ πβ (cid:90) µ −∞ dx e − βr ( x ) + O ( (cid:126) ) (cid:39) (cid:126) √ πβ (cid:34) µ + ∞ (cid:88) k =1 t k k (cid:90) ∞ dr r k − e − βr (cid:35) = 1 (cid:126) √ πβ (cid:34) µ + ∞ (cid:88) k =1 t k k ! β k (cid:35) + O ( (cid:126) ) , (22)where in the second equality we changed the integrationvariable to r and computed the Jacobian using equa-tion (20). We have also assumed the boundary condi-tion r ( −∞ ) = + ∞ that we shall verify shortly.The leading (cid:126) behavior in (22) depends on the param-eters { t k , µ } , and it should match with the supergravityobservable Z ( β ) /γ in equation (17). Clearly, the matrixmodel result (22) has precisely the right structure, andwe find agreement if( t k , µ ) = √ (cid:18) π k k ! , (cid:19) . (23)This unambiguously defines the multi-cut double scaledmodel. With this choice, the spectral density ρ ( q ) of Note that apart from the overall √
2, these agree with the val-ues of the t k parameters found for the complex matrix modeldefinition of the other supergravities discussed in refs. [37, 38]. the matrix model is the correct function given in equa-tion (3) and the definition of r ( x ) in the x < r ( x ) satisfies the constraint √ I (2 πr ) −
1) + x = 0 . (24)Figure 4 shows r ( x ) for x ∈ R (red curve), where we seethat the assumed boundary condition r ( −∞ ) = + ∞ issatisfied. FIG. 4. Plot of the leading solution r ( x ) obtained from (20)after fixing t k according to (23). In the region x < r ( x ) to the operator Q . B. Further comparison with JT supergravity
Fixing the parameters { t k , µ } of the multi-cut modelaccording to equation (23) only ensures the matching be-tween Z ( β ) and γ (cid:104) Tr e − βQ (cid:105) to leading order in (cid:126) . Nowwe must check whether higher trace operators and per-turbative corrections reproduce the all-orders supergrav-ity results (17), after using the identification in equa-tion (1) where H = Q .
1. Leading genus multi-trace observables
We start by extending the leading genus analysis inequation (22) to multi-trace observables. For two andthree Tr e − βQ insertions the computation is reasonablystraightforward, starting from the Appendix A generalformulae (A1) and (A4). Denoting G ( β , . . . , β n ) as theleading (cid:126) behavior of the connected expectation valueof n insertions of Tr e − β i Q , we derive: G ( β , β ) = 2 √ β β πβ T e − β T r ( µ ) ,G ( β , β , β ) = 2 √ β β β π / β T (cid:104) ( (cid:126) ∂ x ) e − β T r ( x ) (cid:105) x = µ (25)where β T = (cid:80) ni =1 β i . Using that r ( x ) vanishes forpositive x and µ> G ( β , β ) precisely matches Z ( β , β ) /γ in equation (17) and G ( β , β , β )=0, suchthat Z ( β , β , β )=0, again in agreement with the su-pergravity result. Since the procedure used in Appendix A becomes in-creasingly tedious when computing higher trace observ-ables, we can instead use a compact formula derived inref. [73, 74] for the leading genus behavior of single-cutHermitian matrix models. While single-cut matrix mod-els are in a different universality class, observables arealso computed from an almost identical effective quan-tum mechanical system [52, 63]. There are only twosubtle differences: For the multi-cut case the functionappearing in Q is positive definite (meaning r ( x ) ≥ n . Takingthis into account, we can apply the general formula ofrefs. [52, 73, 74] to the multi-cut case and find G ( β , . . . , β n ) = 2 n √ β · · · β n π n/ β T (cid:104) ( (cid:126) ∂ x ) n − e − β T r ( x ) (cid:105) x = µ (26)For n =2 , n > G ( β , . . . , β n ) van-ishes for n ≥
3, given that r ( x ) and all its derivativesvanish at x = µ >
0. In summary, we have shown howthe double-cut matrix model exactly reproduces all thesupergravity results to leading order in genus and arbi-trary number of boundaries.
2. Higher genus corrections
Next, we compute (cid:126) corrections to r ( x ) in equa-tion (18), by solving the string equation (13) in pertur-bation theory. There are two different perturbative ex-pansions, one valid for large positive x and the other forlarge negative x . Since µ is positive, the relevant one forcomparing with supergravity is the large positive x ex-pansion. This is completely analogous to choices presentwhen using the type 0A models, as discussed in ref. [37].It is convenient to introduce the parameter c ∈ R + , andadd (cid:126) c to the right-hand side of the string equation (13),giving: ∞ (cid:88) k =1 t k K k + r ( x ) x = (cid:126) c , (27)so that c = 0 corresponds to the case of interest. Insert-ing (18) in this differential equation it is straightforward Again, as explained in ref. [27], there is a factor γ = √ β , β dependence is universal(neither the t k nor µ appear), its normalization ultimately fixesthe conventions that should be used here for the macroscopicloop operator (14). to solve for the first few orders and find: r ( x ) = r ( x ) + (cid:126) cx (cid:20) t (cid:126) (4 − c ) x + (cid:126) (4 − c ) x × ( t (40 − c ) − t (16 − c ) x ) + O ( (cid:126) ) (cid:21) . (28)Notice that the whole perturbative series vanishes when c = 0, the case of interest. This is no accident and followsfrom the observation that r ( x ) = 0 is an exact solutionto the full string equation when c = 0. This means thefunction r ( x ) for the model relevant to JT supergrav-ity receives no perturbative corrections and is given by r ( x ) of equation (20) (with t k as in (23) and plotted inFigure 4) to all orders in perturbation theory. We willcompute and display the non-perturbative contributionsin Section IV.To compute higher order corrections to (cid:104) Tr e − βQ (cid:105) us-ing equation (15) essentially involves computing the sub-leading terms in equation (21). Building on some resultsof Gel’fand and Dikii [75], we derive the following expan-sion in Appendix B (cid:104) x | e − β Q | x (cid:105) (cid:39) e − βr (cid:126) √ πβ (cid:26) − (cid:126) βr r − (cid:126) β × (cid:2) β [( r ) (cid:48) ] + 2 β (12( r r ) − ( r (cid:48) ) ) − r + 2 r r ) (cid:3)(cid:27) (29)where the r n ( x ) are corrections to r ( x ) defined in equa-tion (18), and ‘ (cid:39) ’ means that terms of order (cid:126) have beendropped. Integrating this expansion as in (15), we obtainhigher (cid:126) corrections to (cid:104) Tr e − βQ (cid:105) . In doing so, there isan important distinction that must be made regardingthe leading and subleading terms in equation (29).When integrating (29) only the leading term must beintegrated in the whole range x ∈ ( −∞ , µ ] indicated inequation (15). All subleading (cid:126) contributions can only beintegrated on the region x ∈ [0 , µ ], where the expansionis meaningful (recall that we did a positive x expansion,not a negative x one, and it is meaningless to have bothexpansions present in the same expression). But we havealready shown that the solution r ( x ) = 0 for x > r n ( x ) = 0for n ≥
0. As a result, all (cid:126) corrections to (29) vanishand we conclude that (cid:104) Tr e − βQ (cid:105) = 1 (cid:126) √ πβ (cid:90) µ −∞ dx e − βr ( x ) = e π /β (cid:126) √ πβ . (30)The important difference with the previous computationin equation (22), is that this is an exact result to all ordersin perturbation theory, matching with the supergravityresult in equation (17).Let us emphasize that the vanishing of higher (cid:126) correc-tions to (cid:104) Tr e − βQ (cid:105) does not depend on the special tuningof couplings that define the model, i.e., { t k , µ } in equa-tion (23). Instead, it is a consequence of having r ( x ) = 0for x > t k . With this in mind, consider higher (cid:126) corrections formultiple insertions of the macroscopic loop operatorTr e − β i Q , which via the dictionary (1) will correspond tothe JT supergravity partition function with higher genuscorrections. For the supergravity of interest, all highergenus corrections vanish. It is interesting to see howthis feature emerges from our definition of the double-scaled matrix model. Consider for example the two-pointfunction, (cid:104) Tr( e − β Q )Tr( e − β Q ) (cid:105) . The β “macroscopicloop” can be expanded in terms of insertions of point-likeoperators σ k , the “microscopic loops” [52, 76], giving: (cid:104) Tr( e − β Q )Tr( e − β Q ) (cid:105) = ∞ (cid:88) k =1 β k + (cid:104) Tr( e − β Q ) σ k (cid:105) , (31)(in a particular normalization for the σ k that we will notneed to specify here). However, a σ k -insertion is equiva-lent [22] (fully non-perturbatively) to differentiating thefunction r ( x ) with respect to the general coupling t k ,and so in practical terms, the right hand side of theabove is computed by differentiating the explicit r ( x )-dependence in our result (15). In the genus expansiontherefore, all contributions to the right hand side con-tain t k derivatives of r ( x ), which are given by the mKdVflows: ∂r ( x ) ∂t k ∝ K (cid:48) k [ r ] , (32)where the K k [ r ], polynomials in r ( x ) and its derivatives,are characterized in expressions (11) and (12). Since ithas been established that r ( x )=0 at every order in per-turbation theory for arbitrary t k , we see that ∂r ( x ) /∂t k vanishes, and hence all higher genus contributions to thiscorrelator. This procedure can be iterated to take care ofcorrelators with higher numbers of Tr e − βQ insertions,showing that they also vanish to all orders in perturba-tion theory.In summary, in this Section we have shown how thedouble-cut Hermitian matrix model reproduces all of theperturbative results obtained from supergravity compu-tations in ref. [27]. IV. NON-PERTURBATIVE PHYSICS
While the perturbative expansion of the supergrav-ity observables is surprisingly simple, there can still be This same feature (but in a different language) was also observedin Section 5.2.1 of ref. [27], where the vanishing of the perturba-tion series was shown to be independent of the detailed structureof ρ ( q ) (3) but relied on ρ ( q ) being supported on the whole realline q ∈ R , leaving no cut endpoints to source the relevant resol-vents. An analogous situation also applies to other JT supergrav-ity theories, as shown in ref. [37] for the cases ( α , β ) = ( { , } , non-perturbative contributions that are not captured bythe topological expansion. Although some importantnon-perturbative effects were discussed in ref. [27], themain methods used there cannot derive generic non-perturbative effects, whether using supergravity or therecursive loop equations technology which defines thematrix model order by order in the genus expansion. Asemphasized in refs. [26, 37, 38], the advantage of thealternative matrix model techniques used in this paperis that non-perturbative contributions can be explicitlycomputed because r ( x ) is defined non-perturbatively bythe string equation. Moreover, we shall show that non-perturbative effects are not necessarily small and play acrucial role in the low energy behavior of the model. A. A toy model
Let us start by considering non-perturbative effects ina simple toy model that is relevant for JT supergrav-ity and where everything can be computed analytically.This model is completely analogous to the Bessel modelextensively studied in e.g. refs. [26, 27, 37], and is: r ( x ) = (cid:126) cx . (33)with x >
0. When c = 0 , ± Q ψ q ( x ) = q ψ q ( x ) needed toobtain the spectral density (16). With r ( x ) as the po-tential, the wavefunction can be written in terms of aBessel function ψ q ( x ) = 1 (cid:126) (cid:114) x J a ( | q | x/ (cid:126) ) , (34)where a ≡ (cid:112) c + 1 / ρ ( q ) using equation (16), ρ ( q ) = µζ √ (cid:126) (cid:2) J a ( ζ ) − J a − ( ζ ) J a +1 ( ζ ) (cid:3) , (35)where ζ = µ | q | / (cid:126) and the integration region in (16) wastaken as x ∈ [0 , µ ] given that the potential is defined for x positive. This exact result includes both perturbativeand non-perturbative contributions.The case that is relevant for JT supergravity is when( c, µ ) = (0 , √ ρ ( q ) (cid:12)(cid:12) ( c,µ )=(0 , √ = 1 π (cid:126) − sin(2 √ q/ (cid:126) )2 √ πq . (36)The first term is the only perturbative contribution thatappears in this toy model, and reproduces the q inde-pendent term of ρ ( q ) in (3). The second term is non-perturbative in (cid:126) and is therefore invisible in any per-turbative analysis. In Figure 5 we plot ρ ( q ) and observe0 FIG. 5. Plot of the spectral density of the toy model with( c, µ ) = (0 , √
2) in (36), that is relevant for JT supergravity.While the constant black curve (dashed) corresponds to theperturbative result, in blue we include also non-perturbativeeffects, which are extremely relevant for small q . the dramatic effect of the non-perturbative correctionsfor small q , where there is a complete cancellation at theorigin ρ ( q ) = (4 / π (cid:126) ) q + O ( q ). B. The full model
We are now ready to consider non-perturbative effectsin the full matrix model description of the JT supergrav-ity, determined by fixing { t k , µ } per equation (23). Thefirst step is to numerically solve the string equation (13)for r ( x ). To make sense of it as a finite order differentialequation, we follow ref. [38] and introduce a truncationby only including contributions from t k up to some max-imum value k max . For a high enough order truncation,at the value of x where the solution begins to deviatefrom the true solution, r ( x ) will be closely approximatedby the classical solution r ( x ). Any artefacts due tothe truncation are, at low enough energies, indistinguish-able from other numerical errors due to discretization tofind r ( x ) or when subsequently solving the spectral prob-lem numerically. All of the numerical computations per-formed here are obtained by taking k max =6. While thereis no obstruction in taking higher values of k max , numer-ical investigations show corrections obtained by workingwith higher truncations are small enough for our pur-poses. Similar truncation procedures have been recentlysuccessfully applied to other double scaled matrix modelsin relation to JT gravity and supergravity [39, 72]. Wewill mostly use (cid:126) =1, although it is straightforward to usesmaller (cid:126) , as we shall do later with (cid:126) =1 / r ( x ) for (cid:126) =1, with the dashed curve corresponding tothe leading (cid:126) solution r ( x ) in equation (20), which doesnot receive perturbative corrections. For small valuesof x , non-perturbative effects generate a substantial dif-ference between r ( x ) and r ( x ). For successively smallervalues of (cid:126) , the deviations between the two are containedwithin an increasingly smaller region. -25 -20 -15 -10 -5 0 5 10 15 20 2500.10.20.30.40.50.60.70.8 FIG. 6. The solid black curve corresponds to the full non-perturbative numerical solution r ( x ) to the string equa-tion (13) in the k max =6 truncation with (cid:126) =1. The dashedred curve is the leading genus solution r ( x ) in (20).
1. Spectral density
With the full numerical solution for r ( x ) we can con-struct the operator Q = − ( (cid:126) ∂ x ) + r ( x ) and use it tocompute observables in the matrix model. The eigenfunc-tions ψ q ( x ) of the differential operator Q ψ q = q ψ q are calculated using standard methods, recently appliedto similar situations [26, 37–39, 72]. After numericallycomputing ψ q ( x ), we can solve the integral in equa-tion (15) and obtain the full spectral density of the ma-trix model ρ ( q ). The final result is shown in red inFigure 2 (page 3), for (cid:126) =1 and in Figure 7 for (cid:126) =1 / ρ ( q ) in equation (3). As also noted in thetoy model, non-perturbative corrections are dramatic forsmall q , particularly at the origin where there is a com-plete cancellation ρ (0)=0. (A similar remarkable numer-ical cancellation occurs in the (2,2) JT supergravity [38]using complex matrix models.)It is worth noting that in Appendix E.3 of ref. [27]it was argued that oscillatory non-perturbative contribu-tions to the spectral density would not be present forthis particular JT supergravity theory (in contrast tothe ( α , β ) JT supergravities). The argument was builtaround the fact that the leading perturbative result ρ ( q )has no endpoints associated to the cut, which is wheresuch oscillations develop. However, our construction interms of double-cut matrix models does yield oscillatorycorrections. This is because two separate cut endpoints Since the potential r ( x ) vanishes as x → + ∞ , the eigenfunctions ψ q ( x ) are not normalizable. We explain in Appendix D howto fix the normalization constant by comparing with the (cid:126) → ρ ( q ). -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.402468101214 FIG. 7. Numerical solution (red dots) for ρ ( q ) obtained fromeq. (16) after solving for the spectrum of Q , with r ( x ) com-puted in the k max =6 truncation (see text). The dashed blackline corresponds to the leading genus solution ρ ( q ) in eq. (3).Here (cid:126) =1 / meet and merge at q =0, and hence the intuition aboutoscillations appearing near endpoints applies after all.An interesting question is whether some of the non-perturbative oscillations observed numerically in thespectrum in Figures 2 and 7 can be captured analytically.For the ( α , β =2) JT supergravity theories, an analyticformula for the leading non-perturbative contribution tothe density was suggested in Appendix E.2 of ref. [27]and matched with numerical data from studying com-plex matrix models in ref. [38]. For the (2,2) case, whichalso has vanishing density at the origin and vanishingperturbative corrections beyond the disc, the formula forthe energy spectral density is: ρ ( E ) (cid:39) ρ ( E ) − √ πE sin (cid:20) √ π (cid:90) E ρ ( E (cid:48) ) dE (cid:48) (cid:21) , (37)where ρ ( E )= ρ SJT0 ( E ) /γ with ρ SJT0 ( E ) in equation (3).The matching observed in ref. [38] is especially accuratein a regime where instanton effects (which the formuladoes not capture) are small. This was tested by general-izing the spectral denisty to include more general µ : ρ ( E, µ ) = cosh(2 π √ E ) π (cid:126) √ E + µ − √ √ π (cid:126) √ E , (38)and using it in (37), noting that complex matrix modelinstanton effects scale as exp( − µ √ E/ (cid:126) ). It is interestingto see to what extent a similar story might be true here.In Figure 8 we plot equation (37) and the numeri-cal non-perturbative ρ ( E ), obtained by changing vari-ables to E = q and multiplying by 1 / √ E to convert thedata in Figure 2 to give the energy spectral density. Itis evident that our Dyson-Wigner β =2 JT supergravity FIG. 8. The red dots show the non-perturbative spectral den-sity computed for the β =2 JT supergravity, the focus of thispaper, built from type 0B minimal string models. The bluedotted curve show the approximate analytic formula (37),with the perturbative result (3) corresponding to the dashedblack curve. Here µ = √ (cid:126) =1. physics takes quite a departure from the simple analyticapproximate form (37). The non-perturbative correc-tions result in a very different phase and frequency ofthe undulations, corresponding to a very different under-lying spectrum. The result in this case is also different tothe ( α , β ) = ( { , } ,
2) theories studied in refs. [38, 39],not the least because these other cases have finite or di-vergent value at E =0.It is interesting to study when a formula like (37) mighthave a chance of working for the present case. A key pointis that its overall form arises from taking a WKB approx-imation to the underlying wavefunctions, as is shown ex-plicitly in Appendix C. The remarkable thing about thespecial cases of the ( α , β ) models, is that the form worksextremely well far beyond the small (cid:126) regime where theapproximation usually holds. Evidently the special fea-tures of those cases (discussed [27] in terms of the formof the disc, crosscap, and cylinder terms) are not at playhere. So if there is improvement to be made to the match-ing for this case, it should be found by reducing both in-stanton effects (controlled by µ ) and working at smaller (cid:126) .This can be demonstrated in either variable, q or E , andwe choose q in what follows in order to facilitate a com-parison to the previous Section. Changing variables tothe Q spectrum the WKB-like formula (37) gives: ρ ( q ) (cid:39) ρ ( q ) − √ π | q | sin (cid:20) √ π (cid:90) | q | dq (cid:48) ρ ( q (cid:48) ) (cid:21) . (39)In these variables, the eigenvalue spectral density for2more general µ is: ρ ( q, µ ) = cosh(2 πq ) π (cid:126) + µ − √ √ π (cid:126) . (40)When µ = √
2, we recover the usual expression in equa-tion (3). Different values of µ and (cid:126) were explored andindeed somewhat better matching is achieved at large µ ,and smaller values of (cid:126) , but with the best results atsmall energies. Departures from the formula begin to de-velop at intermediate energies, marking the onset of non-perturbative physics that not cannot be fit by this simpleform. An example is given in Figure 9, where µ =70 and (cid:126) =1 /
5. At low energies and such large µ and/or small (cid:126) ,the system should closely resemble the simple Bessel toymodel of Subsection IV A, and indeed it does (see Fig-ure 5), but deviations set in as q is increased. -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.10102030405060708090100 FIG. 9. The red dots show the non-perturbative spectral den-sity computed for µ =70 and (cid:126) =1 /
5, zoomed in to low q , wherethe classical (black) curve (40) is almost a straight line. Theblue dotted curve is the WKB formula (39).
2. Spectral form factor
Using the non-perturbatively computed eigenfunctions ψ q ( x ) allows the computation of many other observ-ables of interest, including all perturbative and non-perturbative corrections. A quantity that is particularlyinteresting is the spectral form factor, defined as the two-point correlator involving two asymptotic boundaries,written as a sum of disconnected and connected pieces: (cid:104) Z ( β + it ) Z ( β − it ) (cid:105) ≡ Z ( β + it ) Z ( β − it )+ Z ( β + it, β − it ) , (41)where the two boundary lengths β and β have beenanalytically continued to β + it and β − it . This quan-tity is useful for diagnosing certain universal aspects of quantum chaotic behavior [77–79]. Random matrix the-ory predicts the spectral form factor to exhibit “dip”,“ramp” and “plateau” behavior, describing the interac-tion of eigenvalues. In the context of the JT gravity/SYKcorrespondence, these features of SYK describe the infor-mation scrambling properties of black holes [77]. FIG. 10. Full non-perturbative spectral form factor vs. time t ,computed from the matrix model side with β =50 and (cid:126) =1,using (15), (A1), and the methods described in refs. [38, 39]. Using the identification in equation (1), the spectralform factor can be computed from the matrix model sideby inserting two operators Tr e − βQ and performing theanalytic continuation. For other JT supergravity the-ories this has been recently computed in refs. [38, 39](see equation (26) of ref. [38]). Using the eigenfunctions ψ q ( x ) relevant to this case, we compute an example andobtain the result shown in Figure 10, showing the ex-pected features. Non-perturbative effects again play acrucial role in developing the plateau, and even in theslope of the dip. For brevity, we have omitted showingplots of the individual disconnected portions of the spec-tral form factor, but there is a great deal of similarityto the (2,2) case studied in ref. [38], and so we refer thereader there for results and extended discussion. V. CLOSING REMARKS
We have shown how a particular JT supergravitytheory—one which sums only over orientable surfaces andequally weights spin structures in the Euclidean pathintegral—is given a complete non-perturbative descrip-tion by a double-cut Hermitian matrix model ( i.e. a β =2Dyson-Wigner ensemble). While generally the double-cut matrix model is characterized by two string equa-tions associated to the ZS integrable hierarchy, here thetwo string equations collapse to a single string equation,the Painlev´e II/mKdV integrable hierarchy, also known3from unitary matrix models. These equations are knownto describe type 0B minimal strings [45], and we showed aprecise combination of such minimal models reproducesthe JT supergravity. This is completely analogous tothe work of refs. [37–39], which produced similar resultsfor the Altland-Zirnbauer ( α , β )=( { , , } ,
2) JT super-gravities, showing how to construct them from type 0Aminimal strings.It was observed long ago in ref. [42] that there is anon-perturbative map between the string equations ofthe complex matrix models (type 0A minimal strings)and those of the unitary matrix models (the type 0Bminimal strings). It is in fact the Miura transforma-tion [80, 81] that links the parent KdV and mKdV hierar-chies: u ( x )= v ( x ) + (cid:126) v (cid:48) ( x ), where the v ( x ) of that paperis related to r ( x ) by a simple rescaling. (This is differ-ent from the map for the k =1 case noticed in ref. [82],or the connection more recently discussed for unscaledmodels in ref. [83].) In this sense, the ( α , β =2) JT super-gravities, non-perturbatively defined as complex matrixmodels in refs. [37, 38], can straightforwardly be cast interms of unitary matrix model language. This does not mean that such a recasting captures the Dyson-Wigner β =2 JT supergravity discussed in this paper however.The key new ingredient, arising from our construction ofthe alternative loop operator in equation (15), is thatfor the β =2 case, r ( x ) becomes the potential in theSchr¨odinger quantum mechanics, and not the combina-tion r ( x ) + (cid:126) r (cid:48) ( x ), which is appropriate for the otherfamily of JT supergravities. It is a subtle, but crucial dif-ference. In a sense, from a non-perturbative (geometry-independent) high-altitude perspective, the fundamentalvariable is really r ( x ), and the two different (inequiva-lent) uses of it to define potentials amounts to the two(inequivalent) JT supergravity families. This perspectivedeserves further exploration, since this simple differencein choice can change major features of the theory, such aswhether it has non-orientable geometries (preserves T ),respects ( − F , and so on.It is worth noting that there is a language in whichthe two different uses of r ( x ) are both natural. Theintegrable hierarchies in question have an underlying sl (2 , C ) structure [85–90] from which naturally arisestwo “ τ -functions”, τ and τ . In terms of these, define u , =2 (cid:126) ∂ x ln τ , where either choice u , = v ± (cid:126) v (cid:48) givesa KdV-type function u ( x ), and recall that v ( x ) is pro-portional to the r ( x ) of this paper. Their difference gives v = (cid:126) ∂ x ln( τ /τ ) while their sum gives v = (cid:126) ∂ x ln( τ τ ).Perhaps this τ -function language can be aligned with thesupergravity symmetries T and ( − F .There are a number of other interesting potential av-enues for future work. For example, as discussed inref. [27], there are several other JT supergravity theo-ries that are T –invariant (so they include non-orientablesurfaces in the Euclidean path integral) and have notyet been studied non-perturbatively. For example, two of them are classified as double scaled β = 1 , ρ for these modelshas a branch cut comprising the entire real line. It isnatural to conjecture (and we do) that these SJT modelsmay be similarly described by double-cut matrix mod-els. It is a natural guess that the string equations forthose systems will define a function analogous to r ( x )that has a regime where it vanishes to all orders in per-turbation theory. The details will be different however,with important non–perturbative differences, since theunderlying orthogonal polynomials (and hence the stringequations) arising from double scaling will be different.As already noted, the larger system of equations de-scribing double-cut Hermitian matrix models are fromthe Zakharov-Shabat hierarchy, where the function α ( x )and the parameter η (see Subsection II A) are turnedon. In our matching to the JT supergravity we restrictedto the symmetric sector and turned those off. However,ref. [45] gave a type 0B minimal string interpretation tosome of the more general solutions in terms of symmetrybreaking R-R fluxes. It would be interesting to explore ifthose can be understood in terms of the JT supergravity.Notably, the KdV, mKdV, and ZS hierarchies, are allnaturally embedded in a larger hierarchy of differentialequations called the dispersive water wave hierarchy (seeref. [94] and references therein). It was noticed [95, 96]that it is possible to derive string-equation-like ODEsfrom that larger system that have many properties ofminimal string theories, suggesting a large interconnectedweb of minimal string theories of which type 0A andtype 0B are merely a small part. It is tantalising to imag-ine that (by using the procedures used here and elsewherefor building JT gravity models out of minimal strings) aninterconnected web of JT gravity and supergravity theo-ries can similarly be defined.Lastly, it would be worthwhile to use complex matrixmodels and double-cut hermitian matrix models to de-scribe non-trivial deformations of JT supergravity anal-ogous to those described in refs. [97, 98], and also to JTgravity in de Sitter space [99, 100]. Non-perturbative in-sights into such deformations would be useful. Progresshas already been made for deformed JT gravity [72], andis currently underway for JT supergravity. ACKNOWLEDGMENTS
CVJ and FR are partially supported by the DOE grantDE-SC0011687. AS is supported by the Simons Foun-dation through
It from Qubit: Simons collaboration onQuantum Fields, Gravity, and Information . CVJ thanksAmelia for her patience and support, especially duringthe long lockdown due to the pandemic.4
Appendix A: Higher trace operators
In this appendix we show how to compute the expectation value of higher trace operators in the double scaled multi-cut model. The basic idea involves starting from the single trace formula in equation (15) and use the free fermionformalism [71], very nicely explained in Section 10.1 of ref. [52]. While this formalism was originally developed forsingle cut matrix models, the effective quantum mechanical system that arises when computing observables is basicallythe same after replacing r ( x ) → u ( x ) and accounting for the extra factor of √ Double trace operators:
Starting from equation (15), we obtain the following formula for two insertions of Tr e − βQ (cid:10) Tr (cid:0) e − β Q (cid:1) Tr (cid:0) e − β Q (cid:1)(cid:11) c = 2 (cid:90) µ −∞ dx (cid:90) + ∞ µ dx (cid:104) x | e − β Q | x (cid:105) (cid:104) x | e − β Q | x (cid:105) , (A1)where the factor 2 = ( √ is important and comes from the prefactor in (15). Using equation (21), it is quite simpleto evaluate its leading (cid:126) behavior and find (cid:10) Tr (cid:0) e − β Q (cid:1) Tr (cid:0) e − β Q (cid:1)(cid:11) c = 12 π (cid:126) √ β β (cid:90) µ −∞ dx (cid:90) + ∞ µ dx e − β r ( x ) − β r ( x ) − β β β β ( x − x (cid:126) ) + O (1) . (A2)Changing the integration variables to x = (cid:126) ( w − w ) + µ and x = (cid:126) ( w + w ) + µ the integrals decouple after weexpand the exponentials e − β i r ( x i ) (cid:39) e − β i r ( µ ) to leading order in (cid:126) . The remaining ( w , w ) integrals can be easilysolved and we find (cid:10) Tr (cid:0) e − β Q (cid:1) Tr (cid:0) e − β Q (cid:1)(cid:11) c = 2 √ β β πβ T e − β T r ( µ ) + O ( (cid:126) ) , (A3)where we have defined β T = (cid:80) ni =1 β i , in this case with n = 2. Using r ( µ ) = r (1) = 0, the leading behaviorreproduces the supergravity result in equation (17). Triple trace operators:
Let us now consider three insertions of the matrix operator Tr e − βQ . The general formulain this case can be obtain from the neat general expression derived in ref. [101], and written in equation (3.28) ofref. [102]. In our conventions, it is given by (cid:10) Tr (cid:0) e − β Q (cid:1) Tr (cid:0) e − β Q (cid:1) Tr (cid:0) e − β Q (cid:1)(cid:11) c = 2 (cid:90) µ −∞ dx (cid:90) + ∞ µ dx (cid:104) x | e − β Q | x (cid:105)× (cid:20)(cid:90) + ∞ µ dx (cid:104) x | e − β Q | x (cid:105) (cid:104) x | e − β Q | x (cid:105) − (cid:90) µ −∞ dx (cid:104) x | e − β Q | x (cid:105) (cid:104) x | e − β Q | x (cid:105) (cid:21) , (A4)where the factor 2 comes from (15). To evaluate its leading (cid:126) behavior, that we denote as G ( β , β , β ), we setwithout loss of generality β = β = β , as the general answer is then obtained by requiring its symmetric underarbitrary exchanges of β i ↔ β j . Using (21) we find G ( β , β, β ) = 2 (cid:112) β β π / (cid:90) + ∞−∞ dx β β (cid:126) sign( x − µ ) (cid:90) µ −∞ dx (cid:18) (cid:90) + ∞ µ dx e − β r ( x ) − β ( x − x (cid:126) ) × e − βr ( x ) − β ( x − x (cid:126) ) e − βr ( x ) − β ( x − x (cid:126) ) (cid:19) . (A5)Changing the integration variables to x i = (cid:126) ¯ x i + µ we can use the following expansion in (cid:126) e − β i r ( x i ) = e − β i r ( µ ) (cid:104) − β i (cid:2) ( (cid:126) ∂ x ) r ( x ) (cid:3) x = µ ¯ x i + O ( (cid:126) ) (cid:105) . (A6)The first term contributes to (A5) as an integral over the whole real line in ¯ x of the following function F (¯ x ) ≡ sign(¯ x ) (cid:90) −∞ d ¯ x (cid:90) + ∞ d ¯ x e − (¯ x − ¯ x β − (¯ x − ¯ x β − (¯ x − ¯ x β . (A7) In fact, this was done later in ref. [84]. F (¯ x ) = − F ( − ¯ x ), so that it vanishes when integrated over thewhole real line. This means the leading contribution is given by the second term in (A6), so that (A5) can be writtenas G ( β , β, β ) = 2 (cid:112) β β π / ( β + 2 β ) (cid:104) ( (cid:126) ∂ x ) e − ( β +2 β ) r ( x ) (cid:105) x = µ (cid:40) β β (cid:88) i =1 β i I i (cid:41) , (A8)where we have defined I i = (cid:90) + ∞−∞ d ¯ x sign(¯ x ) (cid:90) −∞ d ¯ x (cid:90) + ∞ d ¯ x ¯ x i e − (¯ x − ¯ x β − (¯ x − ¯ x β − (¯ x − ¯ x β . (A9)To evaluate these triple integrals, we first apply a change of coordinates that decouples the exponents x = (cid:112) ββ (2 w + w + 3 w ) , x = (cid:112) ββ (2 w + w − w ) , x = 2 (cid:112) ββ ( w − w ) , (A10)so that the integration region gets mapped to ¯ x ∈ ( −∞ , −→ w ∈ [ − w + 3 w , − w − w ] , ¯ x ∈ [0 , + ∞ ) −→ w ∈ ( −∞ , + ∞ ) , ¯ x ∈ ( −∞ , + ∞ ) −→ w ∈ ( −∞ , . (A11)Applying this transformation we find I i = √
23 (4 ββ ) / (cid:90) −∞ dw e − ( β +2 β ) w (cid:90) + ∞−∞ dw e − β w (cid:90) − w − w − w w dw sign( w − w ) ¯ x i ( w j ) , (A12)where the prefactor comes from the Jacobian in the change of variables. Solving the integral as written in the w i variables is simpler, but still tedious. The final result in each case is given by β I β β = β β + 2 β − β β √ β ( β + 2 β ) / tan − (cid:115) β β + 2 β ,βI β β = ββ + 2 β − β √ β ( β + 2 β ) / tan − (cid:115) β β + 2 β ,βI β β = ββ + 2 β + 2 β ( β + β ) √ β ( β + 2 β ) / tan − (cid:115) β β + 2 β . (A13)Summing these three terms as indicated in (A8), we get a nice cancellation such that the factor between curly bracketsin (A8) is equal to one.This gives an explicit expression for G ( β , β, β ). To get the result for arbitrary values of β i we note that thesymmetry of the observable under arbitrary exchanges β i ↔ β j uniquely fixes ( β + 2 β ) → ( β + β + β ) and β β → β β β . Putting everything together, we arrive at the final result in equation (25). Appendix B: Higher genus perturbative corrections
In this Appendix we compute perturbative (cid:126) corrections to the one point function (cid:104) Tr e − βQ (cid:105) using (15), bycalculating the subleading terms in (21). One way of doing this, is by using the expansion for the resolvent of theoperator Q = − ( (cid:126) ∂ x ) + r ( x ) , worked out long ago by Gel’fand and Dikii [75] and given by (cid:104) x | Q − ξ | x (cid:105) = 1 (cid:126) ∞ (cid:88) p =0 − ξ ) p +1 / (2 p − (cid:101) R p [ r ( x ) ]( − p p !( p − , (B1)6where ξ <
0. The Gel’fand-Dikii functionals (cid:101) R p [ r ( x ) ] are polynomials in r ( x ) and its derivatives computed fromthe following recursion relation 2 p + 12( p + 1) (cid:101) R (cid:48) p +1 = − (cid:126) (cid:101) R (cid:48)(cid:48)(cid:48) p + r (cid:101) R (cid:48) p + 12 ( r ) (cid:48) (cid:101) R p , (B2)with (cid:101) R = 1. Our normalization of the functionals (cid:101) R p [ r ( x ) ] is different from the one used in ref. [75], as we havedefined things differently so that (cid:101) R p [ r ( x ) ] = r ( x ) p + O ( (cid:126) ). Explicit expressions for the first few functionals writtenin terms of u ( x ) = r ( x ) are given in equation (10) of ref. [75]. Applying an inverse Laplace transformation in ξ toequation (B1) we obtain the analogous expansion for (cid:104) x | e − β Q | x (cid:105)(cid:104) x | e − β Q | x (cid:105) = 12 (cid:126) √ πβ ∞ (cid:88) p =0 ( − β ) p p ! (cid:101) R p [ r ( x ) ] . (B3)While this asymptotic series formula is valid to all orders in (cid:126) , we cannot exchange the infinite series with the x integral in (15). Instead we must solve the series order by order in (cid:126) . To do so, we can use the recursion relation (B2)to expand (cid:101) R p in (cid:126) . The expansion becomes simpler when written in terms of u ( x ) = r ( x ) , so that we find (cid:101) R p [ u ] = u p − (cid:126) p ( p − u p − (cid:2) uu (cid:48)(cid:48) + ( p − u (cid:48) ) (cid:3) + (cid:126) p ( p − p − u k − (cid:104) u u (4) + 48( p − u u (cid:48) u (cid:48)(cid:48)(cid:48) + 36( p − u ( u (cid:48)(cid:48) ) + 44( p − p − u ( u (cid:48) ) u (cid:48)(cid:48) + 5( p − p − p − u (cid:48) ) (cid:105) + O ( (cid:126) ) . (B4)Inserting this into (B3) and solving the series order by order in (cid:126) we find (cid:104) x | e − β Q | x (cid:105) = e − βu (cid:126) √ πβ (cid:20) − (cid:126) β (cid:0) u (cid:48)(cid:48) − β ( u (cid:48) ) (cid:1) − (cid:126) β (cid:18) u (4) − βu (cid:48) u (3) − β ( u (cid:48)(cid:48) ) + 44 β ( u (cid:48) ) u (cid:48)(cid:48) − β ( u (cid:48) ) (cid:19) + O ( (cid:126) ) (cid:21) . (B5)The final expansion used in the main text (29) is obtained by replacing u ( x ) → r ( x ) = [ r ( x ) + (cid:80) ∞ n =1 r n ( x ) (cid:126) n ] andexpanding in (cid:126) one last time.To check we did not miss any factor in the expansion (29), let us use it to compute higher (cid:126) corrections for the toymodel defined from equation (33). Using (29) in (15) for the toy model r ( x ) = (cid:126) c/x , we find (cid:104) Tr e − βQ (cid:105) = 1 (cid:126) √ πβ (cid:90) µ dx (cid:20) − (cid:126) βc x + (cid:126) β c ( c − x + O ( (cid:126) ) (cid:21) = µ (cid:126) √ πβ (cid:20) (cid:126) βc µ − (cid:126) β c ( c − µ + O ( (cid:126) ) (cid:21) , (B6)where the integration region is given by x ∈ [0 , µ ] since it is for this range that r ( x ) = (cid:126) c/x is well defined. There isa divergent contribution coming from the x → ρ ( q ) and find ρ ( q ) = µ √ π (cid:126) (cid:20) − (cid:126) c µ q − (cid:126) c ( c − µ q + O ( (cid:126) ) (cid:21) . (B7)Comparing this with the first perturbative terms obtained from expanding the exact expression in (35), we findperfect agreement. The perturbative expansion of (35) becomes particularly simple to work out for values of c suchthat (cid:112) c + 1 / Appendix C: WKB approximation and leading non-perturbative effects
Here, we apply the WKB approximation to compute the eigenfunctions ψ q ( x ) of the operator Q , and then use itto calculate the leading perturbative and non-perturbative contributions to ρ ( q ) in that limit. For small values of (cid:126) ,7the potential of the quantum mechanical system is given by r ( x ) , so that the eigenfunction ψ q ( x ) in the WKBapproximation is given by the standard expression:lim (cid:126) → ψ q ( x ) = A + exp (cid:104) i (cid:126) (cid:82) xx min d ¯ x (cid:112) q − r (¯ x ) (cid:105) + A − exp (cid:104) − i (cid:126) (cid:82) xx min d ¯ x (cid:112) q − r (¯ x ) (cid:105) ( q − r ( x ) ) / , (C1)where A ± are undetermined constants. The constant x min is defined as r ( x min ) = q , so that (C1) is the solutionin the classically allowed region. To fix the constants A ± we use that for the particular case in which r ( x ) = 0 with x >
0, the full eigenfunctions where computed in (34) as ψ q ( x ) (cid:12)(cid:12) r ( x )=0 = 1 (cid:126) (cid:114) x J / ( | q | x/ (cid:126) ) = sin( x | q | / (cid:126) ) (cid:112) π (cid:126) | q | . (C2)Comparing with the general expression (C1) with x min = 0 (since (C2) is obtained from the c → r ( x ) = (cid:126) c/x ) allows us to fix the constants A ± and find an expression for arbitrary r ( x )lim (cid:126) → ψ q ( x ) = sin (cid:104) (cid:126) (cid:82) xx min d ¯ x (cid:112) q − r (¯ x ) (cid:105) √ π (cid:126) ( q − r ( x ) ) / . (C3)We can now use this in equation (16) and compute the spectral density. In the classical limit the eigenfunctions ψ q ( x ) vanish in the classically forbidden region, so that the integral (16) only gets contributions from x ∈ [ x min , µ ].Putting everything together we find:lim (cid:126) → ρ ( q ) = | q |√ π (cid:126) (cid:90) µx min dx (cid:112) q − r ( x ) − | q |√ π (cid:126) (cid:90) µx min dx (cid:112) q − r ( x ) cos (cid:20) (cid:126) (cid:90) xx min d ¯ x (cid:112) q − r (¯ x ) (cid:21) , (C4)where we have used the identity 2 sin ( z ) = 1 − cos(2 z ). The first term gives the usual leading perturbative contribution ρ ( q ), while the second corresponds to the leading non-perturbative term. A nicer expression for the second term canbe obtained when the following relation holds (this will be discussed further below): (cid:90) µx min dx (cid:112) q − r ( x ) cos (cid:20) (cid:126) (cid:90) xx min d ¯ x (cid:112) q − r (¯ x ) (cid:21) = (cid:126) q sin (cid:20) (cid:126) (cid:90) µx min d ¯ x (cid:112) q − r (¯ x ) (cid:21) . (C5)The remaining integral inside the sine can be written more nicely in terms of ρ ( q ) using:2 (cid:126) (cid:90) µx min d ¯ x (cid:112) q − r (¯ x ) = 2 √ π (cid:90) | q | d ¯ q ρ (¯ q ) , (C6)where we have used ρ ( q ) is given by the first term in (C4). Putting everything together, we arrive at the finalexpression for the spectral density in this limit:lim (cid:126) → ρ ( q ) = ρ ( q ) − √ π | q | sin (cid:20) √ π (cid:90) | q | d ¯ q ρ (¯ q ) (cid:21) , (C7)which confirms (after changing variables to E ) that the formula (37) of ref. [27] follows from the WKB form of the Q wavefunctions.This result depends on the relation (C5) being actually true. For r ( x ) = 0 it is straightforward to check that itholds. This corresponds to the Bessel toy models of Section IV A. For general r ( x ) we can differentiate both sideswith respect to µ and also find agreement, but that is only up to a whole function of q and so the final form (C7)might not always be attainable, and the form (C4) should be used more generally. Appendix D: Eigenfunctions normalization
In this Appendix we show how to normalize the eigenfunctions Q ψ q ( x ) = q ψ q ( x ) from their classical (cid:126) → ρ ( q ) from the first line in (22). After applying an inverseLaplace transform we obtain the following expression for ρ ( x ) in terms of r ( x ) ρ ( q ) = | q |√ π (cid:126) (cid:90) µ −∞ dx Θ[ q − r ( x ) ] (cid:112) q − r ( x ) , (D1)8which is precisely the perturbative term obtained from the WKB approximation (C4). Comparing with the fullexpression for ρ ( q ) in (16), we obtain the following condition satisfied by | ψ q ( x ) | lim (cid:126) → | ψ q ( x ) | = 12 π (cid:126) | q | + oscillating , x > , (D2)where we have used that r ( x ) = 0 for x >
0. Since ψ q ( x ) for x → + ∞ behaves like a free particle, we expect to haveadditional oscillating terms which average to zero. Using this condition we can unambiguously fix the normalizationconstant in the eigenfunctions ψ q ( x ).Let us see how this works for the toy model eigenfunctions (34), whose leading order behavior is given in (C2).Computing the norm square we findlim (cid:126) → | ψ q ( x ) | = 12 π (cid:126) | q | − π (cid:126) | q | cos (cid:20) x | q | (cid:126) + π − a ) (cid:21) , (D3)which is precisely the normalization required by equation (D2). [1] R. Jackiw, Nucl. Phys. B , 343 (1985).[2] C. Teitelboim, Phys. Lett. B , 41 (1983).[3] A. Achucarro and M. E. Ortiz, Phys. Rev. D , 3600(1993), arXiv:hep-th/9304068.[4] A. Fabbri, D. Navarro, and J. Navarro-Salas, Nucl.Phys. B , 381 (2001), arXiv:hep-th/0006035.[5] P. Nayak, A. Shukla, R. M. Soni, S. P. Trivedi, andV. Vishal, JHEP , 048 (2018), arXiv:1802.09547 [hep-th].[6] K. S. Kolekar and K. Narayan, Phys. Rev. D , 046012(2018), arXiv:1803.06827 [hep-th].[7] A. Ghosh, H. Maxfield, and G. J. Turiaci, JHEP ,104 (2020), arXiv:1912.07654 [hep-th].[8] S. Sachdev and J. Ye, Phys. Rev. Lett. , 3339 (1993),arXiv:cond-mat/9212030.[9] A. Kitaev, KITP seminars, April 7th and May 27th(2015).[10] K. Jensen, Phys. Rev. Lett. , 111601 (2016),arXiv:1605.06098 [hep-th].[11] J. Maldacena and D. Stanford, Phys. Rev. D , 106002(2016), arXiv:1604.07818 [hep-th].[12] J. Maldacena, D. Stanford, and Z. Yang, PTEP ,12C104 (2016), arXiv:1606.01857 [hep-th].[13] J. Engels¨oy, T. G. Mertens, and H. Verlinde, JHEP ,139 (2016), arXiv:1606.03438 [hep-th].[14] G. Penington, JHEP , 002 (2020), arXiv:1905.08255[hep-th].[15] A. Almheiri, N. Engelhardt, D. Marolf, and H. Max-field, JHEP , 063 (2019), arXiv:1905.08762 [hep-th].[16] A. Almheiri, T. Hartman, J. Maldacena, E. Shaghou-lian, and A. Tajdini, JHEP , 013 (2020),arXiv:1911.12333 [hep-th].[17] G. Penington, S. H. Shenker, D. Stanford, and Z. Yang,(2019), arXiv:1911.11977 [hep-th].[18] P. Saad, S. H. Shenker, and D. Stanford, (2019),arXiv:1903.11115 [hep-th].[19] E. Brezin and V. A. Kazakov, Phys. Lett. B236 , 144(1990).[20] M. R. Douglas and S. H. Shenker, Nucl. Phys.
B335 ,635 (1990).[21] D. J. Gross and A. A. Migdal, Phys. Rev. Lett. , 127(1990). [22] D. J. Gross and A. A. Migdal, Nucl. Phys. B340 , 333(1990).[23] M. Mirzakhani, Invent. Math. , 179 (2006).[24] B. Eynard, JHEP , 031 (2004), arXiv:hep-th/0407261.[25] B. Eynard and N. Orantin, Commun. Num. Theor.Phys. , 347 (2007), arXiv:math-ph/0702045.[26] C. V. Johnson, Phys. Rev. D , 106023 (2020),arXiv:1912.03637 [hep-th].[27] D. Stanford and E. Witten, (2019), arXiv:1907.03363[hep-th].[28] F. Dyson, J. Math. Phys. , 140 (1962).[29] A. Altland and M. R. Zirnbauer, Phys. Rev. B , 1142(1997), arXiv:cond-mat/9602137.[30] W. Fu, D. Gaiotto, J. Maldacena, and S. Sachdev,Phys. Rev. D , 026009 (2017), [Addendum:Phys.Rev.D 95, 069904 (2017)], arXiv:1610.08917 [hep-th].[31] D. Stanford and E. Witten, JHEP , 008 (2017),arXiv:1703.04612 [hep-th].[32] T. Li, J. Liu, Y. Xin, and Y. Zhou, JHEP , 111(2017), arXiv:1702.01738 [hep-th].[33] T. Kanazawa and T. Wettig, JHEP , 050 (2017),arXiv:1706.03044 [hep-th].[34] F. Sun and J. Ye, (2019), arXiv:1905.07694 [cond-mat.str-el].[35] S. Forste and I. Golla, Phys. Lett. B , 157 (2017),arXiv:1703.10969 [hep-th].[36] J. Murugan, D. Stanford, and E. Witten, JHEP ,146 (2017), arXiv:1706.05362 [hep-th].[37] C. V. Johnson, (2020), arXiv:2005.01893 [hep-th].[38] C. V. Johnson, (2020), arXiv:2006.10959 [hep-th].[39] C. V. Johnson, (2020), arXiv:2008.13120 [hep-th].[40] T. R. Morris, Nucl. Phys. B356 , 703 (1991).[41] S. Dalley, C. V. Johnson, and T. Morris, Nucl. Phys.
B368 , 625 (1992).[42] S. Dalley, C. Johnson, T. Morris, and A. Watter-stam, Mod. Phys. Lett. A , 2753 (1992), arXiv:hep-th/9206060.[43] S. Dalley, C. V. Johnson, and T. Morris, Nucl. Phys. B368 , 655 (1992).[44] S. Dalley, C. V. Johnson, and T. Morris, Nucl. Phys. Proc. Suppl. , 87 (1992), hep-th/9108016.[45] I. R. Klebanov, J. M. Maldacena, and N. Seiberg,Commun. Math. Phys. , 275 (2004), arXiv:hep-th/0309168.[46] T. G. Mertens and G. J. Turiaci, (2020),arXiv:2006.07072 [hep-th].[47] T. G. Mertens and G. J. Turiaci, JHEP , 127 (2019),arXiv:1904.05228 [hep-th].[48] G. J. Turiaci, M. Usatyuk, and W. W. Weng, (2020),arXiv:2011.06038 [hep-th].[49] T. G. Mertens, (2020), arXiv:2007.00998 [hep-th].[50] P. Betzios and O. Papadoulaki, (2020),arXiv:2004.00002 [hep-th].[51] J. Polchinski, Cambridge, UK: Univ. Pr. (1998) 531 p.[52] P. H. Ginsparg and G. W. Moore, in Theoretical Ad-vanced Study Institute (TASI 92): From Black Holesand Strings to Particles (1993) pp. 277–469, arXiv:hep-th/9304011.[53] V. Periwal and D. Shevitz, Phys. Rev. Lett. , 1326(1990).[54] V. Periwal and D. Shevitz, Nucl. Phys. B , 731(1990).[55] C. R. Nappi, Mod. Phys. Lett. A , 2773 (1990).[56] C. Crnkovic and G. W. Moore, Phys. Lett. B , 322(1991).[57] T. J. Hollowood, L. Miramontes, A. Pasquinucci, andC. Nappi, Nucl. Phys. B , 247 (1992), arXiv:hep-th/9109046.[58] C. Crnkovic, M. R. Douglas, and G. W. Moore, Int. J.Mod. Phys. A , 7693 (1992), arXiv:hep-th/9108014.[59] M. R. Douglas, N. Seiberg, and S. H. Shenker, Phys.Lett. B , 381 (1990).[60] D. J. Gross and E. Witten, Phys. Rev. D , 446 (1980).[61] G. M. Cicuta, L. Molinari, and E. Montaldi, Mod. Phys.Lett. A , 125 (1986).[62] G. Akemann, J. Baik, and P. Di Francesco, The Ox-ford Handbook of Random Matrix Theory , Oxford Hand-books in Mathematics (Oxford University Press, 2011).[63] P. Di Francesco, P. H. Ginsparg, and J. Zinn-Justin,Phys. Rept. , 1 (1995), arXiv:hep-th/9306153.[64] V. A. Kazakov, Mod. Phys. Lett. A4 , 2125 (1989).[65] B. Eynard, T. Kimura, and S. Ribault, (2015),arXiv:1510.04430 [math-ph].[66] D. Anninos and B. M¨uhlmann, J. Stat. Mech. ,083109 (2020), arXiv:2004.01171 [hep-th].[67] H. Neuberger, Nucl. Phys. B , 689 (1991).[68] E. Brezin, C. Itzykson, G. Parisi, and J. B. Zuber,Commun. Math. Phys. , 35 (1978).[69] D. Bessis, C. Itzykson, and J. B. Zuber, Adv. Appl.Math. , 109 (1980).[70] V. Zakharov and A. Shabat, Funkts. Anal. Prilozhen. , 54 (1974).[71] T. Banks, M. R. Douglas, N. Seiberg, and S. H.Shenker, Phys. Lett. B , 279 (1990).[72] C. V. Johnson and F. Rosso, (2020), arXiv:2011.06026[hep-th].[73] J. Ambjorn, J. Jurkiewicz, and Y. Makeenko, Phys.Lett. B , 517 (1990). [74] G. W. Moore, N. Seiberg, and M. Staudacher, Nucl.Phys. B , 665 (1991).[75] I. Gelfand and L. Dikii, Russ. Math. Surveys , 77(1975).[76] T. Banks, M. R. Douglas, N. Seiberg, and S. H.Shenker, Phys. Lett. B238 , 279 (1990).[77] J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski,P. Saad, S. H. Shenker, D. Stanford, A. Streicher, andM. Tezuka, JHEP , 118 (2017), [Erratum: JHEP 09,002 (2018)], arXiv:1611.04650 [hep-th].[78] T. Guhr, A. Muller-Groeling, and H. A. Weidenmuller,Phys. Rept. , 189 (1998), arXiv:cond-mat/9707301.[79] J. Liu, Phys. Rev. D , 086026 (2018),arXiv:1806.05316 [hep-th].[80] R. M. Miura, J. Math. Phys. , 1202 (1968).[81] R. M. Miura, C. S. Gardner, and M. D. Kruskal, J.Math. Phys. , 1204 (1968).[82] T. R. Morris, FERMILAB-PUB-90-136-T.[83] S. Mizoguchi, Nucl. Phys. B , 462 (2005), arXiv:hep-th/0411049.[84] K. Okuyama and K. Sakai, JHEP , 160 (2020),arXiv:2007.09606 [hep-th].[85] V. G. Drinfeld and V. V. Sokolov, J. Sov. Math. ,1975 (1984).[86] E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, Publ.Res. Inst. Math. Sci. Kyoto , 1077 (1982).[87] M. Jimbo and T. Miwa, Publ. Res. Inst. Math. Sci. Ky-oto , 943 (1983).[88] G. Segal and G. Wilson, Inst. Hautes Etudes Sci. Publ.Math. , 5 (1985).[89] V. G. Kac and M. Wakimoto, Proc. Symp. Pure Math. , 191 (1989).[90] T. J. Hollowood, L. Miramontes, A. Pasquinucci,and C. Nappi, Nucl. Phys. B373 , 247 (1992), hep-th/9109046.[91] G. R. Harris and E. J. Martinec, Phys. Lett. B , 384(1990).[92] E. Brezin and H. Neuberger, Phys. Rev. Lett. , 2098(1990).[93] E. Brezin and H. Neuberger, Nucl. Phys. B , 513(1991).[94] B. A. Kupershmidt, Commun. Math. Phys. , 51(1985).[95] R. Iyer, C. V. Johnson, and J. S. Pennington, J. Phys.A , 015403 (2011), arXiv:1002.1120 [hep-th].[96] R. Iyer, C. V. Johnson, and J. S. Pennington, J. Phys. A44 , 375401 (2011), arXiv:1011.6354 [hep-th].[97] H. Maxfield and G. J. Turiaci, (2020), arXiv:2006.11317[hep-th].[98] E. Witten, (2020), arXiv:2006.13414 [hep-th].[99] J. Maldacena, G. J. Turiaci, and Z. Yang, (2019),arXiv:1904.01911 [hep-th].[100] J. Cotler, K. Jensen, and A. Maloney, JHEP , 048(2020), arXiv:1905.03780 [hep-th].[101] K. Okuyama, JHEP , 037 (2018), arXiv:1808.10161[hep-th].[102] K. Okuyama and K. Sakai, JHEP08