§ INTRODUCTION

Spacetime wormholes have been important in understanding quantum aspects of black holes. For example, for an evaporating black hole, wormhole contributions are important in order to obtain an entropy of Hawking radiation consistent with unitarity [1].

Wormholes also appear in the following context, closer to the present article. The spectrum of a black hole can be computed via the gravitational path integral as pioneered by Gibbons and Hawking [2]. The classical analysis predicts a continuous spectrum, contradicting
the expectation that the black hole spectrum is discrete and has a finite Bekenstein-Hawking entropy.
It is believed that  perturbative quantum effects do not fix this problem [3], but spacetime wormholes have played a role in addressing  some aspects of it.

It was proposed in [4] that a quantum system that describes a black hole has a chaotic spectrum.
Similarly to strongly interacting many-body systems, this means that the statistics of black hole energy levels can be modeled by a random
matrix ensemble. This can be tested by the time dependence of the spectral form factor, introduced in [4] in the context
of black holes. It was later realized in [5] that in order to reproduce the features in the spectral form factor characteristic of
random matrix  level repulsion, it was necessary to include contributions from spacetime wormholes. The wormhole
found in [5] is universal and can be constructed, given the original black hole metric as a starting point, in any dimension
and any theory of gravity.   At late enough times, the contribution from this wormhole reproduces
random matrix behavior,  supporting the idea of black hole chaos.

The connection between black holes and chaos was beautifully illustrated in [6]. The authors studied a solvable theory
of gravity: pure Jackiw-Teitelboim (JT) gravity [7, 8] in a two-dimensional world asymptotic to AdS$_2$ [9, 10, 11, 12]. As a first observation, it is possible to perform
the exact gravitational path integral around the black hole solution, and verify that perturbative quantum effects do not give
access to the discrete nature of the spectrum. The authors of [6] proceeded by evaluating the exact path integral on (oriented)
surfaces with arbitrary topology.   It turned out that the gravitational path integral in this theory is equivalent to a matrix integral; the sum over topologies
corresponds to the $1/L$ expansion of the matrix integral, where $L$ is the rank of the matrices.
The matrix is interpreted as the Hamiltonian of the dual quantum mechanics and therefore, in this particular model, the black hole Hamiltonian really is a random matrix.

The duality between JT gravity and random matrix models has been extended in multiple ways.
Some of the generalizations have been reviewed in  [13] and references therein.
The generalization that is most directly relevant to the present article is the extension from JT gravity to JT supergravity [14].
In this case, depending on certain anomaly coefficients and on whether time-reversal symmetry is assumed, one encounters various
of Altland and Zirnbauer [15], as well as Dyson $\upbeta$ ensembles.

In the present paper, we will extend the comparison between gravity and matrix models to the case of $\N=2$ JT supergravity.  We will identify a matrix model
that agrees with the gravitational path integral of $\N=2$ JT supergravity to all orders in the topological expansion.[The  $\N=0$ and $\N=1$
versions of the duality between JT gravity and a matrix model
are limits of previously conjectured dualities between the minimal (super)string models and matrix integrals [6, 16, 17]. Curiously an $\N=2$ analog of this does not seem to be known.]
We also make some preliminary observations on $\N=4$ JT supergravity.

Before summarizing our results, we would like to motivate the problem. String theory provides multiple concrete examples of quantum systems that at strong coupling describe  black holes, e.g. the D0-brane black hole [18], four dimensional $1/2$-BPS black holes arising from type II string theory compactified on a Calabi-Yau [19], excitations of the $1/16$-BPS black hole in ${\rm AdS}_5\times {\rm S}^5$ [20, 21, 22], among others. All these systems display extended supersymmetry, which affects the corresponding random matrix ensemble.
Most cases can be approximated by JT supergravity models with $\N>1$ supersymmetry coupled to matter.
The last example mentioned  is particularly relevant, since at low temperatures this black hole is described by $\N=2$ JT supergravity coupled to matter [23]. The four dimensional black holes mentioned in the second example are described instead by $\N=4$ JT supergravity [24].

In the remainder of this introduction we will summarize the contents of this article. In section <ref>, we study random matrix ensembles with $\N=2$
supersymmetry (and no time-reversal symmetry) and attempt to motivate the class of random matrix ensembles that we will compare to the gravity calculation.
In the ensembles that we consider, supermultiplets with different $R$-charges are statistically independent and each one is described by an AZ ensemble with $(\upalpha,\upbeta)=(1,2)$.
BPS states can be present and their wavefunctions are random, a property that
was proposed recently [25, 26] as an aspect of black hole chaos.

In section <ref>, we begin our analysis of $\N=2$ JT supergravity. We review the exact calculation of the path integral around the black hole geometry and use it to extract the leading spectral curve of the matrix model. We find a large number of BPS states in a restricted range of $R$-charges. Non-BPS states have a continuous density of states which can be mapped to a choice of matrix potential. In section <ref> we evaluate the path integral on a space with the topology of a cylinder, dual to the spectral form factor. We find the result from gravity to be consistent with supermultiplets being statistically independent and the number of BPS states being fixed (rather than fluctuating as a random variable).

In section <ref>, we perform the gravitational path integral for $\N=2$ JT supergravity on orientable spaces with an arbitrary number of handles and boundaries. Similarly to [6], the calculation reduces to evaluating the volumes of moduli spaces of hyperbolic surfaces, in this case with $\N=2$ supersymmetry.[See [27] for
mathematical work on $\N=2$ hyperbolic geometry.]  The main result of this section is to determine the measure on this moduli space induced by the supergravity path integral, presented in eqn. (<ref>). To obtain this result we evaluate the one-loop determinant arising from JT gravity using the Reidemeister torsion (an alternative, on orientable manifolds,
would be to use the symplectic structure). We compute the integral over the moduli space in the simplest case of a three-holed sphere.

In section <ref>, we complete the evaluation of the $\N=2$ JT supergravity path integral. Having the measure over moduli space of hyperbolic surfaces is not enough, since one needs to efficiently divide by the mapping class group. In section <ref> we review the recursion relation by which Mirzakhani resolved this problem for bosonic surfaces. In section <ref>, after introducing the necessary elements of $\N=2$ supergeometry, we derive a similar recursion relation for volumes of moduli spaces of $\N=2$ hyperbolic surfaces, presented in eqn. (<ref>). We list some interesting properties of these volumes. Importantly, we show in section <ref> that they satisfy the  loop equations of
random matrix theory and use this result to finalize our proof that $\N=2$ JT supergravity is dual to the matrix ensemble proposed in section <ref>.

In section <ref>,  we generalize parts of the story to $\N=2$ quantum mechanics with a time-reversal symmetry, either commuting or anticommuting with the $R$-charge. We identify the correct matrix model ensemble for each case. We also compute the
JT supergravity path integral in the presence of crosscaps and match with the loop equations of the proposed matrix models.  In one case, one has to assume the existence of wormhole
corrections to the spectral curve.

In section <ref>, we generalize some of our findings to $\N=4$ JT supergravity, or alternatively to matrix models with four supercharges and  $\SU(2)$ $R$-symmetry. We evaluate the gravitational path integral around the black hole solution and also on the cylinder that computes the spectral form factor, showing that supermultiplets are statistically independent.  We also
develop the beginnings of $\N=4$ random matrix theory.

In appendix <ref>, we provide a more detailed analysis of the boundary reparametrization mode for $\N=2$ and $\N=4$ JT supergravity. In appendix <ref> we derive the matrix model loop equations in the presence of a logarithmic term in the matrix potential, necessary to match $\N=2$ JT gravity. We also define a notion of “volumes” valid for any matrix model and compute the analog of Mirzakhani's recursion for any spectral curve. This streamlines the derivations in section <ref>. Other technical details are left for appendices <ref> and <ref>.

§ RANDOM MATRICES WITH $\N=2$ SUPERSYMMETRY

§.§ Preliminaries

Our goal in this section is to understand how random matrix theory might apply to $\N=2$ JT supergravity.[Random matrix theory with $\N=2$ supersymmetry was first investigated in [28]. Our approach has some differences in detail.]   We will motivate the idea that multiplets
with different $R$-charges are statistically independent, and  suggest what  type of random matrix ensemble should be relevant.

We assume that the random matrix theory relevant to $\N=2$ JT supergravity is supposed to realize a global $\N=2$ supersymmetry algebra that includes  a Hamiltonian $H$ and
a hermitian conjugate pair of supersymmetry generators $Q$ and $Q^\dagger$ satisfying
Q^2=Q^†^2 =0,.5cm {Q,Q^†}=H,    [H,Q]=[H,Q^†]=0.We also assume the existence of a $\uU(1)$ “$R$-symmetry” group of outer automorphisms of this algebra, generated by the “$R$-charge” $J$, satisfying
[J,Q]=q  Q,     [J,Q^†]=-q   Q^†,     [J,H]=0. What motivates these assumptions is that this global supersymmetry algebra can be naturally preserved in a cutoff version of $\N=2$ JT supergravity, with a cutoff near
the conformal boundary.
The operator $(-1)^\fF$ that distinguishes bosons from fermions is supposed to be
(-1)^=e^±ıπJ.   with either choice of sign.
This implies that $\h q$ is an odd integer, and that $J$ has even (odd) integer eigenvalues for bosonic (fermionic) states.   After possibly exchanging $Q$ and $Q^\dagger$,
one can assume that $\h q>0$.

As explained in section <ref>, there is also an “anomalous” case in which eigenvalues of $J$ are valued not in $\Z$ but in $\delta+\Z$, with a constant $\delta$.
Then $e^{\pm \i \pi J}$ still commutes with bosonic operators, and anticommutes with fermionic ones, but its eigenvalues are not $\pm 1$ and there is not really a natural
notion of which states are “bosonic” and which are “fermionic.”

In the present section and most of the article, we  consider $\N=2$ random matrix models
without time-reversal symmetry.   Time-reversal symmetry will be included in section <ref>.

§.§ Review of $\N=1$

We begin by reviewing  random matrix theory  with $\N=1$ supersymmetry,  relevant to $\N=1$ JT supergravity [14].   Here actually we will only consider the “non-anomalous”
case that arises, for example, as a low energy approximation to an SYK model with an even number $N$ of Majorana fermions.
As will be apparent, the assumption of a $\uU(1)$ $R$-symmetry leads for $\N=2$ to a structure similar to what one finds for $\N=1$ supersymmetry with even $N$.
The odd $N$ case of an $\N=1$ SYK model leads to a very different random matrix theory.

In $\N=1$ random matrix theory, we assume that the Hamiltonian is $H=Q^2$, where $Q$ is a self-adjoint supercharge,
odd under the symmetry $(-1)^\fF$ that distinguishes
bosons from fermions.  In a basis in which this operator is block diagonal
(-1)^=(  1     00     -1),with diagonal blocks of sizes  $L_b$, $L_f$ (respectively the numbers of bosonic and fermionic states), $Q$ is off-diagonal,
Q=(  0     ^†    0).Here $\qQ$ is a complex-valued $L_b\times L_f$ matrix, subject to no constraint;
hermiticity of $Q$ just says that the lower left block in eqn. (<ref>) is the hermitian adjoint of $\qQ$.

This structure is invariant under a unitary group $G=\uU(L_b)\times \uU(L_f)$  with the two factors acting on the bosonic and fermionic states, respectively:
U= (  U_L_b     00     U_L_f).$\qQ$ transforms as
→U_L_b U_L_f^-1, that is, as a bifundamental of $\uU(L_b)\times \uU(L_f).$
In [14], $\N=1$ JT supergravity was related to a random matrix ensemble in which $\qQ$ is treated as a random matrix with a measure of the general form[In this article,  for
any matrix $M$,  we denote by  $\d M$  the product of Euclidean measure for all independent matrix elements of $M$.]
μ̣=  exp(-  F(^†)),with a suitable function $F$.
As usual in random matrix theory, an integral over a random matrix $\qQ$ with such a measure can be conveniently reduced to an integral over its eigenvalues, in an appropriate
sense.  For example, in the present case, if $L_b =L_f=L$, then $\uU(L_b)\times \uU(L_f)$ can be used to diagonalize $\qQ$,
=λ_1                   λ_2                  ⋱                   λ_L  ,with all $\lambda_i\geq 0$.   There remains a residual $\uU(1)^L$ symmetry that preserves this form.

It is then necessary to find the appropriate measure for integration over the $\lambda$'s.    This may be done as follows.  Let $\mathfrak g$ be the Lie algebra of
$G$, $\mathfrak h$ the Lie algebra of the residual symmetry group $\uU(1)^L$, and $\mathfrak g^\perp$ the orthocomplement of $\mathfrak h$ in $\mathfrak g$.
Let $\O_\qQ$ be the $G$ orbit of the matrix $\qQ$ in the space of $L_b\times L_f$ complex matrices, and let $T_\qQ$ be the tangent space to this
orbit at the point corresponding to $\qQ$.   The measure factor that we want is the determinant of the natural map  $W:\mathfrak g^\perp\to T_\qQ$ which expresses the
fact that $\mathfrak g^\perp$ generates the orbit.   In practice,
we will find an orthonormal basis  $e_1,e_2,\cdots \in \mathfrak g^\perp$ that maps to an orthonormal basis $W(e_1),W(e_2),\cdots\in T_\qQ$,
implying that the desired volume factor is the product of “eigenvalues” of $W$, that is, it is the product $\prod_a|W(e_a)|/|e_a|$, where the norm of a matrix $m$ is defined by
$|m|^2=\Tr\,m^\dagger m$.  In evaluating $\prod_a |W(e_a)|/|e_a|$, we will drop constant factors independent of the $\lambda$'s.

First of all, for each eigenvalue $\lambda_i$, there is a generator $e$ of $\uU(L_b)$ (or $\uU(L_f)$; this choice does not matter because of the residual $\uU(1)^L$ symmetry) that rotates
$\lambda_i$ in the complex plane.   The generator $e$, if normalized to have norm 1, acts on $\lambda_i$ by
$\delta_e\lambda_i=\i \lambda_i$, with $|\delta_e\lambda_i|=\lambda_i$.   So $|W(e)|/|e|=\lambda_i$.
By itself  this would lead to a measure $\prod_i \lambda_i\d \lambda_i$.

Now consider a  pair of eigenvalues $\lambda_i,$ $\lambda_j$, and the subgroup $\uU(2)_b\times \uU(2)_f$ of $\uU(L_b)\times \uU(L_f)$ that rotates the corresponding eigenvectors of
$\qQ^\dagger \qQ$ among themselves.  The group $\uU(2)_b\times \uU(2)_f$ is eight-dimensional; we have accounted for four generators (two residual symmetries and two
that rotates the phases of $\lambda_i,\lambda_j$), so there are four more to consider.  Two linear combinations of generators take the form
0         0∈𝔲(2)_b, 1cm    0         0∈𝔲(2)_f,
1cm ∈.These generators transform $\qQ$ by
δ λ_i     0
0    λ_j = 0     -(λ_i-λ_j) (λ_i-λ_j)     0 .For these two generators, we have $\prod_{a=1,2}|W(e_a)|/|e_a|=
(\lambda_i-\lambda_j)^2$.   The other two linear combinations
take the form
0         0∈𝔲(2)_b, 1cm    -0        0∈𝔲(2)_f,
1cm ∈, transforming $\qQ$ by
δ λ_i     0
0    λ_j = 0     (λ_i+λ_j) (λ_i+λ_j)     0 ,and for these generators we get $\prod_a |W(e_a)|/|e_a|= (\lambda_i+\lambda_j)^2$.

At this stage, we have accounted for a full basis of $\mathfrak g^\perp$.
Therefore, for this example with $L_b=L_f$, the measure for the eigenvalues is
∏_i=1^L_b λ_i λ̣_i ∏_1≤i<j≤L_b (λ_i-λ_j)^2(λ_i+λ_j)^2.In random matrix theory, an ensemble with nonnegative “eigenvalues” $\lambda_i$ and measure
∏_iλ̣_i  λ_i^ ∏_i<j(λ_i^2-λ_j^2)^is usually called an Altland-Zirnbauer [15] $(\upalpha,\upbeta)$ ensemble,[Special cases of such ensembles had certainly been considered previously, for example in
[29, 30, 31, 32, 33]. The last two references were in the context of applications to gravity.] so here we have such an ensemble with $(\upalpha,\upbeta)=(1,2)$.

It remains to consider the case $L_b\not= L_f$.   It turns out that the resulting measure depends only on $|L_b-L_f|$, so we lose nothing essential if we assume $L_b>L_f$,
say $L_b=L_f+\nu$.   $\qQ$ is now a rectangular $L_b\times L_f$ matrix.   In its canonical form, it has $\nu$ rows of zeroes:
=λ_1                   λ_2                  ⋱                   λ_L_f0                0               ⋱        ,The residual symmetry group is now $\uU(1)^{L_f}\times \uU(\nu)$, where the second factor acts on the last  $\nu$ rows of $\qQ$.
In addition, for each  $i$, there are $2\nu$ generators of $\mathfrak g^\perp$ that at linearized order map the $i^{th}$ (left) eigenvector of $\qQ$ into the space of null
vectors.   For each of these generators, $|W(e)|/|e|=\lambda_i$.     Overall, for each $i$, we now get $1+2\nu$ factors of $\lambda_i$, and  the measure becomes
∏_iλ̣_i  λ_i^1+2ν ∏_i<j(λ_i^2-λ_j^2)^2, corresponding to   an AZ ensemble with $(\upalpha,\upbeta)=(1+2\nu,2)$.   It will be convenient for later discussions to introduce a parameter $\upalpha_0$ and to write
=_0+ν,1cm (_0,)=(1,2).

§.§ Random Matrices for $\N=2$

Now we will try to generalize this discussion to $\N=2$ supersymmetry.   An immediate difference is that the $\Z_2$-valued symmetry $(-1)^\fF$ of the previous discussion
is now promoted to an additive $R$-charge symmetry, with generator $J$.   For our purposes in this section, it is convenient to assume that $J$ has integer eigenvalues and
that $\h q=1$.   The results we get are straightforwardly adapted to a more general case described in section <ref>.

Let $\H_k$, of dimension $L_k$, be the space of quantum states of $R$-charge $k$.   The full quantum Hilbert space is then $\H=\oplus_k \H_k$.
The supercharge $Q$, since it has $R$-charge 1,  takes the form $Q=\sum_k Q_k$ with $Q_k:\H_k\to \H_{k+1}$ and therefore $Q_k^\dagger:\H_{k+1}\to \H_k$.
Similarly $Q^\dagger=\sum_k Q_k^\dagger$.

The supersymmetry algebra has two types of irreducible multiplet.   First, a single zero-energy state of any $R$-charge, annihilated by $Q$ and $Q^\dagger$, is a multiplet by
itself.  These are the supersymmetric or BPS states.   Second, one can have a pair of states with adjacent values of the $R$-charge, say $\psi_k\in\H_k$, $\psi_{k+1}\in\H_{k+1}$,
with $Q\psi_k=\lambda\psi_{k+1}$, $Q^\dagger\psi_{k+1}=\bar\lambda \psi_k$.   These two states make up what we will call an $(k,k+1)$ supermultiplet.[It will be useful, starting from section <ref>, to label a non-BPS multiplet
of charges $(k,k+\h q\, )$ (where in our present discussion $\hq=1$) by the average charge $q=k+\h q/2$.]

Each $\H_k$
can be decomposed as the direct sum of three pieces:
_̋k=_̋k^0⊕_̋k^+⊕_̋k^-. Here $\H_k^0$ consists of supersymmetric states of $R$-charge $k$, $\H_k^+$ consists of states of $R$-charge $k$ that are in $(k,k+1)$ multiplets, and $\H_k^-$ consists
of states of $R$-charge $k$ that are in $(k-1,k)$ multiplets.  Letting $L_k^0$, $L_k^+$, and $L_k^-$ denote the dimensions of $\H_k^0$, $\H_k^+$, and $\H_k^-$, respectively,
we have clearly
L_k=L_k^0+L_k^++L_k^-. Another obvious implication is that $L_k^{+}=L_{k+1}^-$.

Consider first a simplified model in which the $R$-charge takes only two adjacent values say $\H=\H_{k-1}\oplus \H_{k}$.   We then have simply $Q:\H_{k-1}\to \H_k$,
$Q^\dagger:\H_k\to\H_{k-1}$.   The structure is the same as in the $\N=1$ case discussed in section <ref>, with the nonhermitian supercharge $Q$ playing the
role that was played previously by $\qQ$.    A natural random matrix ensemble would then be an AZ $(\upalpha,\upbeta)=(1+2\nu,2)$ ensemble, with $\nu=|L_k-L_{k-1}|$, as in section <ref>.

However, as soon as the $R$-charge takes more than two adjacent values, we run into apparent difficulty.  For any integer $k$, let $Q_k$ be the restriction of $Q$ to a map from $\H_k$
to $\H_{k+1}$.
Suppose that there are three adjacent values of the $R$-charge,
say $\H=\H_{k-1}\oplus \H_k\oplus \H_{k+1}$.   Now
we have $Q=Q_k+Q_{k-1}$, where $Q_k$ and $Q_{k-1}$ are related by
Q_k Q_k-1=0, which follows from the supersymmetry relation $Q^2=0$.  This constraint means that $Q_{k-1}:\H_{k-1}\to \H_k$ and $Q_k:\H_k\to \H_{k+1}$ cannot be chosen independently.
However, we will argue that nonetheless, it is natural in random matrix theory to expect that the $(k-1,k)$ and $(k,k+1)$ supermultiplets are statistically independent.

As a preliminary, we have to think about what  a generic solution of the constraint $Q_k Q_{k-1}=0$ looks like.   In fact, the space of generic solutions of this constraint
has multiple components, differing in the values of $L_k^0$ and $L_k^\pm$.   There are only two general constraints.  One constraint
comes from the supersymmetric index. If $\H$ has finite dimension with $∑_k L_k<∞$ (implying in particular that $L_k=0$ for all but finitely many $k$), then
\be\label{index} \sum_k (-1)^k L_k^0 = \sum_k (-1)^k L_k, \ee because the $(k,k+1)$ multiplets make no net contribution to the right hand side.
This condition is satisfied for all solutions of $Q^2=0$, not just generic ones.
In JT supergravity, the individual $L_k$ are infinite, for all $k$,
but a regularized version of this condition makes sense, with $L_k$ replaced by $__̋k e^-βH$.
A second constraint holds only for generic solutions of $Q^2=0$.   In a generic solution, for any $k$, either $L_k^0$ or $L_k+1^0$ vanishes, since otherwise
$Q$ could be perturbed slightly to combine a pair of supersymmetric states of $R$-charges $k$ and $k+1$ into a single $(k,k+1)$ multiplet.

These constraints and eqn. (\ref{zn}) are not enough to determine the $L_k^0, L_k^±$ in terms of the $L_k$.
For example, consider a simple case with $L_k-1=L_k=L_k+1=2$ and other $L_k$ vanishing.
The spectrum could consist of two supersymmetric states of $R$-charge $k-1$ and two $(k,k+1)$ multiplets; but it could also consist of two supersymmetric
states of $R$-charge $k+1$ and two $(k-1,k)$ multiplets, or one supersymmetric state of $R$-charge $k-1$ and one of $R$-charge $k+1$, along with
an $(k-1,k)$ multiplet and an $(k,k+1)$ multiplet.

If $∑_k L_k<∞$, then the $L_k^±$ can be uniquely determined from eqn. (\ref{zn})  if the $L^0_k$ are given.
In a matrix model relevant to JT supergravity, we expect that the $L_k$ and $L_k^±$ are all infinite in a double-scaled limit.
But it is reasonable to imagine
starting with a cutoff matrix model with $∑_k L_k<∞$, in which the $L^±_k$ are known in terms of the $L_k$ and the $L_k^0$.  Our strategy will be to understand
what is a reasonable random matrix model in such a cutoff situation, and then to extrapolate to a double-scaled limit.

But what should we assume for the $L_k^0$?
In an appropriate context, one might be interested in a probability distribution governing all possible solutions of $Q^2=0$.  In such a case, the $L^0_k$ would be random variables.
In $=2$ JT supergravity, however, one can just
calculate the $L_k^0$ from the disk partition function (see  [34] or Section \ref{sec:N2JTG}), and they appear to have no fluctuations associated to wormholes.

Since we want to assume that the $L_k^0$ are known, this means that in a cutoff version of the theory with $∑_k L_k<∞$, we should also consider the $L_k^±$ to be known.
Thus we want  to consider a particular component of the space of solutions of $Q^2=0$, characterized by given values of the $L_k^0$ and the $L_k^±$,
and understand what is a reasonable random matrix model in that context.   Hopefully, we will get an answer that makes sense in a double-scaled limit
in which the Hilbert space dimensions become infinite.

Roughly speaking, we want to consider  a cutoff model with $∑_k L_k<∞$ and a measure of the general form
\be\label{roughmeasure} \prod_k \d Q_k   \prod_s \delta(Q_{s+1} Q_s) \prod_r \exp(-\Tr_{\H_r} F_r(Q_r^\dagger Q_r)),\ee
with suitable functions $F_r$. The delta functions are meant to impose the vanishing of the real and imaginary parts of each matrix element of $Q_s+1Q_s$.  The solutions of the delta function constraints consist of a finite number of families, as just explained, and we want to focus on just one family.   There is not
a convenient way to enforce the restriction to one family by including an extra factor in eqn. (\ref{roughmeasure}), and we will just impose that restriction by hand.
For the desired component of solutions of $Q^2=0$,
after taking account of the delta functions and the symmetry group $G=∏_k (L_k)$, we want to reduce the model to an integral over a suitable set of

There are two reasons that the output of this procedure might lead to a model in which the representations of $R$-charges $(k,k+1)$ for different values of $k$ are not statistically
independent.  The $(k-1,k)$ and $(k,k+1)$ representations are potentially coupled both by the fact that the symmetry group $(L_k)$ acts on both $Q_k-1$ and $Q_k$ and by
the constraint $Q_k Q_k-1=0$.   However, it turns out that these two effects cancel, and the multiplets of different charge are statistically independent.

Let us go back to our simplified model with three consecutive charges and first consider the case that all $L_k^0$ vanish, so there are generically no supersymmetric states.
Hence $_̋k+1=_̋k+1^-$, $_̋k=_̋k^+⊕_̋k^-$, $_̋k-1=_̋k-1^+$.    To see the essential cancellation, let us look at the case $L_k+1=L_k-1=1$, $L_k=2$.
By the action of $G$, if we arrange $=̋_̋k+1⊕_̋k⊕_̋k-1$ as a sort of column vector
a  generic $Q$ with $Q^2=0$ and leaving no supersymmetric states can be put in the form
\be\label{canform}Q=\left(\begin{array}{c|cc|c} 0 & \lambda & 0 & 0\\  \hline 0&0&0&0 \\ 0&0&0&\t\lambda\\ \hline 0&0&0&0 \end{array}\right).\ee
We want to consider small fluctuations around this canonical form, and the measure that results from removing these small fluctuations by a combination
of delta functions and symmetries.
A general ansatz for the small fluctuations is
\be\label{cnform}Q=\left(\begin{array}{c|cc|c} 0 & \lambda &u  & 0\\  \hline 0&0&0&v \\ 0&0&0&\t\lambda\\ \hline 0&0&0&0 \end{array}\right),\ee
where the perturbation is parametrized by $u,v$.
A delta function constraint imposing $Q^2=0$ thus reduces in this case to
\be\label{deltafn}\delta^2(\t\lambda u+\lambda v), \ee where, for a complex number $b=b_1+ıb_2$, we define $δ^2(b)=δ(b_1)δ(b_2)$.
We should also analyze the action of $G=(1)_k+1×(2)_k ×(1)_k-1$.   There is a residual $(1)×(1)$ subgroup of $G$ that leaves fixed the canonical
form (\ref{canform}) of $Q$. Also, another $(1)×(1)$ subgroup -- say $(1)_k+1×(1)_k-1$ -- can be used to set $λ,λ≥0$. An argument similar to the one presented in \ref{randomone} implies that the contribution to the measure from these generators is a factor of $λ  λ̃$. The symmetry
generators not yet accounted for are the off-diagonal generators of $(2)_k$, say
\be\label{symgens}\begin{pmatrix} 0 & \w\cr \bar\w & 0 \end{pmatrix}. \ee
The transformation of $(u,v)$ that this generates is
\be\label{transgen} \delta_\w(u,v) =(-\w \lambda,\w\t\lambda). \ee
Transforming to a new  basis by
\be\label{stuv} (u,v)=\frac{s(\t\lambda,\lambda)+ t(-\lambda,\t\lambda)}{\sqrt{\lambda^2+\t\lambda^2}},\ee
normalized such that the Jacobian is trivial, the delta function is
\be\label{deltafn2}\delta^2\left(s\sqrt{\lambda^2+\t\lambda^2}\right)=\frac{1}{\lambda^2+\t\lambda^2}\delta^2(s),\ee
and the transformation generated by the off-diagonal symmetries is
\be\label{residtrans}\delta_\w s=0,~~\delta_\w t = (\lambda^2+\t\lambda^2)^{1/2}\w. \ee
Taking account the real and imaginary parts of $$, the volume factor associated to the off-diagonal symmetries is $λ^2+λ^2$,
and this precisely cancels the dependence on $λ,λ$  in (\ref{deltafn2}).   Hence in this example the measure is the product of a measure for $λ$
and a measure for $λ$, so the $(k-1,k)$ and $(k,k+1)$ multiplets are statistically independent.

Now consider the case of general $L_k+1,L_k,L_k-1$, still with all $L_k^0=0$.   The constraint $Q^2=0$ and the symmetries imply that
$Q_k: _̋k^+
and $Q_k-1:_̋k-1→ _̋k^+
can be put in the canonical form
\be\label{anotherform} Q_k=\left(\begin{array} {ccc|cc} \lambda_1 & 0 & 0 & \,0 & 0\\ 0& \lambda_2& 0 &\, 0 & 0 \cr 0 & 0&\lambda_3&\,0&0\end{array}\right),\hskip1cm
Q_{k-1}=\left(\begin{array}{cc} 0&0\cr 0&0\cr 0&0  \cr \hline\t\lambda_1&0\cr 0&\t\lambda_2\end{array}\right)\ee with all $λ_i,λ_j≥0$
(here we illustrate the case $L_k+1=3,  L_k=5,  L_k-1=2$).   The $λ$'s determine the energies of $(k,k+1)$ multiplets, and the $λ$'s determine the
energies of $(k-1,k)$ multiplets.   The effects that could potentially couple the two types of multiplet are the delta functions and symmetries that have been used to set to 0
the right block of $Q_k$ and the upper block of $Q_k-1$.   Those blocks are $L_k+1×L_k-1$ matrices, with one matrix element for each pair $(λ_i,λ_j)$.
For each  $i,j$, the vanishing of the $i,j$ matrix elements in the right block  of $Q_k$ and the upper block of $Q_k-1$ is determined by a complex delta function
that comes from the constraint $Q_k Q_k-1=0$, as well as a pair of off-diagonal generators of $(L_k)$, generalizing eqn. (\ref{symgens}).   For each $i,j$,
the effects on the measure of the off-diagonal symmetry generators and the   delta function cancel in exactly the same fashion as in the previous paragraph.  The delta function
and symmetry generators that are relevant for the vanishing of the $i,j$ matrix elements are not sensitive to the existence or non-existence of other matrix elements,
so a cancellation occurs precisely as described previously.

Hence the $(k,k+1)$ and $(k-1,k)$ multiplets are statistically independent.  The measure for each multiplet is the same as if the other multiplet did not exist,
and therefore, as in section \ref{randomone}, each multiplet is governed by an AZ  ensemble with $(,)=(1,2)$.

What happens if the $L_k^0$ do not all vanish?   For example, suppose that $L_k^0≠0$ (and therefore $L_k+1^0=L_k-1^0=0$).   The canonical forms
of $Q_k$ and $Q_k-1$ then have extra columns and rows of zeroes.   This does not lead to the existence of any new delta functions.\footnote{The rows and columns of
zeroes can be enforced by gauge-fixing the symmetries, and this should be done before trying to analyze the delta functions associated to the $Q^2=0$ condition.}   It does lead,
for each $λ_i$, to the existence of $2L_k^0$ generators  of $(L_k)$ that rotate the column of $Q_k$ containing $λ_i$ into the columns of zeroes, and
also, for each $λ_j$, to an orthogonal set of $2L_k^0$  generators of $(L_k)$  that rotate the row
of $Q_k-1$ containing $λ_j$ into the $L_k^0$ rows of zeroes.
As in section \ref{randomone},  the effect of these generators is to supply a factor $∏_iλ_i^2 L_k^0$
in the measure for the $λ$'s, and likewise a factor $∏_jλ_j^2 L_k^0$ in the measure for the $λ$'s.   No coupling is induced between the
$λ$'s and $λ$'s because one set of new generators acts only on the $λ$'s and an orthogonal set of new generators acts only on the $λ$'s.
Hence the $(k,k+1)$ and $(k-1,k)$ multiplets are statistically independent AZ ensembles with $(,)=(1+2 L_k^0,2)$.

If instead we assume that $L_k^0=0$ while $L_k+1^0$ and $L_k-1^0$ are potentially both nonzero, then there are $2 L_k+1^0$ new symmetry generators
rotating the column of $Q_k$ containing $λ_i$ for each $i$, and $2L_k-1^0$ new symmetry generators rotating the column of $Q_k-1$ containing $λ_j$ for each $j$.
Now the $(k,k+1)$ multiplets are described by an AZ ensemble with $(,)=(1+2L_k+1^0, 2)$, and the $(k-1,k)$ multiplets are described by an AZ ensemble
with $(,)=(1+2 L_k-1^0,2)$.

We can summarize what we have learned by saying that if there are just three adjacent values of the $R$-charge, then each multiplet, say of $R$-charges $r,r+1$ (where $r$ could be $k-1$ or $k$)
is described by an AZ multiplet with
\be\label{bonven}\upalpha=\upalpha_0+\upbeta\nu, ~~~~(\upalpha_0,\upbeta)=(1,2), \ee
where $ν$ is the number of BPS states with $R$-charge $r$ or $r+1$.

Perhaps surprisingly, something essentially new happens with four consecutive values of the $R$-charge, say $k-1,k,k+1,k+2$.   To explain the basic issue,
let us assume that each of the four Hilbert spaces is of dimension 1.   Then each of $Q_k-1, Q_k$, and $Q_k+1$ is a $1×1$ matrix, say $Q_k-1=a$, $Q_k=b$,
$Q_k+1=c$.   The constraints $Q_k+1Q_k=Q_k Q_k-1=0$ become $ab=bc=0$.    Let us look at the branch of the moduli space of solutions with $a,c≠0$, $b=0$.
This branch describes a $(k-1,k)$ supermultiplet and a $(k+1,k+2)$ supermultiplet.   The delta functions would appear to be
\be\label{zimbo} \delta^2(ab)\delta^2(bc)=\frac{1}{|a|^2}\frac{1}{|c|^2} \delta^2(b) \delta^2(b) =\frac{1}{|a|^2}\frac{1}{|c|^2} \delta^2(b)\delta^2(0).\ee
Clearly this does not make sense as written, because of the $δ^2(0)$.

Concretely, the reason that this happened is that with four consecutive values of the $R$-charge, the constraints are not independent.  For any $R$-charge $k$, the constraint operators
$Z_k =Q_k+1 Q_k$ satisfy $Z_k Q_k-1=Q_k+1Z_k-1$.   In the case under discussion, the constraints are $ab$ and $bc$, and the relation between the constraints
is $(ab)c=a(bc)$.

There is actually a natural procedure to define a measure in such a problem with redundant constraints.   This procedure is known in the context of BV quantization and also in algebraic
geometry, but rather than giving a general explanation, we will just explain how the procedure works in the case at hand.   In doing so, to slightly simplify the story, we will
treat $a,b,c$ as real variables.  In the context of JT supergravity with time-reversal symmetry, there is actually a natural problem in which all the matrices are real and in particular
$a,b,c$ are real variables.  This will be explained in section \ref{boundaryview}.  But it will probably be obvious to the reader that by complexifying all of the objects that
we will consider, including the Lagrange multipliers and the ghosts, one could adapt what we will describe to the case of complex variables.

First of all, to impose a constraint $bc=0$ and also a constraint $ab=0$, we could add two Lagrange multipliers $λ$ and $λ'$ and include a term
$ı(λbc+λ' ab)$ in the exponent of the matrix integral.   Thus we would define an extended version of the matrix integral that contains a factor
\be\label{zolbox}\int \frac{\d\lambda \d\lambda'}{2\pi}\,\:\exp\left(\i(\lambda bc+\lambda' ab)\right). \ee
One can indeed view eqn. (\ref{zolbox}) (and its obvious analog in more general cases) as a more microscopic description of the origin of the delta functions in eqn.
Eqn. (\ref{zolbox}) is  fine if the constraints are independent, but here we are interested precisely in a case in which they  are not independent:  they obey the identity $a(bc)-c(ab)=0$.
This means that the extended matrix model with the Lagrange multipliers included has a gauge invariance
\be\label{olbox}\delta \lambda= -\epsilon a,\hskip1cm \delta\lambda'=\epsilon c, \ee with infinitesimal parameter $ϵ$.
Let us treat the extended theory as a gauge theory with this gauge invariance.   It could be gauge-fixed with a  gauge condition $Λ=0$ for a rather general function $Λ$; let us
consider the simple choice $Λ=λ'$.
Following standard reasoning, the ghost determinant for this gauge condition is\footnote{Faddeev-Popov quantization is often described without specifying that one
should take the absolute value of the ghost determinant, mainly because in many standard applications this determinant is anyway positive.
But in fact, since one is trying to define a gauge-fixed measure, and measures are positive, one does want
the absolute value of the ghost determinant.}  $|Λ̣/ϵ̣|=|c|$.  So in this gauge, the additional factor that we introduced to
deal with the constraints reduces to
\be\label{nolbox} \frac{|c|}{2\pi}\int\d \lambda \exp(\i\lambda bc)=|c|\delta(bc).\ee
On the locus with $c≠0$, we can scale $c$ out of the delta function, with $|c|δ(bc)=δ(b)$.
With this measure, the offending variable $b$ can be eliminated
and forgotten, and we reduce to the case that the two multiplets of charges $(k-1,k)$ and $(k+1,k+2)$ are statistically independent, and described by the ensembles
that one finds by studying them separately.

We will leave our study of random $=2$ supersymmetry here, remarking only that it appears that no essentially new issues arise if one considers Hilbert spaces of higher
dimension or more than four consecutive values of the $R$-charge.    Hopefully, what we have said is sufficient to convince the reader that statistically independent AZ ensembles
with the parameters in eqn. (\ref{bonven}) are a natural candidate for random $=2$ supersymmetry.   In the rest of this article, we will confirm that this kind of ensemble, with the
appropriate spectral curve, does match predictions of $=2$ JT supergravity.

\subsection{Restoring $\h q$}\label{restoring}

In this discussion, we have arbitrarily assumed that the $R$-charge generator $J$ has integer eigenvalues, and that the supercharges $Q,Q^†$ have $R$-charges
1, $-1$.

To see why one would want to relax these assumptions, we can consider $=2$ SYK models [35].   The simplest such models are
constructed from $N$ complex fermions $ψ_a$, each
of $R$-charge 1, with a supercharge $Q$ that is a polynomial in the $ψ_a$ of some odd degree $q≥3$.  $Q$ is then of $R$-charge $q$.
By including complex fermions with different $R$-charges, one can construct an $=2$ SYK model
with $q=1$ [36], and a model in which $Q$ has negative odd $R$-charge can be constructed by just exchanging $Q$ and
$Q^†$.  So in general $q$ is any odd integer.

In addition, quantizing a complex fermion of $R$-charge $n$ gives a pair of states of $R$-charges\footnote{This statement
assumes that the $R$-charge generator $J$ is defined to be odd under a charge conjugation operation that exchanges $\psi_a$ with $\psi_a^\dagger$.  Without that
assumption, one is free to add an arbitrary constant in the definition of $J$.} $±n/2$.   As a result,
depending on the numbers and $R$-charges of the elementary fermions, the spectrum of an $=2$ SYK model may consist of states of integer $R$-charge,
or states with $R$-charge valued in $1/2+$.  The nonintegralilty of the $R$-charge is a sort of anomaly.

Though these values for the $R$-charges of the operator $Q$ and of the physical states
differ from what was assumed in section \ref{randomtwo}, the analysis in that section carries over in an obvious way
to the family of all states whose $R$-charges differ from some given value by integer multiples of $q$.   The states of a supersymmetric SYK model come in $q$ families of that type,
and as we will see, such a spectrum is also natural in JT supergravity.

If one is going to consider anomalous models in which the $R$-charges of physical states are not integer-valued,
it is natural to consider a general case in which the $R$-charges take values in $δ+$ (and $q$ is any positive odd integer), for any $δ∈[0,1)$.
From a boundary point of view, arbitrary $δ$ can be achieved by just adding a constant to the $R$-charge generator.
Roughly speaking, the
parameter $δ$ can be interpreted from a bulk point of view as $θ/2π$, where
$θ$ is the theta-angle of the bulk $(1)$ gauge field $A$, with field strength $F=Ạ$,  that is related to the $R$-charge on the boundary.   This statement reflects the identity
\be\label{welid} \exp\left(\i\frac{\theta}{2\pi}\int_D F\right) =\exp\left(\i\frac{\theta}{2\pi} \oint_{\partial D}A  \right) \ee for an oriented two-manifold $D$,
showing that a bulk topological angle $θ$  induces a charge $θ/2π$ on the boundary of $D$.   However, in the context of $=2$ JT supergravity, one has to take into account
the fact that the super Schwarzian mode, which we discuss in more detail in section \ref{sec:N2JTG}, contains a complex fermion of charge $q$.  Quantization of this fermion gives two
boundary states of charge $±q/2$.  Adding this to the boundary charge associated to a bulk theta-angle, we find that the relation between the charge defect $δ$
in a dual quantum mechanics model that lives on the boundary and the bulk theta-angle is really
\be\label{delfo} \delta =\frac{1}{2} +\frac{\theta}{2\pi}~{\mathrm{mod}}~\Z.\ee
This formula is analogous to the relation between bulk and boundary anomalies that was found for $=1$ JT supergravity in [14]. We will comment more on this later when we analyze the path integral of $=2$ JT supergravity.

\subsection{Check By Simulation of $\N=2$ SYK}\label{check}
\begin{figure}[t!]
\begin{center}
\includegraphics[width=6in]{Fig1Alt.pdf}
\end{center}
\vspace{-0.8cm}
\caption{\footnotesize  Level spacing statistics for $\cN=2$ SYK with $N=10$ fermions, averaging over $400$ instances of the couplings. The plots show the probability distribution of $r=(\lambda_{j+2}-\lambda_{j+1})/(\lambda_{j+1}-\lambda_j)$ for the multiplets of charges $(-4,-1)$, $(-3,0)$, and $(-2,1)$.  We omit multiplets related to these by $\CT$ symmetry
or with only one state. The smooth curves denote the surmise $P_\upbeta(r)$ with $\upbeta=1$ (dashed blue), $\upbeta=2$ (black) and $\upbeta=4$ (blue). The distribution of all multiplets is consistent with $\upbeta=2$. The case $(-4,-1)$ has larger  deviations since the Hilbert space dimension is smaller. \label{fig:N10}}
\end{figure}

In this section, we describe the results of a simulation comparing the local level statistics of $=2$ SYK models [35] to predictions of the
$=2$ random matrix ensemble that was proposed in section \ref{randomtwo}. The simplest $=2$ SYK models are constructed from
$N$ complex fermions $ψ_a$, $a=1,…,N$, of $R$-charge 1,
satisfying ${ ψ_a , ψ^†_b} = δ_ab$, with ${ψ_a , ψ_b} = { ψ^†_a , ψ^†_b} = 0$.
For simplicity, we take $q=3$ and thus a cubic supercharge:
\bea\label{eq:N2SYKQQ}
Q = \i \sum_{1\leq a<b<c\leq N} C_{abc}\,  \psi_{a} \psi_{b} \psi_{c}.
\ea
The $C_abc$ are random complex numbers with Gaussian statistics and we measure time in units such that $⟨C_abc C_abc⟩= 2/N^2$. This supercharge automatically satisfies the algebra $Q^2=Q^†^2=0$ for all values of the couplings (this would not be true if $Q$ depended on both $ψ$ and $ψ^†$). The Hamiltonian is given by $H = { Q, Q^†}$ and $[H,Q]=[H,Q^†]=0$.   The $R$-charge is defined explicitly as $J = ∑_a ψ^†_a ψ_a  - N/2$.

We represent the fermions $ψ_a$ in the Hilbert space $$̋ of $N$ qubits
as $\psi_a=Z_1 \otimes \ldots Z_{a-1}\otimes (X_a+\i Y_a)/2$, where $X_a$,$Y_a$ and $Z_a$ are the Pauli matrices acting on the $a^{th}$ qubit.
Then we construct $Q$ using (<ref>). In this representation $J$ is already diagonal.  The subspace $\H_n$ of $\H$ consisting of states with $J=n$ has dimension
$dim ℋ_n = N N/2+n$.
For moderate values of $N$, exact diagonalization of the model is possible and the level statistics can be compared to expectations.
In more detail, we  compute the ``eigenvalues'' (or singular values) of $Q_k: ℋ_k →ℋ_k+3$.
Here  $k$ ranges from  from $-N/2$ to $N/2-=N/2-3$. We can test the value of $$ by computing the ratio
$r=(λ_j+2-λ_j+1)/(λ_j+1-λ_j)$, evaluated after arranging the spectrum in increasing order. This quantity was introduced in
[37] and does not require ``unfolding'' (rescaling according to the local density of states). In an AZ $(,)$ ensemble, this quantity
depends only on $$, not on $$ (which affects the local behavior only near zero energy).
The analog of the Wigner surmise is given by
$P_(r) = Z_^-1 (r+r^2)^/(1+r+r^2)^1+3/2 $; see [38].
The coefficient $Z_$ is found by normalizing the probability distribution to $∫_0^∞ṛ P_(r) = 1$.
It would be interesting but possibly more difficult to verify the random matrix prediction for $$.
\begin{figure}[t!]
\begin{center}
\begin{center}
\includegraphics[width=4.5in]{Fig2Alt.pdf}
\end{center}
\end{center}
\vspace{-0.6cm}
\caption{\footnotesize  Level spacing statistics for $\cN=2$ SYK with $N=11$ fermions, averaging over $200$ instances of the couplings. The  plots show the probability distribution of $r$ for
multiplets with the indicated $R$-charges. The smooth curves denote the surmise $P_\upbeta(r)$ with $\upbeta=1$ (dashed blue), $\upbeta=2$ (black) and $\upbeta=4$ (blue). The distribution of all multiplets is consistent with $\upbeta=2$, except for the case of charges $(-3/2,3/2)$, which is matched by $\upbeta=1$. \label{fig:N11}}
\end{figure}

We show the results of a simulation  for $N=10$ in fig. \ref{fig:N10} and for $N=11$ in fig. \ref{fig:N11}.  The model has a $$ symmetry (charge conjugation plus
time-reversal)   that acts by $ψ_a ↔ψ_a^†$ and $J ↔- J$.
Multiplets related by $$ symmetry  to the ones shown are omitted in the figures. We also omit the multiplets $(-N/2,-N/2+q  )$, $(N/2-q,N/2)$, which always occur with multiplicity 1.
For  $N=10$, all cases are described by an ensemble with $=2$. For $N=11$,
this is the case except for the $$ self-conjugate $(-3/2,3/2)$ multiplet, for which $=1$.
Some of these features are anticipated in [28].

The results sketched in the figures can be understood as follows.  For generic $k$, the $$  symmetry exchanges the $(k,k+ )$  and $(-k-,-k)$ multiplets,
implying that they have the same spectrum and level statistics,
without changing the random matrix theory prediction $=2$.
when $N$ is odd  there is a $$-invariant multiplet  $(-q/2,q/2)$, and for this multiplet the random matrix prediction is modified in a way we explain in section \ref{boundaryview}.
In the case $N=11$, illustrated in the figure, if we define $$ so that $Q^-1=Q^†$, then $^2=1$, and this leads to $=1$ for the $$-invariant
multiplet, as explained
in section \ref{boundaryview} and as illustrated in the figure.   In a similar model with $Q ^-1=Q^†$ and $^2=-1$, one expects $=4$, as explained in
section \ref{boundaryview}.

The fact that different supermultiplets have the level spacing statistics characteristic of random matrix universality supports the proposal that the supermultiplets are statistically independent. A more precise verification of this claim in an SYK simulation is an open problem.

\section{$\mathcal{N}=2$ JT Gravity Black Hole}\label{sec:N2JTG}
Our next goal is to analyze $=2$ JT gravity in the simplest spaces, the disk and the cylinder. In section \ref{sec:Z01}, we study the disk, which
determines the black hole spectrum to leading order in the topological expansion and provides us with the spectral curve of the random matrix ensemble outlined in section \ref{randomtwo}.
In section \ref{sec:Z02}, we turn to the cylinder, which in random matrix theory  is uniquely fixed by the ensemble and is independent of the details of the spectral curve. By
analyzing the cylinder,  we verify the prediction that different supermultiplets are statistically independent. We conclude in  section
\ref{sec:3bwh} with some preliminary calculations for the three-boundary wormhole, which we explore   more fully in section \ref{sec:GPI}.

\subsection{The Leading Order Black Hole Spectrum}\label{sec:Z01}
We will be brief in describing $=2$ JT supergravity, since this theory has been  studied extensively in the past [35, 34, 39, 40, 41, 42, 43, 25, 26].
The action of this theory consists of two terms. The first is given by a constant $-S_0$ times the Euler characteristic of the spacetime. Since this term is a topological
invariant,  it does not require a supersymmetric completion. In the sum over spacetime topologies,  this term weights surfaces by $(e^S_0)^2-2g-n$, where $g$ is the number of handles, or  spacetime wormholes, and $n$ is the number of boundaries.  The expansion is  thus an asymptotic expansion in $e^-S_0$.

The second term consists of a bulk piece that can be written, in a first order formulation of gravity, as a topological $BF$ theory with gauge supergroup $G=(1,1|1)$, or equivalently\footnote{\label{osptwotwo} Actually, $\OSp(2|2)$ has a center $\Z_2$, while $\SU(1,1|1)$ has trivial center.   The precise relation between the two
supergroups is that $\SU(1,1|1)=\OSp(2|2)/\Z_2$.   We write $\OSp'(2|2)=\OSp(2|2)/\Z_2$.  Similarly a maximal bosonic subgroup of $\SU(1,1|1)$ is really $(\SL(2,\R)\times \uU(1))/\Z_2$
(where $\Z_2$ is generated by the product of  $-1\in \SL(2,\R)$ and a $\pi$ rotation in $ \uU(1)$), not quite $\SL(2,\R)\times \uU(1)$ as stated in the text.  For brevity, we will
omit details such as these when not essential.}  $(2|2)$. This group has $SL(2,ℝ)×(1)$ as a maximal bosonic subgroup, together with four fermionic generators. An explicit construction of this action using the first order formulation can be found in [42], while a construction using two-dimensional $=2$ superspace is developed in [39, 40]. In the second order formulation, the theory describes a two-dimensional spacetime metric together with a complex gravitino, the JT gravity dilaton together with a complex dilatino, and a
$BF$ theory of the $(1)$ $R$-symmetry.
The dual quantum mechanical system lives on a nearly AdS (NAdS) boundary, defined by specifying the proper length of the boundary and the boundary value of the dilaton,
and taking them to both diverge with a fixed ratio.
In the presence of NAdS boundaries,  one needs to supplement the bulk term with boundary counterterms
including the Gibbons-Hawking-York boundary term.
Depending on the boundary conditions,
boundary terms might be required for the $(1)$ gauge field as well.

We normalize the generator $J$ of the $(1)$ $R$-symmetry such that its spectrum consists of the integers $$. Fermionic operators have odd charge and bosonic
operators  have even charge. In the simplest theory,  the gravitino and dilatino have unit charge, but one can consider more general theories where their charge is any odd integer $q$, where we follow the notation of section \ref{randommatrix}. In the $BF$ formulation of the theory, this is equivalent to taking the gauge group to be not
$(1,1|1)$ but its  $$-fold cover obtained  by ``unwrapping'' the $(1)$ subgroup $$ times.

After these preliminaries, let us begin with the evaluation of the $=2$ JT gravity path integral on the disk with NAdS boundary conditions.  We denote
this path integral as\footnote{The reader
will hopefully not confuse the inverse temperature $\beta$ and the $R$-charge fugacity $\alpha$ with the Altland-Zirnbauer parameters $(\upalpha,\upbeta)$.}  $Z_disk(β,α)$.  Here $β$ is the  renormalized circumference of the boundary circle, and $α$ is a fugacity: fermions are twisted in phase by $e^ıαJ$ as they go around the thermal circle. Therefore $μ= ıα/β$ has the interpretation of a  chemical potential for the $R$ symmetry.

The general rules of the gravitational path integral imply that the disk path integral defined in the previous paragraph computes the following quantity for the quantum system describing the $=2$ JT gravity black hole:
\bea\label{eq:zdiskrmtp}
Z_{\rm disk}(\beta,\alpha)=\langle {\rm Tr}\big( e^{-\beta H} e^{\i \alpha J}\big)\rangle_{g=0}.
\ea
$H$ is the Hamiltonian acting on the Hilbert space of the boundary theory, and $J$ generates  the $(1)$ $R$-symmetry. The brackets on the right hand side anticipate that for pure $=2$ JT gravity the quantum mechanical dual is an ensemble average over boundary theories. The label $g=0$ on the right means this is evaluated to leading order in the genus expansion of the matrix integral describing this ensemble.   The trace that appears in eqn. (\ref{eq:zdiskrmtp}) is a trace in the Hilbert space of the boundary theory.

In general, the $BF$ theory partition function reduces to an integral over the moduli space of flat connections modulo gauge transformations. In the case of gravity, corresponding to $SL(2,ℝ)⊂G$, one further restricts to the component with a geometric description and divides by the mapping class group; more details can be found in section \ref{sec:GPI}. This gauge theory procedure is  equivalent to integrating over the moduli space of hyperbolic surfaces, in our case with extended $=2$ supersymmetry (see [27] for mathematical
theory of $=2$ hyperbolic surfaces). The case of the disk is particularly simple, since the moduli space consists of a single configuration up to a boundary reparametrization mode.
This mode is governed by the $=2$ Schwarzian theory introduced in [35]. Its partition function was computed in [34], and we now give a quick overview of the calculation.

We begin with an outline of how the $=2$ Schwarzian mode arises. Parametrize the hyperbolic disk with a complex coordinate $e^-ıx$, such that $x ≅x+2π$. (The precise form of the metric will not be important for the following discussion.) The center of the disk is at $e^-ıx = 0$ or  $Im x=-∞$, while the NAdS boundary is located at $Im  x = T$ for some large $T$. Including the fermions, the disk is described by $x,ψ,ψ̅$ with equivalence
\bea
(x,\psi,\bar{\psi})\cong (x+ 2\pi ,- \psi, -\bar{\psi}).
\ea
In principle there could be a non-trivial holonomy $ψ→- e^ıϕ ψ$ as we go around the thermal circle, but smoothness at
the center of the disk requires $ϕ=0$. We set $x= ıT + y$ and the boundary is parametrized  by $y,ψ,ψ̅$ with real $y$ and identification
\bea
(y,\psi,\bar{\psi})\cong (y+ 2\pi ,- \psi, -\bar{\psi}).
\ea
Now we pick the observable that we want to insert at the boundary, namely the grand canonical partition function $Tr(e^-βHe^ıαJ)$. To compute this observable, the boundary quantum mechanical system is formulated on a supercircle parametrized by $t,θ,θ̅$ with an equivalence relation
\bea\label{eq:equivreltttb}
(t,\theta,\bar{\theta}) \cong (t+\beta, -e^{\i \hq \alpha} \theta, - e^{-\i \hq\alpha} \bar{\theta}).
\ea
After localizing the gravity path integral on the hyperbolic disk, the $=2$ Schwarzian theory is the only leftover gravitational mode. It involves an integral over all $=2$ superconformal identifications between the boundary of the disk, parametrized by $(y,ψ)$ and the worldline of the quantum mechanical system, parametrized by $(t,θ)$, that are compatible with the above equivalences, and such that
the map from $t$ to $y$ has winding number one.
Any such identification can be obtained from a standard one by an invertible superconformal reparametrization of $(y,ψ)$.   Here we are
not just being careless in omitting $ψ̅$ and $θ̅$.
The identification from $(y,ψ,ψ̅)$ to $(t,θ,θ̅)$ and therefore the reparametrization of $(y,ψ,ψ̅)$
is required to be $=2$ superconformal.  For a general number $$ of supersymmetries, this condition is  subtle,
but in the special case $=2$, there is an easy way to implement it: the group of $=2$ superconformal reparametrizations of $(y,ψ,ψ̅)$ is the same as the group of
all reparametrizations of $(y,ψ)$. (This is related to the fact that the $=2$ superconformal algebra can act on a chiral superspace that is parametrized by $y,ψ$ with $ψ̅$ forgotten.)
So we will implement the superconformal condition by writing formulas relating $(t,θ)$ to $(y,ψ)$ with no mention of $ψ̅$, $θ̅$.
As for the restriction to maps of winding number 1, this a simple illustration of the fact that to go from $BF$ theory to gravity we need to restrict the class of flat connections by hand.  Finally,
to  not overcount geometries, it is important to remove the reparametrizations that correspond to the isometries of the $=2$ hyperbolic disk, which form a group $(1,1|1)$.

The integration space $𝒲$ of the Schwarzian theory is therefore the group of superconformal reparametrizations of $(y,ψ)$ divided by the right action of
$(1,1|1)$.   This  is a symplectic supermanifold and the action arising from $=2$ JT gravity is the generator, via Poisson brackets, of a global $(1)$ symmetry of $𝒲$,
with generator $∂/∂t$; see [34].  For these reasons, the integral over the Schwarzian mode can be computed by localization on fixed points of $∂/∂t$. However, in this context, because of the quotient by $(1|1)$, ``fixed point'' means ``fixed point up to a superconformal automorphism of the disk.'' The relevant automorphisms of the disk are generated by $∂/∂y$ and $ı(ψ∂_ψ - ψ̅ ∂_ψ̅)$ (we will include $ψ̅$ in
some formulas, but it can be consistently set to zero).
The coefficients of each generator can be different at different fixed points. Taking into account the winding number constraint, the fixed points take the form
\begin{align}\notag
y(t,\theta) &= \frac{2\pi}{\beta} t,\\
\psi(t,\theta) &= e^{-\i \hq \alpha\, t/\beta} \theta.\label{fixedpt}
\end{align}
These are annihilated by
\bea
D = \frac{\partial}{\partial t} - \frac{2\pi}{\beta} \frac{\partial}{\partial y} + \frac{\hq \alpha}{\beta}\, \i (\psi \partial_\psi - \bar{\psi} \partial_{\bar{\psi}}),
\ea
so they do represent fixed points.   Since the boundary condition only depends on the value of $α$ mod\footnote{The boundary condition on $\theta$ depends on the value
of $\alpha$ mod $2\pi n/\h q$.   But when we introduce the parameter $\h q$, we presume that the theory can be probed by operators of $R$-charge 1.   Taking this into
account, the indeterminacy in $\alpha$ is $\alpha\to \alpha+2\pi \Z$.}  $2π$, eqn. (\ref{fixedpt}) really describes an infinite family of fixed
points, one for each real-valued lift of $α$.

Now we want to study the one-loop integral that arises in expanding around these fixed points. For this, one needs to know the value of the classical Schwarzian action at a fixed point, and also the ``rotation eigenvalues,'' that is, the eigenvalues of the operator $D$ acting on the tangent space of a fixed point. These are evaluated  in appendix \ref{sec:disktrumpet}. The final answer for the disk partition function, after summing over fixed points weighted by a product of functional determinants, is given by
\beq
Z_{\rm disk}(\beta,\alpha) =e^{S_0} \sum_{n\in \mathbb{Z}} \exp(2\pi \i n \delta)\frac{\hq \,\cos\left( \pi \widehat{q}\, (\frac{\alpha}{2\pi}+ n)\right)}{2\pi^3 \left(1-4 \hq^{\, 2}(\frac{\alpha}{2\pi}+ n)^2\right)} e^{\frac{\pi^2}{\beta}(1-4\widehat{q}^{\, 2} (\frac{\alpha}{2\pi}+  n)^2)}.
\eeq
We have weighted the $n^th$ fixed point with a factor  $exp(2πın δ)$.   As we will see in a moment, $δ$ is the charge defect parameter that was introduced in section \ref{restoring} and related to the bulk theta-angle in eqn. (\ref{delfo}).

As we explain in section \ref{randommatrix}, the Hilbert space decomposes into supermultiplets with charges $(k,k+ )$, where $k∈+δ$ and $k,k+$ are the $R$-charges of the states
in a two-dimensional supermultiplet. It will be convenient, in the gravity analysis, to label a supermultiplet by the average charge $q=k+/2$, such that the $R$-charges of the multiplets are $(q-/2,q+/2)$. Since the fermions of the boundary super-reparametrization mode carries $R$-charge $+/2$ and $-/2$, the parameter $q$ can be interpreted as the bulk contribution to the microscopic $R$-charge. In this presentation, the bulk charge takes values $q ∈ℤ+δ- 1/2$, since $$ is always odd.

The partition function on the boundary side can be expanded into  BPS states and two-dimensional supermultiplets  as
\bea\label{eq:spectrumcomm}
\langle {\rm Tr}\big( e^{-\beta H} e^{\i \alpha J}\big)\rangle_{g=0} &=&\sum_{k\in \mathbb{Z}+\delta} e^{\i \alpha k} N_{\rm BPS}(k) \nonumber\\
&&+  \sum_{q\in \mathbb{Z}+\delta-\frac{1}{2}} \big(e^{\i \alpha (q-\frac{\hq}{2})} + e^{\i \alpha (q+\frac{\hq}{2})}\big) \langle {\rm Tr}_{\M,q}\big( e^{-\beta H}\big)\rangle_{g=0}.
\ea
On the left hand side of this formula, the symbol $$ represents a trace over the whole Hilbert space of the boundary theory (including BPS states and both bose and fermi states of
two-dimensional supermultiplets).  On the right hand side, the sum over BPS states of $R$-charge $k$ has been separated out. In the second line, we included a sum over non-BPS multiplets,  labeled by the average $R$-charge $q$,
accounted for by the explicit factor $e^ıα(q-/2) + e^ıα(q+/2)$. Therefore $_,q$ is a matrix model trace in which each  supermultiplet with average charge $q$ is counted once;
in other words, what are counted are the independent ``eigenvalues'' of the matrix model that governs $q±/2$ supermultiplets.  At the end of section \ref{sec:Z02}, we will
justify the claim that the number of BPS states of charge $k$, which we denote
$N_BPS(k)$, does not fluctuate as a random variable and instead can be regarded as a fixed number.

It is convenient to define $ ⟨Tr_,q( e^-βH)⟩_g=0 =  ∫_0^∞Ẹ e^-βE  ρ_q(E)$,
where $ρ_q(E)$ is the spectral density for a supermultiplet of charge $(q-/2,q+/2)$. The disk amplitude encodes both the number of BPS states of a given $R$-charge and the non-BPS density of states. The $=2$ Schwarzian partition function can be expressed
in the following way [41], via an inverse Laplace transform and Poisson resummation:
\bea \label{handyexp}
Z_{\rm disk}(\beta,\alpha)&=& \sum_{k\in \mathbb{Z}+\delta, |k|<\frac{\hat{q}}{2} } e^{ \i \alpha k} ~ \frac{e^{S_0}}{4\pi^2}\cos \Big(\frac{\pi k}{\hq}\Big) \label{diskdos}\\
&+&  \sum_{q\in \mathbb{Z} + \delta-\frac{1}{2}} \big( e^{\i \alpha (q-\frac{\hq}{2})} + e^{\i \alpha (q+\frac{\hq}{2})}\big) \int_{E_0(q)}^\infty dE e^{-\beta E}\frac{e^{S_0}}{2\pi}\frac{\sinh{\big(2\pi \sqrt{E-E_0(q)}\big)}}{4\pi^2 E}.\nonumber
\ea
where $E_0(q) ≡q^2/4^  2$ corresponds to the energy gap for a non-BPS supermultiplet of $R$-charges $q±/2$. This is explained in appendix \ref{sec:disktrumpet}.  Since $E_0(q)$ is monotonically increasing in $|q|$,  only finitely many of the $ρ_q(E)$ are nonzero at any given $E$.    Given
this, the formula (\ref{handyexp}) is uniquely determined.

We can now read off the necessary input to the random matrix ensemble that governs $=2$ JT supergravity:
\bea
N_{\rm BPS}(k) =&\begin{cases} \frac{e^{S_0}}{4\pi^2} \cos \big( \frac{\pi k}{\hq} \big) &\hskip.5cm |k|<\frac{\hq}{2} \cr
0 & \hskip.5cm|k|\geq \frac{\h q}{2}\end{cases}\\
\rho_q(E)=&\begin{cases}\frac{e^{S_0}}{2\pi}\frac{\sinh{\big(2\pi \sqrt{E-E_0(q)}\big)}}{4\pi^2 E}& E>E_0(q)\\
0 & E\leq E_0(q).\end{cases}
\ea

We conclude  with several observations.  The formulas that we have written make sense for any value of $δ$, but
are consistent with a charge conjugation symmetry $$ or a charge conjugation plus time-reversal symmetry $$ precisely if $δ=0$ or $1/2$.  (At $δ=0,1/2$, JT gravity
and the boundary matrix model potentially have both of these symmetries, though  in an $=2$ SYK model, $$ is more natural.)
When $δ=0$, the charges that appear in the spectrum are integers and the theory is anomaly-free.  When $δ=1/2$, the charges are  half-integers
and there is a mixed anomaly between $J$ and $$ or $$, since the expectation that $e^2πıJ=1$ clashes with
the expectation that $J$ is odd under $$ or $$: if $J$ is defined to be odd, then $e^2πıJ=-1$.
For generic $δ$, there is no $$ or $$ symmetry, and
one can preserve the statement $e^2πıJ=1$ by shifting $J$ by $δ$.

Another observation is that since BPS states only occur for  charges $-/2 < k < /2$,
there are never two BPS states differing in charge by $$.   This is expected, since  two BPS states with charges separated by $$ could combine into a non-BPS

Finally, we note that the energy $E_0(q)$ of the lightest non-BPS state is positive except for $q=0$, corresponding to a multiplet with $R$-charges $(- /2,+/2)$ (a value that is available precisely if
The behavior of the density of states near threshold is quite different in the two cases.  For generic $q$,
$ρ_q(E_0(q) +ε) ∼√(ε)$, while for $q=0$, the threshold is at $E=0$ and   $ρ_q=0(ε) ∼1/√(ε)$.
This difference is actually expected based on the random matrix analysis of section \ref{randomtwo}.   There, we claimed that the non-BPS multiplets are governed by an AZ ensemble with $(,)=(1+2ν,2)$ (where $ν$ is the number of BPS states).   In leading order at large $e^S_0$, the difference
between an AZ ensemble $(,)$ ensemble and a Dyson $$ ensemble with a suitable potential is unimportant except near zero energy.
So as long as  $E_0(q)>0$, we expect the typical threshold behavior  $ρ_q(E_0(q) +ε) ∼√(ε)$ of a Dyson ensemble.
However, when the threshold is at zero energy, an AZ $(,)$ ensemble has the threshold behavior $ρ(ε)∼1/√(ε)$,
as long as $$ is of order 1
(not of order $e^S_0$).   This condition is satisfied here, since $N_(±/2)=0$ and the expected ensemble therefore has $(,)=(1,2)$.

We define the spectral curve $y_q(x)$, for a multiplet with $R$-charges $q±/2$, by the condition that $y_q(x±ıε) = ∓ıπe^-S_0ρ_q(x)$.  So
\bea\label{eq:N2sc}
y_q(x) =\frac{1}{2\pi} \frac{\sin \big( 2\pi \sqrt{-x + E_0(q )}\big)}{4\pi x}.
\ea
This function, by definition, has a branch cut along the support of $ρ_q(x)$.  It also has, for $q ≠0$, a pole at $x=0$.
The interpretation of this pole in random matrix theory is explored in appendix \ref{app:Loop}; here we summarize a few facts.
For $|q|<$, the residue of the pole, namely
\beq
{\rm Res}_{x=0}~ y_{q}(x) = \frac{\sin \big( \frac{\pi |q|}{\hq}\big)}{8\pi^2},
\eeq
is determined by the number of BPS states in the multiplet of charges $q±/2$; it is
$1/2 N_(k)$ where $k=q±/2$ with the sign determined such that $|k|</2$ (except that a factor $e^S_0$ present in $N_$ was removed in defining $y_q$).
Random matrix
analysis predicts a pole in $y_q(x)$ with this residue, assuming that the matrix potential is regular at zero energy.
For $|q|≥$, there are no BPS states, but there is still a pole.  To account for this in random matrix theory, one has to assume that the matrix potential
$F(^†)$ defined in eqn. (\ref{genform}) has a logarithmic behavior near the origin, $F(^†)∼log^†$.   For $|q|>$, the residue of the pole
and therefore the coefficient of the logarithm can have either sign, depending on $q$; this coefficient is large, of order $e^S_0$.
When the logarithm has a positive coefficient, the matrix integral is divergent.   This divergence does not affect the loop expansion of the matrix model, because the loop
expansion is only sensitive to the matrix potential at energies of $ E_0(q)$ or greater, and $E_0(q)>0$ whenever the matrix potential has a wrong sign coefficient.   However,
the wrong sign of the logarithm does appear to imply
that the matrix model has a nonperturbative instability, somewhat analogous to the familiar nonperturbative instability of many purely bosonic matrix
models.   It is surprising to run into such an instability in a supersymmetric model.

\subsection{The Two-Boundary Wormhole}\label{sec:Z02}

We now begin our study of the gravitational path integral of $=2$ JT
gravity in geometries with spacetime wormholes. We start with the simplest case: the double trumpet or two-boundary
wormhole, corresponding to geometries of the type displayed in fig. \ref{fig:2bdywh} with the topology of a cylinder with
two boundaries. On each boundary we impose the usual NAdS$_2$ boundary conditions labeled by an inverse temperature $β$ and the $(1)$ fugacity $e^ıα$.
\begin{figure}
\begin{center}
\begin{tikzpicture}[scale=1, baseline={([yshift=0.1cm]current bounding box.center)}]
\node at (-3,0) {\small $\beta_1,\alpha_1$};
\node at (3,0) {\small $\beta_2,\alpha_2$};
\draw[thick] (-2,0) ellipse (0.3 and 1.5);
\draw[thick,dashed] (0,0) ellipse (0.1 and .6);
\node at (0,-1) {\small $b,\phi$};
\draw[thick] (2,0) ellipse (0.3 and 1.5);
\draw[thick] (-1.937,1.47) to [bend right=50] (1.937,1.47);
\draw[thick] (-1.937,-1.47) to [bend left=50] (1.937,-1.47);
\node at (0,-1.985) {};
\end{tikzpicture}
\end{center}
\caption{\footnotesize A cylinder with two NAdS boundaries -- the leading connected  two-boundary wormhole. We label each NAdS boundary by their inverse temperature $\beta$ and $\uU(1)$ $R$-symmetry chemical potential $\alpha$. Each $\N=2$ hyperbolic surface has an interior geodesic, indicated by the dashed line, with length $b$ and $\uU(1)$ holonomy $\phi$, that should be integrated over.}
\label{fig:2bdywh}
\end{figure}

We can anticipate which quantity this gravity path integral corresponds to, in the random matrix ensemble. Each asymptotically NAdS boundary is associated to an insertion of $Tr( e^-βH e^ıαJ)$. The parameters $α,β$ can be chosen independently for the different boundaries.   The path integral over a cylinder with two NAdS boundaries computes the connected genus $0$ contribution to
\beq\label{eq:02rmt}
Z_{0,2}(\beta_1,\alpha_1;\beta_2,\alpha_2)=\big\langle {\rm Tr}\big( e^{-\beta_1 H} e^{\i \alpha_1 J}\big)~{\rm Tr}\big( e^{-\beta_2 H} e^{\i \alpha_2 J}\big)\big\rangle_{{\rm conn.}, g=0},
\eeq
In the total path integral with two boundaries, this contribution is combined with the leading disconnected answer made out of a product of two disconnected disks of order $e^2S_0$. The trace in \eqref{eq:02rmt} can be decomposed in supermultiplets\footnote{In principle there could be contributions from the BPS states as well. In writing \eqref{eq:Z02rmt2} we anticipated that such contributions will turn out to be absent.}
\bea\label{eq:Z02rmt2}
Z_{0,2}(\beta_1,\alpha_1;\beta_2,\alpha_2)&=&\sum_{q_1,q_2\in\mathbb{Z}+\delta-\frac{1}{2}} \big(e^{\i \alpha_1 (q_1-\frac{\hq}{2})} + e^{\i \alpha_1  (q_1+\frac{\hq}{2})}\big)\big(e^{\i \alpha_2  (q_2-\frac{\hq}{2})} + e^{\i \alpha_2  (q_2+\frac{\hq}{2})}\big)\nn
&&~~~\times \big\langle {\rm Tr}_{\M,q_1}( e^{-\beta_1 H})~{\rm Tr}_{\M,q_2}( e^{-\beta_2 H})\big\rangle_{{\rm conn.}, g=0}.
\ea
The expected random ensemble of $=2$ quantum mechanics defined in section \ref{randommatrix} has the property that different supermultiplets are statistically independent. Then, the gravity calculation should reproduce
\beq
\big\langle {\rm Tr}_{\M,q_1}( e^{-\beta_1 H})~{\rm Tr}_{\M,q_2}( e^{-\beta_2 H})\big\rangle_{{\rm conn.}, g=0} \overset{?}{\propto} \delta_{q_1,q_2}
\eeq
In the rest of this section, we verify this expectation. Moreover, we also will see that the function of $β_1$ and $β_2$ on the right hand side also matches with the result from the ensemble proposed in section \ref{randommatrix}.

After these preliminaries, let us go back to gravity and compute the partition function, following [6]. The path integral localizes to an integral over the moduli space of $=2$ hyperbolic spaces.
If we ignore the gluing on the boundaries, the only moduli of the cylinder are $b,ϕ$.
From the point of view of $BF$ theory, these moduli arise as follows.
A flat connection on the cylinder can be described by its holonomy around the interior geodesic.  We denote this holonomy by  $U∈G$. Two holonomies related by conjugation $U→R U R^-1$, $R∈G$, are gauge-equivalent, so we only care about the conjugacy class of $U$.   In order for a flat connection with holonomy $U$ to have a geometric interpretation in terms
of a hyperbolic metric, the conjugacy class of $U$ must be hyperbolic (meaning that, after reducing modulo the odd variables, the $SL(2,)$ component of $U$ is
A hyperbolic element $U$ can be conjugated to a diagonal form:
\bea\label{hyperbolicconj}
-e^{\i\hq\phi}e^{\frac{b}{2}}&0&0 \\
0&-e^{\i\hq\phi}e^{-\frac{b}{2}}&0 \\
\hline
\end{array}\right).\label{eq:Ugeo}
\ea\\
Here  $b$ is the length of the interior geodesic in fig.  \ref{fig:2bdywh} and  $e^ıϕ$ is the $R$-symmetry holonomy around this  geodesic.
The minus sign appearing in the $(2,ℝ)$ block indicates that when $ϕ=0$, fermions are antiperiodic around this geodesic.  At $ϕ=π$, the fermions are periodic.
As opposed to the case of
$=1$ supergravity, we do not need to consider spin structures separately, because changing the
spin structure would be equivalent to shifting $ϕ$ by $π$.

For convenience, we  have represented $U$as a $3 ×3$ supermatrix,
but we remind the reader that the gauge group is really a $$-fold cover of $(1,1|1)$.   This cover has no convenient matrix representation, so it is often convenient to write formulas
projected to $(1,1|1)$.
In the $$-fold cover, the parameter $ϕ$ runs from $0$ to $2π$, although in $(1,1|1)$ itself, a shift of $ϕ$ by $2π/$ would give the same group element.

One can separate the cylinder into two ``trumpets'' by cutting along the interior geodesic in
fig.  \ref{fig:2bdywh}.   Each trumpet is a cylinder with one asymptotically NAdS boundary and one geodesic boundary. An $=2$ Schwarzian mode propagates on each
NAdS boundary.  Taking this into account, we now  have additional moduli that arise from the way we glue the two trumpets along the interior geodesic.
These moduli are present because a global rotation or $R$-symmetry rotation of either trumpet would act non-trivially on the Schwarzian modes of the other one.  Hence
one can make a  rotation or $R$-symmetry transformation of one trumpet relative to the other before gluing.
Again, the $BF$ theory perspective is useful.  Gauge transformations are constrained to be trivial along an NAdS boundary, so with two such boundaries, one can define
a gauge-invariant
``holonomy'' $V$ by parallel transport from one boundary to the other.   Flatness of the connection implies that $V$ must commute with $U$, so they are diagonal in the same
basis, $V=diag (-e^ϱ/2 + ıφ,-e^-ϱ/2 + ıφ, e^2 ıφ)$, with ``gluing parameters'' $ϱ,φ$.    In order not to overcount geometries, we restrict  to $0≤ϱ≤b$,  $0≤φ≤2π$. To summarize, the interior moduli of the cylinder are $b$, $ϕ$ together with the geometrical twist parameter
$ϱ$ and the $R$-symmetry twist parameter $φ$.

In standard approaches to the path integral, as in [6], the integration measure
for these  parameters  is $ḅ   ϕ̣  ϱ̣  φ̣$.   (The torsion, which we use in section \ref{sec:GPI}, suggests a slightly different though ultimately equivalent
gluing procedure.)
The path integral over the bulk is independent of $ϱ$ and $φ$ and therefore they can be integrated out  from the beginning, producing an effective measure $2πb  ḅ  ϕ̣$.

In this example of the trumpet, all interior moduli are bosonic, with fermionic moduli coming only from the Schwarzian modes that we study next. As soon as we
increase the number of boundaries or handles, there are also  bulk  fermionic moduli.

The final ingredient we need to complete the evaluation of $Z_0,2$ is  the path integral of the $=2$ Schwarzian modes.   The Schwarzian action arises in gluing the bulk geometry
to the worldline of the boundary quantum mechanics, so it depends on the parameters $b,ϕ$ of the bulk geometry.   We will focus on the Schwarzian path integral on one NAdS$_2$

\begin{figure}
\begin{center}
\begin{tikzpicture}[scale=0.7, baseline={([yshift=-0.1cm]current bounding box.center)}]
\draw[very thick, blue] (6.5,0) ellipse (0.3 and 1.5);
\draw[very thick,blue,->] (6.8,0) -- (6.8,0.01);
\draw[very thick,gray] (8.5,0) ellipse (0.1 and 0.7);
\draw[very thick,gray,->] (8.6,0) -- (8.6,0.01);
\draw[thick] (6.54,1.49) to [bend right=20] (8.5,0.7);
\draw[thick] (6.54,-1.49) to [bend left=20] (8.5,-0.7);
\node at (5.2,0) {\small $\beta,\alpha $};
\node at (9.5,0) {\small $b,\phi $};
\node at (11,0) {$\to$};
\draw[thick,->] (13.5,-2) to (13.5,4);
\draw[thick,->] (12.8,-1) to (18,-1);
\draw[thick,dashed] (16,-1.1) to (16,3.5);
\draw[very thick, gray,->] (13.5,-1) to  (15,-1);
\draw[very thick, gray] (14.9,-1) to  (16,-1);
\draw[very thick, blue,->] (13.5,3) to  (15,3);
\draw[very thick, blue] (14.9,3) to  (16,3);
\draw[thick,dashed] (13.4,3) to (16.3,3);
\node at (18.3,3.8) {\small $x$};
\draw[thick] (18,4) -- (18,3.5) -- (18.5,3.5);
\node at (16,-1.4) {\small $2\pi$};
\node at (13,3) {\small $T$};
\end{tikzpicture}
\end{center}
\caption{\footnotesize On the left panel we represent the trumpet geometry which is a cylinder with one NAdS boundary (left) with inverse temperature $\beta$ and chemical potential $\alpha$, and one geodesic boundary (right) with length $b$ and $\uU(1)$ holonomy $\phi$. On the right panel we represent the trumpet in the complex $x$ plane with $x\cong x+2\pi$. The NAdS boundary is at $x=\i T + y$, for $T\gg1$, while the geodesic boundary is along the real axis. }
\label{fig:trumpet}
\end{figure}

The calculation is very similar to the one that computes the disk partition function. We view the trumpet as the complex $x$ plane with the equivalence $x≅x+2π$, and a metric of scalar curvature $-2$. Let us put the geodesic at $Im x=0$ and the outer NAdS boundary at $Im x=T$, for some large $T$. Including the fermions, the trumpet is described by coordinates $x,ψ,ψ̅$ with equivalence
\beq
(x,\psi,\bar{\psi} ) \cong (x+2\pi, -e^{\i \hq \phi} \psi, -e^{-\i \hq \phi} \bar{\psi}).
\eeq
On the upper boundary, write $x=ıT+ y$ with real $y$. Thus the upper boundary is described by $y,ψ,ψ̅$ with
\beq
(y,\psi,\bar{\psi})\cong (y+2\pi, -e^{\i\hq  \phi} \psi,- e^{-\i\hq \phi} \bar{\psi}).
\eeq
Just like in the case of the disk, the boundary quantum mechanics system is formulated on a supercircle with coordinates $t,θ,θ̅$ and an equivalence relation \eqref{eq:equivreltttb}.
For the case of the trumpet, the fixed points take the form
\bea
y(t,\theta) &=& \frac{2\pi}{\beta} t,\nn
\psi(t,\theta) &=& e^{\i\hq ( \phi-\alpha) t/\beta} \theta,
\ea
with $α→α+ 2πn$, and $n∈ℤ$. These fixed points are annihilated by an operator similar to the one appearing in the case of the disk
\beq
D= \frac{\partial}{\partial t} - \frac{2\pi}{\beta} \frac{\partial}{\partial y} - \hq \frac{\phi-\alpha}{\beta} \i (\psi \partial_\psi - \bar{\psi}\partial_{\bar{\psi}}).
\eeq
The action on these fixed points can be computed from the bosonic sector only. The rotation eigenvalues from the bosonic sector are also straightforward. The fermionic rotation eigenvalues
are independent of $ϕ$. The calculation is explained in appendix \ref{sec:disktrumpet}. The trumpet partition function with bulk parameters
$b,ϕ$  is given by
\bea\label{trumpet}
Z_{\rm tr}(\beta,\alpha; b,\phi)&=& \sum_{n\in \mathbb{Z}} \exp(2\pi\i n\delta)  \frac{\hq \, \cos\left( \pi \hq \,(n+\frac{\alpha}{2\pi})\right)}{\pi \beta} e^{-\frac{b^2}{4\beta}-\frac{4\pi^2\, \hq^{\, 2}}{\beta}(n+\frac{\alpha}{2\pi}-\frac{\phi}{2\pi})^2},
\ea
The exponential term is precisely the sum of the bosonic Schwarzian action for the trumpet and the bosonic $(1)$ mode action, with the same boundary conditions for bosonic fields as in $=2$ JT gravity. Using Poisson resummation, the expression \eqref{trumpet} can be written as a sum over supermultiplets
\bea
Z_{\rm tr}(\beta,\alpha; b,\phi)=\sum_{q\in \mathbb{Z} + \delta-\frac{1}{2}} \big( e^{\i \alpha (q-\frac{\hq}{2} )} + e^{\i \alpha (q+\frac{\hq}{2})}\big)  e^{- \i \phi q} e^{-\beta E_0(q)} \frac{1}{2\pi} \frac{ e^{-\frac{b^2}{4\beta}}}{\sqrt{4\pi \beta}}.\label{trumpet2}
\ea
This expression is particularly simple. Other than the chemical potential dependence, which reproduces the expectation for the charge spectrum of a supermultiplet, the rest is almost the same as if one considered bosonic JT gravity coupled to a $(1)$ gauge field. We remind the reader that the bosonic JT gravity trumpet is indeed $Z^bos._tr(β, b) = e^-b^2/4β/√(4πβ)$ in our conventions.

Formally, we expect the trumpet path integral to be invariant under $ϕ→ϕ+2π$, but eqn. (\ref{delfo}) generically does not have this property; under $ϕ→ϕ+2π$,
it is multiplied by $exp(-2πı(δ-1/2))$.
We have introduced a charge $δ$ mod $$
in the boundary quantum mechanics.   The bulk theory must describe the same charge propagating on the outer boundary.
The Schwarzian mode on the outer boundary carries a charge $±q/2=1/2$ mod $$.   So it accounts for a half-unit of boundary charge.
To reproduce the boundary quantum mechanics, an
additional charge $δ-1/2$ on the outer boundary is needed; the bulk theory must have a gauge theory theta-angle $2π(δ-1/2)$, as claimed in eqn. (\ref{delfo}).   This
induces a charge $δ-1/2$ mod $$ on the outer boundary, which together with the contribution of the Schwarzian mode reproduces the charge in the boundary
quantum mechanics.   But this theta-angle will also place a charge $1/2-δ$ on the inner boundary.   We have not included the corresponding Wilson operator
on the inner boundary; doing so would give a factor $exp(ıϕ(δ-1/2))$ in the path integral, canceling the anomaly.
When we combine two trumpets to make the double trumpet,  the charges on the inner boundaries  cancel
and the anomaly under $ϕ→ϕ+2π$ disappears.

We are ready now to assemble the pieces by gluing together the two trumpets along their common interior geodesic:
\vspace{-0.3cm}\\
\bea
\begin{tikzpicture}[scale=0.8, baseline={([yshift=-0.1cm]current bounding box.center)}]
\draw[thick] (5.5,0) ellipse (0.3 and 1.5);
\draw[thick] (7.5,0) ellipse (0.1 and 0.7);
\draw[thick] (5.54,1.49) to [bend right=20] (7.5,0.7);
\draw[thick] (5.54,-1.49) to [bend left=20] (7.5,-0.7);
\draw[thick] (11.5,0) ellipse (0.3 and 1.5);
\draw[thick] (9.5,0) ellipse (0.1 and 0.7);
\draw[thick] (11.46,1.49) to [bend left=20] (9.5,0.7);
\draw[thick] (11.46,-1.49) to [bend right=20] (9.5,-0.7);
\draw[thick,red,->] (8.6,0) to [bend right=20] (8.5,0.5)  ;
\node at (8.5,1) {\textcolor{red}{\small $\varrho,\varphi$}};
\node at (7.5,-1.3) {\small $b,\phi$};
\node at (9.5,-1.3) {\small $b,-\phi$};
\end{tikzpicture}
\ea
\vspace{-0cm}\\
Twist or gluing parameters are shown in red.
After integrating over the twist parameters, the path integral on the cylinder is
\bea
Z_{0,2}(\beta_1,\alpha_1 ; \beta_2, \alpha_2 ) &=& (2\pi) \int_0^{2\pi} \d\phi \int_0^\infty b\d b ~Z_{\rm tr}(\beta_1, \alpha_1; b,\phi)Z_{\rm tr}(\beta_2, \alpha_2;b,-\phi).\nonumber
\ea
When we glue together the two trumpets, we reverse the orientation of the inner boundary prior to gluing (differently put, the orientation of the geodesic that comes by viewing it as the boundary of one trumpet is opposite to the orientation that comes from viewing it as the boundary of the other trumpet). This has the effect of reversing the sign of $ϕ$ on one side. In appendix \ref{app:JTU1bos} we recall the analogous result for bosonic JT gravity coupled to a $(1)$ gauge field [44].

To compare with the matrix model, it is convenient to use the representation of the trumpet given in \eqref{trumpet2}. After integrating over the geodesic length, we obtain
\bea
Z_{0,2}(\beta_1,\alpha_1 ; \beta_2, \alpha_2 )  &=& \int_0^{2\pi} \frac{\d\phi}{2\pi} \sum_{q_1,q_2 \in \mathbb{Z}+\delta-\frac{1}{2}}\big( e^{\i \alpha_1 (q_1-\frac{\hq}{2}) } + e^{\i \alpha_1  (q_1+\frac{\hq}{2})}\big) \big( e^{ \i \alpha_2  (q_2-\frac{\hq}{2}) } + e^{\i \alpha_2  (q_2+\frac{\hq}{2})}\big)   \nonumber\\
&&\times \, e^{ \i \phi (q_2-q_1)}\, \frac{\sqrt{\beta_1 \beta_2}}{2\pi(\beta_1+\beta_2)} e^{-\beta_1 E_0(q_1)-\beta_2 E_0(q_2)}.
\ea
The anomaly under $ϕ→ϕ+2π$ has disappeared, as expected.

We now integrate over the internal holonomy. This produces a delta function imposing that $q_1 = q_2 ≡q$. The final answer is
\beq\label{eq:Z0202}
Z_{0,2}=\sum_{q\in \mathbb{Z}+\delta-\frac{1}{2}}  \big( e^{\i \alpha_1 (q-\frac{\hq}{2}) } + e^{\i \alpha_1  (q+\frac{\hq}{2}) }\big) \big( e^{ \i \alpha_2 (q-\frac{\hq}{2}) } + e^{ \i \alpha_2  (q+\frac{\hq}{2}) }\big)   \frac{\sqrt{\beta_1 \beta_2}}{2\pi (\beta_1+\beta_2)}  e^{-(\beta_1+\beta_2) E_0(q)}.
\eeq
This is consistent with the statistical independence of the Hamiltonian on each supermultiplet, a feature present in the ensemble described in section \ref{randommatrix}.
As in the discussion of the disk path integral, the factors $ e^ıα(q-/2)  + e^ıα(q+/2)  $ represent a sum over the two states in a supermultiplet, labeled by the average charge $q$.
Going back to the random matrix interpretation in eqn. \eqref{eq:Z02rmt2}, the result \eqref{eq:Z0202} implies
\bea\label{trumpresult}
\langle {\rm Tr}_{\M,q_1}( e^{-\beta_1 H})~{\rm Tr}_{\M,q_2}( e^{-\beta_2 H})\rangle_{{\rm conn.}, g=0} = \frac{\sqrt{\beta_1 \beta_2}}{2\pi(\beta_1+\beta_2)} e^{-(\beta_1+\beta_2)E_0(q_1)} \delta_{q_1,q_2}.
\ea
This agrees with the prediction based on the random matrix theory analysis of section \ref{randomtwo}, since eqn. (\ref{trumpresult}) is the prediction of a $=2$  AZ ensemble  if
either $E_0(q)>0$ (in which case the value of $$ does not matter) or $E_0(q)=0$ and
$$ is of order 1 (not of order $e^S_0$).   In $=2$ JT supergravity,
generically $E_0(q)>0$, and when $E_0(q)=0$, $=1$.

\begin{figure}[t!]
\begin{center}
\begin{tikzpicture}[scale=0.8, baseline={([yshift=-0.1cm]current bounding box.center)}]
\draw[thick] (0,-2) ellipse (1 and .15);%Boundary
\draw[thick, rotate around={-30:(-3,2.5)}] (-3,2.5) ellipse (.15 and 1);%Boundary
\draw[thick, rotate around={30:(3,2.5)}] (3,2.5) ellipse (.15 and 1);%Boundary
\draw[thick, bend right=30] (-1.8,2.45) to (1.8,2.45);
\draw[thick, bend left=20] (-1.8,2.45) to (-2.5,3.35);
\draw[thick, bend right=20] (1.8,2.45) to (2.5,3.35);
\draw[thick, bend right=20] (-2.2,1.55) to (-3.5,1.65);
\draw[thick, bend left=20] (2.2,1.55) to (3.5,1.65);
\draw[thick, bend left=30] (-2.2,1.55) to (-0.5,-1);
\draw[thick, bend right=30] (2.2,1.55) to (0.5,-1);
\draw[thick, bend left=30] (-0.5,-1) to (-1,-2);
\draw[thick, bend right=30] (0.5,-1) to (1,-2);
\node at (-4,3) {\small $\beta_1,\alpha_1$};
\node at (4,3) {\small $\beta_2,\alpha_2$};
\node at (0,-2.7) {\small $\beta_3,\alpha_3$};
%   \node at (-6,0) {\LARGE $Z_{0,3}~~=$};
\end{tikzpicture}
\end{center}
\caption{\footnotesize
The leading connected three-boundary wormhole: a three-holed sphere with three NAdS boundaries, each of which is labeled by its
inverse temperature $\beta$ and $\uU(1)$ $R$-charge chemical potential $\alpha$. }
\label{fig:3bdywh}
\end{figure}

From \eqref{eq:Z0202}, we see that the cylinder  path integral receives contributions only from the non-BPS multiplets.   Therefore, the numbers
$N_BPS(k)$ of BPS states have trivial dispersion, at least at leading order in $e^S_0$.   This justifies a claim made
in section \ref{sec:Z01}, following eqn. \eqref{eq:spectrumcomm}.   The computations in the rest of this article will show that the absence of dispersion in
$N_BPS(k)$ remains true to all orders in the topological expansion; the path integral for every topology except the disk receives contributions only from the non-BPS
multiplets (see footnote \ref{footnoteBPSfluct}).

\subsection{Preliminaries on the Three-Boundary Wormhole}\label{sec:3bwh}

So far we have considered the simple cases of the disk and the cylinder, which did not require integrals over complicated moduli space of $=2$ hyperbolic surfaces. We will study the general case for an arbitrary number of handles and boundaries in the next two sections.

As an introduction, to familiarize the reader with what the final answers look like, we anticipate the result for the leading
three-boundary wormhole. This corresponds to the gravitational path integral $Z_0,3(β_1,α_1;β_2,α_2;β_3,α_3)$ over a spacetime that is a three-holed sphere with three asymptotically NAdS boundaries labeled by  inverse temperatures $β_1,β_2,β_3$ and chemical potentials $α_1,α_2,α_3$; see fig. \ref{fig:3bdywh}. The goal is to study to what extent the gravity path integral can be identified with
\beq\label{eq:03rmt}
Z_{0,3}(\beta_1,\alpha_1;\beta_2,\alpha_2;\beta_3,\alpha_3)=\big\langle {\rm Tr}( e^{-\beta_1 H+\i \alpha_1 J})~{\rm Tr}( e^{-\beta_2 H+\i \alpha_2 J})~{\rm Tr}( e^{-\beta_3 H+\i \alpha_3 J})\big\rangle_{{\rm conn.}, g=0}.
\eeq
Similar to the case with two boundaries, if supermultiplets are statistically independent, we expect to obtain that the fixed multiplet connected correlator is zero unless the three multiplets coincide
\beq
\big\langle {\rm Tr}_{\M,q_1}( e^{-\beta_1 H})~{\rm Tr}_{\M,q_2}( e^{-\beta_2 H})~{\rm Tr}_{\M,q_3}( e^{-\beta_3 H})\big\rangle_{{\rm conn.}, g=0} \propto \delta_{q_1,q_2} \delta_{q_2,q_3},
\eeq
In the rest of this section we reproduce this result from gravity. Moreover, we also match the function of multiplet and temperature appearing as a prefactor in the equation above with the prediction from the topological recursion of the AZ matrix ensemble.

We begin as in the previous cases.  The first step consists, in $BF$ language, in localizing the path integral on flat $(1,1|1)$ connections that correspond to
hyperbolic geometries. This guarantees the presence of a geodesic homotopic to each asymptotically NAdS boundary. As in  [6], we
can cut along those geodesics and divide the geometry into three trumpets and an internal manifold that is a hyperbolic three-holed sphere with geodesic boundaries.
The path integral on the trumpets was already computed in  \eqref{trumpet2}. We denote the interior or bulk contribution by\\
\beq
~~~~V_{0,3}({\sf b}_1,{\sf b}_2, {\sf b}_3)~~~~  \to~~~~ \begin{tikzpicture}[scale=0.6, baseline={([yshift=-0.1cm]current bounding box.center)}]
\draw[thick, rotate around={-30:(-2,2)}] (-2,2) ellipse (.15 and .5);%Geodesic
\draw[thick, rotate around={30:(2,2)}] (2,2) ellipse (.15 and .5);%Geodesic
\draw[thick] (0,-1) ellipse (.5 and .15); %Geodesic
\draw[thick, bend right=30] (-1.8,2.45) to (1.8,2.45);
\draw[thick, bend left=30] (-2.2,1.55) to (-0.5,-1);
\draw[thick, bend right=30] (2.2,1.55) to (0.5,-1);
\node at (-4.2,2) {\small ${\sf b}_1=(b_1,\phi_1)$};
\node at (4.2,2) {\small ${\sf b}_2=(b_2,\phi_2)$};
\node at (1.6-1.5,-1.8) {\small ${\sf b}_3=(b_3,\phi_3)$};
\end{tikzpicture}
\eeq\\
We introduced a notation we will use throughout, where $b = (b,ϕ)$ combines the length $b$ of a boundary with the $(1)$ holonomy $e^ıϕ$ around the boundary. The final answer for $Z_0,3$ is given by gluing this quantity $V_0,3(b_1,b_2, b_3)$ to the three trumpets, including the integral over the twist parameters,
\bea
Z_{0,3}= \left[ \prod_{j=1,2,3} \int_0^\infty b_j \d b_j \int_0^{2\pi}2\pi \d \phi_j\, Z_{\rm tr} (\beta_j,\alpha_j; \overline{{\sf b}}_j)\right] V_{0,3}({\sf b}_1,{\sf b}_2, {\sf b}_3),\nonumber
\ea
where $b = (b, - ϕ)$ is the orientation reversed version of $b=(b,ϕ)$. As explained before, one needs to reverse the orientation of the $(1)$ holonomy around one of the geodesics involved in the gluing; in this expression we chose to apply this reversal on the trumpet. To avoid clutter, we omitted the arguments of $Z_0,3$. The factors of $b_j$ come from the integral over the length-twist parameter while the factors of $2π$ come from the $(1)$-twist parameter.

We leave a detailed derivation of the gravity path integral on spaces with geodesic boundaries and handles for the following two sections. For now we quote the result derived in section \ref{sec:3hole} for the particular case of $V_0,3$;
\beq\label{ansatz}
V_{0,3}({\sf b}_1,{\sf b}_2, {\sf b}_3)= - \frac{1}{2\pi} \frac{1}{4\hq^{\, 2}}\delta''(\phi_1+\phi_2+\phi_3).
\eeq
The delta function imposes the constraint $ϕ_1 + ϕ_2 + ϕ_3 = 0$.  A flat $(1)$ connection on the three-holed sphere with boundary holonomies $ϕ_1,ϕ_2,ϕ_3$
exists if and only if $ϕ_1+ϕ_2+ϕ_3=0$.   Accordingly, in $BF$ theory with gauge group $(1)$, we would get a delta function $δ(ϕ_1+ϕ_2+ϕ_3)$.
In $BF$ theory of $(1,1|1)$, the three-holed sphere has fermionic moduli which appear as corrections to the argument of the delta function.
Integrating  over those fermionic moduli produces the two derivatives acting on the delta function. Details of this calculation are spelled out in section \ref{sec:GPI}.

The overall prefactor  in \eqref{ansatz} depends on the normalization of the $=2$ JT gravity path integral measure. The calculation in the next section also clarifies our choice of

Accepting for now this result for $V_0,3(b_1,b_2,b_3)$, we are ready to put the pieces together. The answer is actually quite simple. For concreteness, integrate first over the third holonomy $ϕ_3$. Using the delta function appearing in \eqref{ansatz}, the only remaining dependence on the other holonomies is \be\label{output}e^{\i\phi_1(q_1-q_3) + \i \phi_2 (q_2-q_3)}.\ee We observe again that although the individual $q_i$ are valued in $δ+$ for some $δ$,  the differences $q_1-q_3$ and $q_2-q_3$ are always integers.  So the
expression that results from integration over $ϕ_3$ is periodic under $ϕ_1 ∼ϕ_1 + 2π$ and $ϕ_2 ∼ϕ_2 + 2π$. Next we perform the integrals over $ϕ_1$ and $ϕ_2$, which sets $q_1=q_3$ and $q_2=q_3$. So the three supermultiplets propagating in the three trumpets are all the same. The final answer after including the trumpet
partition functions and integrating over the three geodesic lengths is given by
\bea
Z_{0,3}&=& \sum_{q\in \mathbb{Z}+\delta-\frac{1}{2}} \big(e^{\i\alpha_1 (q-\frac{\hq}{2})} + e^{\i \alpha_1 (q+\frac{\hq}{2})} \big)\big(e^{\i\alpha_2 (q-\frac{\hq}{2})} + e^{\i \alpha_2  (q+\frac{\hq}{2})}\big)\big(e^{\i\alpha_3 (q-\frac{\hq}{2})} + e^{\i \alpha_3  (q+\frac{\hq}{2})}\big)\nonumber\\
&&~~~~\times (2\pi) \frac{q^2}{4\hq^{\, 2}} \frac{\sqrt{\beta_1 \beta_2\beta_3}}{\pi^{3/2}} e^{-(\beta_1+\beta_2+\beta_3)E_0(q)}.
\ea
The first line is completely fixed by the structure of the $=2$ supermultiplet and the constraint of the three of
them being the same. The second line has more information regarding the details of the matrix model spectral curve.
The factor of $2π$ is the answer one would obtain from a $(1)$ gauge theory on the three-holed sphere.
The factor of $q^2/4 ^  2= E_0(q)$ comes from the details inherent to the $=2$ supergravity. The temperature
dependence is similar to the one appearing in bosonic JT gravity. We would like to emphasize that these features are
highly non-trivial consequences of the explicit evaluation. This will become clear from the derivation in the following sections.

The gravity prediction for $⟨Tr_,q_1( e^-β_1 H) Tr_,q_2( e^-β_2 H) Tr_,q_3( e^-β_3 H)⟩_conn., g=0 $,
the random matrix partition function correlator  for  fixed supermultiplet, is:
\bea
(2\pi)E_0(q_3) \frac{\sqrt{\beta_1 \beta_2\beta_3}}{\pi^{3/2}} e^{-(\beta_1 + \beta_2 + \beta_3)E_0(q_3)} \delta_{q_1,q_2} \delta_{q_2,q_3}.
\ea
Consistent with the discussion in section \ref{randommatrix}, the different supermultiplets are statistically independent.
The temperature dependence reproduces precisely the matrix model topological recursion for an $(_0,)=(1,2)$
ensemble, with $ν$ either zero or proportional to $e^S_0$, applied to the spectral curve \eqref{eq:N2sc}. We leave it  for the reader to verify this, by a
straightforward application of the matrix model loop equation presented in appendix \ref{app:Loop}.
In the next two sections we will provide a procedure to compute the gravitational path integral of $=2$ JT gravity for any topology and
prove to all orders in the genus expansion the match with the random matrix topological recursion. This check will therefore become a particular case of a more general analysis.

\section{Gravity Path Integral for Spaces with Wormholes}\label{sec:GPI}

\subsection{The Path Integral of $BF$ Theory}\label{pbf}
With the exception of the disk, which we have already studied,
any oriented NAdS two-manifold can be built by gluing trumpets onto an ``interior'' manifold, a hyperbolic  two-manifolds with geodesic boundary.
This decomposition leads to a convenient evaluation of the path integral of JT gravity [6].   This analysis has been  to $=1$ JT supergravity [14],
and here we will describe the generalization to $=2$.

The  path integral on the trumpet requires special treatment, but in any event we have already discussed this, in arriving at  eqn. \eqref{trumpet}.
The remaining JT gravity path integral on the interior manifold  $C$ is naturally formulated in $BF$ theory.   The action of $BF$ theory with group $G$,
on a surface $Y$, is $I= - ı∫_Y Tr   BF$. $B$ is an adjoint-valued spin zero field and  $F=Ạ + A∧A$ is the field strength of a gauge connection  $A$.
Integrating over $B$ produces a delta function $δ(F)$, localizing the path integral to the moduli space\footnote{We use the same letter $\T$ for the full moduli
space of flat connections, and for the geometric component relevant for gravity. We hope this will not cause confusion. The motivation for the name $\T$ is that in the case of
gauge group $\SU(1,1|1)$, $\T$ is an $\N=2$ version of Teichm\"{u}ller space.}  $$ of flat connections on $C$ modulo gauge transformations.
In defining $$, one wants to specify the conjugacy classes of the geodesics around the boundaries of $C$; then in gluing onto the external trumpets, one integrates
over those holonomies.
In the $BF$ description, the boundary holonomy is specified by fixing the gauge connection $A$ along the boundary.
In gravity one has to further reduce to a component of $$ that is related to hyperbolic metrics, and also mod out by the mapping class group.
In the present section, we do not consider such global questions, and merely aim to
determine the appropriate measure on $$, deferring global questions to section \ref{sec:RR}.

To derive the measure on the moduli space, one can expand the path integral around a chosen flat connection $A_0$, which represents a  point in $𝒯$. After
integrating over $B$ to get the delta function $δ(F)=δ(Ạ+A∧A)$, one integrates over fluctuations in $A$ and over the ghosts that are introduced
for gauge-fixing.   With a standard gauge choice, this leads to a measure $μ$ on $$ that differs from a classical measure $μ_0$ by a certain ratio of determinants:
\bea\label{bfrmeasure}
\mu =  \mu_0 \cdot \frac{\sqrt{{\rm det}'\Delta_0}}{\sqrt{{\rm det}'\Delta_2}}.
\ea
In eqn. (\ref{bfrmeasure}),  $Δ_q = D^* D  + D D^*$ is the Laplacian acting on adjoint-valued $q$-forms, constructed from the exterior derivative $D= +̣ [ A_0, ·]$ and its adjoint $D^*$.
The numerator arises from the ghost path integral, while the denominator arises from the integral over $A$.  The primes denote the removal of zero-modes.
One way to describe the classical measure $μ_0$ is the following: $$ is naturally a Riemannian manifold,\footnote{For compact $G$, if
one imposes the harmonic gauge condition $\d\star \delta A=0$ on a zero-mode $\delta A$ of $A$ (where $\star$ is the Hodge star),
then the  metric on the space of zero-modes of $A$ and the Riemannian structure of $\T$ are defined
by  $|\delta A|^2=\int_C\Tr\,\delta A\wedge \star\delta A$.  For non-compact $G$, this definition needs to be slightly modified.}
and $μ_0$ is the Riemannian measure on $$;
physically, $μ_0$ is the natural measure on the space of zero-modes of $A$.

On an oriented two-manifold,  an alternative description of $μ_0$ is generally more useful.
If $C$ is oriented, then $$ is a symplectic manifold and $μ_0$ is its natural symplectic measure.
In addition, if $C$ is oriented, then
the ratio $ √(det'Δ_0)/√(det'Δ_2)$ is equal to 1 because Hodge duality between 0-forms and 2-forms implies
that $Δ_0$ and $Δ_2$ have the same spectrum.   Therefore, in this case,  the quantum measure $μ$ coincides with the classical measure $μ_0$.

Yet another way to look at the measure $μ$ that comes from $BF$ theory is as follows.
On any two-manifold, orientable or not, this measure  is the analytic torsion of Ray and Singer  [45].  This observation, or more precisely its generalization to
any spacetime dimension, was made by Schwarz [46].
In any dimension,  analytic torsion is a certain ratio of determinants times a classical measure $μ_0$ on the space of zero-modes; in two dimensions,
the definition reduces to precisely what we have written
in eqn. (\ref{bfrmeasure}).
Analytic torsion is equivalent to combinatorial torsion,   originally defined by Reidemeister [47], as was proved in
[48, 49, 50, 51]. In physical terms, this means that $BF$ theory has the unusual property that it can be defined by a lattice theory with
a result that is completely independent of the choice of lattice.   By taking a very fine lattice, one can motivate the idea that the lattice theory has a continuum limit (namely $BF$ theory).
On the other hand, one can choose a very crude lattice so that the path integral can be directly calculated, leading to explicit formulas.

Although the measure $μ_0$ in eqn. (\ref{bfrmeasure})
can be understood as a Riemannian measure, as far as we know this does not lead to an effective method of calculation.  On an orientable two-manifold,
the symplectic interpretation of $μ_0$ definitely does lead to an effective method of calculation.  However, as far as we know, such calculations are not  simpler than
calculations based on the torsion.  We have chosen to base our calculations on the relationship of the measure to the torsion, because this is directly related to the physics,
as the derivation of eqn. (\ref{bfrmeasure}) showed, and because this approach is equally valid on orientable or unorientable two-manifolds.

In this section and the next one, we consider orientable two-manifolds only.   Unorientable two-manifolds are considered in section \ref{timereversal}.
Any orientable two-manifold can be constructed by gluing three-holed spheres along their boundaries.   By determining the moduli space measure for the three-holed sphere
as well as the procedure for gluing, we can determine the moduli space measure for an arbitrary oriented two-manifold.

The computation of  the torsion of a three-holed sphere
and the gluing procedure were described in [14], section 3, in a way that is valid for any possibly non-compact gauge group or supergroup.   (The examples considered in that
paper were $(2,)$ and $OSp(1|2)$.)    Here we will not repeat those general explanations and will just apply the procedure for the gauge group  $(1,1|1)$ that
is relevant to $=2$ JT supergravity.

There is a subtlety in the statement that the symplectic structure and the torsion define the same measure.   On an oriented two-manifold $C$ without boundary, this is true as stated:
if $_g$ is the moduli space of flat connections with gauge group $G$ on a closed surface of genus $g$, then the symplectic structure and the torsion define the same measure
$μ$ on $_g$ (assuming the symplectic structure is properly normalized).  On the other hand, if $C$ is an oriented two-manifold of genus $g$ with $n$ boundary circles, then the symplectic structure and the torsion define measures on two closely related but slightly different spaces.
The symplectic structure determines a measure $μ$ on what we will call $_g,n$ (or $_g,n(w⃗)$ if we wish to be more precise), the moduli space of flat bundles with
prescribed conjugacy classes $w⃗=(w_1,⋯,w_n)$ of the holonomies
around the boundaries.   The torsion defines a measure $τ$ on what we will call $_g,n$, the moduli space of flat bundles
over $C$ without a restriction on the boundary holonomies.   (In the case of gravity, we restrict the holonomies to be hyperbolic, but $τ$  can be defined without this restriction.)
The two spaces and measures are closely related, and in the case of $=2$ JT gravity, we will ultimately determine the precise relationship in eqn. (\ref{comparison}).

\subsection{Measure On Moduli Space of $\cN=2$ Hyperbolic Surfaces}\label{sec:3hole}
In this section, after some preliminaries, we determine the torsion for a  three-holed sphere, the torsion for a circle and the associated gluing procedure, and the final answer for
the measure that comes from
the $BF$ path integral on an arbitrary orientable surface.

\subsubsection{JT Gravity as a $BF$ Theory}\label{JTBF}

We start by reviewing some aspects of $BF$ theory with gauge group $SL(2,)$  on an oriented  two-manifold $C$. We study flat connections on  $C$,
with the proviso that if $C$ has a non-trivial boundary, then
the holonomy  of the flat connection around any boundary component is a hyperbolic
element of $(2,)$, that is, an element that is conjugate to
\bea\label{eq:Ul}
U_b^\pm = \pm \left(\hspace{-1mm}\begin{array}{cc}
e^{b/2} & 0 \\
0& e^{-b/2}
\end{array}\hspace{-1mm}\right),
\ea
with some $b>0$.   With this specification, the
moduli space of flat $SL(2,ℝ)$ connections on $C$ has several disconnected components, one of which parametrizes hyperbolic metrics on $C$ that have
curvature $R=-2$ and such that, for any given boundary, the corresponding parameter $b$ is the circumference of that boundary.\footnote{The group $\SL(2,\R)$ is contractible onto
its maximal compact subgroup $\uU(1)$, so topologically any $\SL(2,\R)$ bundle over $C$
is equivalent to a $\uU(1)$ bundle and therefore has a first Chern class.  (This first Chern class can be expressed as an integral over $C$, once one
picks a reduction of the structure group of the bundle from $\SL(2,\R)$ to $\uU(1)$.)   If $C$ is a closed oriented surface of genus $g$, then the values of the first Chern class of a flat
$\SL(2,\R)$ bundle over $C$
range from $g-1$ to $-(g-1)$, and hyperbolic metrics on $C$ are related to the component of first Chern class $g-1$.   The first Chern class can still be defined if $C$
is an oriented two-manifold with boundary and the flat connection on $C$ is constrained to have hyperbolic conjugacy around the boundary, because hyperbolic holonomy around
a given boundary component reduces
the structure group of the flat bundle along that boundary  to a contractible subgroup (the one-parameter subgroup of $\SL(2,\R)$ that contains the holonomy around
the given boundary).   Again the component of $\T$
with maximum possible first Chern class is the one related to hyperbolic metrics on $C$.} That is the component that we will study.

In purely bosonic JT gravity, the sign in eqn. (\ref{eq:Ul})  is irrelevant.   One can take the gauge group to be $PSL(2,)$ and disregard this sign.
In a theory with fermions, one wants to endow $C$ with a spin structure as well as a hyperbolic metric.    In that case, one must take the gauge group to be $(2,)$
and the sign in eqn. (\ref{eq:Ul}) is meaningful: it determines the spin structure on the given boundary component of $C$.

\begin{figure}
\centering
\includegraphics[scale=0.3]{Gluing.png}
\caption{\footnotesize Decomposition of a genus 2 surface as a union of three-holed spheres $Y_1,$ $Y_2$, glued on circles $S_1,S_2,S_3$.}
\label{fig:pair of pants}
\end{figure}
The flat connections on $C$ that are related to hyperbolic metrics on $C$ have hyperbolic  holonomy around any homotopically nontrivial embedded
loop in $C$ (not just the boundaries
of $C$).   As a result, if we ``cut'' $C$ along some embedded circles to decompose it into a union of three-holed spheres, there will be hyperbolic
holonomy around each of the boundaries of these three-holed spheres.   The loops along which we cut can always be chosen to be geodesics, so the cutting decomposes $C$ into
a union of three-holed spheres with geodesic boundaries.  In brief, we will call these geodesic three-holed spheres. In fig. \ref{fig:pair of pants}, we give an example of
building a genus 2 surface by gluing together two three-holed spheres.

In $=2$ JT supergravity,  the gauge group is $(1,1|1)$ (or more precisely a $$-fold cover thereof).   Since a maximal
bosonic subgroup of $(1,1|1)$ is $(1,1)×(1)≅(2,)×(1)$, we will briefly discuss $BF$ theory of $(2,)×(1)$ before
studying $(1,1|1)$.
$BF$ theories based on  $(2,)×G$, with any compact $G$, were studied in [44], the main result being a match
with a random matrix theory in which the different representations of $G$ are statistically independent.
The case $G=(1)$, relevant to $=2$ JT supergravity, is important for the present article and has special properties because $(1)$ is abelian.
In $(1)$ gauge theory, the holonomy around a given boundary can be an arbitrary element $e^ıϕ∈(1)$.   If $C$ has $n$ boundary components, then
the holonomies $e^ıϕ_1,⋯,e^ıϕ_n$ around these boundaries cannot be chosen arbitrarily.   A flat connection on $C$ with these holonomies
exists if and only if
\be\label{holconstraint} \sum_{i=1}^n \phi_i = 0 ~{\mathrm {mod}}~2\pi \mathbb{Z}.\ee
This constraint, which is slightly deformed in the case of $(1,1|1)$,  will be important in what follows.

$(1,1|1)$, the gauge group relevant to $=2$ JT supergravity,
can be  presented\footnote{$\SU(1,1|1)$ is isomorphic to ${\mathrm{OSp}}(2|2)/\Z_2$, so it is also related to the group of linear transformations of
a vector space of dimension $2|2$ that preserves a certain symplectic form.    This will be important in section \ref{sec:RR} when we derive the $\N=2$ analog of
Mirzakhani's recursion  relation for volumes. But except in that application, we use the $\SU(1,1|1)$ description, since it is easier to calculate with $3\times 3$ matrices.} as the group of linear transformations of a vector space $V$ of dimension $2|1$
that preserves a certain hermitian inner product.   This inner product can be described by saying that for
\bea\label{eq:innprod}
|(a,b|\eta)|^2 =\i( a \bar{b} - b \bar{a})+ 2 \i \bar{\eta} \eta.
\ea
For  bosonic generators of $𝔰𝔲(1,1|1)$, we can choose
\bea\label{bosegen}
{\sf e}=\left(\begin{array}{cc|c}
0&1\hspace{.5mm}&\hspace{.5mm}0 \\
0&0\hspace{.5mm}&\hspace{.5mm}0 \\
\hline 0&0\hspace{.5mm}&\hspace{.5mm}0
\end{array}\right),
~{\sf f}=\left(\begin{array}{cc|c}
0&0\hspace{.5mm}&\hspace{.5mm}0 \\
1&0\hspace{.5mm}&\hspace{.5mm}0 \\
\hline 0&0\hspace{.5mm}&\hspace{.5mm}0
\end{array}\right),
~{\sf h}=\left(\begin{array}{cc|c}
1&0\hspace{.5mm}&\hspace{.5mm}0 \\
0&-1\hspace{.5mm}&\hspace{.5mm}0 \\
\hline 0&0\hspace{.5mm}&\hspace{.5mm}0
\end{array}\right),
~{\sf z}=\left(\begin{array}{cc|c}
\i&0\hspace{.5mm}&\hspace{.5mm}0 \\
0& \i\hspace{.5mm}&\hspace{.5mm}0 \\
\hline 0&0\hspace{.5mm}&\hspace{.5mm}2\i
\end{array}\right).\label{slgenerators}
\ea
The first three generators satisfy the $𝔰𝔩(2)$ algebra $[h,e]=2e$, $[h,f]=-2f$ and $[e,f]=h$, while $z$ is the $𝔲(1)$ generator  and
accordingly commutes with $e,f$, and $h$.   The $R$-charge generator $J$ that was introduced in section \ref{randomtwo} is represented in this basis by $J=-ız$ (the
following formulas will be slightly simplified by using $z$ rather than $J$).  We have chosen $e,f,h, $ and $z$ so that they generate symmetries of the hermitian form
(\ref{eq:innprod}).   Therefore, they are generators of $(1,1|1)$; in Lorentz signature, they are represented by anti-hermitian operators, whose exponentials are unitary.

The four fermionic generators of $(1,1|1)$ are complex linear combinations of
\bea\label{fermigen}
{\sf q}_1=\left(\begin{array}{cc|c}
0&0\hspace{.5mm}&\hspace{.5mm}0 \\
0&0\hspace{.5mm}&\hspace{.5mm}0 \\
\hline 0&1\hspace{.5mm}&\hspace{.5mm}0
\end{array}\right),
~{\sf q}_2=\left(\begin{array}{cc|c}
0&0\hspace{.5mm}&\hspace{.5mm}1 \\
0&0\hspace{.5mm}&\hspace{.5mm}0 \\
\hline 0&0\hspace{.5mm}&\hspace{.5mm}0
\end{array}\right),
~{\sf q}_3=\left(\begin{array}{cc|c}
0&0\hspace{.5mm}&\hspace{.5mm}0 \\
0&0\hspace{.5mm}&\hspace{.5mm}0 \\
\hline 1&0\hspace{.5mm}&\hspace{.5mm}0
\end{array}\right),~
{\sf q}_4=\left(\begin{array}{cc|c}
0&0\hspace{.5mm}&\hspace{.5mm}0 \\
0&0\hspace{.5mm}&\hspace{.5mm}1 \\
\hline 0&0\hspace{.5mm}&\hspace{.5mm}0
\end{array}\right).
\ea   One can work out the commutators of the $q_i$ with the bosonic generators defined in eqn. (\ref{bosegen}) and their anticommutators with each other, but we omit
these formulas.

In eqn. (\ref{fermigen}),  we have simply picked a convenient basis for the four-dimensional space of fermionic linear transformations of the vector space $V$.   Certain complex
linear combinations of the $q_i$ preserve the quadratic form (\ref{eq:innprod}) and therefore are generators of $(1,1|1)$.   However, for the Euclidean signature calculation
that we will perform, the existence of such a real basis for the fermionic generators is not important, though the real structure of the maximal bosonic subgroup $(2,)×(1)$ is
important.    We will pause to explain this point, which sometimes causes  confusion.

First of all, in any quantum field theory with or without gravity,
in Lorentz signature, any group or supergroup $G$ of symmetries always has a real structure. The reason for this is that elements of $G$ act on the quantum
Hilbert space $$̋ of the theory as unitary transformations.  This statement only makes sense if $G$ has a real structure.   One can (with some restrictions on the states
a given group element can act on)
analytically continue the  action of $G$ on $\H$  to an action of the complexification $G_$ of $G$, but only elements of $G$ are unitary; elements of $G_$ that are not
in $G$ act on $$̋ as (densely defined) unbounded and nonunitary operators.

This remark applies to any theory, supersymmetric or not.   In the case of a supersymmetric theory, after Wick rotation to Euclidean signature, generically the fermionic
generators do not carry any real structure.   For example, consider $\N=1$ supersymmetry in four dimensions.   This superalgebra has four fermionic generators, and,
although they carry a real structure in Lorentz signature, they do not admit any real structure in Euclidean structure  (they take values in a pseudoreal rather than real
representation of the rotation group ${\mathrm{Spin}}(4)$).

More broadly, even without supersymmetry, in Euclidean signature, fermionic integration variables in general do not carry any real structure.   For example, in the standard model of particle physics – because of the chiral nature of the weak interactions – after Wick rotation to Euclidean signature, there is no way to define a real structure for the fermionic
variables.   This causes no difficulty, because fermionic integration is a purely algebraic operation.   When one says that $\int \d \psi \,\psi = 1$, $\int \d\psi\,1=0$,
there is no need to know whether $\psi$ can be considered “real” (and if the answer to that question is “no,” as it is in the standard model or many four-dimensional
models with $\N=1$
supersymmetry, there is no need to apologize and add a complex conjugate variable $\psi^*$).

By contrast, for the bosonic variables, one does need a real structure in Euclidean signature in order to define the appropriate integration cycle.   This statement generalizes
the following fact about 1-dimensional integrals: if $F(z)$ is a holomorphic function on $\C$ (possibly with some singularities), then to make sense of an integral
$\int_\gamma F(z)\d z$, one needs to pick an integration cycle $\gamma\subset \C$.   For example, in  general, the integral of $F(z)\d z$ on the real $z$-axis gives a different result
from the “same” integral on the imaginary $z$-axis, even if both of these integrals converge.

The measure  that we will compute using the relation of $BF$ theory to torsion makes sense as a holomorphic measure on the moduli space of flat  connections
on $C$ with structure group $\SL(1,1|1)_\C$, the complexification of $\SU(1,1|1)$.
To integrate this measure, we need a real form for the bosonic integration variables but we do not need a real form for the fermionic integration variables.
When we say that the gauge group we work with is $\SU(1,1|1)$, we are really specifying the complex Lie algebra of the gauge group and the real form
that we will use in integrating over the bosons, which will be  a maximal bosonic subgroup $\SU(1,1)\times \uU(1)\cong \SL(2,\R)\times \uU(1)$ of $\SU(1,1|1)$.

At this stage, the reader may wonder why the supergroup $\SU(1,1|1)$ that is appropriate for $\N=2$ JT supergravity in Euclidean signature admits a real structure
for the fermionic generators, given that this does not happen in a generic supersymmetric theory.   This fact can be viewed as an accident
of two dimensions.   The connected component of the symmetry group of $\mathrm{AdS}_2$ in Lorentz signature is ${\mathrm{SO}}(2,1)$; Wick rotation converts this
to $\mathrm{SO}(1,2)$, which happens to be isomorphic to $\mathrm{SO}(2,1)$. As a result of this equivalence, after including spin and fermionic generators,
one gets the same supergroup $\SU(1,1|1)$ in either Lorentz or Euclidean signature.  Since it is always possible to define a real structure in Lorentz signature,
it follows that in this theory one can also do so in Euclidean signature.    In dimension $d$, the two groups would be $\mathrm{SO}(2,d-1)$ and $\mathrm{SO}(1,d)$,
which are no longer isomorphic.  For some values of $d$, a superextension of this symmetry group will not have any real structure in Euclidean signature.

Going back to $\SU(1,1|1)$, although this supergroup does admit a real structure for the fermionic generators, it will be more convenient in our calculations to ignore
this fact and make no use of this real structure.

In this discussion, we have  implicitly assumed that $\hq=1$.  To incorporate $\hq$, we want to take a $\hq$-fold cover of  $\SU(1,1|1)$ and of its maximal
bosonic subgroup $\SL(2,\R)\times \uU(1)$.  The cover is defined by taking a $\hq$-fold cover of the second factor $\uU(1)$.  Taking this cover means that the fermionic generators
of $\SU(1,1|1)$ have $R$-charge that is greater than the minimum possible $R$-charge of any operator by a factor $\pm \hq$.

§.§.§ The Three-Holed Sphere

The ultimate goal of this section is to compute the path integral measure for the component of the moduli of flat $\SU(1,1|1)$ connections on an orientable surface $C$
associated to hyperbolic surfaces with $\cN=2$ supersymmetry.   Any orientable surface can be constructed by gluing together three-holed spheres along some of their boundaries, which are circles.   So to obtain a general result, we will
need to know the torsion of a flat connection on a three-holed sphere or a circle, and the procedure for gluing.

[scale=0.8, baseline=([yshift=-0.1cm]current bounding box.center)]
[thick, rotate around=-30:(-2,2)] (-2,2) ellipse (.15 and .5);
[thick, rotate around=30:(2,2)] (2,2) ellipse (.15 and .5);
[thick] (0,-1) ellipse (.5 and .15);
[thick, bend right=30] (-1.8,2.45) to (1.8,2.45);
[thick, bend left=30] (-2.2,1.55) to (-0.5,-1);
[thick, bend right=30] (2.2,1.55) to (0.5,-1);
at (-2.6,2.2) $U$;
at (2.6,2.2) $V$;
at (0,-1.6) $W$;
at (0,1) $Y$;

[scale=0.9, baseline=([yshift=-0mm]current bounding box.center)]
[thick,fill=gray!60] (0,0) ellipse (2.5 and 1.5);
[thick,fill=white] (0,-1.5) to [out=150, in=-90] (-1.3,-1.5+1.5) to [out=90, in=90] (0,-1.5) ;
[thick,fill=white] (0,-1.5) to [out=30, in=-90] (1.3,-1.5+1.5) to [out=90, in=90] (0,-1.5) ;
[fill,black] (0,-1.5) circle (0.1);
[thick,->] (1.85,-1.01) to [bend right=5] (1.85+0.1,-1.01+0.09)  ;
[thick,->] (-1,-.8) to [bend right = 30] (-1,-.8+0.01)  ;
[thick,->] (1,-.8) to [bend left= 20] (1,-.8-0.01)  ;
at (-5,0) $\to$;
at (2.5,-1) $W$;
at (-1.7,0) $U$;
at (1.7,0) $V$;
at (0,-2) $P$;
at (0,0.5) $Y$;

On the left we show the hyperbolic three-holed sphere $Y$ with geodesic boundaries labeled by $\SU(1,1|1)$ holonomies $U$, $V$ and $W$. On the right we show a topological space given by the interior of the curves labeled by the same three holonomies. The combinatorial torsion is conveniently calculated using the picture on the right.

We denote as $\mathcal{R}$ the moduli “space” of flat $\SU(1,1|1)$ connections on a three-holed sphere $Y$, with the boundary holonomies required to be hyperbolic
but not otherwise restricted.   As we will see, $\cR$  is a rather subtle sort of space, not a supermanifold.
We denote the holonomies of a flat $\SU(1,1|1)$ connection on $Y$ around the boundaries as $U,V,W$; see fig. <ref>.
Actually, to define the holonomies as group elements (not just conjugacy classes), one needs to pick a base-point in the definition of the fundamental group and to choose, for each boundary circle, a corresponding element of the fundamental group.   A way to do this is given in the right panel of the figure.   From this picture, it is hopefully obvious that, as $UVW$ is the holonomy around
a contractible loop in $Y$, any flat connection on $Y$ will satisfy

U V W = 1.
We can solve this relation for $W$: $W=V^{-1}U^{-1}$.   $U$ and $V$ are not subject to any constraint, so we can pick an arbitrary pair $U,V\in \SU(1,1|1)$, and this
will determine a flat $\SU(1,1|1)$ connection on $Y$.   This parametrization is redundant, however:
a conjugation  $(U,V,W) \cong (R U R^{-1}, R V R^{-1}, R W R^{-1})$, which one can think of as resulting from a gauge transformation that equals $R$ at the base-point $P$
that was used in defining the fundamental group,  preserves constraint (<ref>) and is associated to  a gauge-equivalent flat connection.

We now present the parametrization of $\mathcal{R}$ which we use in the torsion computation.   Given a pair $(U,V)\in \SU(1,1|1)$, we cannot conjugate them
to be simultaneously diagonal, since they do not commute, but
we can conjugate them to a pair $(U_0,V_0)$ such that $U_0$ is upper triangular and $V_0$ is lower triangular:

U_0 =  (
-e^ıϕ_ae^a/2   -e^ıϕ_aκ   0

0   -e^ıϕ_ae^-a/2   0

0   0   e^2ıϕ_a
) e^ψ_2 q_2 e^ψ_4 q_4,
V_0 =  (
-e^ıϕ_be^-b/2   0   0

-e^ıϕ_b   -e^ıϕ_be^b/2   0

0   0   e^2ıϕ_b
)e^ψ_1 q_1 e^ψ_3 q_3.

Every pair $(U,V)$ is conjugate to a unique such pair $(U_0,V_0)$.   The triangular nature of $U_0,V_0$ will simplify the calculation considerably.[It is to achieve this simplification
that, as discussed in section <ref>, we ignore the real structure that could be imposed on the fermionic generators.   $\SU(1,1|1)$ actually has a real Borel
subgroup that could be used to get somewhat similar simplifications in a real basis. (For this, one would define “upper triangular” or “lower triangular” generators to be those with
a positive or negative eigenvalue of ${\sf h}$.)    It does not appear that this would lead to simpler calculations.]
The boundary parameters $(a,b,\phi_a,\phi_b,\kappa)$  are the bosonic moduli of $\cR$.   $\cR$ also has four “internal”  fermionic moduli $\psi_1,\cdots,\psi_4$.  The torsion $\tau_Y$ will be holomorphic in all of these
parameters.  When ultimately we integrate over moduli (usually after gluing $Y$ onto other three-holed spheres and/or external trumpets)  $a,b,\phi_a,\phi_b,$ and $\kappa$ are all  taken to be real,[It is actually more natural to say that the parameters  $a,b,\cdots$ are real modulo nilpotents.   The reason is that in general the same oriented two-manifold $C$
can be decomposed in three-holed spheres in many different ways.   The different sets of variables $a,b,\cdots$ and $a',b',\cdots$ associated to these
decompositions are related by a change of variables
that is real modulo nilpotents  but not actually real.
So we cannot
in an invariant way say that the bosonic parameters are real, but we can say that they are real modulo nilpotents, which is good enough so that the moduli
space integrals make sense.  See for example  [52]  for discussion of such issues.] but we do not invoke a real structure for the fermionic parameters $\psi_1,\cdots,\psi_4$.
As usual, we actually  work with a $\hq$-fold cover of $\SU(1,1|1)$, so $\phi_a$ and $\phi_b$ are defined
mod $2\pi$  (not $2\pi/\hq$ as the formulas in $3\times 3$ matrices suggest).

If two boundary holonomies are $U_0$ and $V_0$, then the third is $W_0=V_0^{-1}U_0^{-1}$.
In the parametrization (<ref>) of the moduli space,
$a$ and $b$ are the geodesic lengths of the $U_0$ and $V_0$ boundaries while $\phi_a$ and $\phi_b$ parametrize their $\uU(1)$ holonomies. But what is the interpretation of $\kappa$? We would like to replace it instead by the geodesic length of the boundary with holonomy $W_0$; we denote this length by $c$. The length and $\uU(1)$ holonomy of $W_0$
can be determined by conjugating $W_0$ to be diagonal.  Thus, if there is $D\in \SU(1,1|1)$ such that

D W_0 D^-1 =  (
-e^ıϕ_ce^-c/2   0   0

0   -e^ıϕ_ce^c/2   0

0   0   e^2ıϕ_c

then the $W_0$ boundary has circumference $c$ and $\uU(1)$ twist parameter $\phi_c$.
The straightforward but tedious calculation to determine $c$ and $\phi_c$ is presented
in appendix <ref>. Before quoting the final result, it is instructive to consider the simpler case with vanishing fermionic moduli $\psi_1 = \cdots= \psi_4 =0$. In this case the matrix $W_0=V_0^{-1}U_0^{-1}$ is

W_0 |_ψ= 0 =  (
e^-ı(ϕ_a + ϕ_b)e^b-a/2   -e^-ı(ϕ_a + ϕ_b)e^b/2κ   0

-e^-ı(ϕ_a+ ϕ_b)e^-a/2   e^-ı(ϕ_a + ϕ_b)(e^a-b/2+κ)   0

0   0   e^-2 ı(ϕ_a + ϕ_b)

This case simplifies since $W_0$ is block diagonal.   From the bottom right matrix element of eqn. (<ref>), we can read off that $\phi_c=-\phi_a-\phi_b$.  Equivalently,

ϕ_a + ϕ_b + ϕ_c = 0    for  ψ_i =0, i=1,…,4.
Thus the boundary holonomies cannot be specified independently, or the moduli space will be empty.
We essentially already met this constraint in eqn. (<ref>); it reflects the fact that when we set the odd moduli to zero, the structure group of a flat
$\SU(1,1|1) $ connection reduces to the maximal bosonic subgroup $\SL(2,\R)\times \uU(1)$, in which the second factor is abelian.
The $2\times 2$ block in $W_0$ is
e^-ı(ϕ_a + ϕ_b)e^b-a/2    -e^-ı(ϕ_a + ϕ_b)e^b/2κ

-e^-ı(ϕ_a + ϕ_b)e^-a/2    e^-ı(ϕ_a + ϕ_b)(e^a-b/2+κ)
) =  -e^ıϕ_c (
-e^b-a/2    e^b/2κ

e^-a/2    -(e^a-b/2+κ)

The matrix on the right, after removing a factor of $-e^{\i \hq\phi_c}$, has eigenvalues $e^{\pm c/2}$, so its trace is $2\cosh\frac{c}{2}$.
Computing the trace,  we learn  that with all fermionic moduli set to zero, $\kappa=\kappa_0$ with

κ_0=- 2 cosh(c/2) - 2 cosh( b-a/2),   for  ψ_i =0, i=1,…,4.
When the odd moduli vanish, these relations determine the parameters $c,\phi_c$ that parametrize the conjugacy class of $W_0$.

Conceptually, with non-vanishing fermionic moduli, one proceeds  in the same way. The only modification is that now the right hand sides of (<ref>) and (<ref>) get nilpotent contributions proportional to the fermionic moduli.
This will modify both the result for $\kappa$ and the constraint $\phi_a+\phi_b+\phi_c=0$.
The most general possible form of this modification is

κ   =    κ_0 + ∑_i<j κ_ij ψ_i ψ_j + κ_4 ψ_1 ψ_2 ψ_3 ψ_4,

ϕ_a +ϕ_b + ϕ_c   =   ∑_i<j f_ij ψ_i ψ_j + f_4 ψ_1ψ_2ψ_3ψ_4.
In appendix <ref>, we derive explicit expressions for the  coefficients $\kappa_{ij}$ and $f_{ij}$, $i,j=1,\ldots,4$, that appear in these relations. The non-zero $f$'s are
f_12 = -ıe^-ıϕ_a/4(cosh(c/2) + cos(ϕ_a+ϕ_b)),    f_14=ı(e^-b/2-ıϕ_a-e^-a/2+ıϕ_b)/4(cosh(c/2) + cos(ϕ_a+ϕ_b)),

f_23 =-ı(e^b/2-ıϕ_a-e^a/2+ıϕ_b)/4(cosh(c/2) + cos(ϕ_a+ϕ_b)),    f_34=ıe^ıϕ_b (cosh(c/2) + cosh(a-b/2))/2(cosh(c/2) + cos(ϕ_a+ϕ_b)),

f_4=-ıe^-2ıϕ_a (1+e^ı(ϕ_a+ϕ_b)cosh(c/2) )/8(cosh(c/2) + cos(ϕ_a+ϕ_b))^2.
and the nonzero $\kappa$'s are
κ_12 =e^ıϕ_b+e^-ıϕ_acosh(c/2)/2(cosh(c/2) + cos(ϕ_a+ϕ_b)),      κ_14=-e^-a/2-ıϕ_a+e^-b/2+ıϕ_b+(e^-b/2-ıϕ_a+e^-a/2+ıϕ_b)cosh(c/2)/2(cosh(c/2) + cos(ϕ_a+ϕ_b))       κ_23 =e^a/2-ıϕ_a+e^b/2+ıϕ_b+(e^b/2-ıϕ_a+e^a/2+ıϕ_b)cosh(c/2)/2(cosh(c/2) + cos(ϕ_a+ϕ_b)) ,        κ_34 =(cosh(c/2)+cosh(a-b/2))(e^ıϕ_bcosh(c/2)+e^-ıϕ_a)/cosh(c/2) + cos(ϕ_a+ϕ_b).      κ_4 = e^ı(ϕ_b-ϕ_a)(1+cos(ϕ_a+ϕ_b)cosh(c/2))/4(cosh(c/2) + cos(ϕ_a+ϕ_b))^2.
It will be convenient for future reference to define the nilpotent function
K= ∑_i<j k_ij ψ_i ψ_j+k_4 ψ_1 ψ_2ψ_3ψ_4,    k_ij = -∂_c κ_ij/sinh(c/2),  k_4 = - ∂_c κ_4/sinh(c/2).
The coefficients on the right hand side can be easily extracted from (<ref>). We define this function $K$ since it will appear in the final expression for the torsion.

At this point, however, we should point out why $\cR$ is not a supermanifold and instead is a rather strange kind of “space.”   Because $\uU(1)$ is abelian,
a  flat $\SU(1,1|1)$ connection on $Y$ with
$\psi_1=\cdots \psi_4=0$ is always invariant under the $R$-symmetry group $\uU(1)\subset \SU(1,1|1)$.   Accordingly, let $\cR'$ be the supermanifold of dimension $5|4$
parametrized by $a,b,\phi_a,\phi_b,\kappa|\psi_1,\psi_2,\psi_3,\psi_4$.   Then, formally, $\cR$ is the quotient $\cR'/\uU(1)$, with $\uU(1)$ acting only on $\psi_1,\cdots,\psi_4$.
The trouble with this is that dividing a supermanifold, in this case $\cR'$, by a continuous group, in this case $\uU(1)$, that acts only on odd variables is not a nice operation,
and the quotient $\cR=\cR'/\uU(1)$ is definitely not a supermanifold.  We do not want to define directly what should be meant by a volume form on $\cR$ or by integration on $\cR$.