aainstitutetext: Dipartimento SMFI, Università di Parma and INFN Gruppo
Collegato di Parma, Viale G.P. Usberti 7/A, 43100 Parma, Italybbinstitutetext:
Galileo Galilei Institute for Theoretical Physics, INFN, Largo Enrico Fermi,
2, 50125 Firenze, Italyccinstitutetext: Dipartimento di Fisica, Università di
Firenze and INFN Sezione di Firenze, via G. Sansone 1, 50019 Sesto Fiorentino,
Italy

# The resurgence of the plateau in supersymmetric ${\cal N}=1$ Jackiw-
Teitelboim gravity

Luca Griguolo b Jacopo Papalini c Lorenzo Russo c and Domenico Seminara
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Significant progresses have been made recently in understanding the spectral
form factor of Jackiw-Teitelboim gravity, particularly at late times where
non-perturbative effects are expected to play a dominant role. By focusing on
a peculiar regime of large time and fixed temperature, called $\tau$-scaling
limit, it was found that it is possible to analytically investigate the late-
time plateau directly through the gravitational genus expansion. We extend
this analysis to the supersymmetric $\mathcal{N}=1$ generalization of the
bosonic theory, revealing an interesting structure. First, we notice that in
the $\tau$-scaling limit the perturbative sum over genera truncates after a
single term, which solely accounts for the ramp behaviour. Instead a non-
perturbative completion, responsible for the plateau, is encoded into an exact
formula coming from the properties of the chiral gaussian ensemble, governing
the spectral properties of the supersymmetric theory. We are able to recover
the non-perturbative contributions by slightly deforming the genus of the
involved surfaces and using resurgence theory. We derive a closed-form
analytical expression for the late-time plateau and a trans-series expansion
that captures the ramp-plateau transition.

##  1 Introduction

An important challenge in contemporary theoretical physics is to explain the
discrete spectrum of microstates of black holes, as predicted by a complete
theory of quantum gravity. In recent years, significant progress has been made
in this direction, mainly exploiting non-perturbative quantum properties of
low-dimensional gravities. In this context wormholes geometries have played a
prominent role: they enter in the derivation of the Page curve Almheiri_2020 ;
Penington:2019kki  and elucidate various aspects related to late-time
correlators Blommaert_2019 ; Saad:2019pqd ; Blommaert:2020seb . Wormholes also
appear in direct analysis of the discreteness of energy levels of quantum
black holes. A notable example is the spectral form factor (SFF), a toy model
for the thermal two-point function which is particularly valuable in measuring
level statistics in quantum chaotic systems PhysRevLett.56.2449 .

In a dynamical theory of gravity, Maldacena Maldacena:2001kr  pointed out that
correlation functions measured in black hole backgrounds should not decay at
late times, as black holes themselves represent finite entropy systems.
Consequently, the behavior of the SFF serves as an excellent diagnostic for
discerning the discreteness of energy levels Cotler:2016fpe . A concrete
application of these ideas, in the context of quantum gravity and holography,
is the inspection of the SFF in the Sachdev-Ye-Kitaev (SYK) model
Sachdev:1992fk ; Kitaev:2017awl : this observable displays a peculiar ramp and
plateau structure as a function of time Cotler:2016fpe ; You_2017 ; Garcia-
Garcia:2016mno  and supports the conjecture that statistical properties of the
energy levels of quantum chaotic systems are universally described by random
matrix models Bohigas:1983er . As a matter of fact, Jackiw-Teitelboim (JT)
gravity Jackiw:1984je ; Teitelboim:1983ux , which is holographically dual to
the low-energy sector of the SYK model was later found to be described by a
double-scaled random matrix model Saad:2019lba . In JT gravity the form of the
SFF in not completely understood from the gravitational perspective, as the
ramp arises from the perturbative contribution of a wormhole Saad:2018bqo
connecting the two boundaries of the spacetime, but the plateau necessitates
of additional non-perturbative effects Cotler:2016fpe ; Saad:2019lba .
Instead, the complementary picture from the matrix model side is more
encouraging and in Johnson:2020exp  it was demonstrated how the correct
structure of the SFF, incorporating essential non-perturbative contributions,
can be numerically obtained by interpreting JT gravity as a combination of
minimal string models.

Nonetheless, it is of utmost importance to gain some insights into the physics
that governs the transition from the ramp to the plateau directly from a bulk
geometrical perspective. Remarkably, it was recently realized Saad:2022kfe ;
Blommaert:2022lbh ; Okuyama:2023pio  that the perturbative topological
expansion of JT gravity contains far more information than initially believed.
By taking a suitable rescaling for the late time, known as the $\tau$-scaling
limit, it becomes possible to sum the entire asymptotic perturbative series
and to obtain an expression that not only captures the ramp but also accounts
for the plateau.

Encouraged by these results, we perform here the analysis of the SFF in the
$\mathcal{N}=1$ Super JT gravity (SJT), a supersymmetric generalization of the
bosonic theory Stanford:2019vob , in the $\tau$-scaling limit. Previous
studies of the SFF in this theory were conducted in Johnson:2020exp
exploiting RMT tools, an approach that required significant numerical efforts.
We argue that it becomes instead possible to investigate the plateau behavior
of the theory analytically, thereby highlighting the form of the necessary
non-perturbative contributions that complement the gravitational topological
expansion.

#### Summary of results

The paper is organized as follows. In Section 2, we review the main results
concerning the $\tau$-scaling limit of the SFF in JT gravity and its low-
energy regime, the Airy model. By exploiting the universal behavior of the
density-density correlator in matrix models of the unitary class, we obtain an
integral representation for the $\tau$-scaled SFF and demonstrate that the
bulk topological expansion becomes convergent. Interestingly, after the
summation, the result features a ramp and a late-time plateau, allowing for
the investigation of seemingly non-perturbative phenomena directly from the
bulk.

In Section 3.1 we explore the correspondent genus expansion for the
$\tau$-scaled SFF in SJT gravity. With a simple proof based on the
mathematical properties of the Weyl-Petersson super-volumes, we show that only
the genus zero term contributes to our observable. In detail, only a linear
piece in $\tau$ survives. This term arises from the cylinder surface built by
gluing two asymptotic trumpets with all the corrections coming from the
surfaces of higher genus vanishing. Using this result, we argue that in SJT
gravity only the ramp can be accessed perturbatively and it becomes imperative
to incorporate certain non-perturbative effects to witness the emergence of
the plateau.

Guided by this observation, in Section 3.2 we begin the investigation of the
non-perturbative structure of the SFF of SJT gravity drawing upon the
correspondence of the theory to an Altland-Zirnbauer matrix model, the so
called chiral Gaussian unitary ensemble. The statistics governing the
correlation between close eigenvalues here has been recently studied in
Akemann:2021uhf : as long as the eigenvalues are in the bulk of the spectral
support, the repulsion between them is governed by the universal sine kernel,
appearing also in the bosonic case. On the other hand, as one moves towards
the hard edge of the spectrum, the eigenvalue statistics is controlled by the
Bessel kernel, that controls the low-energy limit of SJT. Moreover, in
Akemann:2021uhf  it has been shown that, as the level spacing goes to zero,
the Bessel kernel becomes well approximated by the sine kernel. Relying on
this finding, we argue that the integral representation, obtained in the
previous section, can be employed to compute the $\tau$-scaled SFF for both
the Bessel model and SJT gravity. By explicitly evaluating the observable in
the case of the Bessel model, we find that the perturbative series in $\tau$
terminates after the linear term, as expected from the genus expansion.
However, a non-perturbative completion emerges, precisely determining the
late-time plateau.

The remaining part of the paper is dedicated to deduce the analytic form of
these non-perturbative contributions in SJT gravity from a more geometrical
perspective. More precisely, our objective is to obtain a trans-series
representation for the $\tau$-scaled SFF relying on the techniques of
resurgence theory ecalle1985fonctions . This approach provides a general
mathematical framework to extract the non-perturbative pieces necessary to
have a well-defined physical observable solely in terms of perturbative data.
Resurgence techniques have been already applied to JT gravity: we stress that
without the $\tau$-scaling limit the genus expansion is in fact asymptotic and
resurgence is an ideal approach to recover its non-perturbative completion. In
a beautiful series of papers Gregori:2021tvs ; Eynard:2023qdr ;
Schiappa:2023ned  the resurgent nature of JT gravity has been carefully
studied, starting both from its minimal string embedding and its Weil-Peterson
volumes interpretation. As a result instanton-actions and Stokes data for
transseries have been computed and tested, while the calculations have been
systematized into the framework of a non-perturbative extension of topological
recursion relation Eynard:2007kz . Remarkably, in the present case, only the
perturbative contributions have a clear geometrical interpretation and it
would be insightful to derive some links between the genus expansion and the
non-perturbative matrix model results. As already noticed, in SJT gravity only
one perturbative coefficient is non-vanishing in the $\tau$-scaling limit and
it would seem impossible to successfully complete the resurgence program, that
is usually based on the properties of asymptotic series. Our central
observation is that the perturbative contribution to the SSF is encoded in a
particular coefficient $P_{g}^{(\rho)}(\beta)$, weighting the contribution of
a genus $g$ topology and depending on the original spectral density $\rho$. In
Section 4.1 we find that by formally deforming the parameter $g$, the
perturbative series in $\tau$ defined through $P_{g}^{(\rho)}(\beta)$
immediately becomes asymptotic and admits a resurgent analysis. This
particular realization of resurgence theory is known in the literature as
Cheshire cat resurgence article1 ; Dorigoni:2017smz ; Dorigoni:2019kux , and
it has been already exploited in previous papers to complete models with
”trivial” perturbative series Fujimori:2022qij . Thanks to this method we are
able to recover the exact form of the non-perturbative correction in the low-
energy regime of SJT gravity, i.e. the Bessel model, fixing the Stokes
coefficient that could not be determined by resurgence alone.

Leveraging this result, in Section 4.2, we investigate the SFF in a
mathematical example that interpolates between the Bessel and the Airy models.
This artificial construction, emerging from the low-energy expansion of the
SJT spectral density, not only serves as a toy model for the most general case
but will also provide a generating function for the computation of the SFF in
SJT gravity. Using our method we are able to determine a trans-series
expression for the $\tau$-scaled SFF and determine an analytic form for its
late-time plateau. We also perform a numerical analysis highlighting a
remarkable agreement between the trans-series and the exact result.

In Section 4.3 we finally generalize the analysis to the theory of SJT
gravity. In this case, the technical part of the computation is much more
intricate, but we are still able to obtain a trans-series expression for the
$\tau$-scaled SFF. Our conclusion are contained in Section 5, while we defer
in Appendix C some technical aspects of our computations.

##  2 Review of $\tau$-scaling limit of $\mathrm{SFF}$

In this section, we will briefly summarize the main results of the SFF in the
$\tau$-scaling limit. So far, at least to our knowledge, this analysis was
carried out for the JT gravity case and its low energy limit, the Airy model
Saad:2022kfe ; Okuyama:2023pio . Interestingly, it was discovered that the
spectral form factor for these two theories exhibits a simple form with a
convergent genus expansion Saad:2022kfe .

The spectral form factor is defined as the expectation value of the product of
two analytically continued partition functions, i.e.

$\mathrm{SSF}(\beta,t)\equiv\langle Z(\beta+it)Z(\beta-it)\rangle,$ (1)

where $Z(\beta)=\Tr e^{-\beta H}$. We can express the r.h.s of (1) as a double
Laplace transform over the energies $E_{1}$ and $E_{2}$ of the pair-
correlation density111We recall that
$\rho(E_{1},E_{2})=\expectationvalue{\tilde{\rho}(E_{1})\tilde{\rho}(E_{2})}$,
where $\tilde{\rho}(E)=\sum_{k}\delta(E-E_{k})$ is the spectral
density.$\rho(E_{1},E_{2})$:

$\mathrm{SSF}(\beta,t)=\int_{0}^{\infty}\mathrm{d}E_{1}\int_{0}^{\infty}\mathrm{d}E_{2}e^{-\beta(E_{1}+E_{2})}e^{-it(E_{1}-E_{2})}\rho(E_{1},E_{2}).$
(2)

Below, we shall focus on the connected part of the $\mathrm{SSF}(\beta,t)$,
which is defined as

$\mathrm{SSF}_{\rm c}(\beta,t)\equiv\langle Z(\beta+it)Z(\beta-
it)\rangle_{c}=\langle Z(\beta+it)Z(\beta-
it)\rangle-\expectationvalue{Z(\beta+it)}\expectationvalue{Z(\beta-it)}.$

In particular, we are interested in its behavior at late times. Using the
stationary phase method for large values of $t$, the integral (2) is dominated
by contributions from regions where $E_{1}$ and $E_{2}$ are very close to each
other. In this regime we can express $\mathrm{SSF}_{\rm c}(\beta,t)$ as

$\mathrm{SSF}_{\rm
c}(\beta,t)\underset{t\rightarrow+\infty}{\approx}\int_{0}^{\infty}\mathrm{d}E_{1}\int_{0}^{\infty}\mathrm{d}E_{2}e^{-\beta(E_{1}+E_{2})}e^{it(E_{1}-E_{2})}\rho_{\rm
c}^{\mathrm{eff}}(E_{1},E_{2})$ (3)

where $\rho_{\rm c}^{\mathrm{eff}}(E_{1},E_{2})$ is an effective pair-
correlation density experienced by neighboring energy levels. For the cases
under scrutiny, it is given by the well-known sine kernel:

$\rho_{\rm
c}^{\mathrm{eff}}(E_{1},E_{2})=\delta\left(E_{-}\right)\rho(E)-\frac{\sin^{2}\left(\pi\rho(E)E_{-}\right)}{\pi^{2}E_{-}^{2}},$
(4)

where $E_{-}\equiv E_{1}-E_{2}$ and $E\equiv\frac{E_{1}+E_{2}}{2}$. The result
(4) actually holds for any random matrix potential $V(H)$ in the unitary
universality class222This is a direct manifestation of random matrix
universality, governing universal level statistics at small energy
separations..

Performing the integration over $E_{-}$ in (4) in the large $t-$limit leads to
the following integral representation of $\mathrm{SSF}_{\rm c}(\beta,t)$ in
terms of the usual eigenvalue density $\rho(E)$:

$\mathrm{SSF}_{\rm
c}(\beta,t)\underset{t\rightarrow+\infty}{\approx}\int_{0}^{\infty}\mathrm{d}Ee^{-2\beta
E}\min\left(\rho(E),\frac{t}{2\pi}\right)$ (5)

In Saad:2022kfe  it was argued that the uncontrolled approximation (5) can be
made rigorous and ultimately exact in the so-called $\tau$-scaled limit. It is
a double-scaling limit where $t$ and $S_{0}$ go to $\infty$ while
$\tau=e^{-S_{0}}t$ is held fixed. Then the spectral form factor in this limit,
$\mathrm{SFF}_{\infty}(\beta,\tau)$ is defined as

$\mathrm{SFF}_{\infty}(\beta,\tau)\equiv\lim_{S_{0}\to+\infty}e^{-S_{0}}\langle
Z(\beta+i\tau e^{S_{0}})Z(\beta-i\tau e^{S_{0}})\rangle_{\rm c}$ (6)

and it is computed by the r.h.s. of (5). This fact was first checked for the
Airy model Saad:2022kfe , which provides the low energy approximation of JT
gravity on orientable Riemann surfaces. In fact, if one exploits the disk-
level spectral density $\rho_{\rm Airy}(E)=\frac{\sqrt{E}}{2\pi}$, the
integral (5) yields:

$\begin{split}\mathrm{SFF}_{\infty}^{\mathrm{Airy}}(\beta,\tau)&=\int_{0}^{\infty}\mathrm{d}Ee^{-2\beta
E}\min\left(\frac{\sqrt{E}}{2\pi},\frac{\tau}{2\pi}\right)\\\
&=\frac{1}{2\pi}\frac{\pi^{\frac{1}{2}}}{2^{\frac{5}{2}}\beta^{\frac{3}{2}}}\erf\left(\sqrt{2\beta}\tau\right)=\frac{\tau}{4\beta\pi}-\frac{\tau^{3}}{6\pi}+\frac{\beta}{10\pi}\tau^{5}+...\end{split}$
(7)

Strikingly, this result perfectly reproduces the $\tau$-scaled limit evaluated
from the exact expression of the spectral form factor within the Airy model
Okuyama:2021cub . Furthermore, an insightful interpretation of the terms
appearing in the expansion (7) in terms of a semiclassical theory of
“encounters” of periodic orbits was discussed in Saad:2022kfe .

When applied to JT gravity, whose spectral density is $\rho_{\rm
JT}(E)=\frac{1}{4\pi^{2}}\sinh\left(2\pi\sqrt{E}\right)$, the integral
representation (5) of the spectral form factor in the $\tau$-scaled limit
yields

$\begin{split}\mathrm{SFF}_{\infty}^{\mathrm{JT}}&(\beta,\tau)=\frac{1}{2\pi}\int_{0}^{\infty}\mathrm{d}Ee^{-2\beta
E}\min\left(\frac{1}{2\pi}\sinh\left(2\pi\sqrt{E}\right),\tau\right)\\\
&=\frac{e^{\frac{\pi^{2}}{2\beta}}}{16\sqrt{2\pi}\beta^{\frac{3}{2}}}\left[\erf\left(\frac{\frac{\beta}{\pi}\mathrm{arcsinh}(2\pi\tau)+\pi}{\sqrt{2\beta}}\right)+\erf\left(\frac{\frac{\beta}{\pi}\mathrm{arcsinh}(2\pi\tau)-\pi}{\sqrt{2\beta}}\right)\right].\end{split}$
(8)

While the usual genus expansion of the complete SFF in JT gravity is an
asymptotic series, expanding (8) in powers of $\tau$ produces a convergent
series333One can immediately realize the series in $\tau$ corresponds to the
genus expansion, by undoing the $\tau$ scaling.. This result was significant,
as it paved the way for a perturbative treatment of the late-time plateau.
Unlike the Airy model, an exact closed expression for the SFF in JT gravity
remains elusive. However one can check the validity of the result (8) using
the topological recursion techniques. In Saad:2022kfe  it was verified that
the expansion of (8) accurately reproduces the $\tau$-scaled genus expansion
up to order $\tau^{13}$.

Additionally, in cases where the spectral density is a monotonically
increasing function of energy, as occurs in JT gravity and the Airy model, the
coefficient of $\tau^{2g+1}$ for $g\geq 1$ in the Taylor-expansion of the
$\tau-$scaled spectral form factor can be computed using the subsequent
formula:

$P_{g}^{(\rho)}(\beta)=-\frac{1}{g(2g+1)(2\pi)^{2g+1}}\oint\frac{\mathrm{d}E}{2\pi
i}\frac{e^{-2\beta E}}{\rho_{0}(E)^{2g}}.$ (9)

Here the integral is evaluated on a contour around the origin. In Saad:2022kfe
it was demonstrated that this prescription, upon summation, reproduces the
integral representation (5), i.e.

$\begin{split}\mathrm{SFF}(\beta,\tau)&=\frac{\tau}{4\pi\beta}+\sum_{g=1}^{\infty}P_{g}^{(\rho)}(\beta)\tau^{2g+1}=\int_{0}^{\infty}\mathrm{d}Ee^{-2\beta
E}\min\left(\rho_{0}(E),\frac{\tau}{2\pi}\right).\end{split}$ (10)

The aim of the upcoming sections is to broaden the application of this
formalism within the framework of SJT gravity. A pivotal technical step in
achieving this goal will entail extending the definition of (9) to accommodate
a non-monotonically increasing spectral density.

##  3 The $\tau$-scaling limit for SJT gravity

###  3.1 A perturbative bulk expansion

This section is devoted to understanding the general form of all terms in the
genus expansion of the Spectral Form Factor in SJT gravity444In this paper we
only consider the $\mathcal{N}=1$ supersymmetric generalization of the bosonic
theory where time reversal is not gauged and each spin structure of the
manifolds appearing in the path integral is weighted by $(-1)^{\zeta}$, with
$\zeta$ the Atiyah-Singer index.  from the bulk point of view. This derivation
closely mirrors the one conducted in Blommaert:2022lbh  for JT gravity. While
in the cases of JT gravity, the cancellation of certain terms exhibiting
unfavorable growth with respect to $\tau$ in the genus $g$ allows for
perturbative access to the plateau during the $\tau$-scaling limit, our
findings in the SJT scenario reveal a different outcome. Specifically, we
observe that all higher genus terms vanish in the $\tau$-scaling limit.

By employing the standard topological decomposition of the gravitational path
integral, the key ingredients for our analysis consist of the volumes of
moduli space of super Riemann surfaces of genus $g$ and $n$ boundaries having
lengths $b_{1},\cdots,b_{n}$. These volume are denotes as
$\tilde{V}_{g,n}(b_{1},...,b_{n})$. Furthermore we need the partition function
$Z_{\mathrm{SJT}}^{\mathrm{tr}}(\beta,b)$ on the trumpet spacetime. In fact,
these two objects serve as the building blocks to construct the partition
function on any topology through a simple gluing procedure.

The partition function of SJT on the basic topologies, namely the disk and the
trumpet, was explicitly computed in [22] using the super-Schwarzian model of
the boundary theory. The expressions for these partition functions are as
follows:

$Z_{\mathrm{SJT}}^{\mathrm{disk}}(\beta,b)=\sqrt{\frac{2}{\pi\beta}}e^{\frac{\pi^{2}}{\beta}}\qquad
Z_{\mathrm{SJT}}^{\mathrm{tr}}(\beta,b)=\frac{1}{\sqrt{2\pi\beta}}e^{-\frac{b^{2}}{4\beta}}$
(11)

To investigate the SFF, we have to compute the connected two-boundary
correlator of the theory. This correlator exhibits the following genus
expansion:

$Z^{\mathrm{SJT}}(\beta_{1},\beta_{2})=\sum_{g=0}^{+\infty}Z_{g}^{\mathrm{SJT}}(\beta_{1},\beta_{2})+[\mathrm{nonperturbative\
terms}].$ (12)

The investigation of the possible nonperturbative completion appearing in the
expansion (12) will constitute the topic of the next Section. In (12) we
denoted with $Z_{g}^{\mathrm{SJT}}(\beta_{1},\beta_{2})$ the genus-g
gravitational wormhole amplitude that can be evaluated by gluing together two
trumpets through the insertion of a Weil-Petersson volume
$\tilde{V}_{g,2}(b_{1},b_{2})$. Achieving this involves integrating over all
possible lengths, $b_{1}$ and $b_{2}$, of the geodesic boundaries, i.e.

$Z_{g}^{\mathrm{SJT}}(\beta_{1},\beta_{2})=e^{-2gS_{0}}\int_{0}^{\infty}b_{1}\mathrm{d}b_{1}\int_{0}^{\infty}b_{2}\mathrm{d}b_{2}\tilde{V}_{g,2}(b_{1},b_{2})Z_{\mathrm{SJT}}^{\mathrm{tr}}(\beta_{1},b_{1})Z_{\mathrm{SJT}}^{\mathrm{tr}}(\beta_{2},b_{2}).$
(13)

Above the genus-counting topological factor $e^{-2gS_{0}}$ accounts for the
contribution of the Einstein-Hilbert action, which weights the various terms
in the genus expansion. To compute the volumes of the moduli space of super
Riemann surfaces, one could use the analogous of the recurrence relations
introduced by Mirzakhani for the bosonic case Mirzakhani:2006fta ;
Mirzakhani:2006eta  and generalized to the superspace in Stanford:2019vob .
However, for the upcoming discussion, it will not be necessary to have the
explicit expressions of the volumes. We will only need to know that these
volumes are polynomials in $b_{i}^{2}$ of degree $g-1$ for $g>0$
Norbury:2020vyi . Thus we formally write

$\tilde{V}_{g,2}(b_{1},b_{2})=\sum_{d_{1},d_{2}=0}^{d_{1}+d_{2}=g-1}\frac{\tilde{V}_{g,n}^{d_{1},d_{2}}}{4^{d_{1}}4^{d_{2}}d_{1}!d_{2}!}b_{1}^{2d_{1}}b_{2}^{2d_{2}}.$
(14)

The unknown coefficients $\tilde{V}_{g,n}^{d_{1},d_{2}}$ could be computed
explicitly exploiting the results in Norbury:2020vyi . Plugging the polynomial
(14) into (13), we obtain the following representation for the genus-$g$
contribution to the wormhole amplitude:

$Z_{g}^{\mathrm{SJT}}(\beta_{1},\beta_{2})=\frac{e^{-2gS_{0}}}{2\pi\sqrt{\beta_{1}\beta_{2}}}\sum_{d_{1},d_{2}=0}^{d_{1}+d_{2}=g-1}\frac{\tilde{V}_{g,n}^{d_{1},d_{2}}}{4^{d_{1}}4^{d_{2}}d_{1}!d_{2}!}\int_{0}^{\infty}b_{1}\mathrm{d}b_{1}\int_{0}^{\infty}b_{2}\mathrm{d}b_{2}\
b_{1}^{2d_{1}}b_{2}^{2d_{2}}\
e^{-\frac{b_{1}^{2}}{4\beta_{1}}}e^{-\frac{b_{2}^{2}}{4\beta_{2}}}$ (15)

By integrating over $b_{1}$ and $b_{2}$, and subsequently analytically
continuing the expression to complex values of $\beta_{1,2}$, i.e.,
$\beta_{1,2}=\beta\pm it$, we obtain

$Z_{g}^{\mathrm{SJT}}(\beta_{1}-it,\beta_{2}+it)=\frac{2}{\pi}e^{-2gS_{0}}\sum_{d_{1},d_{2}=0}^{d_{1}+d_{2}=g-1}\tilde{V}_{g,2}^{d_{1},d_{2}}\left(\beta+it\right)^{d_{1}+\frac{1}{2}}\left(\beta-
it\right)^{d_{2}+\frac{1}{2}}.$ (16)

The leading behavior of (16) in the limit $t\rightarrow+\infty$ only comes
from the terms with $d_{1}+d_{2}=g-1$. Thus the leading contribution to
$Z_{g}^{\mathrm{SJT}}$ takes the form:

$Z_{g}^{\mathrm{SJT}}(\beta+it,\beta-
it)\underset{t\rightarrow+\infty}{\approx}C_{g}e^{-2gS_{0}}t^{g}$ (17)

for some unknown constant coefficient $C_{g}$. If we now switch to the
rescaled time variable $\tau=e^{-S_{0}}t$ and we perform the limit
$S_{0}\rightarrow\infty$, we get for $g>0$

$\lim_{S_{0}\to\infty}e^{-S_{0}}Z_{g}^{\mathrm{SJT}}(\beta+it,\beta-
it)=\lim_{S_{0}\to\infty}C_{g}e^{-(g+1)S_{0}}\tau^{g}=0.$ (18)

Clearly, all the subleading terms that we have omitted in (17) will also
become negligible in this regime.

This proves that, in the context of SJT, all contributions to the SFF arising
from surfaces of genus $g>0$ vanish in the $\tau$-scaling limit. The only
exception to this scenario is the genus-zero contribution, i.e. the cylinder
that is built by directly gluing two trumpets together. Its expression is
given as

$Z_{g=0}^{\mathrm{SJT}}(\beta_{1},\beta_{2})=c\frac{\sqrt{\beta^{2}+t^{2}}}{2\pi\beta}\underset{t\rightarrow+\infty}{\approx}c\frac{t}{2\pi\beta}.$
(19)

Here, following the conventions in Stanford:2019vob , we have incorporated a
factor $c$ that can take on different values depending on whether we sum over
spin structures and orientation reversal. In our specific case, where we sum
over one but not the other and thus we set $c=2.$ Consequently, after
substituting $t=\tau e^{S_{0}}$ and multiplying by a scaling factor of
$e^{-S_{0}}$ as indicated in (18), we arrive at a finite contribution of
$\frac{\tau}{\pi\beta}$ from the double-trumpet topology in the $\tau$-scaling
limit. This remains the only term that survives in the genus expansion (12).

It is worth highlighting some key differences between the current analysis and
the corresponding calculations in JT gravity. In the latter, the volumes of
the moduli space of Riemann surfaces with two boundaries are polynomials of
degree $3g-1$ with respect to the variables $b_{1}^{2}$ and $b_{2}^{2}$. If
one were to extrapolate the current analysis to the JT gravity case, one might
naively anticipate a leading term scaling as $t^{3g}$ for large $t$. However,
as detailed in Blommaert:2022lbh , a remarkable series of highly nontrivial
cancellations occur in the large time expansion of the sum:

$\sum_{d_{1},d_{2}=0}^{d_{1}+d_{2}=3g-1}\tilde{V}_{g,2}^{d_{1},d_{2}}\left(\beta+it\right)^{d_{1}+\frac{1}{2}}\left(\beta-
it\right)^{d_{2}+\frac{1}{2}},$ (20)

This effectively reduces the maximal power of $t$ from $t^{3g}$ to $t^{2g+1}$.
Consequently, the dominant term for JT gravity at genus $g$ is:

$e^{-S_{0}}Z_{g}^{\mathrm{JT}}(\beta+it,\beta-
it)\approx\hat{C}_{g}e^{-(2g+1)S_{0}}t^{2g+1}=\hat{C}_{g}\tau^{2g+1},$ (21)

This term survives the $\tau$-scaling limit for each genus $g>0$, as
substantiated in Blommaert:2022lbh .

###  3.2 The nonperturbative structure

In the previous section, we observed that, with the exception of the linear
term, that causes the ramp growth in the spectral from factor, all
perturbative terms in the genus expansion of the SSF for SJT gravity vanish in
the $\tau$-scaling limit. However, as anticipated in (12), this fact does not
preclude the possibility of nonperturbative contributions appearing in the
rescaled time $\tau$. Indeed, if the genus-zero contribution to the
$\tau$-scaled spectral form factor is meaningful on its own, we expect some
nonperturbative term to be responsibile of compensating the linear growth of
the ramp and generating a late-time plateau.

In this section, we start looking into the nonperturbative structure of the
SFF, focusing firstly on the low-energy regime of SJT gravity, where a
description in terms of the more manageable Bessel model is accessible. In the
same way that the pure Airy case serves as a prototype for the
Saad–Shenker–Stanford description of JT gravity Saad:2019lba , this Bessel
case should in fact play a similar role for SJT gravity. The disk-level
spectral density for SJT gravity, obtained by perfoming the inverse Laplace
transform of the disk result in (11), is in fact given by

$\rho_{0}^{(\mathrm{SJT})}(E)=\sqrt{2}\
\frac{\cosh(2\pi\sqrt{E})}{\pi\sqrt{E}}$ (22)

which reduces indeed to the Bessel spectral density as $E\rightarrow 0$.

In order to provide a sensible nonperturbative completion of the SFF,
sensitive to the underlying discreteness of the energy eigenvalues, we will
exploit the matrix model description of JT supergravity, which was classified
by Stanford and Witten in Stanford:2019vob  by generalizing the relationship
between JT gravity and double-scaled random matrix theory Saad:2019lba . 555In
this work, we will only consider SJT gravity without time reversal, meaning
that we do not have to gauge the time reversal symmetry in the bulk. It was
stated that, when fermions are introduced to the boundary theory, it is
necessary to sum appropriately over the spin structures of orientable surfaces
in the bulk. There are two ways to accomplish this while obeying the
topological field theories’ general requirements Stanford:2019vob : one can
weight the different spin structures with 1 or with $(-1)^{\zeta}$, where
$\zeta$ is the Atiyah-Singer index. These two prescriptions define two
distinct gravitational theories and consequently different dual random matrix
theories, which are distinguished by the properties of their supercharge $Q$.
Weighting with 1 in the bulk corresponds to a theory with a random supercharge
sampled from a Gaussian Unitary Ensemble (GUE), instead weighting with
$(-1)^{\zeta}$ induces a supercharge of the form
$Q=\begin{pmatrix}0&C^{\dagger}\\\ C&0\end{pmatrix}$ where $C$ is a $L\times
L$ complex square matrix transforming as a bifoundamental of $U(L)\times U(L)$
and obeying the Altland-Zirnbauer statistic with $\alpha=1$ and $\beta=2$. In
this work, we will focus on this last case, which in the literature is also
known as chiral Gaussian unitary ensemble. From now on, we will adopt the
standard notation $e^{-S_{0}}\equiv\hbar$ used in the matrix model context.

In the following, we will be interested in the statistics governing the
correlation between close eigenvalues in the aforementioned chiral Gaussian
unitary ensemble. In this context, it has been recently observed
Akemann:2021uhf  that, as long as the eigenvalues are in the bulk of the
spectral support, the repulsion between them is governed by the universal sine
kernel, already introduced in (4). On the other hand, as we move towards the
hard edge of the spectrum at $E=0$, the eigenvalue statistics for this model
is controlled by the Bessel kernel, which we derive in detail in (71).
Moreover, in Akemann:2021uhf  it was also shown that, as the level spacing
goes to zero, the Bessel kernel becomes well approximated by the sine kernel.
In A we review and rephrase the derivation by applying it specifically to the
tau-scaling limit, where this uncontrolled approximation becomes exact.

As a conclusion, by virtue of the same steps that led to (5) through the
Fourier transform of the sine kernel, we claim that the $\tau$-scaled spectral
form factor for SJT gravity is given by 666From now on, we will simply call
SJT gravity the theory corresponding to $(1,2)$ Altland-Zirnbauer without time
reversal. :

$\mathrm{SFF}_{\mathrm{SJT}}=\int_{0}^{\infty}\mathrm{d}Ee^{-2\beta
E}\min\left(\frac{\cosh\left(2\pi\sqrt{E}\right)}{\pi\sqrt{2E}},\frac{\tau}{2\pi}\right)$
(23)

As a first observation, we remark that we have not been able to perform the
integral over $E$ in (23) in closed analytic form, since this would imply
finding the solutions $E^{*}$ to the trascendental equation:

$\frac{\cosh\left(2\pi\sqrt{E^{*}}\right)}{\pi\sqrt{2E^{*}}}=\frac{\tau}{2\pi}$
(24)

Although efficient results can be easily obtained numerically, our focus is to
derive a characterization for the nonperturbative contributions hidden in the
above expression and their possible geometrical interpretation. To this aim,
as a first step, we can see what happens in the Bessel model, where the
solution to the low energy limit of (24) is readily found to be
$E^{*}=2/\tau^{2}$.

$\tau$$\mathrm{SFF}_{\mathrm{Bessel}}(\tau)$ Figure 1: The blue line
corresponds to the result for SFF obtained by approximating the Bessel kernel
with the sine kernel, i.e. formula (25). The other lines correspond to
numerical evaluations of the SFF in (69) using (71) for different values of
$\hbar$. In particular: red line corresponds to $\hbar=\frac{1}{10}$, orange
line to $\hbar=\frac{1}{30}$, green line to $\hbar=\frac{1}{50}$. Finally,
dark dots correspond to $\hbar=\frac{1}{100}$ and are almost perfectly
overlapped with the analytic result, valid as $\hbar\rightarrow 0$. In this
graph, $\beta=1$.

In this case, the integral over $E$ is very easy to perform

$\begin{split}\mathrm{SFF}_{\mathrm{Bessel}}(\beta,\tau)&=\frac{\tau}{2\pi}\int_{0}^{\frac{2}{\tau^{2}}}\mathrm{d}Ee^{-2\beta
E}+\frac{1}{\pi}\int_{\frac{1}{\tau^{2}}}^{+\infty}\frac{\mathrm{d}E}{\sqrt{2E}}e^{-2\beta
E}\\\
&=\frac{\tau}{4\pi\beta}-\frac{\tau}{4\pi\beta}e^{-\frac{4\beta}{\tau^{2}}}+\frac{\text{erfc}\left(\frac{2\sqrt{\beta}}{\tau}\right)}{2\sqrt{\pi}\sqrt{\beta}}\end{split}$
(25)

We note something remarkable: the only perturbative term in $\tau$ appearing
in (25) is the linear one. This result is consistent with what we have found
in Section 3.1 from the bulk point of view, where we proved that for SJT
gravity all higher orders in $\tau$ vanish identically in the
$\hbar\rightarrow 0$ limit. If this property holds for SJT, it was in fact
expected to hold also for its low energy limit, as we confirm here.
Furthermore, the coefficient of the linear term in (25) exactly agrees with
the one computed in (19) from the gravitational perspective 777Actually, there
is a mismatch factor of $1/4$ between the linear term in (25) and the double-
trumpet, that is due to the relation
$Z_{\mathrm{MM}}^{\mathrm{tr}}(\beta,b)=\frac{1}{2}Z_{\mathrm{bulk}}^{\mathrm{tr}}(\beta,b)$
between the matrix model and the bulk normalizations of the trumpet partition
function..

The novelty is that the matrix model description has provided us with a
nonperturbative completion of this bulk picture, represented by the last two
terms in (25). As we can appreciate in Figure 1, these nonperturbative terms
determine a late time plateau. In Figure 1, we have also represented some
numerical evaluations of the SFF (69), by inserting the exact expression (71)
for the Bessel kernel for different values of $\hbar$: we notice that, as
$\hbar$ becomes smaller, the numerical lines tend to the analytic result (25),
as we expected from our discussion above.

A final comment is now in order. While for JT gravity and the Airy model the
$\tau$-scaling limit was responsable for a perturbative plateau (7) (8), in
this case the power series in $\tau$ is completely annihilated, except for the
linear term, and a plateau is produced from purely nonperturbative effects. On
general ground, one would expect that nonperturbative contributions to an
observable arise from the completion of ”standard” perturbation theories, that
are usually asymptotic in quantum systems. The peculiar behavior of (25) makes
us wonder whether there is still some kind of mechanism through which we can
make the nonperturbative part of the result to resurge, so that to find a
connection to a genus -like expansion. A particular form of resurgence theory
will enable us to realize this program, providing us with a analytic
transeries approximation of the SJT $\tau$-scaled spectral form factor (23).

##  4 Resurgence for the $\tau$-scaled SFF

###  4.1 A new contour for $P_{g}^{(\rho)}(\beta)$ and the Bessel model

As anticipated in 2, it was proven in Saad:2022kfe  that the general
coefficient (9) of $\tau^{2g+1}$ generates, upon summation over $g$, the
formula (10) for the spectral form factor in the $\tau$-scaling limit. The
demonstration relies on the fact that the spectral density $\rho_{0}(E)$ is a
monotonic function of the energy, i.e. on the fact that there is a unique
solution $E_{*}$ that satisfies $\rho(E_{*})=\frac{\tau}{2\pi}$.

In this Section, we present an extension of (9) that allows to recover (10)
for more general spectral densities, such as the one relevant for SJT gravity:
in this last case, $\rho_{\mathrm{SJT}}$ is not monotonic and (24) has two
non-trivial solutions $E_{1}^{*}(\tau)$ and $E_{2}^{*}(\tau)$. We propose then
the following generalization of (9):

$P_{g}^{(\rho)}(\beta)=-\frac{1}{g(2g+1)(2\pi)^{2g+1}}\int_{\gamma}\frac{\mathrm{d}E}{2\pi
i}\frac{e^{-2\beta E}}{\rho_{0}(E)^{2g}}$ (26)

where $\gamma$ is the contour diplayed in Figure 2, which encircles the origin
and wraps the whole positive real axis888Mirroring the derivation of
Saad:2022kfe , one can in fact perform the summation over $g$ inside (26) and
then deform $\gamma$ into a contour surrounding the emerging branch cut
between $E_{1}^{*}(\tau)$ and $E_{2}^{*}(\tau)$, the two solutions of the
trascendental equation (24)..

O$E_{1}^{*}$$E_{2}^{*}$$\gamma$$\operatorname{Re}\,E$$\operatorname{Im}\,E$
Figure 2: The new contour $\gamma$ used in (26) is displayed. In the case of
SJT, where after summation a branch cut emerges between $E_{1}^{*}(\tau)$ and
$E_{2}^{*}(\tau)$, $\gamma$ can be deformed into a new contour surrounding the
branch cut.

As a first step, we notice that by simply sobstituting the spectral density of
SJT gravity into (26) it is immediate to realize that the integral vanishes.
This occurrence arises due to the integrand being an entire function, allowing
the contour $\gamma$ to be closed without encircling any singularity along the
real axis. Consequently, the integral evaluates to zero as dictated by
Cauchy’s theorem. This observation indeed aligns with the findings of Section
3.1, where a bulk calculation showed that each higher genus perturbative
contribution to the SFF inevitably vanishes in the $\tau$-scaling limit: the
coefficients (26) correctly encode the perturbative data of the bulk
calculations but the information appears somewhat trivial. However, if we
insist in assuming the definition (26) as the geometrical input for the
perturbative series, we can still gain nontrivial information out of it. For
instance, we can reconstruct from it the tower of nonperturbative
contributions: resurgence theory suggests to extract some asymptotic series
from the SSF constructed in terms of these coefficients. To do this, we employ
a slick trick: we deform the parameter $g$, initially constrained to be a
natural number, to a real value by introducing a small parameter $\epsilon$,
such that $g\to g+\epsilon$. In the literature, this strategy is known as
Cheshire cat resurgence Kozcaz:2016wvy ; Dorigoni:2017smz ; Fujimori:2022qij
999However this is the first instance, at least in our knowledge, that
Cheshire cat resurgence is used in the context of JT gravity.: although the
perturbative series truncates when we send the deformation parameter
$\epsilon$ to zero, we can still reconstruct non-perturbative physics out of
the perturbative data using this method.

Moreover, with this deformation, the implementation of the contour $\gamma$
becomes very natural. This is a direct consequence of introducing in (26) a
branch cut along the positive real axis, which acts as an obstruction to
closing the contour and obtaining a vanishing result.

Let us see what happens concretely in the case of the Bessel model 101010To
simplify numerical factors in the calculation, we find it convenient to
renormalize the spectral density as $\rho_{0}(E)=\frac{1}{2\pi\sqrt{E}}$.. By
sending $g\rightarrow g+\epsilon$ in (26), we get the following form for the
coefficient $P_{g+\epsilon}^{(\rho_{\mathrm{Bessel}})}(\beta)$:

$P_{g+\epsilon}^{(\rho_{\mathrm{Bessel}})}(\beta)=-\frac{1}{2\pi(g+\epsilon)(2(g+\epsilon)+1)}\int_{\gamma}\frac{\mathrm{d}E}{2\pi
i}e^{-2\beta E}E^{g+\epsilon}.$ (27)

The integral can be performed by taking the discontinuity across the real axis
and yields

$P_{g+\epsilon}^{(\rho_{\mathrm{Bessel}})}(\beta)=-\frac{1}{2\pi^{2}(g+\epsilon)(2(g+\epsilon)+1)}e^{i\pi\epsilon}\sin{(\pi\epsilon)}\frac{\Gamma(g+1+\epsilon)}{(2\beta)^{g+1+\epsilon}}$
(28)

Crucially the coefficients of the series no longer vanish and exhibit the
appropriate asymptotic behavior for large genus, growing like $g!$, which is
indicative of an asymptotic series. We now proceed to perform a Borel
transform of the series in $\tau$ and we get: 111111The usual prescription
would be
$\mathcal{B}\left[\sum_{g}a_{g}z^{g}\right](\xi)=\sum_{g}\frac{a_{g}}{g!}\xi^{g}$
but in our case we find it convenient to define it as
$\mathcal{B}\left[\sum_{g}a_{g}z^{g+\epsilon}\right](\xi)=\sum_{g}\frac{a_{g}}{\Gamma(g+1+\epsilon)}\xi^{g+\epsilon}$
to explicitly remove the $\Gamma(g+1+\epsilon)$ factor.

$\begin{split}\mathcal{B}\left[\sum_{g=1}^{\infty}P_{g+\epsilon}^{(\rho_{\mathrm{Bessel}})}(\beta)\
\tau^{2g+2\epsilon+1}\right](\xi)&=\frac{e^{i\pi\epsilon}\sin{(\pi\epsilon)}}{2\pi^{2}}\
\frac{\tau}{2\beta}\sum_{g=1}^{\infty}\frac{\xi^{g+\epsilon}}{(g+\epsilon)(2(g+\epsilon)+1)}\\\
&=C_{\epsilon}\
\frac{\tau}{2\beta}\xi^{1+\epsilon}\left(\Phi(\xi,1,\epsilon+1)-\Phi\left(\xi,1,\epsilon+\frac{3}{2}\right)\right)\end{split}$
(29)

where we defined
$C_{\epsilon}=\frac{e^{i\pi\epsilon}\sin{(\pi\epsilon)}}{2\pi^{2}}$ and we
denoted with $\Phi(\xi,s,a)=\sum_{k=0}^{\infty}\frac{\xi^{k}}{(k+a)^{s}}$ the
Lerch Trascendent. A crucial property of this function is that it has a branch
cut in the complex $\xi$-plane from $\xi=1$ to $\xi=\infty$. Because of that,
we cannot naively perform an inverse Borel transform without running into an
ambiguity, since the real axis represents now a Stokes line in the Borel
plane. Nonetheless, according to Resurgence Theory, we can deal with this
issue through the directional Laplace Transform
$\mathcal{S}_{\vartheta}\left[\mathcal{B}(\xi)\right]$ 121212It is defined as
$\mathcal{S}_{\vartheta}\left[\mathcal{B}(\xi)\right](x)=\frac{1}{x}\int_{0}^{e^{i\vartheta\infty}}e^{-\xi/x}\mathcal{B}(\xi)\mathrm{d}\xi$
: in particular, the nonperturbative contribution we are looking for will be
given by the difference between the directional Laplace Transform performed
slightly above and slightly below the Stokes line at $\vartheta=0$. Writing
the integrals explicitly, we need to evaluate

$\begin{split}\left(\mathcal{S}_{0^{+}}-\mathcal{S}_{0^{-}}\right)\left[\mathcal{B}(\xi)\right]&=\frac{2\beta}{\tau^{2}}\int_{1}^{\infty}\mathrm{d}\xi\,e^{-\frac{2\beta}{\tau^{2}}\xi}\left[\mathcal{B}(\xi+i\delta)-\mathcal{B}(\xi-i\delta)\right]\\\
&=\frac{2\beta}{\tau^{2}}e^{-\frac{2\beta}{\tau^{2}}}\int_{0}^{\infty}\mathrm{d}\xi\,e^{-\frac{2\beta}{\tau^{2}}\xi}\mathrm{Disc}\left(\mathcal{B}\right)(\xi+1)\end{split}$
(30)

where in the second step we shifted the integration variable.

$\frac{\tau}{2\pi}$$\rho_{0}(E)$$E$$\rho_{\mathrm{Bessel}}$$\rho_{\mathrm{SJT}}$$\rho_{\mathrm{LE}}$$E_{1}^{*}(\tau)$$E_{2}^{*}(\tau)$$E_{\mathrm{min}}$
Figure 3: Plot of the spectral density $\rho_{0}(E)$ as a function of the
energy: the blue line represents SJT gravity, the green represents the Bessel
model, while the red line correponds to low energy expansion of SJT, capturing
the first order correction to the Bessel model and the object of 4.2. The
horizontal line, corresponding to $\tau/2\pi$, determines the intersections
$E_{1}^{*}(\tau)$ and $E_{2}^{*}(\tau)$ with $\rho_{\mathrm{SJT}}$.

The final ingredient we need is the jump of $\mathcal{B}(\xi)$ across the
branch cut. The latter can be evaluated using the following expressions for
the discontinuity of the Lerch Phi functions, which hold true when
$\real(\xi)>1$ and $\imaginary(\xi)=0$:

$\begin{split}&\Phi(\xi-i\delta,s,a)=\Phi(\xi,s,a)\\\
&\Phi(\xi+i\delta,s,a)=\Phi(\xi,s,a)+\frac{2\pi
i\xi^{-a}}{\Gamma(s)}\log^{s-1}(\xi)\end{split}$ (31)

Using this result, the discontinuity of the function $\mathcal{B}(\xi)$ across
its branch cut is given by $\mathrm{Disc}(\mathcal{B}(\xi))=2\pi
i(1-\xi^{-\frac{1}{2}})$. Here, we choose to consider the cut of the function
$\xi^{1+\epsilon}$ from $\xi=0$ to $\xi=\infty$ along the negative axis.
Performing the integral (30), we obtain

$\begin{split}\left(\mathcal{S}_{0^{+}}-\mathcal{S}_{0^{-}}\right)\left[\mathcal{B}(\xi)\right]&=C_{\epsilon}\
\frac{2\pi
i}{\tau}e^{-\frac{2\beta}{\tau^{2}}}\int_{0}^{\infty}\mathrm{d}\xi\,e^{-\frac{2\beta}{\tau^{2}}\xi}\left(1-(\xi+1)^{-\frac{1}{2}}\right)\\\
&=-(1-e^{2\pi
i\epsilon})\left(\frac{\tau}{4\pi\beta}e^{-\frac{2\beta}{\tau^{2}}}-\frac{1}{2^{\frac{3}{2}}\sqrt{\pi\beta}}\text{erfc}\left(\frac{\sqrt{2\beta}}{\tau}\right)\right)\end{split}$
(32)

We highlight two observations regarding this result. Firstly, resurgence has
revealed the hidden nonperturbative contribution in (32), which has exactly
the form we were expecting. Secondly, as $\epsilon$ approaches zero, the
overall phase jump in (32) causes the entire result to vanish. This is known
as a Cheshire cat point, that occurs when $g$ is restored to be an integer.
However, it is important to note that resurgence alone does not allow us to
determine the real part of the transseries parameter. Therefore, we are unable
to identify the remaining real contribution as we approach this Cheshire cat
point. The best we can do is to introduce an undetermined transseries
parameter, denoted as $\sigma$ 131313In the context of resurgence, these
constants are referred to as Stokes coefficients., which must be determined
using external input not provided by resurgence. We express this as follows:

$\begin{split}\left(\mathcal{S}_{0^{+}}-\mathcal{S}_{0^{-}}\right)\left[\mathcal{B}(\xi)\right]=\sigma\left(\frac{\tau}{4\pi\beta}e^{-\frac{2\beta}{\tau^{2}}}-\frac{1}{2^{\frac{3}{2}}\sqrt{\pi\beta}}\text{erfc}\left(\frac{\sqrt{2\beta}}{\tau}\right)\right)\end{split}$
(33)

Fortunately, in this case, we have the correct result (25), which we computed
in 3.2. By carefully comparing the different normalizations chosen for the
spectral density 141414In Section 3.2, we considered the spectral density
$\rho_{\mathrm{Bessel}}=\frac{1}{\pi\sqrt{2E}}$, whereas here we considered
$\rho_{\mathrm{Bessel}}=\frac{1}{2\pi\sqrt{E}}$. This amounts to redefining
the constant $e^{S_{0}}\rightarrow\frac{1}{\sqrt{2}}e^{S_{0}}$. Consequently,
we need to rescale $\tau\rightarrow\sqrt{2}\tau$. Additionally, recalling that
$\mathrm{SFF}=\lim_{S_{0}\rightarrow\infty}e^{-S_{0}}\left<ZZ\right>$, there
is an overall relative factor of $1/\sqrt{2}$., we can establish $\sigma=1$.

Thus far, resurgence has enabled us to recover a result that was already
available. However, in the following sections, we will apply this machinery to
SJT gravity, where we lack an exact result, unlike the Bessel model. It is in
this scenario that resurgence will truly demonstrate its full potential.

###  4.2 Low-energy limit: a simplified model

Before delving into the application of this formalism to SJT gravity, it is
valuable to examine a simplified model, specifically its low-energy expansion.
In the subsequent discussion, our focus will be on the spectral density given
by:

$\rho_{\mathrm{LE}}(E)=\frac{1+\alpha E}{2\pi\sqrt{E}}$ (34)

By setting $\alpha=2\pi^{2}$, we precisely obtain the low-energy expansion of
the spectral density of SJT gravity, including the first correction to the
Bessel model. Figure 3 showcases the Bessel model, the full SJT spectral
density, and the newly introduced $\rho_{\mathrm{LE}}(E)$. We note that (34)
interpolates between a Bessel behaviour at low energy and an Airy-like
behaviour at large energies. It is not the first time that a similar behaviour
is investigated in the literature Johnson:2019eik .

Before commencing our analysis, we wish to emphasize that, given the current
spectral density (34), it is indeed possible to derive a closed-form solution
for the spectral form factor by explicitly evaluating the integral in (23)
with $\rho_{\mathrm{SJT}}(E)\rightarrow\rho_{\mathrm{LE}}(E)$: the computation
is contained in Appendix B, where also the relevant expansions are presented.
Conversely, we have found more interesting to describe the resurgence method
in the main text, providing insights into the transeries expansion and
capturing the nonperturbative aspects of the problem in a general framework.
Furthermore, the approach we are about to present will be more transparent in
computing the spectral form factor of SJT, where an explicit analytic
expression through the aforementioned integral representation is not feasible.

With that in mind, we proceed to compute the coefficients
$P_{g}^{(\rho_{\mathrm{LE}})}(\beta)$ associated with this spectral density by
substituting (34) into our master formula (26):

$P_{g}^{(\rho_{\mathrm{LE}})}(\beta)=-\frac{1}{g(2g+1)(2\pi)}\int_{\gamma}\frac{\mathrm{d}E}{2\pi
i}e^{-2\beta E}\frac{E^{g}}{\left(1+\alpha E\right)^{2g}}$ (35)

Next, we utilize the following expansion:

$\frac{1}{\left(1+\alpha
E\right)^{2g}}=\sum_{k=0}^{\infty}\frac{\Gamma(2g+k)}{\Gamma(2g)}\frac{(-1)^{k}\alpha^{k}E^{k}}{k!}$
(36)

After setting $g\rightarrow g+\epsilon$, we can perform the integral for each
term of the series separately, yielding:

$P_{g+\epsilon}^{(\rho_{\mathrm{LE}})}(\beta)=\frac{e^{i\pi\epsilon}\sin(\pi\epsilon)}{2\pi^{2}}\sum_{k=0}^{\infty}\frac{(-1)^{k}\alpha^{k}}{k!\left(2\beta\right)^{g+\epsilon+k+1}}\
\frac{\Gamma(2g+2\epsilon+k)}{\Gamma(2g+2\epsilon)(g+\epsilon)(2g+2\epsilon+1)}\Gamma(g+\epsilon+k+1)$
(37)

$\tau$$3$$2$$1$$\mathrm{SFF}_{\mathrm{LE}}(\tau)$ Figure 4: We show
$\mathrm{SFF}_{\mathrm{LE}}(\tau)$, for a choice of $\beta=\frac{1}{2}$ and
$\alpha=\frac{1}{2}$. The blue line represents
$\mathrm{SFF}_{\mathrm{LE}}^{\mathrm{exact}}\left(\tau\right)$, which comes
from the direct evaluation of (23) with
$\rho_{\mathrm{SJT}}(E)\rightarrow\rho_{\mathrm{LE}}(E)$. We denote with
$\mathrm{SFF}_{\mathrm{LE}}^{k_{\mathrm{max}}}$ the expression (44) where the
sum over $k$ on the second line is truncated to $k_{\mathrm{max}}$. The green
line corresponds to $k_{\mathrm{max}}=2$, while dashed line is the Bessel
model result (33), or just (44) after setting $\alpha=0$.

It is noteworthy that each term of the series in $k$ continues to display the
desired asymptotic behavior with respect to the genus $g$. This indicates that
we can once again leverage the tools of resurgence, just as we did before. We
then perform the Borel resummation over $g$, for each term of the series over
$k$ separately, i.e.

$\begin{split}&\mathcal{B}\left[\sum_{g=1}^{\infty}\tau^{2g+2\epsilon+1}P_{g+\epsilon}^{(\rho_{\mathrm{LE}})}(\beta)\right](\xi)=\\\
&\frac{C_{\epsilon}}{\tau}\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!}\left(\frac{\alpha}{\tau^{2}}\right)^{k}\mathcal{B}\left[\sum_{g=1}^{+\infty}\left(\frac{\tau^{2}}{2\beta}\right)^{g+\epsilon+k+1}\
\frac{\Gamma(2g+2\epsilon+k)\Gamma(g+\epsilon+k+1)}{\Gamma(2g+2\epsilon)(g+\epsilon)(2g+2\epsilon+1)}\right](\xi)\end{split}$
(38)

The Borel transform can be computed in terms of an hypergeometric
${}_{3}F_{2}$ function that, for each $k$, develops once again a singularity
on the real axis of the $\xi$-plane. Therefore, proceeding as before, we
evaluate its discontinuity across the Stokes line at $\vartheta=0$. The
expression reads as follows:

$\begin{split}&\left(\mathcal{S}_{0^{+}}-\mathcal{S}_{0^{-}}\right)\left[\mathcal{B}(\xi)\right]=\eta\sum_{k=0}^{\infty}\frac{(-1)^{k}\alpha^{k}}{\tau^{1+2k}k!}\times\\\
&\times\int_{0}^{+\infty}d\xi
e^{-\frac{2\beta}{\tau^{2}}\xi}\mathrm{Disc}\left[\xi^{\epsilon+k}\frac{\xi\Gamma(k+2\epsilon+2)\,_{3}F_{2}\left(1,\frac{k}{2}+\epsilon+1,\frac{k}{2}+\epsilon+\frac{3}{2};\epsilon+2,\epsilon+\frac{5}{2};\xi\right)}{(\epsilon+1)(2\epsilon+3)\Gamma(2(\epsilon+1))}\right]\end{split}$
(39)

where we have introduced an unfixed Stokes coefficient $\eta$ associated with
the discontinuity since, arguing as before, the real part of the jump cannot
be determined as we send $\epsilon\rightarrow 0$. An important criterion for
fixing this ambiguity is the limit $\alpha\rightarrow 0$, which leads to the
recovery of the Bessel model (33) with only the term $k=0$ surviving in the
sum. Consistency with that result allows us to set $\eta=\frac{i}{4\pi^{2}}.$

Once the Stokes ambiguity has been resolved, all the information regarding the
nonperturbative completion should be encoded in the remaining integral. The
latter can now be evaluated by taking the limit $\epsilon\rightarrow 0$ at
this stage, where the hypergeometric function simplifies drastically. We
obtain:

$\begin{split}&\left(\mathcal{S}_{0^{+}}-\mathcal{S}_{0^{-}}\right)\left[\mathcal{B}(\xi)\right]=\frac{i}{4\pi^{2}}\sum_{k=0}^{\infty}\frac{(-1)^{k}\alpha^{k}}{\tau^{1+2k}k!}\int_{0}^{+\infty}d\xi
e^{-\frac{2\beta}{\tau^{2}}\xi}\xi^{k}\mathrm{Disc}\left[f_{k}(\xi)\right]\end{split}$
(40)

where we have defined $f_{k}(\xi)$ as:

$\begin{split}f_{k}(\xi)\\!=\\!\frac{\left(\\!\left(\sqrt{\xi}+1\right)^{k}\\!\\!-\\!\left(1-\sqrt{\xi}\right)^{k}\\!\\!-\\!\sqrt{\xi}\left(2(k-1)(1-\xi)^{k}\\!+\\!\left(1-\sqrt{\xi}\right)^{k}\\!+\\!\left(\sqrt{\xi}+1\right)^{k}\right)\\!\right)\\!\Gamma(k+2)}{k\left(k^{2}-1\right)\sqrt{\xi}(1-\xi)^{k}}\end{split}$
(41)

We observe that this function exhibits a branch cut singularity at $\xi=1$ for
$k=0,1$, whereas for $k\geq 2$ it only has a pole singularity at $\xi=1$. The
case $k=1$ deserves separate treatment, so we will evaluate it explicitly by
integrating the discontinuity of $f_{1}(\xi)$ 151515We observe that
$f_{k}(\xi)$ is not defined for $k=1$, however $\lim_{a\rightarrow
1}f_{a}(\xi)$ is well defined, so we adopt the sloppy notation
$f_{1}(\xi)\equiv\lim_{a\rightarrow 1}f_{a}(\xi)$ for semplicity.  across the
branch cut at $\xi\in\left[1,+\infty\right.\left.\right)$:

$\begin{split}\left(\mathcal{S}_{0^{+}}-\mathcal{S}_{0^{-}}\right)\left[\mathcal{B}(\xi)\right]_{k=1}&=\frac{i}{4\pi^{2}}\frac{(-1)\alpha}{\tau^{3}}\int_{1}^{+\infty}d\xi
e^{-\frac{2\beta}{\tau^{2}}\xi}\xi\text{Disc}\left(\frac{2\tanh^{-1}\left(\sqrt{\xi}\right)}{\sqrt{\xi}}-2\right)\\\
&=\frac{\alpha}{2\pi\tau}\frac{4\beta
e^{-\frac{2\beta}{\tau^{2}}}+\sqrt{2\pi}\sqrt{\beta}\tau\text{erfc}\left(\frac{\sqrt{2}\sqrt{\beta}}{\tau}\right)}{8\beta^{2}}\end{split}$
(42)

$\tau$$4$$3$$2$$5$$\log\left|\Delta_{\mathrm{rel}}\right|$$k_{\mathrm{max}}=5$$k_{\mathrm{max}}=4$$k_{\mathrm{max}}=3$$k_{\mathrm{max}}=2$$k_{\mathrm{max}}=1$
Figure 5: In this Figure we show we show the increasing level of precision of
our transeries (44), by plotting the relative error
$\Delta_{\mathrm{rel}}=\frac{\mathrm{SFF}_{\mathrm{LE}}^{\mathrm{exact}}\left(\tau\right)-\mathrm{SFF}_{\mathrm{LE}}^{\mathrm{k_{\mathrm{max}}}}\left(\tau\right)}{\mathrm{SFF}_{\mathrm{LE}}^{\mathrm{exact}}\left(\tau\right)}$
compared to the exact result. With $k_{\mathrm{max}}=1$ we mean the first line
of (44). Here again $\beta=\frac{1}{2}$ and $\alpha=\frac{1}{2}$.

All higher orders in the expansions, for $k\geq 2$, can be obtained by
evaluating the residue of $e^{-\frac{2\beta}{\tau^{2}}\xi}\xi^{k}f_{k}(\xi)$
at $\xi=1$. It can be expressed in terms of generalized Laguerre Polynomials
as:

$\begin{split}\mathrm{Res}_{\xi=1}\left(e^{-\frac{2\beta}{\tau^{2}}\xi}\xi^{k}f_{k}(\xi)\right)=&2(k-2)!(-1)^{k-1}\exp\left(-\frac{2\beta}{\tau^{2}}\right)\sum_{n=0}^{k}\
(-1)^{n}\binom{k}{n}L_{k-1}^{\left\lfloor\frac{n}{2}\right\rfloor+1}\left(\frac{2\beta}{\tau^{2}}\right)\end{split}$
(43)

Collecting all pieces together, the result for the spectral form factor reads
as follows:

$\begin{split}\mathrm{SFF}_{\mathrm{LE}}(\beta,\tau,\alpha)&=\frac{\tau}{4\pi\beta}+\theta\left(\tau-2\sqrt{\alpha}\right)\left\\{\frac{\left(\alpha-\tau^{2}\right)e^{-\frac{2\beta}{\tau^{2}}}}{4\pi\beta\tau}+\frac{(\alpha+4\beta)\text{erfc}\left(\frac{\sqrt{2}\sqrt{\beta}}{\tau}\right)}{8\sqrt{2\pi}\beta^{3/2}}\right.\\\
&\left.-\frac{e^{-\frac{2\beta}{\tau^{2}}}}{\pi\tau}\sum_{k=2}^{\infty}\frac{1}{k(k-1)}\left(\frac{\alpha}{\tau^{2}}\right)^{k}\sum_{n=0}^{k}\
(-1)^{n}\binom{k}{n}L_{k-1}^{\left\lfloor\frac{n}{2}\right\rfloor+1}\left(\frac{2\beta}{\tau^{2}}\right)\right\\}\end{split}$
(44)

Some comments about the result (44) are in order:

  * •

The term in brackets encodes the nonperturbative completion extracted by
resurgence. In front of this, we have also added a $\theta$ function to
account for the fact that when
$\frac{\tau}{2\pi}<\rho_{\mathrm{LE}}(E_{\mathrm{min}})=\frac{\sqrt{\alpha}}{\pi}$
only the linear term of the transeries is expected to remain. In particular,
the nonperturbative terms in the first line, arising from a branch cut
singularity in the Borel plane, play a crucial role in determining the shape
and position of the plateau in the spectral form factor. On the other hand,
the terms in the second line, originating from poles in the Borel plane,
accurately capture the transition from the ramp to the plateau region, since
they vanish in the $\tau\rightarrow\infty$ limit.

  * •

In Figure 4, one can appreciate the blue line as the exact result for the low
energy model (34) under examination. The dashed line represents the Bessel
model approximation, while the green line corresponds to the first line of
(44) 161616Without considering the $\theta$ function constraint.. Notably,
even at this level of approximation, we observe a remarkable agreement with
the exact result. To demonstrate the improvement achieved by including the
terms in the second line of (44), we provide Figure 5, which illustrates the
relative error of the transseries with respect to the exact result in a
logarithmic scale. The plot depicts the increasing number of corrections
incorporated.

  * •

An important comment arises from confronting the resurgence result with the
expansion of the analytical solution presented in Appendix B. We notice that
the transeries provided by resurgence, as one can see in (44), misses a term
of order $\mathcal{O}\left(e^{-\frac{2\beta\tau^{2}}{\alpha}}\right)$ compared
to the exact result. We can understand the origin of this mismatch by noticing
that the radius of convergence of the expansion (36) is $1/\alpha$ and the
integral over $E$, overcoming the radius of convergence of the series, will in
general lose nonperturbative terms in $\alpha$. The presence of this
additional pieces can be traced back to the fact that $E=1/\alpha$ is exactly
the point at which the spectral density (34) reverses its monotonicity.
Because of that, our resurgence analysis will in general not be able to
capture the second solution $E_{2}^{*}(\tau)$ at which $\rho_{\mathrm{LE}}(E)$
intersecates with $\frac{\tau}{2\pi}$. In other words, in our approach this
solution appears to be infinitely far from $E=0$, i.e.
$E_{2}^{*}(\tau)\rightarrow+\infty$. However, as we can appreciate in Figure
5, the contribution arising from having the second solution $E_{2}^{*}(\tau)$
located at finite distance is suppressed on the plateu, compared to the
contributions of the resurgence transeries. As we move towards the ramp
instead, this nonperturbative effects in $\alpha$ become more evident. In
order to capture the contributions related to the second solution one should
presumably perform resurgence in the $\alpha$ parameter.

###  4.3 JT supergravity

With the results of the previous section at hand, we are now ready to
generalize our analysis to SJT. In this case, we must examine the following
spectral density:

$\rho_{0}(E)=\frac{\cosh\left(2\pi\sqrt{E}\right)}{2\pi\sqrt{E}}$ (45)

To extend the previous procedure, we need to use the power series expansion
around $E=0$ for the expression:

$\left(\frac{2\pi\sqrt{E}}{\cosh(2\pi\sqrt{E})}\right)^{2g}=(2\pi)^{2g}E^{g}\sum_{n=0}^{\infty}\mathcal{E}_{2n}^{(2g)}(g)\frac{(4\pi)^{2n}}{(2n)!}E^{n},$
(46)

where $\mathcal{E}_{2n}^{(2g)}(g)$ represents the higher-order Euler
polynomials. We limit the summation to even integers due to the property
$\mathcal{E}_{2n+1}^{(2g)}(g)=0$ article2 . It is worth noting that an
explicit representation of these polynomials is available in terms of the
Stirling numbers of the second kind article2 :

$\mathcal{E}_{2n}^{(2g)}(g)=\sum_{k=0}^{2n}\binom{2n}{k}g^{2n-k}\sum_{l=0}^{k}(-1)^{l}\binom{2g+l-1}{l}2^{-l}l!S(k,l)\;.$
(47)

Exploiting the expansion (46), we can compute the coefficients
$P_{g+\epsilon}^{(\rho_{\mathrm{SJT}})}(\beta)$ as:

$\begin{split}P_{g+\epsilon}^{(\rho_{\mathrm{SJT}})}(\beta)&=-\frac{1}{(g+\epsilon)(2g+2\epsilon+1)(2\pi)^{2}i}\sum_{n=0}^{\infty}\mathcal{E}_{2n}^{(2g+2\epsilon)}(g+\epsilon)\frac{(4\pi)^{2n}}{(2n)!}\int_{\gamma}\mathrm{d}Ee^{-2\beta
E}E^{n+g+\epsilon}\\\
&=-\frac{e^{i\pi\epsilon}\sin(\pi\epsilon)}{2\pi^{2}(g+\epsilon)(2g+2\epsilon+1)}\sum_{n=0}^{\infty}\mathcal{E}_{2n}^{(2g+2\epsilon)}(g+\epsilon)\frac{(4\pi)^{2n}}{(2n)!}\frac{\Gamma(n+g+\epsilon+1)}{(2\beta)^{n+g+\epsilon+1}}\;.\end{split}$
(48)

Recalling the definition (47), we can bring the sums over $n,k,l$ outside and
obtain the following asymptotic series:

$\sum_{g=1}^{\infty}P_{g+\epsilon}^{(\rho_{\mathrm{SJT}})}(\beta)\tau^{2g+2\epsilon+1}=\lim_{\nu\to
0}\mathcal{D}\left(\sum_{l=0}^{k}(-2)^{-l}S(k,l)\mathcal{I}_{n,l}(\nu,\epsilon)\right)$
(49)

where we have defined the formal differential operator $\mathcal{D}$ as:

$\mathcal{D}=\sum_{n=0}^{\infty}\frac{(4\pi)^{2n}}{(2n)!}\sum_{k=0}^{2n}\binom{2n}{k}\
\frac{\text{d}^{2n-k}}{\text{d}\nu^{2n-k}}$ (50)

and the quantity $\mathcal{I}_{n,l}(\nu,\epsilon)$ as:

$\begin{split}\mathcal{I}_{n,l}(\nu,\epsilon)&=-\frac{e^{i\pi\epsilon}\sin(\pi\epsilon)}{2\pi^{2}\tau^{2n+1}}\sum_{g=1}^{\infty}\frac{\Gamma(n+g+\epsilon+1)\Gamma(2(g+\epsilon)+l)}{(g+\epsilon)(2(g+\epsilon)+1)\Gamma(2(g+\epsilon))}\
e^{(g+\epsilon)\nu}\left(\frac{\tau^{2}}{2\beta}\right)^{g+\epsilon+n+1}\end{split}$
(51)

and we will focus on $\mathcal{I}_{n,l}(\nu,\epsilon)$ to conduct the
resurgence analysis. Remarkably, we note the insertion of the differential
operator has made $\mathcal{I}_{n,l}(\nu,\epsilon)$ look very similar to the
series (38) that we encountered in the previous section. Because of this
observation, we will manage to perform the Borel resummation in terms of the
same special functions.

$\tau$$\mathrm{SFF}_{\mathrm{SJT}}(\tau)$ Figure 6: In this plot we study
$\mathrm{SFF}_{\mathrm{SJT}}(\tau)$ focusing on the region near the late-time
plateau, by comparing the numerical analysis with the analytical results
obtained in (55) and (56), for a choice of $\beta=10$. The dots represent
$\mathrm{SFF}_{\mathrm{SJT}}^{\mathrm{num}}\left(\tau\right)$, which is the
numerical evaluation of (23). We denote with
$\mathrm{SFF}_{\mathrm{SJT}}^{n_{\mathrm{max}}}$ the expression (55) where the
sum over $n$ on the second line is truncated to $n_{\mathrm{max}}$. Red line
corresponds to $n_{\mathrm{max}}=3$, while blue line to $n_{\mathrm{max}}=5$:
we note that, as $n_{\mathrm{max}}$ increases, the approximation gets better.
The dotted line corresponds to the late-time approximation
$\mathrm{SFF}_{\mathrm{SJT}}^{\mathrm{late\
time}}(\tau)\simeq\frac{\tau}{4\pi\beta}\left(1-e^{-\frac{2\beta}{\tau^{2}}}\right)+\frac{e^{\frac{\pi^{2}}{2\beta}}}{2\sqrt{2\pi\beta}}\,\text{erfc}\left(\frac{\sqrt{2\beta}}{\tau}\right)$.

Following the same logic that brought to the result (39), we arrive at the
following expression for the discontinuity of the Borel resummation of
$\mathcal{I}_{n,l}(\nu,\epsilon)$:

$\begin{split}&\left(\mathcal{S}_{0^{+}}-\mathcal{S}_{0^{-}}\right)\left[\mathcal{B}(\xi)\right]=\frac{i}{4\pi^{2}\tau^{2n+1}}e^{-\nu(n+1)}\times\\\
&\times\int_{0}^{+\infty}d\xi
e^{-\frac{2\beta}{\tau^{2}}e^{-\nu}\xi}\mathrm{Disc}\left[\frac{\Gamma(l+2\epsilon+2)\xi^{n+\epsilon+1}\,_{3}F_{2}\left(1,\frac{l}{2}+\epsilon+1,\frac{l}{2}+\epsilon+\frac{3}{2};\epsilon+2,\epsilon+\frac{5}{2};\xi\right)}{(\epsilon+1)(2\epsilon+3)\Gamma(2(\epsilon+1))}\right]\end{split}$
(52)

Once again, it turns out that in the cases $l=0$ and $l=1$ the function in the
brackets features a branch cut, while for $l\geq 2$ the function has a pole.
Performing the same steps as in (42) and (43) and taking the limit
$\epsilon\to 0$, we are left with a purely nonperturbative term
$\tilde{\mathcal{I}}_{n,l}(\nu)$ given by:

$\tilde{\mathcal{I}}_{n,l}(\nu)=\begin{cases}-\frac{1}{2\pi}\left[\tau\left(\frac{1}{2\beta}\right)^{n+1}\Gamma\left(n+1,\frac{2\beta
e^{-\nu}}{\tau^{2}}\right)-e^{-\frac{\nu}{2}}\left(\frac{1}{2\beta}\right)^{n+\frac{1}{2}}\Gamma\left(n+\frac{1}{2},\frac{2\beta
e^{-\nu}}{\tau^{2}}\right)\right],&l=0\\\
-\frac{1}{2\pi}e^{-\frac{\nu}{2}}\left(\frac{1}{2\beta}\right)^{n+\frac{1}{2}}\Gamma\left(n+\frac{1}{2},\frac{2e^{-\nu}\beta}{\tau^{2}}\right),&l=1\\\
-\frac{e^{-\nu(n+1)}}{\pi\tau^{2n+1}}(-1)^{l}(l-2)!e^{-\frac{2\beta
e^{-\nu}}{\tau^{2}}}\sum_{m=0}^{l}(-1)^{m}\binom{l}{m}\
L_{l-1}^{\left\lfloor\frac{m}{2}\right\rfloor-l+n+1}\left(\frac{2\beta
e^{-\nu}}{\tau^{2}}\right),&l\geq 2\\\ \end{cases}$ (53)

Therefore, collecting all pieces together, we arrive at the following
expression for the SFF of SJT gravity:

$\begin{split}\mathrm{SFF}_{\mathrm{SJT}}(\beta,\tau)&=\frac{\tau}{4\pi\beta}+\lim_{\nu\to
0}\sum_{n=0}^{\infty}\frac{(4\pi)^{2n}}{(2n)!}\sum_{k=0}^{2n}\binom{2n}{k}\
\frac{\text{d}^{2n-k}}{\text{d}\nu^{2n-k}}\left(\sum_{l=0}^{k}(-2)^{-l}S(k,l)\tilde{\mathcal{I}}_{n,l}(\nu)\right)\end{split}$
(54)

The above expression already displays the full result but the underlying non-
perturbative ingredients are not apparent in this form. In order to obtain a
more explicit expression for $\mathrm{SFF}_{\mathrm{SJT}}(\beta,\tau)$, it is
necessary to know how the derivative of order $2n-k$ acts on the fundamental
blocks appearing in $\tilde{\mathcal{I}}_{n,l}$. In C we compute their general
form and below we will use it to characterize the nonperturbative structure of
$\mathrm{SFF}_{\mathrm{SJT}}(\beta,\tau)$.

In fact, by manipulating (54) as shown in C, we finally arrive at the final
trans-series expansion for the $\mathrm{SFF}_{\mathrm{SJT}}$:

$\begin{split}\mathrm{SFF}_{\mathrm{SJT}}(\beta,\tau)&=\frac{\tau}{4\pi\beta}\left(1-e^{-\frac{2\beta}{\tau^{2}}}\right)+\,\frac{1}{2\sqrt{2\pi\beta}}\text{erfc}\left(\frac{\sqrt{2\beta}}{\tau}\right)+\\\
&+\frac{1}{\pi}\sum_{n=1}^{\infty}\frac{1}{(2n)!}\left(\frac{4\pi}{\sqrt{2\beta}}\right)^{2n}\left[\frac{\Gamma\left(n+\frac{1}{2}\right)}{2^{2n+1}\sqrt{2\beta}}\
\text{erfc}\left(\frac{\sqrt{2\beta}}{\tau}\right)+\frac{e^{-\frac{2\beta}{\tau^{2}}}}{\tau}\mathcal{Z}_{n}\left(\frac{2\beta}{\tau^{2}}\right)\right]\end{split}$
(55)

where $\mathcal{Z}_{n}\left(\frac{2\beta}{\tau^{2}}\right)$ is a polynomial of
order $2n-2$ whose explicit form is presented in (95). The first few
polynomials are displayed in (C). In the first line of (55) we recognize the
contribution of the Bessel model (33), while the second line presents an
infinite series of novel nonperturbative corrections. In Figure 6 we show the
remarkable agreement between the numerical evaluation of the spectral form
factor and some approximations of (55) with an increasing number of terms
included. Moreover, in Figure 7 one can quantitatively appreciate the quality
of the approximation by noticing how the relative error becomes smaller and
smaller.

By performing the sum over $n$ for the term
$\propto\text{erfc}\left(\frac{\sqrt{2\beta}}{\tau}\right)$ in (55), one can
finally rewrite the above as:

$\begin{split}\mathrm{SFF}_{\mathrm{SJT}}(\beta,\tau)&\\!=\\!\frac{\tau}{4\pi\beta}\left(1-e^{-\frac{2\beta}{\tau^{2}}}\right)\\!+\\!\frac{e^{\frac{\pi^{2}}{2\beta}}}{2\sqrt{2\pi\beta}}\,\text{erfc}\left(\frac{\sqrt{2\beta}}{\tau}\right)\\!+\\!\frac{e^{-\frac{2\beta}{\tau^{2}}}}{\pi\tau}\sum_{n=1}^{\infty}\frac{1}{(2n)!}\left(\frac{4\pi}{\sqrt{2\beta}}\right)^{\\!2n}\\!\\!\\!\mathcal{Z}_{n}\\!\left(\frac{2\beta}{\tau^{2}}\right)\end{split}$
(56)

$\tau$$\log\left|\Delta_{\mathrm{rel}}\right|$$n_{\mathrm{max}}=3$$n_{\mathrm{max}}=5$
Figure 7: In this Figure we show the increasing level of precision of our
transeries (55) for SJT, by plotting the relative error
$\Delta_{\mathrm{rel}}=\frac{\mathrm{SFF}_{\mathrm{SJT}}^{\mathrm{num}}\left(\tau\right)-\mathrm{SFF}_{\mathrm{SJT}}^{\mathrm{n_{\mathrm{max}}}}\left(\tau\right)}{\mathrm{SFF}_{\mathrm{SJT}}^{\mathrm{num}}\left(\tau\right)}$.
The dotted line corresponds to the relative error of
$\mathrm{SFF}_{\mathrm{SJT}}^{\mathrm{late\ time}}(\tau)$, while the orange
line represents the improved late-time approximation (57). Here again
$\beta=10$. Spikes in the plot correspond to points where
$\Delta_{\mathrm{rel}}\rightarrow 0$, i.e the transeries approximation exactly
coincides with the numerical result.

In this new form of the result, we recognize the second term as the leading
contribution $\tau\rightarrow\infty$, since all other terms gets suppressed in
this regime, influencing instead the transition between ramp and plateau
171717We point out that, as it happened for the simplified model in (44), all
the non-perturbative terms in (56) should be multiplied by a $\theta$
function. In fact, below a certain threshold for the value of $\tau$, only the
ramp is expected to remain in the expansion. . Actually, exploiting the
expression (97), we can perform the sum over $n$ for the zeroth order term in
$\mathcal{Z}_{n}\left(\frac{2\beta}{\tau^{2}}\right)$ and thus obtain an
improved late-time approximation given by:

$\begin{split}\mathrm{SFF}_{\mathrm{SJT}}(\beta,\tau)&\\!=\\!\frac{\tau}{4\pi\beta}\left(1-e^{-\frac{2\beta}{\tau^{2}}}\right)\\!+\\!\frac{e^{\frac{\pi^{2}}{2\beta}}}{2\sqrt{2\pi\beta}}\,\text{erfc}\left(\frac{\sqrt{2\beta}}{\tau}\right)\\!+\\!\left(e^{\frac{\pi^{2}}{2\beta}}-1\right)\frac{e^{-\frac{2\beta}{\tau^{2}}}}{\pi\tau}\\!+\\!\mathcal{O}\left(\tau^{-3}e^{-\frac{2\beta}{\tau^{2}}}\right)\end{split}$
(57)

We can appreciate the quality of this approximation by looking at Figure (7).

We conclude by remarking that also in the SJT case we expect that the
resurgence approach is not able to capture the contribution from the second
solution $E_{2}^{*}(\tau)$ at which $\rho_{\mathrm{SJT}}(E)$ intersecates with
$\frac{\tau}{2\pi}$: at variance with the simplified model previously
discussed, here we do not have a parameter controlling the missing
contributions and determining their exact behaviour is much more difficult. We
did not attempt their evaluation but we are nevertheless confident that their
presence should not affect significantly the formation of the plateau, as it
can be appreciated from the numerical analysis.

##  5 Conclusions and Outlook

The principal aim of this paper was to explicitly examine the spectral form
factor in a supersymmetric version of Jackiw-Teitelbom gravity, using the
$\tau$-scaling limit proposed in Saad:2022kfe . The main virtue of this
approach, in the purely bosonic case, was to recover the expected plateau
structure resumming the perturbative genus contributions to the SFF (see also
Blommaert:2022lbh  for a similar computation); the emerging series in the
$\tau$ variable has a finite radius of convergence and suggests a truly
geometric interpretation for the late-time behaviour of SFF in JT-gravity. In
the supersymmetric case we have observed that a straightforward generalization
of the $\tau$-scaling limit seems to lead nowhere, because the structure of
the supermoduli volumes stops the perturbative series after the first term,
producing just the ramp behaviour. On the other hands, exploiting the relation
between SJT gravity and the chiral gaussian ensemble, we argued that the SFF
in the $\tau$-scaling limit can be extracted from the same integral formula
governing the bosonic case, simply adapting to the SJT spectral density. We
expected therefore the appearance of truly nonperturbative contribution in
$\tau$. Studying the low-energy description of super JT-gravity, the Bessel
model, and its first order correction we learnt that non-perturbative
contributions in $\tau$ are responsible for flattening the ramp and the
correct non-perturbative contributions were obtained using Cheshire cat
resurgence article1 ; Dorigoni:2017smz . Basically we deformed the coefficient
of the perturbative expansion, encoded into the functions $P_{g}^{(\rho)}$, by
an analytic continuation in the genus, generating an asymptotic series that
has been completed according resurgence theory. This procedure accurately
reproduced the exact behaviour of our toy models and we applied the same
strategy to the full super JT-gravity. Our final result is (56), expressing
analytically the spectral form factor in the $\tau$-scaling limit: taking into
account the non-perturbative completion we were also able to describe the
transition between the ramp and the plateau. Although based on some
assumptions suggested by the low-energy description, our computations seem to
fit perfectly with the physical expectations: on the other hand, it would be
nice to check our results against different methodological approaches. The
spectral form factor in super JT-gravity has been previously studied in
Johnson:2020exp , where explicit numerical computations have been performed
using a non-perturbative formulation of the theory. It would be very
interesting to compare, if possible, our analytical calculations with the
numerical curves presented in Johnson:2020exp . Another aspect that should be
explored in the future concerns the geometrical interpretation of the non-
perturbative contributions extracted from resurgence theory. In particular the
coefficient of the exponentially suppressed terms in (56) are polynomials in
$1/\tau$, suggesting the relevance of geometries with negative Euler
characteristic: this feature could be explained as the effect of disconnected
surfaces, as sometimes invoked in the literature Saad:2019lba . In order to
better understand the analytic deformation in the genus parameter, it would be
nice to obtain the formula of the coefficients $P_{g}^{(\rho)}$ solely using a
geometrical approach: this could probably be easier in the bosonic JT case. A
natural follow-up of these investigations would be the study of the spectral
form factor in ${\cal N}=2$ super JT-gravity: in this theory the perturbative
topological expansion in terms of Euclidean surfaces has been discussed
recently by Turiaci and Witten Turiaci:2023jfa  and some nonperturbative
properties have been examined Johnson:2023ofr . It would be interesting to
explore the $\tau$-scaling limit in this case.

##  Appendix A Validity of the sine-kernel statistics for SJT gravity

In general, the Altland-Zirnbauer $(\alpha,\beta)$ matrix model is
characterized by the following integral measure:

$D\lambda=\prod_{i<j}\left|\lambda_{i}-\lambda_{j}\right|^{\beta}\prod_{i}\lambda_{i}^{\frac{\alpha-1}{2}}e^{-LV\left(\lambda_{i}\right)}\mathrm{d}\lambda_{i}\qquad,\quad\lambda_{i}>0$
(58)

where $\lambda_{i}$ are the doubly-degenerate eigenvalues of the Hamiltonian
$H=Q^{2}$. In our case of interest, it reduces to:

$D\lambda=\prod_{i<j}\left|\lambda_{i}-\lambda_{j}\right|^{2}e^{-LV\left(\lambda_{i}\right)}\mathrm{d}\lambda_{i}\qquad,\quad\lambda_{i}>0$
(59)

We note this is just the usual measure for unitary matrix models, except for
the requirement of positivity of the eigenvalues.

Actually, the infinite support of the spectral density in (22) suggests that
the suitable random matrix theory dual to SJT gravity has to be a double-
scaled matrix model where we send the highest eigenvalue
$\lambda_{\mathrm{max}}$ and the matrix dimension $L$ to infinity, while
keeping fixed the ratio $\frac{L}{\lambda_{\mathrm{max}}}$. As in the case of
ordinary JT gravity, the potential $V(H)$ must then be tuned in such a way to
obtain the desired leading order spectral density (22).

In particular, in Okuyama:2020qpm  it was observed that the matrix model of JT
supergravity without time-reversal symmetry is equivalent to a Brezin-Gross-
Witten model, once that we introduce an infinite number of couplings $t_{k}$
to the matrix model potential. This derivation was parallel to the one
previously perfomed in Okuyama:2019xbv ; Okuyama:2020ncd , where it was shown
that the JT gravity matrix model is a special case of the Kontsevich-Witten
topological gravity, in the background where infinitely many couplings are
turned on to a specific value.

The Airy and Bessel model, which are respectively the low-energy limits of JT
and SJT, can be easily implemented in this formalism with a further
restriction of the non-vanishing background couplings. In the following, we
will briefly review the derivation of Okuyama:2020qpm  in this framework and
finally perform the $\tau$-scaling limit.

We consider the free energy of the BGW matrix model $F(t_{k})$, as a function
of the unspecified background couplings $t_{k}$. We then define for later use:

$u:=\hbar^{2}\partial_{0}F({t_{k}}),\qquad\partial_{k}:=\frac{\partial}{\partial
t_{k}}$ (60)

The fundamental equation for our purpose is the so called string equation, a
highly non linear differential equation for the potential $u$:

$\frac{\hbar}{4}[(\partial_{0}\mathcal{R})^{2}-2\mathcal{R}\partial_{0}^{2}\mathcal{R}]-2u\mathcal{R}^{2}=0$
(61)

which originally arose from double–scaling limits of complex matrix models
Morris:1990bw ; Dalley:1991qg ; Dalley:1992br ; Klebanov:2003wg . The quantity
$\mathcal{R}$ in (61) is a linear combination of Gelfand-Dickii polynomials in
$u$

$\mathcal{R}=\sum_{k=0}^{\infty}\hat{t}_{k}R_{k}[u]\qquad\hat{t}_{k}=t_{k}-\delta_{k,0}$
(62)

As anticipated before, in order to obtain the SJT gravity theory, we have to
set the couplings to the specific values
$t_{0}=0,t_{k}=\frac{(-1)^{k-1}}{(k-1)!}$, while the Bessel model would
correspond to the trivial background $t_{n}=0$ for each $n\geq 0$. However, as
proposed in Okuyama:2020qpm , it is useful to leave $t_{0}\neq 0$ free and fix
it at last. In the latter case, the string equation (61) admits an exact
solution for the potential, namely:

$u(x)=\frac{\hbar^{2}}{8x^{2}},$ (63)

with $x=1-t_{0}$. With a known $u(x)$, it is useful to define an auxiliary
quantum mechanical system by introducing an Hamiltonian of this form:

$\mathcal{H}=-\frac{\hbar^{2}}{4}\partial_{x}^{2}-u(x)$ (64)

The operator $\mathcal{H}$ can be diagonalized and its eigenfunctions
$\psi(x,E)$, called Baker-Akhiezer functions, allow us to extrapolate the
fully nonperturbative physics of the model. In the case of our interest, the
Bessel model, they are given by:

$\psi(x)=\braket{x}{E}=\frac{\sqrt{x}}{\hbar}J_{0}\left(\frac{x\sqrt{2E}}{\hbar}\right)$
(65)

where $J_{0}(x)$ is the Bessel function of the first kind. We can now define
the projector

$\Pi:=\int_{0}^{1}\ket{x}\bra{x}\mathrm{d}x$ (66)

through which one can easily express the partition function and the connected
two-point function Okuyama:2020qpm :

$Z(\beta)=\Tr\left(e^{-\beta\mathcal{H}}\Pi\right)\qquad
Z(\beta_{1},\beta_{2})_{c}=\Tr\left[e^{-\beta_{1}\mathcal{H}}\left(1-\Pi\right)e^{-\beta_{2}\mathcal{H}}\Pi\right]$
(67)

In our discussion, we will be considering the second object which, after
analytical continuation of the inverse temperatures as $\beta_{1,2}=\beta\pm
it$, will provide us with the desidered spectral form factor as a function of
the time $t$. We are interested in studying the late-time behavior of this
observable in the $\tau$ scaling limit, so we set $t=\tau/\hbar$ and we will
keep only the leading term in the $\hbar\rightarrow 0$ limit. In terms of the
rescaled time variable $\tau$, the two-point function in (67) reads:

$\begin{split}Z\left(\beta+i\tau/\hbar,\beta-i\tau/\hbar\right)&=\Tr\left(e^{-2\beta\mathcal{H}}\Pi\right)-\Tr\left(e^{-(\beta+i\frac{\tau}{\hbar})\mathcal{H}}\Pi
e^{-(\beta-i\frac{\tau}{\hbar})\mathcal{H}}\Pi\right)\end{split}$ (68)

Introducing the new variables $E=\frac{E_{1}+E_{2}}{2}$,
$\omega=\frac{E_{2}-E_{1}}{\hbar}$, the above is rewritten in this way:

$\begin{split}Z\left(\beta+i\tau/\hbar,\beta-i\tau/\hbar\right)&=\int_{0}^{+\infty}\mathrm{d}Ee^{-2\beta
E}\rho(E)\\\ &-\hbar\int_{0}^{+\infty}\mathrm{d}Ee^{-2\beta
E}\int_{-\frac{2E}{\hbar}}^{\frac{2E}{\hbar}}\mathrm{d}\omega
e^{i\omega\tau}K\left(E+\frac{\hbar\omega}{2},E-\frac{\hbar\omega}{2}\right)^{2}\end{split}$
(69)

where we denoted the spectral density and Christoffel- Darboux kernel as:

$\begin{split}\rho(E)&=\bra{E}\Pi\ket{E}=\int_{0}^{1}\mathrm{d}x\
\left|\psi(x;E)\right|^{2}\\\
K\left(E_{1},E_{2}\right)&=\bra{E_{1}}\Pi\ket{E_{2}}=\int_{0}^{1}\mathrm{d}x\
\psi(x;E_{1})\psi(x;E_{2})\end{split}$ (70)

By plugging into (70) the eigenfunctions (65), we obtain an explicit form for
the Bessel kernel:

$K_{\mathrm{Bessel}}\left(E_{1},E_{2}\right)=\frac{\sqrt{E_{1}}J_{1}\left(\frac{\sqrt{2E_{1}}}{\hbar}\right)J_{0}\left(\frac{\sqrt{2E_{2}}}{\hbar}\right)-\sqrt{E_{2}}J_{0}\left(\frac{\sqrt{2E_{1}}}{\hbar}\right)J_{1}\left(\frac{\sqrt{2E_{2}}}{\hbar}\right)}{\sqrt{2}\hbar(E_{1}-E_{2})}$
(71)

We observe that when $\hbar$ is small, the argument of the Bessel functions
goes to infinity; we can thus exploit the asymptotic expansion for the Bessel
functions in this limit. In particular, one has

$J_{1}\left(x_{1}\right)J_{0}\left(x_{2}\right)=\frac{1}{\pi\sqrt{x_{1}x_{2}}}\left[\sin(x_{1}-x_{2})-\cos(x_{1}+x_{2})+\mathcal{O}(1/x)\right]$
(72)

when $x_{1},x_{2}$ are large. Therefore the Bessel kernel in this regime
becomes

$K_{\mathrm{Bessel}}\left(E_{1},E_{2}\right)\simeq\frac{1}{2\pi\left(E_{1}E_{2}\right)^{\frac{1}{4}}}\left[\frac{\sin\left(\frac{\sqrt{2E_{1}}}{\hbar}-\frac{\sqrt{2E_{2}}}{\hbar}\right)}{\sqrt{E_{1}}-\sqrt{E_{2}}}-\frac{\cos\left(\frac{\sqrt{2E_{1}}}{\hbar}+\frac{\sqrt{2E_{2}}}{\hbar}\right)}{\sqrt{E_{1}}+\sqrt{E_{2}}}\right]$
(73)

When setting $E_{1,2}=E\pm\frac{\hbar\omega}{2}$, we immediately note the
second term in (73) is suppressed by a power of $\hbar$ compared to the first
one and therefore can be neglected in the $\hbar\rightarrow 0$ limit. When
expanding the first term in the same regime, the leading contribution yields

$K_{\mathrm{Bessel}}\left(E+\frac{\hbar\omega}{2},E-\frac{\hbar\omega}{2}\right)=\frac{\sin\left(\frac{\omega}{\sqrt{2}\sqrt{E}}\right)}{\pi\hbar\omega}+\mathcal{O}(\hbar)$
(74)

We have demonstrated that the Bessel kernel reduces to the sine kernel
$K_{\mathrm{sin}}=\frac{\sin\left(\pi\rho_{0}(E)\omega\right)}{\pi\omega}$ in
the $\tau$-scaling limit, where we identify the disk-level Bessel spectral
density $\rho_{\mathrm{Bessel}}(E)=\frac{1}{\pi\sqrt{2E}}$ 181818One can also
obtain the full spectral density
$\rho(E)=\frac{J_{0}\left(\frac{\sqrt{2E}}{\hbar}\right){}^{2}+J_{1}\left(\frac{\sqrt{2E}}{\hbar}\right){}^{2}}{2\hbar^{2}}$,
computed by plugging (65) into (70). By performing the limit $\hbar\rightarrow
0$ and using once again the asymptotic expansion for the Bessel functions, we
recover the disk-level spectral density. . We note it exactly matches with the
low-energy limit of the SJT disk spectral density (22), once we divide the
latter by a factor of 2 to switch from the bulk to the matrix model notation
191919In fact each energy eigenvalue for SJT is double-dengerate..

Thus, the genus-zero contribution 202020With genus-zero contribution we mean
that we are moltiplying by a factor of $\hbar=e^{S_{0}}$ in order to isolate
the leading term after $\tau$-scaling. to the spectral form factor for the
Bessel model in the $\tau$-scaling limit yields:

$\mathrm{SFF}_{\mathrm{Bessel}}=\lim_{\hbar\to 0}\hbar\langle
Z\left(\beta+i\tau/\hbar\right)Z\left(\beta-i\tau/\hbar\right)\rangle=\int_{0}^{\infty}\mathrm{d}Ee^{-2\beta
E}\min\left(\frac{1}{\pi\sqrt{2E}},\frac{\tau}{2\pi}\right)$ (75)

where we performed the integral over $\omega$ in (69), which as
$\hbar\rightarrow 0$ is just given by the Fourier transform of
$K_{\mathrm{sin}}^{2}$.

We can actually go beyond the Bessel model approximation and argue that the
structure (75) is still valid for the full SJT gravity as well. In fact we saw
in (59) that, far from the spectral edge, the Altland-Zirnbauer $(1,2)$ matrix
model is governed by the same integral measure characterizing matrix models of
unitary type. Interestingly, when $E_{1}-E_{2}\sim\mathcal{O}(\hbar)$, it
turns out the two-body eigenvalue correlator is universal for all unitary
matrix models Brezin:1993qg ; Dyson:1962es , the repulsion between close
energy levels being always captured by the sine kernel in this limit. As a
consequence, we will argue any possibile deviation from the universal sine
kernel for the $(1,2)$ Altland-Zirnbauer might only emerge as one approaches
the spectral edge, a regime where the potential is wildly varying. However, we
have just proven in (74) that in the $\tau$-scaling limit the sine kernel
behavior keeps holding for the Bessel model, which indeed probes the regime of
SJT near the spectral edge.

##  Appendix B Exact $\tau$-scaled SFF for the simplified model

In this Appendix we derive the exact form of the $\tau-$scaled SFF for the toy
model of section 4.2 and we compare the result obtained with the trans-series
(44).

We recall that the spectral density of this model is
$\rho_{\mathrm{LE}}(E)=\frac{1+\alpha E}{2\pi\sqrt{E}}$, therefore according
to the discussion of section 3.2 its $\tau-$scaled SFF admits the following
integral representation:

$\begin{split}\mathrm{SFF}_{\mathrm{LE}}&=\int_{0}^{\infty}\mathrm{d}Ee^{-2\beta
E}\min\left(\frac{1+\alpha E}{2\pi\sqrt{E}},\frac{\tau}{2\pi}\right)=\\\
&=\frac{\tau}{4\pi\beta}-\theta(\tau-2\sqrt{\alpha})\int_{\mathrm{E}_{1}(\tau)}^{\mathrm{E}_{2}(\tau)}\mathrm{d}Ee^{-2\beta
E}\left(\frac{\tau}{2\pi}-\frac{1+\alpha E}{2\pi\sqrt{E}}\right),\end{split}$
(76)

where in the second line we used the identity $\min(x,y)=x-(x-y)\theta(x-y)$
and we denoted with $\mathrm{E}_{1,2}^{*}(\tau)$ the solutions to the equation
$\rho_{\mathrm{LE}}(E)=\frac{\tau}{2\pi}$ when they exist (i.e. when
$\tau>2\sqrt{\alpha}$).

Unlike the case of SJT gravity, $\mathrm{E}_{1,2}^{*}(\tau)$ can be readily
obtained in this model and they have the following form:

$\mathrm{E}_{1}^{*}(\tau)=\frac{\tau\left(\tau-\sqrt{\tau^{2}-4\alpha}\right)-2\alpha}{2\alpha^{2}},\quad\mathrm{E}_{2}^{*}(\tau)=\frac{\tau\left(\tau+\sqrt{\tau^{2}-4\alpha}\right)-2\alpha}{2\alpha^{2}}$
(77)

Using these expressions and observing that

$\int\mathrm{d}Ee^{-2\beta E}\left(\frac{\tau}{2\pi}-\frac{1+\alpha
E}{2\pi\sqrt{E}}\right)=-\frac{4\sqrt{\beta}e^{-2\beta
E}\left(\tau-\alpha\sqrt{E}\right)\\!+\\!\sqrt{2\pi}(\alpha+4\beta)\text{erf}\left(\sqrt{2\beta
E}\right)}{16\pi\beta^{3/2}}+C,$ (78)

it is immediate to arrive at the exact result for $\tau-$scaled SFF:

$\begin{split}&\mathrm{SFF}_{\mathrm{LE}}=\frac{\tau}{4\pi\beta}-\frac{1}{16\pi\beta^{3/2}}\theta(\tau-2\sqrt{\alpha})\left[4\sqrt{\beta}e^{-2\beta\mathrm{E}_{1}^{*}(\tau)}\left(\tau-\alpha\sqrt{\mathrm{E}_{1}^{*}(\tau)}\right)+\right.\\\
&-\\!\left.4\sqrt{\beta}e^{-2\beta\mathrm{E}_{2}^{*}(\tau)}\left(\tau-\alpha\sqrt{\mathrm{E}_{2}^{*}(\tau)}\right)\\!+\\!\sqrt{2\pi}(\alpha+4\beta)\left(\text{erf}\left(\sqrt{2\beta\mathrm{E}_{1}^{*}(\tau)}\right)-\text{erf}\left(\sqrt{2\beta\mathrm{E}_{2}^{*}(\tau)}\right)\\!\right)\\!\right]\end{split}$
(79)

This formula looks quite different from the trans-series (44) of the main
text, however it can be recast in a form more similar to (44) by performing an
expansion around $\tau=+\infty$.

We observe that when $\tau\to+\infty$,
$\mathrm{E}_{1}^{*}(\tau)\sim\left(\frac{1}{\tau}\right)^{2}$ and
$\mathrm{E}_{2}^{*}(\tau)\sim\frac{\tau^{2}}{\alpha^{2}}$, therefore
$\mathrm{SFF}_{\mathrm{LE}}$ consists of two pieces in this limit: one piece
coming from $\mathrm{E}_{1}^{*}(\tau)$ which analytical around $\tau=+\infty$
and the other coming from $\mathrm{E}_{2}^{*}(\tau)$ which is non-
analytical.212121Actually, the constant term of (80) comes from the expansion
of $\text{erf}\left(\sqrt{2\beta\mathrm{E}_{2}^{*}(\tau)}\right)$ and it is
the only analytical term pertaining to a function of
$\mathrm{E}_{2}^{*}(\tau)$. Explicitly:

$\begin{split}\mathrm{SFF}_{\mathrm{LE}}&=\frac{\alpha+4\beta}{8\left(\sqrt{2\pi}\beta^{3/2}\right)}-\frac{1}{2\pi\tau}+\frac{\beta-2\alpha}{6\pi\tau^{3}}+O\left(\frac{1}{\tau^{5}}\right)+\\\
&-\left(\frac{\alpha^{2}e^{\frac{4\beta}{\alpha}}}{16\pi\beta^{2}\tau}-\frac{\alpha^{2}e^{\frac{4\beta}{\alpha}}\left(\alpha^{2}-8\beta^{2}\right)}{64\left(\pi\beta^{3}\right)\tau^{3}}++O\left(\frac{1}{\tau^{5}}\right)\right)e^{-\frac{2\beta\tau^{2}}{\alpha^{2}}}\end{split}$
(80)

By comparison we can verify that the first line of (80) perfectly matches with
the large $\tau$ expansion of the trans-series in (44) obtained in the main
text, and we checked this for the first $15$ terms. Instead, the second line
features a non analytical term in $\alpha$ around $\alpha=0$ and can not be
captured by the methods of section 4.2. The reason for this fact is that in
(36) we performed a Taylor expansion around $\alpha=0$ which for its nature
can not reproduce non-analytical terms such as
$e^{-\frac{2\beta\tau^{2}}{\alpha^{2}}}$.

##  Appendix C More detailed calculations for SJT gravity

In this Section we explicitly compute the general derivatives that are needed
in the main text. The fundamental mathematical tool that we will use is the
the Faa di Bruno’s formula, a generalization of the chain rule for higher
order derivatives:

$\frac{\text{d}^{n}}{\text{d}x^{n}}f(g(x))=\sum_{k=0}^{n}f^{(k)}(g(x))\
B_{n,k}(g^{\prime}(x),g^{\prime\prime}(x),...,g^{(n-k+1)})$ (81)

where $B_{n,k}(x_{1},x_{2},...,x_{n-k+1})$ are exponential Bell polynomials.
Many properties are known for these polynomials but for our purposes it is
sufficient to know the two following relations:

$\begin{split}B_{n,k}(ab\,x_{1},ab^{2}\,x_{2},...,ab^{n-k-1}\,x_{n-k+1})&=a^{k}b^{n}B_{n,k}(x_{1},x_{2},...,x_{n-k+1})\\\
B_{n,k}(1,1,...,1)&=S(n,k)\end{split}$ (82)

The last ingredient we are missing is the following derivative of the
incomplete Gamma function and generalized Laguerre polynomial:

$\frac{\partial^{n}}{\partial
z^{n}}\Gamma(a,z)=z^{-n}\sum_{s=0}^{n}(-1)^{n}\binom{n}{s}\frac{\Gamma(-a+s)}{\Gamma(-a)}\Gamma(a-s+n,z),\quad\frac{\partial^{m}L_{n}^{\lambda}(z)}{\partial
z^{m}}=(-1)^{m}L_{n-m}^{m+\lambda}(z)$ (83)

Exploiting the above relations, one can write down the following master
formulae:

$\begin{split}&\lim_{\nu\rightarrow
0}\frac{\text{d}^{2n-k}}{\text{d}\nu^{2n-k}}\left(\Gamma\left(n+1,\frac{e^{-\nu}2\beta}{\tau^{2}}\right)\right)=\\\
&=\sum_{b=0}^{2n-k}(-1)^{k+b}S(2n-k,b)\sum_{c=0}^{b}(-1)^{c}\binom{b}{c}\frac{\Gamma(n+2)}{\Gamma(n+2-c)}\Gamma\left(b-c+n+1,\frac{2\beta}{\tau^{2}}\right)\end{split}$
(84) $\begin{split}&\lim_{\nu\rightarrow
0}\frac{\text{d}^{2n-k}}{\text{d}\nu^{2n-k}}\left(e^{-\frac{\nu}{2}}\Gamma\left(n+\frac{1}{2},\frac{e^{-\nu}2\beta}{\tau^{2}}\right)\right)=\\\
&\frac{\Gamma\left(n+\frac{3}{2}\right)}{(-2)^{2n-k}}\sum_{s=0}^{2n-k}2^{s}\binom{2n-k}{s}\sum_{b=0}^{s}(-1)^{b}S(s,b)\sum_{c=0}^{b}(-1)^{c}\frac{\binom{b}{c}}{\Gamma\left(n+\frac{3}{2}-c\right)}\
\Gamma\left(b-c+n+\frac{1}{2},\frac{2\beta}{\tau^{2}}\right)\end{split}$ (85)
$\begin{split}&\lim_{\nu\to
0}\frac{\mathrm{d}^{2n-k}}{\mathrm{d}\nu^{2n-k}}\left(e^{-\nu(n+1)}e^{-\frac{2\beta}{\tau^{2}}e^{-\nu}}\sum_{m=0}^{l}(-1)^{m}\binom{l}{m}L_{l-1}^{\left\lfloor\frac{m}{2}\right\rfloor+n-l+1}\left(\frac{2\beta}{\tau^{2}}e^{-\nu}\right)\right)=e^{-\frac{2\beta}{\tau^{2}}}\times\\\
&\sum_{t=0}^{2n-k}A_{n,k,t}\sum_{b=0}^{t}\binom{t}{b}B_{b}\left(-\frac{2\beta}{\tau^{2}}\right)\sum_{r=0}^{t-b}\left(-\frac{2\beta}{\tau^{2}}\right)^{r}\\!S(t-b,r)\sum_{m=0}^{l}(-1)^{m}\binom{l}{m}L_{l-r-1}^{n-l+r+\left\lfloor\frac{m}{2}\right\rfloor+1}\\!\\!\left(\frac{2\beta}{\tau^{2}}\right)\end{split}$
(86)

with $A_{n,k,t}=\frac{\binom{2n-k}{t}(-1)^{k}}{\left(n+1\right)^{k-2n+t}}$ and
$B_{n}\left(x\right)$ the Bell polynomial. We will also need the following
identities:

$\begin{split}\Gamma\left(n+\frac{1}{2},z\right)&=\text{erfc}\left(\sqrt{z}\right)\Gamma\left(n+\frac{1}{2}\right)+(-1)^{n-1}e^{-z}\sqrt{z}\sum_{s=0}^{n-1}(-z)^{s}\left(\frac{1}{2}-n\right)_{-s+n-1}\\\
\Gamma\left(n,z\right)&=(n-1)!e^{-z}\sum_{s=0}^{n-1}\frac{z^{s}}{s!}\end{split}$
(87)

We will now manipulate the expression (54) of the main text. First of all,
exploiting the properties of the Stirling numbers of the second kind
$S(k,0)=\delta_{k,0}$ and $S(k,1)=1-\delta_{k,0}$, we reorganize the
transeries as:

$\begin{split}\mathrm{SFF}_{\mathrm{SJT}}(\beta,\tau)&\\!=\\!\frac{\tau}{4\pi\beta}\\!+\\!\lim_{\nu\to
0}\left\\{\\!\sum_{n=0}^{\infty}\frac{(4\pi)^{2n}}{(2n)!}\
\frac{\text{d}^{2n}\tilde{\mathcal{I}}_{n,0}(\nu)}{\text{d}\nu^{2n}}\\!-\frac{1}{2}\sum_{n=1}^{\infty}\frac{(4\pi)^{2n}}{(2n)!}\sum_{k=1}^{2n}\binom{2n}{k}\frac{\text{d}^{2n-k}\tilde{\mathcal{I}}_{n,1}(\nu)}{\text{d}\nu^{2n-k}}+\right.\\\
&\left.+\sum_{n=1}^{\infty}\frac{(4\pi)^{2n}}{(2n)!}\sum_{k=2}^{2n}\binom{2n}{k}\frac{\text{d}^{2n-k}}{\text{d}\nu^{2n-k}}\left(\sum_{l=2}^{k}(-2)^{-l}S(k,l)\tilde{\mathcal{I}}_{n,l}(\nu)\right)\right\\}\end{split}$
(88)

Thanks to the identities (84), (85) and (87), one can write:

$\begin{split}&\lim_{\nu\rightarrow 0}\
\frac{\text{d}^{2n}}{\text{d}\nu^{2n}}\tilde{\mathcal{I}}_{n,0}(\nu)=\\\
&=\frac{1}{\pi
2^{2n+1}}\left(\frac{1}{2\beta}\right)^{n+\frac{1}{2}}\Gamma\left(n+\frac{1}{2}\right)\text{erfc}\left(\frac{\sqrt{2\beta}}{\tau}\right)+\frac{e^{-\frac{2\beta}{\tau^{2}}}}{\pi\tau}\frac{1}{(2\beta)^{n}}\left[\mathcal{T}_{n}\left(\frac{2\beta}{\tau^{2}}\right)+\mathcal{S}_{n}\left(\frac{2\beta}{\tau^{2}}\right)\right]\end{split}$
(89)

where $\mathcal{T}_{n}\left(\frac{2\beta}{\tau^{2}}\right)$ and
$\mathcal{S}_{n}\left(\frac{2\beta}{\tau^{2}}\right)$ are polynomials in
$\frac{2\beta}{\tau^{2}}$ of degree $3n-1$, defined by:

$\begin{split}\mathcal{S}_{n}\left(\frac{2\beta}{\tau^{2}}\right)&=\frac{\Gamma(n+2)}{2}\sum_{b=0}^{2n}(-1)^{b+1}S(2n,b)\sum_{c=0}^{b}(-1)^{c}\binom{b}{c}\frac{\Gamma(b-c+n+1)}{\Gamma(-c+n+2)}\sum_{t=0}^{b-c+n}\frac{1}{t!}\left(\frac{2\beta}{\tau^{2}}\right)^{t-1}\\\
\mathcal{T}_{n}\left(\frac{2\beta}{\tau^{2}}\right)&=\mbox{\small$\displaystyle\frac{(-1)^{n+1}}{2^{2n+1}}\sum_{s=0}^{2n}2^{s}\binom{2n}{s}\sum_{b=0}^{s}S(s,b)\sum_{c=0}^{b}\frac{\binom{b}{c}\Gamma\left(n+\frac{3}{2}\right)}{\Gamma\left(n-c+\frac{3}{2}\right)}\\!\\!\sum_{t=0}^{b-c+n-1}\\!\\!\left(-\frac{2\beta}{\tau^{2}}\right)^{\\!t}\\!\\!\frac{\Gamma\left(-t-\frac{1}{2}\right)}{\Gamma\left(\frac{1}{2}-c-b-n\right)}$}\end{split}$
(90)

Thanks to the identities (85) and (87), one can then write:

$-\frac{1}{2}\lim_{\nu\rightarrow 0}\sum_{k=1}^{2n}\binom{2n}{k}\
\frac{\text{d}^{2n-k}}{\text{d}\nu^{2n-k}}\tilde{\mathcal{I}}_{n,1}(\nu)=\frac{e^{-\frac{2\beta}{\tau^{2}}}}{\pi\tau}\frac{1}{(2\beta)^{n}}\mathcal{Q}_{n}\left(\frac{2\beta}{\tau^{2}}\right)$
(91)

where $\mathcal{Q}_{n}\left(\frac{2\beta}{\tau^{2}}\right)$ is a polynomial in
$\frac{2\beta}{\tau^{2}}$ of degree $3n-2$, whose explicit form is given by:

$\displaystyle\mathcal{Q}_{n}\left(\frac{2\beta}{\tau^{2}}\right)$
$\displaystyle=\frac{(-1)^{n+1}\Gamma\left(n+\frac{3}{2}\right)}{4}\sum_{k=1}^{2n}\frac{\binom{2n}{k}}{(-2)^{2n-k}}\sum_{s=0}^{2n-k}2^{s}\binom{2n-k}{s}\times$
(92)
$\displaystyle\times\sum_{b=0}^{s}S(s,b)\sum_{c=0}^{b}\frac{\binom{b}{c}}{\Gamma\left(n+\frac{3}{2}-c\right)}\sum_{t=0}^{b-c+n-1}\left(-\frac{2\beta}{\tau^{2}}\right)^{t}\left(\frac{1}{2}-n-b+c\right)_{b-t-c+n-1}$

Finally, thanks to the identities (86) and (87), we arrive at:

$\begin{split}&\lim_{\nu\rightarrow 0}\sum_{k=2}^{2n}\binom{2n}{k}\
\frac{\text{d}^{2n-k}}{\text{d}\nu^{2n-k}}\left(\sum_{l=2}^{k}(-2)^{-l}S(k,l)\tilde{\mathcal{I}}_{n,l}(\nu)\right)=\frac{e^{-\frac{2\beta}{\tau^{2}}}}{\pi\tau}\frac{1}{(2\beta)^{n}}\mathcal{P}_{n}\left(\frac{2\beta}{\tau^{2}}\right)\end{split}$
(93)

where $\mathcal{P}_{n}\left(\frac{2\beta}{\tau^{2}}\right)$ is a polynomial in
$\frac{2\beta}{\tau^{2}}$ of degree $3n-2$, expressed as:

$\begin{split}\mathcal{P}_{n}\left(\frac{2\beta}{\tau^{2}}\right)&=(-1)^{n+1}\
\sum_{k=2}^{2n}\binom{2n}{k}\sum_{l=2}^{k}\frac{S(k,l)(l-2)!}{2^{l}}\sum_{t=0}^{2n-k}A_{n,k,t}\sum_{b=0}^{t}\binom{t}{b}B_{b}\left(-\frac{2\beta}{\tau^{2}}\right)\times\\\
&\times\sum_{r=0}^{t-b}\left(-\frac{2\beta}{\tau^{2}}\right)^{r+n}S(t-b,r)\sum_{m=0}^{l}(-1)^{m}\binom{l}{m}L_{l-r-1}^{-l+n+r+\left\lfloor\frac{m}{2}\right\rfloor+1}\left(\frac{2\beta}{\tau^{2}}\right)\end{split}$
(94)

Because of the common structure of (89), (91) and (93), it is useful to
introduce the polynomial:

$\begin{split}\mathcal{Z}_{n}\left(x\right)=\mathcal{T}_{n}\left(x\right)+\mathcal{S}_{n}\left(x\right)+\mathcal{Q}_{n}\left(x\right)+\mathcal{P}_{n}\left(x\right)\end{split}$
(95)

through which one can arrive at the result (56) of the main text. Below, we
list the first few polynomials $\mathcal{Z}_{n}\left(x\right)$:

$\displaystyle\mathcal{Z}_{1}\left(x\right)$
$\displaystyle=\frac{1}{8}\qquad\mathcal{Z}_{2}\left(x\right)=-\frac{3x^{2}}{16}+\frac{x}{32}+\frac{3}{64}\qquad\mathcal{Z}_{3}\left(x\right)=\frac{15x^{4}}{32}-\frac{105x^{3}}{64}\\!+\\!\frac{x^{2}}{128}\\!+\\!\frac{5x}{256}+\frac{15}{512}$
$\displaystyle\mathcal{Z}_{4}\left(x\right)$
$\displaystyle=-\frac{105x^{6}}{64}+\frac{1995x^{5}}{128}-\frac{7623x^{4}}{256}+\frac{x^{3}}{512}+\frac{7x^{2}}{1024}+\frac{35x}{2048}+\frac{105}{4096}$
(96) $\displaystyle\mathcal{Z}_{5}\left(x\right)$
$\displaystyle\\!=\\!\frac{945x^{8}}{128}-\frac{34335x^{7}}{256}\\!+\\!\frac{351855x^{6}}{512}-\frac{986205x^{5}}{1024}\\!+\\!\frac{x^{4}}{2048}\\!+\\!\frac{9x^{3}}{4096}\\!+\\!\frac{63x^{2}}{8192}\\!+\\!\frac{315x}{16384}\\!+\\!\frac{945}{32768}$

We notice that, although $\mathcal{T}_{n}\left(x\right)$,
$\mathcal{S}_{n}\left(x\right)$, $\mathcal{Q}_{n}\left(x\right)$ and
$\mathcal{P}_{n}\left(x\right)$ have degree at least $3n-2$, the resulting
polynomial $\mathcal{Z}_{n}\left(x\right)$ has a degree $2n-2$, because of
highly non trivial cancellations occurring among the constituent polynomials.
Although we have determined in principle a close analytical form for
$\mathcal{Z}_{n}$, it is quite intricated. Because of that, we have
empirically analyzed the structure of the first $20$ polynomials and we have
been able to partially determine a more manageable expression for
$\mathcal{Z}_{n}$, i.e.

$\begin{split}\mathcal{Z}_{n}\left(x\right)&=\sum_{k=0}^{n-1}\frac{(2n-1)\text{!!}x^{k}}{(2k+1)\text{!!}2^{3n-k}}+\sum_{k=n}^{2n-3}\frac{(-1)^{k+n+1}\
a_{k}}{2^{3n-k}}x^{k}+\left(1-\delta_{n,1}\right)\frac{(-1)^{n+1}(2n-1)!!}{2^{n+2}}x^{2n-2},\end{split}$
(97)

where $a_{k}$ are coefficients that we have not been able to fix in a direct
way but can be deduced from (89), (91) and (93).

We finally point out that, by exploiting the form (97) for the polynomials
$Z_{n}$, it is also possible to rewrite the transeries expansion (55) of the
main text in another useful form:

$\begin{split}\mathrm{SFF}_{\mathrm{SJT}}(\beta,\tau)\\!=\\!&\frac{\tau}{4\pi\beta}-\frac{\tau}{4\pi\beta}e^{-\frac{2\beta}{\tau^{2}}}+\frac{1}{2\sqrt{2\pi\beta}}\\!\left[\mathrm{erfc}\left(\frac{\sqrt{2\beta}}{\tau}\right)+\sum_{n=1}^{+\infty}\frac{\left(\frac{\pi^{2}}{2\beta}\right)^{n}}{n!}\frac{\Gamma\left(\frac{1}{2}+n,\frac{2\beta}{\tau^{2}}\right)}{\Gamma\left(\frac{1}{2}+n\right)}\right]\\!+\\!\mathcal{R},\end{split}$
(98)

where we define $\mathcal{R}$ as:

$\begin{split}\mathcal{R}=\frac{e^{-\frac{2\beta}{\tau^{2}}}}{\pi\tau}\sum_{n=1}^{+\infty}\frac{\left(\frac{4\pi}{\sqrt{2\beta}}\right)^{2n}}{(2n)!}\sum_{k=n}^{2n-3}\frac{(-1)^{k+n+1}}{2^{3n-k}}a_{k}\left(\frac{2\beta}{\tau^{2}}\right)^{k}+\frac{\tau^{3}}{16\pi\beta^{2}}e^{-\frac{2\beta}{\tau^{2}}}\left(1-e^{-\frac{8\pi^{2}\beta}{\tau^{4}}}\right)-\frac{\pi
e^{-\frac{2\beta}{\tau^{2}}}}{2\beta\tau}.\end{split}$ (99)

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