Degenerate operators in JT and Liouville (super)gravity

We derive explicit expressions for a specific subclass of Jackiw-Teitelboim (JT) gravity bilocal correlators, corresponding to degenerate Virasoro representations. On the disk, these degenerate correlators are structurally simple, and they allow us to shed light on the 1/C Schwarzian bilocal perturbation series. In particular, we prove that the series is asymptotic for generic weight $h\notin - \mathbb{N}/2$. Inspired by its minimal string ancestor, we propose an expression for higher genus corrections to the degenerate correlators. We discuss the extension to the $\mathcal{N}=1$ super JT model. On the disk, we similarly derive properties of the 1/C super-Schwarzian perturbation series, which we independently develop as well. As a byproduct, it is shown that JT supergravity saturates the chaos bound $\lambda_L = 2\pi/\beta$ at first order in 1/C. We develop the fixed-length amplitudes of Liouville supergravity at the level of the disk partition function, the bulk one-point function and the boundary two-point functions. In particular we compute the minimal superstring fixed length boundary two-point functions, which limit to the super JT degenerate correlators. We give some comments on higher topology at the end.


Introduction
Jackiw-Teitelboim (JT) gravity is a remarkable solvable theory of 2d quantum theory [1,2,3,4,5,6]. The recent understanding of the significance of higher genus [7] and the relation to the black hole information paradox [8,9,10] have shown that one needs to understand and solve the gravitational theory in quite some detail to fully grasp the fundamental questions in quantum gravity. In this sense, JT gravity is relatively unique and it would be very beneficial if we could extend our knowledge to related models and deformations to learn of the generality of the proposed resolutions. In this sense, we refer to the exciting recent papers [11,12,13].
In this work, we study a useful subset of boundary correlation functions in JT gravity that is technically simpler to handle and that can be used to understand more deeply some of the structural aspects. This class of correlators plays the same role as degenerate Virasoro representations in 2d Virasoro CFT. We search for and find the same structure in the supersymmetric N = 1 version of JT gravity.
Let us first review the general structure of boundary correlation functions within JT gravity. As well-known, 2d gravity has no bulk degrees of freedom, and by suitable choice of boundary term, gets all of its dynamics from a fluctuating wiggly boundary curve, representing a reparametrization of the boundary circle F (τ ) [3,4,5,6]. The Lagrangian reduces to the Schwarzian derivative, and one can study correlators (at lowest topology) fully from just this Schwarzian system. Schwarzian quantum mechanics can be described as the 0+1 dimensional theory, described by a higherderivative action of the form: where F (τ ) describes a time reparametrization subject to specific boundary/periodicity conditions to describe the physics of interest. The coupling constant C has units of length and is inversely proportional to the 2d Newton constant C ∼ 1/G N . 1 Most studied is the thermal theory where one writes F (τ ) = tan π β f (τ ) where f (τ + β) = f (τ ) + β describes a reparametrization of S 1 . This theory has been studied and solved by several different techniques, both for the partition sum as for a certain class of correlation functions, composed of bilocal operator insertions of the type: (

1.2)
This operator can be viewed as a reparametrized matter CFT two-point function, labeled by the real number h, the weight of the matter CFT operator. One can study correlation functions of these operators by perturbing f (τ ) = τ + (τ ) for a periodic function (τ ) and then study the 1/C expansion in the Schwarzian coupling constant. This corresponds physically to an expansion in boundary graviton fluctuations . One-loop results and the first subleading corrections to the four-point function and its chaotic Lyapunov behavior were studied in [5]. Schwarzian perturbation theory has applications also for higher-point functions [14] and for matter correlators in 2d de Sitter space [15]. Higher loop corrections were recently analyzed in [16]. By relating this system as a particular limit of known dynamical systems, one also has access to the exact answers for the correlation functions. In particular, we have the well-known results for the one-loop exact partition function [5,17,18]: the one-loop exact Schwarzian derivative (or stress tensor) expectation value [18,19]: 2 , (1.4) and the bilocal disk correlators [20,21,19,22,23,24,25,26,27]: where dµ(k) = dkk sinh(2πk) and the ±-notation denotes the product of all cases. The semiclassical (large C) gravitational content of correlators like this was studied in detail in [28], see also [29]. The bilocal correlators get contributions from all orders in G N ∼ 1/C, and have nonperturbative content as well of the order ∼ e #/G N . The last statement will be proven in this work.
In [8], see also [30,31], the contributions of including higher genus handles to the disk geometry was argued to lead to the replacement: , (1.6) in terms of the energy variable E i = k 2 i and where the only new thing is the pair density correlator ρ(E 1 , E 2 ) coming from the matrix ensemble underlying JT gravity [7]. This pair density correlator has a genus expansion: ρ(E 1 , E 2 ) = g ρ g (E 1 , E 2 ) + (non-pert), (1.7) where ρ g (E 1 , E 2 ) ∼ e χS 0 is weighted by the Euler character χ and the (double-scaled) matrix parameter L ≡ e S 0 . Since S 0 can be understood in gravity language as the extremal black hole entropy with S 0 ∼ 1/G N , these perturbative higher genus effects are actually non-perturbative in G N , of the same order as the non-perturbative corrections to the disk correlator. Next to all of this, there are further non-perturbative corrections in e −S 0 that are very important to understand the very late-time physics, but will not play a major role in this work.
In order to gain a better understanding of all of these features, we focus in this work on the special bilocal operators where h ∈ −N/2. They originate from non-unitary matter insertions, but structurally they are special, corresponding to the degenerate Virasoro representations, or to the finite dimensional (non-unitary) representations of sl (2). In this case, both numerator and denominator of the ratio of gamma-functions in (1.5) or (1.6) diverge. The numerator does so only along the codimension 1 subspace of the integrals where k 1 = k 2 ± im for some (half)integer m. This simplified structure will allow us to investigate structural aspects that are more hard to access for generic h, both at the disk level and for the role and nature of higher genus corrections.
In this work, we focus on the following aspects.
Exploing the knowledge of the h ∈ −N/2 bilocal correlators, we will demonstrate the general series expansion in the time separation τ between both endpoints of the bilocal operator of the finite-temperature bilocal correlator for generic h has the structure: 6C τ + (a 0 T + b 0 ) τ 2 + (# T + b 1 ) τ 3 + a 1 T 2 + # T + b 2 τ 4 + # T 2 + # T + b 3 τ 5 + a 2 T 3 + # T 2 + # T + b 4 τ 6 + . . . , (1.8) with T n the thermal piece of the renormalized (or point-split) multi-Schwarzian derivative correlation function. We give expressions below. The structure of this expansion is readily generalized to higher orders in the τ -expansion. The coefficients a n are determined by the small τ expansion of the semi-classical (C → +∞) answer, and the b n can be determined through the zero-temperature result. The other coefficients can in principle also be deduced by exploiting the fact that the coefficients are polynomials in h, and hence knowing a small set of datapoints is sufficient to fix the polynomial. The usual correlators (1.5) cannot be used as datapoints since we are not able to analytically write down this expansion, but the degenerate bilocals (h ∈ −N/2) can. We will prove that the 1/C Schwarzian perturbative series on the disk is asymptotic for any real h / ∈ − N 2 , with non-perturbative effects in G N of order e #/G N that go beyond the boundary graviton expansion. The actual proof is contained in appendix B. The situation is summarized in Figure 1. JT gravity can be viewed as a double-scaling limit of the minimal string, consisting of the 2d Liouville CFT, a matter minimal model, and the bc ghosts to cancel the conformal anomaly. This was first noticed in [7,32], and preliminary remarks concerning boundary correlators were made in the conclusion of [33]. We presented a detailed treatment of this in [34]. Related recent results can be found in e.g. [35,36,37,38]. Since the minimal string has a matrix model description, the same has to be true for its JT limit, and indeed this was the main result of the impressive work [7]. 3 Within this framework, the natural bulk and boundary operators are minimal model primaries dressed by the gravitational Liouville piece. Within the JT limit, is was shown in [34] that the minimal string (genus zero) boundary two-point correlator precisely limits to the degenerate JT correlators we determine in this work. Because of this, they correspond to an integrable subsector of the JT gravity operator insertions.
In JT gravity at higher genus, we will hence draw inspiration from the overarching minimal string framework to define the bilocal correlator. We will show that it only has the same kind of higher topology than the partition function itself, signaling a departure from the generic h bilocal correlator studied in [8]. Diagrammatically, we find: + +... + (1.9) for the genus expansion of the degenerate bilocal correlator.
It is interesting to try to see to which extent this structure remains intact when going to related solvable models. To that effect, we generalize most of these results to the N = 1 super-JT theory in section 5. This model arises in the universal low-energy description of supersymmetric versions of the SYK model [41]. As such it is important for a restricted class of condensed matter systems that might be realizable in the lab, and from this perspective should be considerd of similar importance as the bosonic JT model. We take the opportunity to develop the perturbative treatment of the boundary super-Schwarzian model as well in section 6.2. This allows a discussion on the self-energy of a matter field, getting contributions both from the graviton and the gravitino. A byproduct of this development is a quick proof that the leading Lyapunov behavior of the out-of-time-ordered 4-point function saturates the chaos bound, just as in the bosonic JT model.
In order to address higher genus contributions to the degenerate bilocals in super JT gravity, we will first develop our understanding in its overarching Liouville supergravity model. Roughly paralleling the developments in [34], we construct the fixed length partition function, bulk onepoint function and boundary two-point functions, focussing on the presence of a (super) JT limit. Armed with these results, we develop the matrix model perspective on the higher genus degenerate boundary two-point function of the minimal superstring, and find very analogous results to that of the bosonic case.
We next present a short summary of our results on the Liouville supergravity amplitudes. The Cardy boundary states for the non-degenerate super-Virasoro representations with Liouville momentum s can be labeled as: |NS, s, η , |R, s, η , (1.10) in terms of a representation label s, related to the boundary FZZT cosmological constant µ B (s) by (7.43) [42], a sign η = ±1 representing a local fermion boundary condition, and a global fermionic boundary condition Neveu-Schwarz (NS) or Ramond (R). Associated to each of these FZZT boundary states, will be a fixed-length amplitude that we will determine in the following.
The salient features of our construction can be summarized as follows: • For the R-sector FZZT-branes |R, s, η , the integral transform to transfer from the µ B -basis to the length basis is the following: The contour C is a section of a hyperboloid. This rule will be motivated from the Lagrangian perspective in section 7.2 and applied explicitly to the partition function and the bulk onepoint function in sections 8.1 and 8.2 respectively. The contour is deformed to wrap the negative µ 2 B axis, where the role of taking the discontinuity across the branch cut at negative µ 2 B is played by adding instead of subtracting the two contributions, leading to the effective transformation: For the NS-brane |NS, s, η , one uses the integral transform: . . , C = µ 2 B − ηκ 2 : −i∞ → +i∞, (1.13) leading to +∞ 0 dse − 4π (cosh(2πbs)−η) sinh 2πbs . . . (1.14) where the discontinuity is just as in the bosonic case happening by subtraction. The NSbranes play a somewhat minor role in our story as they behave largely as in the bosonic Liouville story, and in particular they have the same semi-classical (Schwarzian) limit.
• Neveu-Schwarz boundary operator insertions that behave naturally in the fixed-length basis combine Liouville boundary exponentials e β 2 φ and their superpartners as where O b acts as the identity operator in the fixed-length basis. These results are studied in sections 8.4 and 8.3, with their two-point functions in the fixed-length basis obtained in (8.51) and (8.52).
• Ramond boundary operators of the form (σ =+1 +σ =−1 )e β 2 φ change the fermionic boundary condition η of the brane segment in the fixed-length basis. We compare these operator insertions to those in Liouville theory between a pair of ZZ-branes where branch cuts from the spin fields signal a change in boundary condition η.
This work is structured as follows. In section 2 we provide the results on the degenerate JT bilocal correlators. Section 3 provides an application in terms of the Schwarzian 1/C perturbation series for generic weight h. We investigate the addition of higher topology to these degenerate correlators from the matrix model and Liouville gravity perspective in section 4. Sections 5 and 6 provide details on the N = 1 supersymmetric extensions. In order to understand better the supersymmetric cases, we make a detour into Liouville supergravity and the minimal superstring in the next few sections. We start in section 7 with a summary of Liouville supergravities to set the stage, and to initiate our study of fixed length amplitudes. Section 8 then apply this to the partition function, bulk one-point function, and boundary two-point functions. In section 9, we look into matrix model perspective on some of the previous results. The minimal superstring correlators are determined there and generalized to arbitrary higher genus. Section 10 investigates how Ramond boundary operators work in the fixed length basis. We mention some aspects of the story that are left for the future in the concluding section 11. Some complementary material is presented in the appendices. In particular, appendix A contains the degenerate 6j symbol when a degenerate and a non-degenerate line cross in the bulk, given essentially by a Wilson polynomial in the external labels. Appendix B provides a thorough investigation on the perturbative content of generic bilocal correlators, in particular it contains a proof that the 1/C series is asymptotic generically.

Degenerate bilocal correlators in JT gravity
In this section, we determine closed formulas for the degenerate values of h ∈ −N/2 of the bilocal operators.

Disk level: Schwarzian bilocals
As mentioned in the introduction, Schwarzian correlation functions can be computed in several ways. One approach is to use a minisuperspace limit of Liouville CFT between identity branes with Liouville vertex operators [19]. Another is by using SL(2, R) group theoretical methods to describe JT gravity in its first-order BF formulation, described independently in [23,24] and [27]. 4 At a technical level, the computation that is required is the same: the Liouville minisuperspace wavefunctions can be viewed as solutions to the sl(2, R) Casimir eigenvalue equation upon imposing the gravitational constraints (the Hamiltonian reduction). This makes it a so-called parabolic matrix element or Whittaker function. From either approach, one finds the wavefunction: 5 (2.1) The operator insertion is easily deduced from the Liouville perspective as the Liouville primary vertex operator e 2αφ . From the group-theoretical perspective, one needs the matrix elements of the discrete highest weight irreps of SL(2, R), which turn out to be also given by (2.1).
In order to compute a bilocal correlation function (1.5), we include two such wavefunctions, and one operator insertion, and integrate over the auxiliary variable φ. To sum up, the vertex functions that are present in the bilocal correlators (1.5), are found by doing the auxiliary φ-integral as follows: For h ∈ −N/2, the operator insertion is special in the following sense.
To find a well-defined classical limit b → 0, we set m = 1 and n ≥ 1, which with n = 2j + 1 becomes V 2j+1,1 = e −2jφ , where j = 1 2 , 1, 3 2 . . .. Comparing to the non-degenerate primaries, this corresponds to effectively setting the weight h = −j. From group theory, setting the irrep label h = −j ∈ − N 2 is selecting the finite-dimensional (but non-unitary) irreps of SL(2, R) of dimension 2j + 1. Some more details on this were presented in appendix D of [33]. From both perspectives it is clear that this choice will have a special structural significance.
Instead of (2.2), the vertex functions reduce to a linear combination of Dirac delta-functions: with a set of momentum-dependent and combinatorial prefactors c j m (k 1 , k 2 ). The explicit form can be determined using the 1d fusion property 4) and the orthonormality property: The evaluation of (2.2) for h ∈ −N/2 requires using repeated use of (2.4), leading to the schematic After what is essentially a tedious combinatorial exercise, one finds the coefficients: where the last factor contains the Pochhammer symbol (x) n ≡ Γ(x + n)/Γ(x). This last factor represents the inverse of a polynomial in k of order 2h. Despite appearances, the vertex function (2.3) is symmetric under k 1 ↔ k 2 . Inserting these into the finite-temperature two-point function, we get: (2.7) instead of the generic (1.5). We first write some explicit examples, and afterwards we will discuss some properties of this expression.
The simplest example of h = −1/2 = −j was explicitly written in [33]. In this case the k-integral can be done in terms of elementary functions: A second example is that of h = −1: .
Let us now make some remarks on this result.
• The appearance of a binomial expansion in (2.7) is no surprise since the bilocal operator itself can be expanded as such: (2.10) A further indication of the simplified structure is found by transforming to the free field variable F = e φ [20,21], 7 transforming the Schwarzian action (1.1) into with constraint β 0 dτ e φ = +∞ and operator insertion: which since 2j is integer, is only a product of (complex) plane waves in a free theory. One can readily evaluate the φ path integral explicitly in this way and make contact with our main expression (2.7). Since we have just plane waves in a free theory (up to the constraint), these operators can be viewed as an integrable subclass of the bilocal operators in JT gravity.
• The zero-temperature result, and its small-separation expansion can be obtained by expanding (2.7) as k → 0, yielding e.g.
• In the semi-classical regime where C τ, β, the integral in (2.7) is dominated by large k at its saddle k ≈ 2Cπ β . The expression then evaluates to: , (2.14) by evaluating the binomial expansion, and taking the largest term in the k-polynomial.
• To perform the combinatorial manipulation at higher values of j, an alternative diagrammatical option is to deconstruct it into the elementary j = +1/2 bilocals: The identity that ensures equivalence of this holds for h > 0, and is an analogue of the Barnes lemmas. For h ∈ −N/2, this requires an explicit check, and one can readily see that it is true in this case as well. Such diagrams are closely related to the loop gas diagrams of Kostov [43].
Given these degenerate vertex functions (2.3), the diagrammatic language of Schwarzian correlators developed in [19] immediately extends to diagrams including degenerate bilocal lines. A particularly interesting diagram is that of crossing bilocal lines, corresponding to an out-of-time ordered correlator. Each such crossing carries a SL(2, R) 6j-symbol. If we take one operator pair to be a degenerate pair, we require the degenerate 6j-symbols. Since this development is a bit orthogonal to our main story, we develop this in Appendix A. On a technical level, the main conclusion is that the 6j symbol is given in terms of a Wilson polynomial [44] (the non-degenerate 6j-symbol was a Wilson function). On a physical level, the main conclusion is that the degenerate 6j symbol does not encode gravitational shockwaves and chaos, which matches indeed with these operators representing an integrable subsector of the JT model.
It is apparent from the above formulas (especially (2.8) and (2.13)) that the degenerate twopoint function has a convergent 1/C perturbative expansion. In fact, these degenerate value of h ∈ −N/2 are the only values of h for which a convergent perturbative series is achieved. All other values of h / ∈ −N/2 correspond to asymptotic series in 1/C. Since this is somewhat outside our main investigation, we present the proof in Appendix B.
3 Application: Schwarzian perturbation theory and small τ expansion As one of our main applications, we will show that knowledge of the degenerate bilocal correlators on the disk, combined with the structure of the Schwarzian 1/C perturbation expansion, allows us to learn a few lessons on the small τ series expansion for any value of h.

Review: Schwarzian perturbation theory
In this subsection, we provide a brief recap of the perturbative treatment of Schwarzian QM. We need only one elementary result from the Schwarzian perturbative expansion, which is that the coefficient of each term in the series expansion is a polynomial in the weight h of the bilocal operator. Setting f (τ ) = τ + (τ ), and expanding in , one writes for the Lagrangian: 2) and vertices from the cubic and higher powers in the -expansion. Notice that the vertices are β-independent. The bilocal operator is also expanded as From (3.1), one reads that each propagator carries a factor of 1/C and each vertex a factor of C.
where u = 2πτ /β. The zero-temperature limit of this formula is readily taken: The leading correction is ∼ h(h − 1) ∼ m 2 , the mass 2 of the bulk field dual to the inserted boundary operators. We can interpret it as the one-loop self-energy of the bulk particle due to graviton interactions. We will give an analogous interpretation for the one-loop self-energy contribution from the gravitino in the supersymmetric case in section 5.
Contemplating the expansion (3.3) one also finds that a diagram with n connections to the external endpoints is contributing a polynomial in h of order n without constant term. E.g. the diagrams contributing to the zero'th, first and second order term for the bilocal correlator G(τ 1 , τ 2 ) are schematically drawn as: The first line contains the Schwarzian diagrams contributing at leading order 1 (free result) and the first subleading correction ∼ 1/C corresponding to the one-loop gravitational self-energy. The second line represents the six 1/C 2 diagrams. Each contribution is a polynomial in h and we have indicated to which monomials in h they contribute. Dashed lines represent virtual fermions ψ(τ ) coming from the non-trivial Schwarzian path integral measure. The solid lines are CFT matter lines that are external to the actual Schwarzian theory; we choose to draw them nonetheless to emphasize the physical process. Achieving a more systematic understanding of higher loop corrections is complicated by the following facts. The Schwarzian model has a non-trivial path integral measure [18]. 9 One can flatten the measure by exponentiating it and integrating in an additional fermionic variable ψ(τ ) that contributes to loop diagrams as illustrated above. Secondly, the number of vertices increases with each order. Both of these cause the perturbative series to be extremely unwieldy beyond leading order. 10

Small τ -expansion
In order to shed light on the perturbative expansion at higher orders, we will exploit the degenerate result (2.7). The strategy is simple: we can Taylor expand the integrand of (2.7) as a power series in τ /C and then perform the momentum k-integral exactly. This directly gives the small τ -series expansion. In terms of the (renormalized) multi-stress tensor Schwarzian expectation values: 11 one obtains for the specific examples of h = −1/2 and h = −1: The multi-Schwarzian derivative correlators T n always appear in this specific structural fasion. Since this applies to any h ∈ −N/2, and since at every fixed order in the τ -expansion one has a polynomial in h as coefficient, this is sufficient to prove that one has the same expansion structure for any h ∈ R, where only the numerical coefficients change. This leads to the structural expansion of (1.8).
As a classical function, the bilocal (1.2) can be series-expanded in τ 1 − τ 2 before quantizing. By SL(2, R) invariance the resulting expansion needs to be structured into local SL(2, R) invariants. This is indeed true in (1.8), since it is only constructed from expectation values of powers of the Schwarzian derivative. However, with our explicit expressions, we can now assess to what extent it is true in more detail. The answer is quite subtle and depends crucially on renormalization issues of the multi-stress tensor composite operators. We present the analysis in appendix C.
Let us end with a short remark on the time-ordered four-point function. A priori one would expect a four-point function to depend on three time parameters, as the Schwarzian theory is time-invariant, removing the overall shift. However, we know from various perspectives that it only depends on the time differences within each bilocal operator [19]. The small τ -expansion provides yet another way of appreciating this, and it is given strength by our understanding of the expansion of the classical bilocal in appendix C. Performing a double series expansion for the classical bilocals, they are expandable into Schwarzian derivatives, their powers, and their derivatives. 12 We write schematically: where τ ij ≡ τ i − τ j . This depends explicitly on the time differences τ 12 and τ 34 within each bilocal. Quantum-mechanically, the correlators of Schwarzian derivatives would now provide the link between both bilocals, such that also τ 24 appears. However, due to the fact that correlation functions of Schwarzian derivatives are always time-independent, this does not happen, and the four-point correlator only depends on two instead of three time parameters. 13 For out-of-time ordered correlators this simplification does not happen, and we cannot just perform the small τ -expansion on any bilocal bridging other operators.

A proposal for higher topology
Our next goal is to analyze this degenerate two-point function on surfaces of higher topology.
One can consider bilocal correlators that include higher topological corrections and the doubly non-perturbative random matrix completion. Let us first review the story for a generic h > 0 correlator. The rule is to slice up the initial disk region along each of the bilocal lines, and then to add higher topology to the remainder [8]. Working in units where C = 1/2, the higher genus expansion of the bilocal correlator was argued to be of the form (1.6), which we retake here: where the two-level spectral density is replaced by the random matrix answer: representing respectively the disk, the annulus connecting both sides of the bilocal line, and a contact term that has no a priori geometric origin. This geometric connection was made in [7]. The disconnected piece ρ(E 1 )ρ(E 2 ) at genus zero goes back to the sinh measure and we get back to (1.5). In the regime where e S 0 1, higher genus corrections are suppressed, except when E 1 ≈ E 2 which can compensate for the e −S 0 suppression.
For degenerate bilocal lines, we propose to not slice up the surface along the line, just as the vacuum h = 0 line. This means e.g. that a single degenerate bilocal line on the disk has the same higher genus contributions as the disk itself. Let us first present a qualitative argument. Supposing one starts with (4.2) for a degenerate line, then the effects are not important if S 0 1.
If j is half-integer, then since m = 0, one never has E 1 ≈ E 2 , and higher topology is always suppressed (provided S 0 1): only the disconnected disks contribute. If j is integer, then the m = 0 term in the sum is the dominant one: the first two terms of (4.2) cancel by level repulsion in random matrix theory, and the last (plateau) term diverges as ∼ δ(0). This illustrates that the pair correlator ρ(E 1 , E 2 ) does not seem well-behaved when considering degenerate operator insertions.
In order to properly understand how to treat higher genus corrections to these amplitudes, we will take inspiration from the minimal string theory of which JT gravity is a parametric double-scaling limit [34]. Minimal string theory consists of a (p, q) minimal model combined with the Liouville CFT and bc ghosts to form a non-critical string theory with c L + c M + c gh = 0. The Liouville central charge is parametrized as c L = 1 + 6Q 2 , where Q = b + b −1 and b 2 = p/q. This and related 2d models have a long history as toy models for quantum gravity, see e.g. [46,47,48,43,49,50]. The minimal string only has degenerate Virasoro primary matter operators that are dressed by the Liouville sector into physical vertex operators. E.g. the tachyon boundary vertex operators can be written as in terms of the degenerate matter operator O r,s , the Liouville vertex operator e β 2 φ and the bc ghost c. For the particular case of p = 2 and q = 2m − 1, a single-matrix description is possible, and we only have B 1,s for 1 ≤ s ≤ q − 1. The JT limit corresponds to taking m → ∞ (b → 0), for which one obtains an infinite discrete set of physical operator insertions. It was proven in that work that the minimal string two-point function is essentially only captured by the Liouville boundary two-point function: where we define the quantity 14 with the boundary FZZT cosmological constant µ B related to the label s as µ B ≡ x = κ cosh 2πbs where κ = √ µ/ √ sin πb 2 . In our case β = b − β M , in terms of the matter label β M : For the (2, 2m − 1) minimal string with m ∈ N, we hence have β M = −jb for j ∈ 0, 1 2 , . . . which can be identified with a discrete irrep sl(2) spin label j. We now focus on this particular minimal string model, with these operator insertions.
For such boundary operators, we have the Liouville parameter β = b+bj. The Liouville boundary 14 There is an implicit product over all four sign combinations of the S b in this and in subsequent similar equations.
two-point function (4.5) can then be rewritten in terms of elementary functions: .

(4.7)
It is elementary to check that the r.h.s. is symmetric under s 1 ↔ s 2 . We identify the numerator with genus zero resolvents as R g=0,1 (x) = cosh 2π b s where x ≡ µ B = κ cosh 2πbs. Following common conventions, we will sometimes denote µ B by x to streamline the notation.
Next, we transform this expression to the fixed length basis by applying the integral transform for both s 1 and s 2 . Inserting (4.7), we have two terms for which in both cases the first integral is contour deformed to pick up the poles in the denominator. The second integral is then deformed across the branch cut of the cosh 2π b s 2 resp. cosh 2π b s 1 resolvent. We obtain in the end: which is the same expression as that found in [34]. The prefactors are chosen to match the UV behavior → 2j 2 with unit coefficient. The advantage of organizing the calculation like this is that now the resolvent and spectral density of the underlying matrix integral make explicit appearances, where the discontinuity of the resolvent is the spectral density ρ(x(s)) and where x = κ cosh 2πbs. We will use this suggestive way of writing the amplitude as the definition of an insertion in the random matrix integral. In particular, this expression gets its higher genus contributions only from the single-boundary resolvent contributions R g,n=1 (x), computable e.g. from Eynard's topological recursion relations [51,52,53]. This is a strong restriction on the allowed topologies contributing to the amplitude. Next to this restriction, the resulting amplitude has a similar structural flavor than the proposed JT bilocal correlator at higher genus [8], where indeed also only the spectral densities are adjusted to accommodate for different topology. It would be interesting to verify this proposal by an explicit computation of the genus one result in the minimal string continuum language. In matrix language, where the resolvent is R(x) = Tr 1 x−M , we can then identify (4.7) as the genus zero contribution of the matrix computation: 15 11) where the left expression is valid for j ∈ N and the right expression for j ∈ N + 1/2. We dropped overall factors here, and used the notation x 2+inb ≡ κ cosh 2πb(s 2 + inb).
Notice the presence of the ( 1 ↔ 2 ) second term in (4.9). This is in unison with the fact that the amplitude (4.8) is manifestly invariant under swapping 1 and 2 . To understand its meaning, we can write this second term suggestively as: where x +n = cosh 2πb(s + inb). At genus zero we have ρ 0 (x(s + inb)) = sinh 2π b (s + inb) = (−) 2n ρ 0 (x(s)) and both terms in (4.9) are equal. At higher genus, this is no longer the case. Within the s-coordinates, this means one can interpret the full answer as with an effective density ρ eff (s, n) = 1 2 sinh 2πbs ρ(x(s)) + (−) 2n ρ(x(s + inb)) , ρ eff,g=0 (s, n) = sinh 2πbs sinh 2π b s.
(4.14) When viewed gravitationally, this can be interpreted as adding higher topology to each of the possible sectors of the diagram, leading to a symmetrized result in the end. As an example, the genus one correction is gravitationally computed by considering There are obviously other options that agree on the lowes genus zero result but differ beyond that. We believe this is the most natural one as it will have all the expected properties. The matrix operator used in [46] is slightly different, but would also only give single-boundary resolvent contributions Rg,n=1(x), which is our main point.
Since we sum over handles in each sector of the diagram (instead of multiplying), it is conceptually convenient to lump the contributions together when drawing the diagram. The gravitational interpretation is then geometrically: .. + (4.16) where the meaning of the handles is encoded in (4.14). For practical computations, it is more convenient to compute (4.9) directly and add the 1 ↔ 2 contribution by hand.

JT limit
In the JT limit, we set s i = bk i for finite k i . The denominator of (4.7) becomes cosh 2πbs → 1 + 2π 2 b 4 k 2 . (4.17) For the resolvents in the numerator of (4.7), we focus on the spectral region close to the edge at Plugging this in the resolvent, and focusing on the (2, 2m − 1) minimal string for which b 2 = 2/(2m − 1), we get: where z 2 JT = −k 2 . In terms of the uniformizing coordinate k, we can hence write the JT resolvent at lowest order in the genus expansion as: The full one-boundary JT resolvent is denoted as R(k), and has a genus expansion: Hence the JT limit of (4.7) is written as: . (4.21) The label k and the energy variable E are related as E = k 2 , where k is the uniformizing coordinate.
Transforming (4.21) to the length basis is then done by: Deforming the first contour picks up all of the poles. Deforming the second contour picks up the discontinuity across the branch cut, where the discontinuity of R between −k 2 + i and −k 2 − i is the spectral density, at genus zero given by the expression: Finally setting k 2 → −k 2 to have the integral over R + , we obtain: with ρ 0 (k) = k sinh 2πk. One can check that this formula matches with (2.7) upon identifying 1 = β − τ and 2 = τ . We have done this in [34].
Written as (4.22), the inclusion of higher genus and non-perturbative random matrix effects is straightforward since we again only adjust the resolvents R(k) in (4.22) into the exact answer, in the end only replacing (4.24) by: The only effect of higher genus contributions in the fixed length amplitude is hence to replace the spectral measure from the seed value ρ 0 (k) to the random matrix result ρ(k) ; the vertex functions and structure of the amplitude remains the same. The only topological corrections to the degenerate bilocal correlators are hence in one-to-one to those of the partition function; in particular there is no pair density correlator ρ(k 1 , k 2 ) contribution, and hence no handles connecting opposites sides of the bilocal line in this case.
To be slightly more explicit, it is convenient to relate the single-density expectation value ρ to the resolvent W in the notation of [7]. We can write: the 's playing the important role of regulators in the Laplace integrals. For instance, at genus 1 we have: where the principal value Pv is taking the real part of this expression, and is effectively removing divergences as k → 0: ). E.g. for j = 1/2, we have the genus one correction (C = 1/2): The sum of the genus zero and one result for this particular j = 1/2 is plotted in figure 2. Since the genus zero correlator vanishes as τ = 0 or τ = β, this is where the higher genus corrections are quite visible and non-trivial. In terms of bilocal lines connecting the two boundary operators, the reason is the possibility of the line encircling higher topology and becoming noncontractible, effectively becoming sensitive to non-UV physics due to the minimal distance to wrap around the higher topology defect.
Beyond some fixed genus amplitudes, the full non-perturbative spectral density for the JT matrix model was recently computed numerically in [38]. The above then shows that this information is sufficient to determine this class of degenerate boundary correlators.
At late real times t = −iτ , the degenerate correlators (2.7) increase exponentially without bound. In particular, the role of strongly suppressed (e S 0 1) higher topological corrections is less interesting here as it can never stabilize the late time behavior.

Degenerate bilocal correlators in JT supergravity
We generalize the discussion to N = 1 JT supergravity and its boundary super-Schwarzian quantum mechanics. We first review the action and the bilocal operator. We then study the correlator for degenerate bilocal operators and exploit these to reach a similar conclusion on the structure of the perturbative small τ -series (1.8) on the disk.

Set-up: N = 1 super-JT disk correlators
JT supergravity on the disk can be analogously written in terms of a boundary N = 1 super-Schwarzian theory [54,55], with action [41] The super-Schwarzian is defined in N = 1 (τ, θ) superspace by: . This action describes the dynamics of the superframe (f, η) of a boundary super-clock. Written in component fields (the reparametrization f (τ ) and its superpartner η(τ )), one writes The action (5.1) is then written in bosonic space as: Considering the thermal disk theory, one is interested in the set of superreparametrizations of the supercircle. This is defined by the Euclidean path integral: describing a circle time reparametrization f (t), satisfying f (t + β) = f (t) + β and its fermionic superpartner η(t) satisfying antiperiodic boundary conditions η(t + β) = −η(t), as required for fermions around the thermal circle.
Under super-reparametrizations, the inverse superdistance, which is at the same time the elementary h = 1/2 superspace two-point function, is transformed as [41] 1 For higher h, the classical bilocal operator is of the form: expandable in its four components by expanding in the θ i 's. The two fermionic components give zero in a single bilocal correlator by fermion conservation. The bottom and top components are non-zero. The (one-loop) exact partition function Z and bilocal correlation functions for this model are known.
Again the n multi-Schwarzian derivative correlator is n-loop exact. The N = 1 super-Schwarzian correlation functions of bilocal operators were determined in [19] using super-Liouville techniques.
The answer for the (bottom component) two-point function is and its superpartner (i.e. top component) is Some more details and properties are described in appendix E.1.

Disk level: super-Schwarzian bilocals
The computation of the degenerate bilocals in this case proceeds along similar lines as before.
We focus here on the bottom component of the bilocal operator. The top component is treated similarly and discussed in Appendix E.2. Compared to the bosonic theory in section 2, one only replaces the continuous irrep wavefunctions of SL(2, R) by those of OSp(1|2) [23]. Equivalently, we employ the N = 1 super-Liouville minisuperspace wavefunctions [19]. We define the supersymmetric wavefunctions as: The vertex functions for JT supergravity in (5.12) are then found by performing the auxiliary integral: Mirroring the bosonic case in section 2, to determine this expression for h ∈ −N/2, we combine the following normalization constraint [56]: with the 1d fusion property of the modified Bessel function (2.4), leading to e.g. the relation: playing the same role as (2.5) above. In the last equality, the Dirac delta functions are to be interpreted in k-integrals where one move the contour from the real axis to ±i/2.
Applying this relation consecutively, one arrives at the supersymmetric degenerate vertex functions: where we defined both bottom (B) and top (T) contributions: 16 Inserting these for the vertex functions in (5.12), we get the degenerate two-point functions.
Let us give some examples. For h = −1/2, one has .

(5.22)
At zero temperature β → ∞, one finds: Let us make some comments on these formulas.
• The super-Schwarzian bilocal operator (5.9) for j = −h ∈ N/2 has for its bottom component: and is expandable as two terms that each form a binomial expansion of order 2j and 2j − 1 respectively, matching with the expansions in (5.19) and (5.20) respectively.
• If one takes the semi-classical C → +∞ limit, we again take k large in the inverse polynomials in (5.19) and (5.20). Since c j mT has one extra power of 1/k, this piece is subdominant, and the bottom part c j mB gives the expected answer β π sin π β τ 2j .
• One can also determine the degenerate superpartner correlator, this is the top component of the bilocal operator (5.9). We give the results in Appendix E.2.
• As for the bosonic case, the 1/C expansions of these equations yield convergent series. The argument in appendix B should be easily generalizable to provide a proof that for all other values of h the 1/C series is asymptotic only.
6 Application: super-Schwarzian perturbation theory and small τ expansion 6.1 Small τ -expansion As explicit applications, the supersymmetric degenerate bilocal correlators (5.21) and (5.22) have the following small τ expansions: are the (renormalized) thermal super-Schwarzian multi-stress tensor correlators. This leads to the same structure of the small τ -expansion (1.8) as in the bosonic case. Since the expansion coefficients are polynomials in h by the perturbative expansion, and we can determine this structure for all h ∈ −N/2, this is sufficient to uniquely determine these polynomials. So this structure of the perturbative expansion holds for generic real values of h.
Notice that no simplification in this perturbative series occurs due to the presence of supersymmetry: N = 1 supersymmetry in 1d is not sufficient to argue for non-renormalization theorems. We will very explicitly see this at one-loop in the next subsection.

Super-Schwarzian perturbation theory
From the two explicit expressions (6.1) and (6.2), and the fact that the coefficient of the second term in that expansion is a quadratic homogeneous polynomial in h, we can write the answer for the one-loop self-energy Σ for arbitrary real h as: In the second equality, we have split the self-energy into a contribution from gravity, and a contribution from the gravitino (Figure 3). Figure 3: Two Feynman graphs contributing at order 1/C in the supersymmetric theory.
To substantiate this result, we now reproduce this same term from perturbation theory within the super-Schwarzian system directly.
The quadratic piece of the boundary gravitino action contained in (5.3) is For ease of notation, we set β = 2π from here and restore it in the end. The zero-modes of this action satisfy η + 1 4 η = 0: with C = 0 due to antiperiodicity η(τ + 2π) = −η(τ ). The orthonormal eigenmodes are given by ψ n (τ ) = 1 √ 2π e inτ , with eigenvalue λ n = 4iCn(n 2 − 1/2) and hence the zero-modes n = ±1/2. The propagator is then given by: Choosing a contour C that encircles all half-integers = ±1/2, one can write this as: Deforming the contour to encircle the poles 0, ±1/2 and the (vanishing) piece at infinity, one evaluates the residue immediately to find: 17 with special cases: The bottom component of the bilocal operator for generic h (5.9) is explicitly (6.10) The 1/C correction to a single bilocal operator is then readily found. At this order, the bosonic and fermionic contributions just add up; the bosonic contribution found by using the propagator (3.2) in (3.3), while the new fermionic contribution is found by using (6.8) in (6.10). We get: where u = 2πτ /β. The first line is the bosonic answer (3.4) from the N = 0 Schwarzian, and the second line contains the gravitino contribution. Taylor-expanding this in τ to find the lowest correction in the τ /C series, we find the answer: where the gravitino gives a positive contribution to the self-energy, indeed matching with the result we got from analysing the general structure above in (6.3).

Lyapunov behavior in N = 1 JT supergravity
As an aside, starting with (6.10), one can also analyze the lowest fermionic correction to the fourpoint function. One readily sees that this contains two η-propagators, and is hence suppressed as 1/C 2 , unlike the graviton which has a single propagator and only 1/C suppression ( Figure 4). Since this is true for any ordering of the four points, this is also immediately so for the boundary out-of-time ordered four-point function, and hence the leading Lyapunov behavior ∼ β C e 2π β t , and the Lyapunov exponent in particular is maximal λ = 2π/β in N = 1 JT supergravity. When evaluating the 1/C 2 correction in the N = 1 case, the bosonic contribution was evaluated in [16]. The fermionic contribution can be explicitly evaluated using the propagator (6.8) twice for each of the possible contractions of the right graph of figure 4. Note that the bosonic partner of the measure fermion ψ only appears at higher order. We will not evaluate it explicitly.

Liouville supergravities
Next we aim to understand higher genus corrections to these degenerate correlators. In order to do so, and paralleling the bosonic treatment, it is useful to first understand the ancestral minimal superstring for which JT supergravity is found in a parametric limit. The next few sections will have the goal to investigate N = 1 Liouville supergravities in their own right and to find out precisely how one obtains a (super) JT parametric limit. We come back to the degenerate bilocal correlators in section 9 and in particular in subsection 9.1.
In this section we will initiate our study of N = 1 Liouville supergravity. Subsection 7.2 provides a Lagrangian treatment to prove the transformation to fixed length amplitudes in both the Ramond and the Neveu-Schwarz sector. After that, in subsection 7.3, we collect the known super-Liouville amplitudes in the FZZT brane basis, to be used later on.

Liouville supergravity and minimal superstring
We consider the N = 1 supersymmetric Liouville model [58,59,60,61] with central charge on a manifold with a circular boundary. We couple this to a matter sector with c M < 3/2 parametrized by c M = 3 2 − 3q 2 . Demanding a cancellation of the superconformal anomaly requires: Neveu-Schwarz (NS) boundary vertex operators are of the form with ce −ϕ the superghost contribution and e β 2 φ the boundary super-Liouville vertex operator, dressing the matter operator O β M . We have the constraint ∆ β + ∆ β M = 1/2. With 18 These choices are related by applying a boundary reflection transformation β → Q − β, and represents a freedom in dressing the matter operator with given β M . We hence focus on the first case only: Ramond (R) boundary vertex operators are of the form with the constraint ∆ β + ∆ β M = 5/8 and solvable by the same relation (7.4).
If we consider for the matter theory a (p, q) superminimal model, then we have only a finite set of tachyon vertex operators, corresponding to taking the degenerate super-Virasoro primaries as matter operators, and dressing these with the appropriate super-Liouville operators.
Open string tachyon vertex operators in the minimal superstring are of the form: where we left the legpole factors implicit. The operator O r,s is a superminimal model primary operator. This is dressed by the super-Liouville primary vertex operator with parameter: For r + s even, this operator is in the NS sector of the theory, whereas for r + s odd it is in the R sector. 18 The parametrization and definition of βM is in parallel to the bosonic case. It follows from consider timelike super-Liouville to describe the cM < 3/2 matter CFT. This essentially boils down to taking an ordinary super- In the special case where p = 2, we have an even more restricted class: where all half-integer j give Ramond boundary operators, and all integer j give NS boundary operators.
We will not be bothered too much by the precise matter sector in the remainder of this and the next section, since we only focus on cases where the amplitude factorizes into a Liouville piece, a matter piece and a superghost piece. The only length dependence comes from the super-Liouville piece and this will be our focus. The main effect of the matter and ghosts sectors is a cancellation of the dependence on the worldsheet coordinates, much like happens in the bosonic minimal string. However, in section 9 we will investigate the matrix model interpretation of certain of the quantities we computed, and for this we will restrict to the (2, q) superminimal models, with a particular emphasis on q = 4k, with k ∈ N.

Lagrangian approach
In order to motivate the transformation formulas to the fixed length basis mentioned in the introduction, and to develop intuition, we first provide a discussion of the classical Lagrangian of N = 1 Liouville theory, its boundary terms and the transition to fixed length amplitudes.
We study N = 1 Liouville theory on a manifold with a circular boundary. We set the coordinates as z = x + iy, with the boundary at y = 0, and x ∼ x + 2π periodically identified. To preserve half of the supersymmetry, one has the superconformal boundary conditions on the UHP: T (z) =T (z) and T F (z) + ηT F (z) = 0 which allows a sign-factor η = ±1 and hence gives two types of branes. Next to this, the fermionic fields can be either periodic (R) or antiperiodic (NS) upon circling the boundary: T F (z + 2π) = ±T F (z) (see Figure 5). For most of the discussion y x Figure 5: Boundary in N = 1 Liouville theory. Preserving half of supersymmetry requires left-and right movers to be related as T F (z)+ηT F (z) = 0. One also has a global boundary condition T F (z+2π) = ∓T F (z) labeling the boundary state as NS or R respectively. in this work, we will work with fixed fermionic boundary condition. When making contact with higher topology and with matrix model techniques to compute minimal superstring amplitudes, one has to sum over different spin structures, but this will not be important for our purposes and is deferred to future work.
Defining a scalar superfield Φ in terms of the scalar fields φ and F and the two Majorana-Weyl fermions ψ andψ as Φ = φ + iθψ + iθψ + θθF, (7.11) the supersymmetric Liouville model, on a manifold with a boundary that preserves half of the supersymmetry (labeled by η), is given in terms of the superspace Lagrangian We follow the notations and conventions of [62] in this subsection. Enforcing the boundary variation to vanish, we can distinguish two boundary conditions.
• One can find a classical ZZ-brane solution with the Dirichlet boundary conditions • Imposing Neumann (free) boundary conditions is done by imposing These are FZZT branes, with boundary cosmological constant ηµ 0 fixed by supersymmetry.
These cases can be neatly summarized in superspace as We can still preserve supersymmetry by also adding the following boundary term in terms of the fermionic boundary superfield Γ [63,64,62,65]: The fields γ and f are boundary degrees of freedom that need to be quantized or integrated out to obtain the theory in its original variables, see also [63,64,65]. The addition of this term S FZZT is necessary to find regular classical solutions with generalized Neumann boundary conditions.
Including the term S FZZT , the Neumann boundary condition (7.18) is adjusted into the superspace expression: which resembles the bosonic Liouville FZZT boundary condition. In components: The boundary equations of motion derived from (7.19) Plugging these back into the boundary conditions (7.21),(7.22), we get: 26) The first equation shows that if µ B = 0 we return to Neumann boundary conditions for the fermions ψ − ηψ y=0 = 0, whereas when µ B → ∞, we get Dirichlet boundary conditions ψ + ηψ y=0 = 0.
Beyond classical field theory, we integrate out the boundary fields γ and f . The path integral over f yields: The final term is a contact contribution within correlation functions that can be neglected. It will be important however when transforming to the fixed length basis.
The boundary (Grassmann) fermion γ(x) can be canonically quantized by the Clifford algebra {γ, γ} = 1, leading to a description in terms of a 2-dimensional Pauli matrix γ = 1 √ 2 σ. This doubles the dimension of the boundary state Hilbert space. This is the usual description used in the super-Liouville literature, where γ 2 = 1. Within the path integral, the boundary fermion field can have periodic (Ramond) boundary conditions: γ(x + 2π) = γ(x) or antiperiodic (Neveu-Schwarz) boundary conditions: γ(x + 2π) = −γ(x). Since the Lagrangian needs to be single-valued, the last term of (7.19) shows that this boundary condition has to match with that of the field ψ and hence with the label NS or R of the boundary state under consideration.
The boundary fermion field γ needs to be path-integrated out in the Liouville amplitude, much like the boson f . This can either be done by its equation of motion solvable as where the zero-mode γ 0 has been extracted. Or by finding the Green's function of ∂ x G(x|x ) = δ(x − x ) as G(x|x ) = 1 2 sign(x − x ), upon first removing the zero-mode for invertibility again. Using this expression (7.28) together with the boundary conditions (7.21), we find This results in the total boundary action Within a canonical framework, the fermionic zero-mode satisfies γ 2 0 = 1. 19 This reduces the description to the boundary action of [66,67] used in the (super)conformal bootstrap method used to determine the correlation functions.
Taking a derivative of this action with respect to µ B , brings down a factor of which we call the global marking operator M 1 . We will verify this explicitly in section 8.3 further on. Notice that no contribution ∼ µ B e bφ is written as such a term comes from the auxiliary field f which has a contact-term propagator f (x)f (x ) ∼ δ(x − x ). The resulting marking operator would have a second piece which contributes only contact terms to quantum correlation functions and is usually neglected by suitable choice of external operator dimensions.
Next to this, we will need a second local marking operator that combines bosonic and fermionic contributions as This second (local) marking operator M 2 (x) will play the role of no insertion at all, i.e. the h = 0 limit of the boundary two-point function, much like the bosonic marking operators of [34].
Within the path integral, γ 0 is a Grassmann number. Performing the γ 0 -integral by bringing down a factor of γ 0 µ B × 1 2π (ψ + ηψ)e b 2 φ , and then putting it back in the exponent by inserting a dummy γ 0 -integral again, gives the suggestive way of writing the path integrand as Transforming to the length basis is done by the integral transform along the hyperboloid C where µ 2 B − ηµ 0 : −i∞ → +i∞. Inserting (7.34), we can evaluate the µ B -integral into the fixed-length amplitude: For NS-branes, the zero-mode γ 0 does not exist. Instead one finds the total boundary action leading to the fixed length prescription and correlator: For the NS-brane, one can mark the boundary by differentiating an amplitude wrt µ 2 B , bringing down e bφ , which is of the same structure as in the bosonic Liouville CFT.
If one consider only brane segments, connected through boundary operator insertions, each brane segment has its own µ Bi and a different fermion zero-mode γ 0i (which can be zero). Transforming to the length basis is then done by individually transforming each segment to the length basis, using either (7.35) or (7.38) depending on whether the fermion zero-mode is present or not in the segment of interest.

Super-Liouville amplitudes
In this subsection, we summarize the super-Liouville amplitudes we will need.
As always in a supersymmetric worldsheet theory, bulk and boundary vertex operators fall in Ramond and Neveu-Schwarz types. The N = 1 Liouville bulk one-point functions have been determined through the conformal bootstrap in [66,67] as: where the parameter s is related to the boundary cosmological constant µ B by the relations We leave implicit the worldsheet coordinate dependence of this correlator. It will disappear in any case in the end when we include the (super)ghost and matter contributions. We used the notation V α = e αφ for the Neveu-Schwarz bulk operator insertion. The Ramond puncture contains a spin field insertion: The NS (R) sector one-point function is only non-zero provided the fermions satisfy antiperiodic (NS), respectively periodic (R) boundary conditions around the boundary circle, directly relating the fermionic sector of the boundary to the type of bulk insertion.
The boundary two-point function for two vertex operators V β in N = 1 super-Liouville CFT is of the form: where the reflection coefficient d(β|s 1 s 2 ) is the dynamical information that will be discussed below. It multiplies a δ(0) factor corresponding to the infinite Liouville volume. Within the full theory, we mod by the (super)conformal Killing group of the twice-punctured disk and get a finite factor from this. This will be discussed later on. Moreover, the worldsheet coordinate dependence will be cancelled by similar factors coming from the matter-and superghost boundary two-point functions. First, we focus solely on the super-Liouville amplitudes themselves.
The NS-sector primary boundary vertex operators are denoted by The boundary two-point functions with FZZT-boundary parameters s η and s η are denoted as: and are characterized by a (super)Virasoro representation label s and the sign η. To find non-zero, one needs η and η of equal sign.
The two R-sector vertex operators are constructed by applying the spin field σ : and the R-sector two-point function is: One needs η and η to be opposite sign to find non-zero: R-sector boundary operators change the chirality of the boundary. With these definitions, Fukuda and Hosomichi found the following boundary two-point functions [66]: 20 and the Ramond boundary two-point function: where the special functions are defined in Appendix D.

Fixed length amplitudes
In this section, we discuss the above transformation to find the fixed-length disk amplitudes. We first present a derivation of the marked disk partition function and the bulk one-point functions, paralleling the bosonic treatment of [34]. Then we describe local marking operators and how they act indeed as the identity insertion in the fixed length basis, followed by our treatment of the boundary two-point function. This last part is our main result in this section, and in particular equations (8.51) and (8.52).

Marked partition function
We first apply the above transform to the marked partition function, and derive fixed-length disk amplitudes.
The bulk one-point function for FZZT boundary condition s is given by: 1) in terms of the superghost contributions cce −ϕ−φ that we will suppress, and the super-Liouville fields ψ,ψ and φ, where the dependence on the fermionic boundary condition η is implicit in the relation between µ B and the brane parameter s in (7.43). This equation is the superpartner of (7.40). Integrating w.r.t. µ, and choosing a more convenient normalization of the amplitude, Seiberg and Shih found the following unmarked FZZT disk partition functions [68]: We will analyze the two fermionic boundary conditions η = ±1 separately.
The marked partition function is The integration contour (7.35) in the µ B plane is a single leaf of a hyperboloid with top at µ B = +κ ( Figure 6). 21 The contour is initially along the vertical line µ 2 B − ηκ 2 : −i∞ → i∞. We can contour deform it to hug the imaginary axis. 22 Deforming this contour to the imaginary axis, and evaluating, we can write it as: The difference here compared to there is in a (re)normalization of µ and µB, effectively mapping µ0 → κ 2 . 22 The small real segment (0, κ) cancels between top and bottom part of the contour, just like it did in the bosonic case. Note that one cannot contour deform to the right since the integral diverges there. Note also that the case with η = −1 is a disconnected contour in the µB-plane.
where we used: Interpreting the boundary length = β as the inverse temperature, we can read this as a disk contribution of the thermal partition function with energy E = µ 2 B and density of states: starting at E = 0 with a hard spectral edge ρ ∼ 1/ √ E and asymptotically following a power law as ρ ∼ E 1/2b 2 −1/2 . For the particular case of the (2, 4k) minimal superstring, the series expansion of ρ R,η=+1 (E) in powers of √ E truncates at E k−1/2 . At the thermodynamical saddle, we can write the first law as at high energies (UV) going as E ∼ 1/β, just like in the bosonic case [34], and following the (super)JT black hole relation √ E ∼ β −1 at lower energies. The UV behavior means we expect to find a holographic bulk that deviates from the asymptotic AdS boundary conditions. It would be interesting to learn of a dilaton supergravity model and (super)potential that is able to generate this law from a black hole solution in the bulk. Substituting we get: The quantity κ 2 /2 limits to µ as b → 0 by (7.43), and can be viewed as the effective bulk cosmological constant µ eff . Note that deforming the contour in the above way has effectively swapped the local fermionic boundary condition η = ±1 → η = ∓1. We will see this happen for the correlation functions below as well. The integral (8.10) can be evaluated in terms of modified Bessel functions of the second kind as: One can view the transform +∞ 0 dse −κ 2 (cosh(2πbs)−1) cosh πbs . . . as the correct supersymmetric version of the transform to the length basis [42,62]. We will apply this kernel also for correlation functions below.
The marked partition function is in this case given by Looking at the branchcut of arcsinh, we have the discontinuity-like relation: with density of states: starting at E = κ 2 with a spectral edge ρ ∼ √ E − κ 2 and asymptotically a power law as ρ ∼ E 1/2b 2 −1/2 . For the (2, 4k) minimal superstring, the series expansion of ρ R,η=−1 (E) in powers of √ E − κ 2 truncates at (E − κ 2 ) k−1/2 . This model generates a semi-classical first law We now set µ B = κ cosh πbs and hence the fixed-length marked disk amplitude is given by the expression: The integral can be done and leads to So in both cases we have to take the sum of both terms of the contour across the cut. To make a distinction with the bosonic case, where we take the genuine discontinuity by subtraction, here we will denote the sum by the sdiscontinuity, or sDisc for short. This will distinguish the R-sector and the NS-sector branes, the latter requiring taking the genuine discontinuity by subtracting both sides.
In both cases, the final result for the fixed length partition functions matches with the R-sector minisuperspace Liouville QM computation results [62]. This generalizes the observation made by the Zamolodchikovs [69] to the supersymmetric case that the Liouville disk partition function at fixed length (8.10) and (8.17) is exactly computed by the minisuperspace Liouville problem, details of the latter can be found in [62].

Bulk one-point function
Let us insert a (gauge-fixed) bulk operator T α = cc e −ϕ−φ e αφ and transform the bulk one-point function to the fixed length basis. We will parametrize α = Q 2 − θ 2b = Q 2 + iP , in terms of what turns out to be a conical defect θ, or a macroscopic primary P .
Starting with an NS brane with an NS bulk operator insertion (7.40) for η = +1, we first mark the bulk one-point function, by differentiating w.r.t. µ 2 B as: We transform this to the fixed length basis using the NS transformation (7.38). Deforming the contour to µ B : −i∞ + → +i∞ + , 23 we get again two contributions that have to be added. The 2µ B -factor in the NS transform measure (7.38) cancels with the explicit factor above in (8.19). Also, the square root factors in the denominator creates a relative minus sign between both parts of the contour, effectively leading to a subtraction of the pieces on both sides of the branch cut.
We hence see that the NS computation is in effect doing the same discontinuity calculation as in the bosonic case of [34]. We end up with: Substituting µ B = κ sinh πbs, we get 24 The case η = −1 is entirely analogous. We have the discontinuity relation: Setting similarly µ B = κ cosh πbs, we write finally: Once again, following the contour deformation argument, we are effectively swapping η = ±1.
Both of these integrals (8.21) and (8.23) can be readily evaluated explicitly yielding: (8.24) 23 One checks explicitly that there is effectively no branch cut in the region (0, κ). 24 The normalization is somewhat arbitrary. We have chosen it to find the JT bulk defect one-point function in the JT limit.
The bulk one-point functions of the Ramond sector operators (7.41) and (7.42) can be transformed to the length basis in the same way. We do not mark these boundaries further since the Ramond operator creates a branch cut that already marks the boundary. The calculation is identical to the one for the Ramond partition functions in 8.1, and we end up with: The integrals are done as: Setting θ = 2iP b, these bulk insertions are macroscopic holes with label P from the (super)Liouville geometry perspective. These specific bulk insertions are required when gluing disks together. We will write some formulas in the concluding section 11.

Neveu-Schwarz partition function
In the bosonic Liouville gravity, it was illustrated in [34] that the partition function Z can be found from the bulk one-point function by letting θ → 1, and simultaneously removing a single marking. Starting with (8.24), we can remove a single marking by dividing by . Letting θ → 1 defines the partition function. We get with the spectral densities: For the particular case of the (2, 4k + 2) minimal superstring, the series expansions of these quantities in powers of √ E and √ E − κ 2 respectively, truncate at E k+1/2 and (E − κ 2 ) k+1/2 respectively.

Ramond partition function revisited
We can get the Ramond brane partition functions also by applying a single bulk spin field Θ ,¯ α with θ = 1 or α = b/2. Indeed, inserting this value in (8.25) and (8.26), we reproduce the fixed length partition functions (8.10) and (8.17). This means this bulk operator is not really doing anything besides marking the boundary: the amplitudes are marked since the branch cut from the spin field hits the boundary somewhere. These results illustrate the equivalence: This also shows that applying a single boundary marking operator M 1 is effectively changing the fermionic boundary condition from NS to R.
The full situation is summarized in figure 7. Figure 7: Left diagrams: partition function Z NS and bulk one-point functions T α . The blue dots on the boundary represent the markings. Right diagrams: Z R and Θ α . The NS partition function is oncemarked, whereas the NS bulk one-point function is twice marked. All R-sector cases have a single marking due to the branch cut of the spin field.
In the bosonic case, this same mnemonic applies to the bulk one-point function ∼ cosh π(2α − Q)s, where indeed setting α = b/2 in e 2αφ gives the s-dependence of the marked partition function.

Marking operators
The limit where β M = 0 or h = 0 corresponds to inserting a local marking operator M 2 (x) (7.33) on the boundary, with the matter operator the identity 1 M . Here we show, in analogy with the bosonic case [34], that in the fixed-length basis these insertions are indeed identity insertions. Using the shift identities (D.9) leading to we can set β = b in (7.50) and get (up to a prefactor that is chosen with hindsight): As a check on these expressions within N = 1 Liouville supergravity, take the s 1 → s 2 limit of the second and fourth expressions: This amplitude corresponds to two insertions of (7.32) 25 where we have written the operator both in its gauge-invariant and in its gauxe-fixed form, including the superghosts. Since insertions of M 1 correspond to marking the boundary by taking derivatives w.r.t. µ B , this is to be compared to the twice-marked Ramond boundary amplitudes, obtained by marking (8.4) and (8.12) once more: which indeed agree.
Let us now transform these to the fixed-length basis. We will use this paragraph to argue that to find the fixed length operator insertion of interest, we should take a linear combination of both d(β|s + s + ) and d (β|s + s + ) amplitudes. Contour deforming both s and s -integrals in the same fashion as in subsection 8.1, one has the relation: where µ i is shorthand notation for µ Bi . Since these are equal, there is no contribution to d(β|s + s + ) − d (β|s + s + ) when s = s . However, a more careful treatment is required when s = s . By first subtracting both terms in d(β|s + s + ) − d (β|s + s + ) and only in the end evaluating the discontinuity across the cut, we get sDisc d(b|s + s + ) − d (b|s + s + ) = 8π cos π 2b 2 cosh giving indeed back the original partition function in the η = +1 sector (8.10): The combination d(β|s + s + ) − d (β|s + s + ) can be found as the two-point function of the linear combination of operators Λ b + iB b , defined in (7.46). This means that inserting the operator ( b 2 (ψ + ηψ) + i)e b 2 φ has no effect after transforming to the fixed length basis. This local boundary operator is to be identified with M 2 (x) (7.33) upon using that γ 2 0 = 1 in the two-point function. 25 The γ0 insertions in M1 cancel pairwise due to γ 2 0 = 1.
This also holds for more than 2 marking operators. We discuss this using the matrix description in section 9.

Boundary two-point function
Here we will consider the boundary two-point function in general Liouville supergravities. We first consider the super-Liouville sector of the boundary two-point function, since this one will carry almost all of the interesting information. The relevant equations were obtained in [66]. Just as above, it turns out one obtains suggestive answers by pairing up these formulas. Transitioning to the fixed-length amplitudes proceeds by applying the integral transform (7.35). Deforming the contour as for the partition function, we will evaluate the sdiscontinuity of these expressions. We define a boundary (NS) operator combining the bosonic vertex operator with its superpartner (7.46): and a boundary spin (R) operator summing both chiralities of (7.48): The latter will be used in section 10.
For the NS sector operator insertions, we have two possible interesting combinations.
• For a η = +1 boundary, the boundary two-point function O β O β ++ is given by D β s,s ≡ d (β|s + s + ) − d(β|s + s + ). Define the double discontinuity with all plus-signs as: as we are instructed to do in the contour deformation of NS-sector operator insertions of section 8.1. Using the supersymmetric shift relations (D.9), one explicitly evaluates this to: We notice two things. Firstly, the rhs contains the same linear combination of two-point functions, but with a shift in β → β + 1/b. Secondly, one has in effect changed fermionic boundary conditions from η = +1 to η = −1 during this process.
• Secondly, we consider a η = −1 boundary, for which the boundary-two point function Using the same definition (8.45), we obtain: with a similar qualitative interpretation. Notice the appearance of two sinh measure factors in the second line, to be contrasted with the cosh appearing in (8.46).
Next we combine this super-Liouville sector with the matter sector, the superghosts and the overall normalization factor (the legpole factor). The NS boundary tachyon vertex operators are defined as: We have included the (super)ghost contribution explicitly here. Including the superghost and matter contributions and imposing the Virasoro constraints in the full theory has two additional effects. Firstly, normalizing the matter boundary two-point function as 1, the main effect of the superghost and matter boundary two-point function is to cancel the dependence on the worldsheet coordinates x in the final result.
Secondly, just as in the bosonic case, the Liouville piece of the boundary two-point function diverges as δ(0) from the Liouville zero-mode, which is cancelled by the volume of the (super)conformal Killing group of the two-punctured disk. For two NS boundary operators, the ratio is finite and equals 2(Q − 2β) precisely like in the bosonic case. For the boundary Ramond two-point function in string theory, the ratio gives a factor independent of the vertex operator labels. 26 The prefactors in the final result are of three kinds: the explicit legpole factors in the vertex operators (8.48), the prefactors coming from the contour rotation argument in (8.46) and its cousins, and finally the ratio of Γ NS factors present in the super-Liouville two-point functions of (7.50). It is not difficult to show that these conspire to the simple result of 1/S NS (2β M ) for the NS-operator insertion, and 1/S R (2β M ) for the R-operator insertion. 27 26 It would be interesting to explicitly derive this by integrating the N = 1 Liouville boundary three-point function B b Θ α 1 Θ α 2 w.r.t. µB. 27 One way to derive this, is to use the shift relations (D.10) twice for the numerator Γ NS/R (2β − Q), then write it in terms of S NS/R and apply its shift relation twice again (D.9). Use is made throughout of the gamma-function reflection formula in the form . (8.50) We arrive at the expression for η = +1: = ds 1 ds 2 ρ R,η=+1 (s 1 )ρ R,η=+1 (s 2 )e −κ 2 1 sinh 2 πbs 1 e −κ 2 2 sinh 2 πbs 2 (8.51) with ρ R,η=+1 (s) = cosh π b s cosh πbs. For η = −1, we get the slightly simpler: with ρ R,η=−1 (s) = sinh π b s sinh πbs. These are our final results for the boundary two-point function in Liouville supergravity.

JT supergravity limit
Here we discuss how a JT (super)gravity is achieved and we make contact with amplitudes in these limiting models.

Partition functions
The JT limit of the partition functions (8.10) and (8.17) is readily evaluated, by letting s → 0 and → +∞ as in terms of a finite length JT and finite momentum k. We obtain (8.55) which is for η = +1 indeed the N = 1 supersymmetric Schwarzian partition function [18,19]. In the JT limit, the NS-boundary partition functions (8.29) for both η = ±1 produce the bosonic JT density of states ρ(k) = k sinh 2πk.
Bulk one-point function This corresponds to a conical defect of deficit 2π(1 − θ). Analogously,

Boundary two-point function
Finally we take the Schwarzian limit of these expressions. Setting β M = 2bh and s = 2bk 1 , s = 2bk 2 , and using (D.5), the building blocks (7.50) reduce to: (8.62) and the denominators in (8.51) and (8.52) have the limit Use is made of formulas contained in appendix D. The results can now be compared with the N = 1 super-Schwarzian bilocal correlators (5.12) found in [19]. The NS operator insertion with η = +1 (8.51) precisely limits to this super-JT correlator. In [19], the super JT bilocal correlators were constructed by considering N = 1 super-Liouville theory on a cylinder in the NS sector between a pair of ZZ identity branes [69]. This means one uses periodic (R) boundary conditions around the circumference of the cylinder, identifying this with the η = +1 boundary condition on the Liouville supergravity disk studied in this work. From the ZZ-ZZ brane perspective, choosing anti-periodic (NS) boundary conditions around the cylinder circumference, leads to a removal of all fermions in the JT limit and one retrieves only the bosonic Schwarzian system. This is then identified with the η = −1 fermionic boundary condition on the Liouville supergravity disk. This corresponds to (8.61) and (8.62). The fourth line can be viewed as the reparametrized operator 1/τ 2h+1 inserted into the bosonic Schwarzian system, indeed giving just the h → h + 1/2 result. We will come back to this interpretation in the ZZ-ZZ system in subsection 10.3.

Matrix model description
We will focus on the case where p = 2 for which we have a single-matrix description, and it is possible to describe some of the properties derived above more cleanly from the perspective of this matrix model. We first make some observations on the partition functions and marking operators, and then in subsection 9.1 we proceed with our main goal of constructing the minimal superstring boundary two-point functions using the matrix model perspective as a guide towards writing down the analogous proposal for higher genus corrections.
For any random matrix integral, the resolvent is defined as R(x) = Tr 1 x−M and equals the singly marked disk amplitude. For the N = 1 case, and in the Ramond sector, the random matrix in question is a "supercharge" matrix Q, with the Hamiltonian matrix related to it as H = Q 2 . We define the Ramond sector resolvent by where the second way of writing it shows that it is also the usual resolvent of a random matrix model with matrix Q. 28 With this definition, the R-sector resolvent produces the "charge" spectral density by an sdiscontinuity: In terms of x = µ B , transforming in the R-sector to the fixed length amplitude through (7.35), we get indeed: For the (2, 4k) one-matrix superminimal models with k ∈ N, we have an analogous story as for the bosonic (2, 2m−1) models. Let us be more explicit for the case η = +1, related to JT supergravity as found in (8.54). The resolvent in this case is determined in terms of the uniformizing coordinate s as: The sdiscontinuity of the resolvent is twice its value along either side of the cut and we find for the resolvent for the matrix H: In the SJT limit, we set x = 4π 2 b 4 x SJT as b → 0, and get: up to a prefactor, reproducing the super-JT spectral curve if we set x SJT = −z 2 SJT .
We can write the marking two-point function, composed of the difference of the two equations in (8.34), for either case η = ±1 in the following ways: illustrating that this inverse Laplace transforms to This is a simple proof that marking operators act as the identity operator in the fixed-length basis. Notice that it is only by combining e b 2 φ and ψe b 2 φ , that we get a function without any branch cuts (only a function of x 2 ).
Multiple local marking operators are then readily accommodated by the expression: For a NS-sector brane boundary, the treatment is similar to the bosonic minimal string, as we discussed in [34]. Defining we transform to fixed length using (7.38) (9.11) where now the spectral density is obtained by a genuine discontinuity: This NS sector is related to the (2, 4k + 2) (one-matrix) superminimal models. The discontinuity of the resolvent is twice its value along either side of the cut and we get e.g. for η = +1: giving the spectral density of (8.30). In the JT limit, we set x = 4π 2 b 4 x JT as b → 0, and get: (9.14) reproducing the bosonic JT spectral curve if we set x SJT = −z 2 SJT .

Amplitudes
Next, we consider how, from the matrix integral description, we would insert the tachyon vertex operators (7.7) in the minimal superstring. We follow a strategy very similar to the bosonic case in section 4, where we write the Liouville amplitudes in terms of solely the matrix resolvents.
η = +1 28 The regular terms do not have poles or branch cuts in the physical region of interest and are immaterial for our purposes.
For j ∈ N = 1, 2, . . ., the super-Liouville boundary two-point expressions (7.50) simplify to: 29 . (9.16) to the fixed-length basis, we will contour transform both µ B integrals. The integrand contains both poles and branch cuts. However, just like in the bosonic case in section 4, we first do one of the integrals by evaluating the residues. Deforming the second contour, we effectively set s i → s i ± i 1 2b , and we obtain: where ρ R η=+1 (s) = cosh πbs sDisc[R R η=+1 (s)] = cosh πbs cosh π b s. The numerical prefactors are chosen such that The JT limit is found by letting s = 2bk. One can check that one reproduces the weight h = −j/2 super-Schwarzian degenerate bilocal correlator. Very explicitly, for j = 1 and j = 2, we reproduce the −1/2 and −1 super-Schwarzian results, given by (5.21) and (5.22) respectively.
Higher genus effects are now incorporated fully into the spectral density, and one only has the same kind of higher topology as in the partition function, just like we found in the bosonic theory in section 4. In the JT limit, the spectral curve is cos 2πz z and its higher genus effects were studied in [70]. This realizes our original goal of structurally understanding higher genus effects to degenerate super JT bilocal correlators.

Ramond operators
In this section, we consider Ramond boundary operators. They cause a change in fermionic sector between η = +1 to η = −1. Morever they cause also an effective change from NS to R sector in terms of how the transform to the fixed length basis should be implemented, (1.11) or (1.13), in the different segments. We first discuss the generic weight Ramond insertion from the continuum approach. Then we discuss the special class of operator insertions corresponding to the minimal superstring Ramond operators. Finally, we present a way of getting them starting with super-Liouville CFT between a pair of identity ZZ branes.

Continuum approach
We consider Σ β Σ β +− on a boundary that changes fermionic boundary conditions between η = ±1. We combine the super-Liouville two-point functions from (7.51) as D β s,s ≡d(β, +|s + s − )+ d(β, −|s + s − ) and take as the sdiscontinuity a combination of plus and minus signs: This corresponds to picking the brane with label s to be in the R-sector, but the s -brane is in the NS-sector. Using this, one computes: where one sees that the s and s' branes generate a different spectral factor, signaling indeed that the Ramond vertex operators cause transitions between them. Note that the discontinuity has minus signs for the s -parameter, which signals a spin field insertion is causing a transition from a R boundary brane segment (having a fermionic zero-mode) into a NS segment (without a fermionic zero-mode), recalling the bosonic Liouville theory has minus signs in the discontinuity.
Within Liouville supergravity, the Ramond boundary tachyon vertex operators are defined as which include explicit legpole factors, now in the Ramond sector. The insertion of a pair of these Ramond operators leads, after an analogous computation as for the NS operators of subsection 8.4, to the answer: where ρ NS η=−1 (s) = sinh 2πbs sinh π b s.
Within the super-JT limit, we can usẽ In the JT limit, one changes between the super JT and the bosonic JT theory upon crossing the bilocal line. It would be interesting to understand whether this result can be understood directly in the JT theory, without going through its embedding in Liouville supergravity. For h ∈ −N/2 − 1 4 , the Ramond sector bilocal (10.5) becomes degenerate. These degenerate values of h need to be considered separately. Within the framework of the minimal superstring, we consider them next.
which, after picking up the residues and the discontinuity, leads to in terms of the densities of states: which describes a transition from an R-sector to an NS-sector. We indeed recognize the NS density of states from (8.29).

Spin fields from the ZZ-ZZ brane perspective
In this complementary section, we elaborate on a perspective that was developed in [19] and [33] on Schwarzian amplitudes. In that work, the strategy was to consider the cylinder amplitude between a pair of vacuum branes in solely the Liouville CFT. Within that construction, a doublescaling limit where the central charge c → +∞, the amplitude reduces to the Schwarzian partition function. The extension to Schwarzian bilocal operators is made by taking the same double-scaling limit of the cylinder amplitude with a single Liouville primary vertex operator inserted in the middle. One of the main benefits of this approach is that several generalizations are readily studied, exploiting the many investigations of Liouville and super-Liouville CFT done during the past decades. Indeed, we computed the super-Schwarzian bilocal correlators by inserting a super-Liouville primary vertex operator e αφ between two Ramond identity branes ZZ R (Figure 9). In this work, ZZ R ZZ R T Figure 9: Super-Liouville primary field insertion e αφ between two vacuum R-sector branes. The Schwarzian bilocal correlators are obtained in the double scaling limit where the circumference of the cylinder T → 0 while c → +∞ keeping the product cT 24π = C fixed as the Schwarzian coupling constant.
we have seen that there is also a boundary spin field insertion in Liouville supergravity, and the resulting bilocal correlator (10.5) was not obtained before in [19]. In this section we remedy this and study spin field insertions in the super-Liouville CFT between vacuum branes and match these to expressions we found in this work using the Liouville supergravity approach.
We compute the amplitude with the insertion of a super-Liouville spin field operator between one ZZ NS brane and one ZZ R brane (Figure 10 left). Within the minisuperspace limit, we expand ZZ NS ZZ R ZZ R ZZ R = Figure 10: Left: Insertion of spin operator between a ZZ NS and a ZZ R brane. Right: Insertion of Verlinde loop (topological defect) allows one to view this as a single defect consisting of the combination WΘ ±± α2 .
the boundary states as where the Ishibashi states are expanded in primary states and their descendants as: |P, NS = V Q/2+iP + (descendants) (10.13) Within the double-scaling limit where the cylinder circumference T → 0, the (closed-channel) Ishibashi states above are dominated by their primaries. For a single super-Liouville vertex operator insertion between the two branes, we can then compute the amplitude as the three-point function on the sphere. In our case, the operator is in the Ramond sector and can be written as Θ ¯ α ≡ σ ¯ V α where V α ≡ e αφ are the Liouville primaries and σ ¯ is the spin field. We then denote the sphere three-point functions of interest as: 14) and the amplitude of interest becomes: The N = 1 DOZZ formula is given by [71,72,66] 31 , (10.16) . (10.17) In order to implement the double-scaling limit to JT gravity, we need to parametrize the 2d CFT parameters in the following way [19]: α 1 = Q/2 + 2ibk 1 , α 2 = Q/2 + 2ibk 2 and α 3 = 2bh, where k 1 , k 2 and h are kept finite as b → 0. Using the limits: we find the limiting DOZZ-formulas become: , (10.20) which matches with (10.6), with the correspondence of the boundary spin operator Θ −β ↔C 1 and Θ +β ↔C 2 . Taking the sumC 1 +C 2 is then identified with the sum in the quantityd(β, −|s + , s − )+ d(β, +|s + , s − ).
Using the limits of the brane wavefunctions the amplitude (10.15) is then with a change in spectral density from the NS density for k 1 to the R density for k 2 . Notice the presence of the Γ 2h + 1 2 factor, signaling the presence of Ramond degenerate boundary operators when h = − 1 4 , − 3 4 . . .. This expression matches the JT limit of the Liouville supergravity expression we found in (10.5). Setting h = 0 gives an insertion of a pure spin field, and this can be viewed as the most basic operator required to change fermionic boundary conditions.
In [33] we framed this approach of computing JT correlators in the larger geometric bulk picture, and it is insightful to generalize that discussion to include the spin operators discussed here. The general picture described there is as follows. The Liouville cylinder is first doubled into a chiral torus. The double-scaling limit then shrinks that torus into a long narrow tube that degenerates into the boundary of the disk. The exterior region of the cylinder is in the process projected onto its angular zero-mode and can be identified as the interior of the 2d disk region. We checked this picture in [33] by mapping defects in the JT disk bulk to Chern-Simons Wilson loops encircling the cylindrical tube to Verlinde loops [73] contracted onto the Liouville cylindrical tube. This story now gets additional ingredients as follows.
There is a branch cut between the spin field insertion and the NS-brane ( figure 10 left). The latter can also be found by applying a Verlinde loop operator W on the usual ZZ R brane, i.e. |ZZ NS = W |ZZ R . Since Verlinde loops can be viewed as topological defects in CFT [74], one can view the entire construction as the application of a combined operator on the CFT system, consisting of R spin field plus topological defect to soak up the jump on the branch cut at the other end (figure 10 right). One can even move the defect up to the vertex operator. The topological defect operator insertion D(k) itself is given by the expression: 24) to be inserted in the amplitude in the momentum k sector. A summary of the situation is given in figure 11. The periodicity / antiperiodicity of the fermions around the small circle in the ZZ-ZZ picture, maps into the local fermionic boundary conditions η = + / η = −1 respectively in the Liouville disk supergravity picture.
For higher-point functions, we can determine the intermediate set of operators from the fusion rules ( figure 12).  E.g. for the left figure 12 we have V P × V → V Q , whereas for the middle figure we get Θ P × Θ → V Q and the right figure has V P × Θ → Θ Q . Each time we have a fusion in the NS sector V Q we have a k sinh 2πk density and a branchcut, whereas when we have an R fusion Θ Q , we get cosh 2πk and no branchcut.
This readily generalizes to out-of-time ordered correlators.

Concluding remarks
In this work, we have considered the JT bilocal correlators of a special integrable class of operators with weight h ∈ −N/2, corresponding to degenerate Virasoro representations. We have shown that their structure is simpler than that of the generic h correlator and have exploited this to understand more deeply some aspects of the 1/C and small τ perturbation series on the disk that would otherwise be hard to distill. We have also exploited its minimal string embedding to gain an understanding on how higher topology interplays with this class of correlators, illustrating that no handles crossing the bilocal line are generated in this case. We generalized most of these statements to JT supergravity and its degenerate bilocal correlators. In order to analyze the minimal superstring embedding here, we first analyzed the more general Liouville supergravities and how JT limits are obtained in those amplitudes. These results are of interest in their own right. The structure of the degenerate correlators and higher genus corrections mirrors that of the bosonic case in the end. An identical small τ expansion structure was found with no additional cancellations in the series expansion. It would be interesting to understand higher supersymmetric versions such as N = 2 and to see whether one satisfies perturbative non-renormalization theorems in those cases, simplifying the expansion. We also analyzed how Ramond boundary operators work in this case, and how they change fermionic boundary conditions. In the JT limit, this corresponds to sectors of the diagrams changing between the supersymmetric and the non-supersymmetric model.
We end with some remarks and points that deserve further study that we did not mention up to this point.

Massive bulk fields and HKLL
Whereas the degenerate bilocals are largely part of a structural discussion, we should point out a genuine appearance of degenerate bilocals in a direct physical context. When constructing bulk observables and their correlators associated with mass m scalar fields in JT gravity in a diff-invariant way, the Schwarzian path integral can be performed exactly in terms of multiple bilocal operator insertions [30,75], interpretable as a product of HKLL kernels [76,77]. Several of these are of negative weight 1 − ∆ where m 2 = ∆(∆ − 1), and for ∆ = 3/2, 2, 5/2 . . . these are degenerate bilocals for which the results in this work have to be applied. It would be interesting to understand how they alter the deep IR bulk physics of [78,31].

Beyond boundary gravitons
We have established in appendix B that the perturbative 1/C disk expansion is only asymptotic for h / ∈ −N/2. This means that it contains non-perturbative physics even on the disk topology.
We already know the full answer (1.5), but can we get a more physical picture on precisely what it contains that goes beyond boundary gravitons?
Other Liouville supergravity amplitudes In [34], we additionally determined the bosonic fixed length amplitudes for the bulk-boundary two point function and the boundary three-point function. It would be interesting to provide these also for the supergravities studied here. However, this requires knowledge of these amplitudes in N = 1 super-Liouville CFT, which as far as we know, have not been rigourously established.

Quantum group interpretation
The authors of [79,80] constructed a set of self-dual representations of the modular double of U q (osp(1|2)) with Casimir, and supercharge: related as C = Q 2 −1. This Casimir matches structurally with the Hamiltonian in the propagation amplitudes of e.g. (8.10) and subsequent equations (up to shifts and rescalings). Likewise, the spectral densities ρ(s) are to be identified as super-Virasoro vacuum modular S-matrices and presumably Plancherel measures on these representations. It would be interesting to understand this group theoretic perspective better. For the bosonic Liouville gravity models, we showed in [34] that the vertex functions S b (β M ±is 1 ±is 2 ) can be identified as the square of a 3j symbol of the quantum group associated to the modular double of U q (sl(2, R)), where two entries are mixed parabolic matrix elements or Whittaker functions in the principal series representation, and the last entry is a discrete representation insertion. It would be interesting to perform the analogous computation with the Whittaker function of (the modular double of) U q (osp(1|2)) and interpret the vertex functions in the second lines of (8.51) and (8.52) in this same manner.
On a related front, the authors of [81] found a solution to the double-scaled supersymmetric SYK model, related to structure found in a certain quantum group. In the bosonic case, it is known that the double-scaled SYK model and the Liouville gravities are associated to different quantum deformations of SL(2, R), namely SU q (1, 1) and SL q (2, R) respectively. It would be interesting to learn whether this distinction persists in the supersymmetric case.

Other topology
Given the recently understood relevance of higher topology [7,82], including multi-boundary amplitudes (see also [24,83]), it is of interest to understand the structure of the multi-boundary and higher genus amplitudes in Liouville supergravity. Restricting to the (p, q) minimal superstring, we have a matrix approach at our disposal as well. For the (p, q) minimal superstring, annulus amplitudes were extensively discussed from the continuum approach in [84,85]. The annulus amplitude for two NS branes with fermionic labels η and η , and with matter vacuum branes, is given by: Restricting to p = 2, and transforming both loops to the length basis in units where κ = 1, we have: (11.3) with the same gluing measure dνν tanh πν as the bosonic minimal string. Transforming to the length basis and setting p = 2, we get the results: ( 2 )).
(11.6) Both (11.3) and (11.6) are interpretable as summing over all intermediate labels of two bulk onepoint functions (8.24) or (8.27) and (8.28) where α = Q/2 + ibλ, with a certain gluing measure dλρ glue (λ) that can be read from these equations. Diagrammatically, we read this as where we drew the bulk one-point function diagrams from figure 7, and where the branch cuts from the spin fields (drawn as dashed lines) are glued together by this procedure. It would be interesting to understand this structure better for more involved topology, and to tie it to some of the known results in JT supergravity and its super Weil-Petersson volumes [70]. In order to achieve this, we need to get a better grasp on the underlying matrix model. Suppose we start with the genus zero density of states (8.7) for the (2, 4k) minimal superstring. Then the string equation at genus zero of the resulting matrix model is written in terms of an auxiliary quantity f (u), defined as: We can find f (u) by expanding ρ 0 (E) as a polynomial in √ E as: from which, following the steps outlined in e.g. [36,37,38], we can find (11.11) where P (1,0) k−1 is a Jacobi polynomial of order k − 1. 33 This indeed satisfies the relation: The string equation is then found by integrating (11.11) and taking care of the E −1/2 contribution, leading to: At large k, these become the super JT result derived in [37] 34 The multi-loop genus zero amplitudes are then fully determined by the knowledge of the relation u(x) from inverting the string equation (11.13), with a general formula written in e.g. [86]. We leave a more elaborate study to future work. 33 In the bosonic minimal string, this was a Legendre polynomial Pn ≡ P (0,0) n [68,34]. 34 To find both equations (11.15), one can use the Mehler-Heine relation (11.14)

A Degenerate braiding and OTOCs
In this appendix, we examine the crossed four-point function in the case where one of the bilocal lines has a label in the degenerate Virasoro representation j and one non-degenerate bilocal line: We will see that the amplitudes for both uncrossed and crossed diagrams can be accommodated by the following vertex function rule: where we add that the last rule is only written once for each combination of k 1 and k 2 . E.g. the two-point function has two such vertices but we only write the contribution once. We will see that this indeed gives the correct bookkeeping also for crossed lines. For a bulk crossing between a non-degenerate and degenerate line, we use the degenerate 6j-symbol: An explicit expression is written in (A.18), with the particular case of j = 1/2 in (A.11). We will determine these using two complementary methods: the first takes a Schwarzian limit from the degenerate braiding matrix in 2d Virasoro CFT, while the second method deforms the contour of the SL(2, R) 6j symbols directly. The second method is more powerful but it is useful to have both perspectives.
It is known how the crossed four-point function can be obtained by applying the Virasoro braiding kernel within the Schwarzian limit to the uncrossed case [19]. The braiding kernel R realizes the following diagrammatic operation: It is an object defined in 2d CFT by the following operation on conformal blocks: The braiding matrix R itself is related to the fusion matrix F by where is the sign of the braiding. To be explicit, we focus here on the first degenerate primary with j = 1/2. For a degenerate primary α 2 = −b/2 (denoted by a dashed line in (A.5)), the fusion rules require α s = α 1 ± b/2 and α t = α 3 ± b/2. Denoting this choice of sign in the subscript of the fusion matrix, the degenerate fusion matrix is of the form: 35 , Parametrizing α 1 = Q/2 + ibk 1 , α 4 = Q/2 + ibk 2 , α 2 = −b/2 and α 3 = bh, and taking the Schwarzian limit where b → 0, we obtain: 36 Let us work this out for the ++ case as an example, for which ik s = ik 1 + 1/2 and ik t = ik 2 + 1/2. Following the logic of [19], the braiding procedure results in the following combination of Schwarzian vertex functions and braiding matrix: where we take the vertex functions from the uncrossed diagram (the left diagram of (A.4)). This expression can be rewritten suggestively as: where the first factors are the diagrammatic rule for the vertex functions (A.2) of the rightmost diagram of (A.4), already evaluating the delta-functions and cancelling the measures for k s and k t . The last factor can be interpreted as the degenerate 6j-symbol. 35 These expressions can be found in many works. Some particularly convenient ones are [87,88,89]. 36 The phase factor in (A.6) evaluates to (−) , and is an overall sign factor which is ignored.
The other three cases −+, +− and −− are worked out analogously, resulting in the 6j-symbols: 37 Denoting the boundary lengths of the four segments JT,i , where i = 1, 2, 3, 4, we can write the full amplitude for the crossed four-point function as: From a 2d CFT point of view, higher degenerate fusion matrices get significantly more complicated. Instead, there is a second way of getting the expressions (A.11) for the 6j-symbols, without resorting to 2d CFT. For non-degenerate labels, the 6j-symbol has an integral expression written in [19]: where k i+j ≡ k i + k j etc. This was obtained as the Schwarzian limit of the well-known fusion kernel of Virasoro CFT, obtained by Ponsot and Teschner in [90]. The expression can be viewed as an integral representation of the Wilson function of Groenevelt's work [44].
Suppose the label h 1 becomes degenerate, then the vertex function of the h 1 bilocal line contains the piece 1/Γ(2h 1 ) bringing the entire amplitude to zero, unless the 6j symbol (A.13) contains a pole that precisely compensates the zero of the vertex function. This can happen when multiple poles coincide in the degenerate limit. A sketch of the poles of the integrand of (A.13) is given in figure 13. u u Figure 13: Poles series from the 6 Gamma-functions in the numerator of (A.13), with the poles on the rhs of the contour colored in red. Left: starting situation where h 1,2 > 0 and k i ∈ R. Right: As one changes h 1 → −1/2 and k s,t ∈ R ± i/2 to correspond to the degenerate insertion and fusion rules, some of the red poles cause the contour to pinch off. One picks up the residue from these poles which give singular contributions (due to coincidence with some of the other (black) poles).
This leads to the contour getting pinched off between the poles. We can deform the contour by picking up only those residues which give divergent contributions due to multiple poles coinciding.
Within the full amplitude, this process results in only some of the residues of the u-integral that have to be considered. In this case, we only need the pole series where u = h 1 + ik s − ik 1 + n (from the fifth gamma in the numerator of the integrand of A.13). The integer n ranges only over a short range to give a divergent pole contribution. In fact, we can keep the entire range n ∈ N since the residues of all other poles in this series turn out to vanish as we show below. The resulting contribution was evaluated in [19] in terms of a 4 F 3 hypergeometric function and yields: where the hypergeometric function is defined by the series expansion: in terms of Pochhammer symbols (a) n ≡ a(a + 1) . . . (a + n − 1). Combining the vertex functions for the j bilocal line with the first line of (A.14), we have the h 1 → −1/2 limit as: where use is made of (2.2) and (2.5), evaluating to Dirac delta functions giving the degenerate fusion rules. Across a higher j bilocal line, we get (2.3), and the momentum labels are related as ik s = ik 1 + m and ik t = ik 2 +m where −j ≤ m,m ≤ j. With these values of k s and k t in terms of k 1 and k 2 , we can see that the first and fourth entry of the 4 F 3 in (A.14) are negative integers. This means the 4 F 3 truncates to a polynomial, that is called the Wilson polynomial P n (x; a, b, c, d). In the notation of [44], we can write: With the value of k s and k t in terms of k 1 and k 2 , the polynomial is of order min(j − m, j +m) in k 1 , k 2 . This leads to the final form of the degenerate 6j-symbols, where we stripped off the pieces that are associated to the vertex functions: Very explicitly, for h 1 = −j → −1/2, we can evaluate the Wilson polynomial (or compute the relevant residues in (A.13)), 38 and we obtain for (A.18) which matches with (A.11), with matching ± signs.
One could also generalize to the case where both operator pairs are degenerate. We have not done so here, and leave this to future work.
Physically, the 6j symbol for non-degenerate matter weights h 1 and h 2 contains the semi-classical information for gravitational shockwave scattering in the bulk of particles with masses m 2 i ∼ h i (h i − 1), and with four black hole regions of energies k 2 i [91,19]. Since for degenerate operators, two of these k i 's are complex, the black hole shockwave interpretation is not valid here. This is in accord with our interpretation in section 2 in terms of an integrable subclass of operators.

B Precision testing the two-point function and perturbative expansion
In this appendix, we wish to show that the perturbative 1/C expansion of the bosonic Schwarzian bilocal correlator (1.5) is always asymptotic except at negative half-integers h ∈ −N/2. Our strategy is to investigate explicitly the particular case of h = +1/2 and h = +1, and illustrative that the perturbative series is asymptotic for these cases, developing the technical tools we need further on. Then we provide a general proof that it is asymptotic for any real value of h, excluding the cases h ∈ −N/2 which we investigated above in section 2. The zero-temperature Schwarzian bilocal correlator is given by taking the β → +∞ limit of (1.5), which effectively projects onto the k 2 = 0 sector: .
In order to distill lessons on the perturbative 1/C expansion of this object, we focus on the cases where h ∈ +N/2 first. Using the asymptotic expansions one can write a (formal) asymptotic series expansion of the h = 1 and h = 1/2 result in terms of Bernoulli numbers as: This is done in a multi-step process: first we insert the above asymptotic expansion (B.2) for the hyperbolic functions. Then we Taylor-expand the Boltzmann factor e − τ 2C k 2 , perform the kintegral and finally resum the series-expansion of the coth and tanh in terms of Bernoulli numbers This series expansion is asymptotic as one can check very explicitly: the coefficients of these series diverge at large orders. This means the correlators contain non-perturbative (in G N ∼ 1/C) contributions. We will later show that the non-perturbative contribution to the correlator we have just discussed on the one hand, and the higher-genus contributions on the other hand, are both contributing at the same order in the G N ∼ 1/C expansion, both as ∼ e − # C .

B.1 Explicit form of zero-temperature expansion
Generalizing to higher value of h ∈ N/2, each coefficient at any fixed order in the expansion is larger than the h = 1 coefficient, meaning these series are even more divergent. An explicit expression can be obtained in terms of combinations of Bernoulli numbers, but is not very revealing. Instead we focus on obtaining generic expressions for the first few lowest-order terms. The trick is to start with h ∈ N, for which we can write We expand the product as a power series in k: with coefficients explicitly determined by the combinatorics as: These coefficients have closed expressions, and doing the k-integral, one can write down the expansion: While derived only for integer h, the structure of the Schwarzian perturbation series at this order, reviewed in section 3.1, shows that for any h the quantity g m has to be a polynomial of m'th order in h, and hence the above formula is correct for any real h. 39 The third term is positive for h > 0 and the fourth for h > 2. Some other lessons are that all singular terms as τ → 0 are always positive, and integer h always has no 1/τ -term, reflected in the presence of h(h − n) factors at the levels τ odd . The positive terms in the expansion ∼ τ >0 are always alternating in sign, as is seen by (B.10). 40

B.2 General proof of asymptotic series
In this section, we will show explicitly that the 1/C Schwarzian perturbative expansions are asymptotic series. We will do this by working up from the zero-temperature case, via the fixed energy eigenstate, to the thermal ensemble.

Ground state
For any real value of h, one can write the zero-temperature two-point function (B.1) as: This consists of a finite number of terms for which e − τ 2C k 2 cannot be Taylor-expanded (the negative-order n < 0 terms ∼ 1 τ >0 in the τ -expansion) and an infinite number of terms (n > 0) where it can. The latter is the important contribution to test convergence of the power series. For any fixed value of k, the function f k (h) = Γ(h±ik) 2 Γ(1±ik) 2 has the following properties: 39 Indeed, evaluating this correction for h = 1, h = 1/2 and h = −1/2 agrees with the second-and third-order terms in all of the above series expansions (B.3), (B.4) and (2.8). 40 This also resolves a small puzzle we faced in section 4 of [19]. We performed the small-τ expansion of the h = 1 bilocal and matched the first correction to the 1/τ 2 pole to 1 6 T , but we didn't explain the zero-temperature additive constant. With formula (B.8) and our general understanding of the τ -expansion, we can indeed check numerically that the zero-temperature piece is indeed captured by the third term in (B.8).
• It is positive and monotonically increasing as a function of h.
• For any real value of h, this function is bounded by a polynomial in k; this is because for h ∈ N it is polynomial. Since any h has an integer on top of it, and since the function is monotonically increasing, this result follows.
Hence, the coefficient at a fixed positive order n in the τ -expansion, n C n (h)τ n , in the expansion is given by: The integral is convergent by the polynomial boundedness of f k . The series itself is an alternating series with the following important property. If the series is asymptotic for some value h * , then it is automatically so for higher values of h > h * ; indeed, due to the monotonicity properties of f k (h), the size of the coefficient |C n | increases for higher values of h. We hence investigate the limiting case where h → −∞, for which we can use the asymptotic Stirling result: Γ(h ± ik) ∼ 2π |h| 2h−1 e −2πk e −2h . (B.11) The coefficient C n in this limit becomes: 41 (B.12) One checks explicitly that this diverges as C n ∼ n as n → +∞, meaning even the h → −∞ series is asymptotic. By the above argument, this then happens for any h ∈ R.

Energy eigenstate
For an energy eigenstate with k 2 = M , one only replaces the function f k (h) by: which is likewise positive, monotonically increasing as a function of h and polynomially bounded in k. The expansion coefficient is now: (B.14) with C n (h, M = 0) ≡ C n (h). As before, it suffices to look at the relevant h → −∞ asymptotics: It is readily seen that C n (h → −∞, M ) decreases for increasing M , but increases for increasing n sufficiently large. 43 This behavior with n immediately shows that the resulting τ /C expansion is again asymptotic, for any value of h.

Thermal ensemble
For the thermal ensemble, one then has to take the final convergent Laplace M -integral has larger coefficients D n h, β C for n sufficiently large, and is thus asymptotic only. This is a property of the series expansion (1.8) written in the introduction. 44 If one imagines creating the thermal background in the bulk through gravitational effects, then it is natural to take β ∼ C. The above result then shows that if we think of the coherent background as created through gravitational fluctuations, then that perturbative series is asymptotic. Now let's not assume β proportional to C, and interpret β as an independent length scale. One can rearrange the series into a power series in 1/C, by writing: n D n h, β C τ n C n ≡ pD p h, τ β τ p C p , (B.20) 42 Notice the presence of the 1/C n coefficient which has to be here for dimensional reasons. 43 This last property can be checked numerically. 44 From the perspective of this section it seems harder to make explicit the form of the coefficients Dn h, β C in terms of multi-stress tensor correlators T n , which is something where the degenerate cases h ∈ −N/2 are more convenient for.
in term of new coefficientsD p h, τ β , and this series must hence also be divergent. We now prove that the coefficientsD p h, τ β appearing in the 1/C expansion (B.20) are welldefined, and that the divergence of the series (B.20) only comes from the summation, making it asymptotic as well. The coefficientD p h, τ β in this series can be found in the Schwarzian 1/C perturbation series generically by using the higher vertices of the Schwarzian action (3.1) (which only contribute a τ /β-independent factor), the propagator (3.2) and the expansion of the bilocal itself (3.3). A crucial property is that the coefficient has no singularity as τ /β → 0 (due to the existence of a well-defined zero-temperature limit). The coefficientD p h, τ β at fixed order p is a (complicated) analytic function of τ /β within its convergence radius 0 ≤ τ < β, with a convergent Taylor series expansion within that radius. The divergence of the series (B.20) can then only arise from the summation over p, meaning the 1/C Schwarzian perturbation series is an asymptotic series as well. We have hence proved rather generically that the series expansion is asymptotic both at zero temperature, for an energy eigenstate, and at finite temperature.
However, there is an exception to this analysis: if Γ(2h) = ∞, and hence h = 0, −1/2, −1 . . ., the above k-integrals diverge from the k 1 − k 2 ≈ im region (for a (half)integer m) but this is cancelled by a similar divergence from Γ(2h). Hence this case requires separate treatment, and this is our main story in section 2.

Non-perturbative corrections
It is interesting to assess the size of the non-perturbative contributions, that goes beyond boundary gravitons, by looking at the growth of the coefficients of this asymptotic series. We will only discuss the zero-temperature case, as we don't expect finite temperature to change this. For both h = 1 and h = 1/2 the coefficients at large n grow as 45 C n ∼ n! √ n(2π 2 ) n .

(B.21)
To estimate the size of non-perturbative corrections, a useful diagnostic is to Borel transform this series as [92]: where the pole in the Borel t-plane closest to the origin signals the size of the leading nonperturbative corrections. Since this Borel series has its convergence properties from the geometric series, we find the radius of convergence t * ∼ 2π, leading to non-perturbative effects of order It is interesting to note that this is of the same order in G N ∼ 1/C as the perturbative genus expansion of the matrix integral in powers of e −#S 0 = e −#C .

C Expansion of classical bilocal in SL(2, R) invariants
Starting with a classical bilocal (1.2), one can perform a small τ -expansion with coefficients that have to be, due to local SL(2, R) invariance, composed of the Schwarzian derivative, its derivatives and its powers. Can we understand such small time expansions within correlation functions, We will take up the simplest degenerate correlator with h = −1/2 as a case study. Denoting the Schwarzian derivative T (τ ) ≡ {F, τ }, the classical bilocal (1.2) for h = −1/2 has the small time expansion: 160T − 160(T ) 2 − 208T T + 8T 3 0 τ 7 + . . .
Comparing (C.2) and (3.6), several terms match, such as the T contribution at τ 3 and the T 2 at τ 5 . However, other terms are more puzzling: such as the τ 2 term and τ 4 terms.
In order to properly identify contributions, a crucial point is the renormalization of the composite stress tensors, which start at O(τ 5 ). 46 The Schwarzian derivative two-point function is: and diverges at τ = 0 due to the contact terms. The last term written here is a zero-temperature contribution that depends on precisely how one defines the operator. Note that even renormalizing by subtracting the zero-temperature answer still isn't enough to renormalize this correlator. These terms actually cause a breakdown of the perturbative expansion (C.2). Nonetheless, we would like to find out which renormalization is implicitly used in the correlation functions (3.6). We propose to replace the contact terms by a pole instead using the role δ (n) (τ ) → 1/τ n+1 , (C.4) replacing the n-fold derivative of the delta-function by a pole. We were not able to determine the precise coefficient, but will check now that this rule explains the general structure in (3.6). 47 Using this rule, these contribute to the perturbative expansion at lower orders in the τ expansion. This is dimensionally correct, and we illustrate here how the previous missing terms and the particular structure in (3.6) can be explained.
The τ 6 -term only contributes giving a zero-temperature τ 2 -term and a T τ 4 -term.

C.1 Aside: normal-ordered bilocal operators
Within the gravitational quantum theory, one can alternatively define the normal-ordered bilocal operator by entirely removing all singular contact term contributions studied above. Since these contact terms arise from poles in the OPE when coming from 2d CFT [19], this is the same definition of normal-ordering of an interacting 2d CFT [93]. We will denote this bilocal operator as: : (C.9) For this case of h = −1/2 (2.8), one has a simple adjustment of the bilocal operator that sets to zero all contact term contributions, by setting: 8Cβ . (C.10) One checks explicitly that indeed stripping off this factor of e τ 8C , one finds solely the T n contributions to the series expansion. We will call this the normal-ordered bilocal operator. Explicitly (−) n+1 Γ(2n)2 n−1 T n−1 τ 2n−1 . (C.11) This means one has a simple procedure for computing an arbitrary n-point correlator of Schwarzian derivatives, by considering the normal-ordered h = −1/2 bilocal as a generating function. The generalization of the correlator to arbitrary h ∈ R, including the supersymmetric N = 1 case, is readily made: : where one has dµ(k) = dkk sinh(2πk) for the bosonic theory, and dµ(k) = dk2 cosh(2πk) for the N = 1 case. This has a small τ expansion of the same structure as (C.11).

D Special Functions
The b-deformed Gamma-function is defined by where Γ 2 (z| 1 , 2 ) is the Barnes double gamma function. We collect several useful formulae.
The b-deformed gamma function Γ b (x) has the shift properties: One defines the double sine function S b (x) as which has the properties: (D.4) and the small b-limits: The supersymmetric extensions are defined as: and have the properties:

E Details on supersymmetric correlators
This appendix contains some technical details on the N = 1 boundary two-point functions on the disk.