But why are we supposed to integrate over an $R$-symmetry gluing parameter $\varphi$?  The point is that, at the asymptotic boundary of the trumpet (or of any NAdS$_2$
spacetime), gauge transformations are not allowed.    So if $A$ is the $R$-symmetry gauge field, we can define a gauge-invariant parameter $\vartheta=\int_\ell A$, where
the integration path $\ell$ starts on the asymptotic boundary of the trumpet, heads in to the geodesic $\gamma$, winds once around $\gamma$, and then heads back out to
the asymptotic boundary.    In our previous analysis with $\rR$ symmetry rather than $\CR$, the parameter $\vartheta$ would have no gauge-invariant content (beyond what is contained in
the monodromy $V$ around $\gamma$) because the parts of the integral from the asymptotic boundary to $\gamma$ and back again would cancel.  However, with $\CR$ symmetry,
because the monodromy around $\gamma$ involves the action of $\cC$, under which $A$ is odd, the “incoming” and “outgoing” parts of the integral add, instead of canceling.
Since these integrals do add, one can identify $\vartheta$ as $2\varphi$, where $\varphi$ is the $R$-symmetry gluing parameter that we have to integrate over, according to
eqn. (<ref>).  The range of the $\uU(1)$ holonomy parameter $\vartheta$ is $0\leq\vartheta\leq 2\pi$, so the range of $\varphi=\vartheta/2$ is $0\leq \varphi\leq\pi$.
Integration over $\varphi$ with measure $\d\varphi$ just gives a constant factor $\pi$, since the integrand does not depend on $\varphi$.

Finally, we discuss some topological details that will be important in understanding and completing the calculations in section <ref>.
In the case of $\tT$ symmetry, $U$ and $V$ are unique, up to conjugation.   However, with $\CT$ symmetry, $U$ has an arbitrary $R$-symmetry monodromy $e^{\i\hq \phi}$ and
for each choice of $\phi$, there are two possible choices of $V$.   For given $\phi$, each choice of $V$ corresponds to a flat $\uU(1)_R$ bundle $\L\to C$
with holonomy $e^{\i\hq\phi}$ around the boundary of $C$.   The two flat bundles differ by the value of the first Chern class  $c_1(\L)$. This is a relative
first Chern class, because $C$ has a boundary; one cannot say which choice of $V$ corresponds to $c_1(\L)=0$ and which  corresponds to $c_1(\L)\not=0$,
but one can say that they are different.[The topology is easier to describe if we turn $C$ into a compact manifold without boundary  by gluing on a disc.   In that
way we make a closed, unorientable manifold which is a copy of ${\mathbb {RP}}^2$.  A flat $\uU(1)$ bundle over this manifold is either trivial, or else has holonomy $-1$ around the
generator of $\pi_1({\mathbb{RP}}^2)\cong \Z_2$.   The first Chern class of this nontrivial flat bundle is the nonzero element of $H^2({\mathbb{RP}}^2,\Z)\cong\Z_2$.   This
is a torsion element, so it cannot be detected by curvature.   In general the first Chern class of a line bundle with connection is determined by the curvature of the connection
only modulo torsion.]  $\CT$ invariance implies that
the $\uU(1)_R$ theta-angle has the possible values $0$ and $\pi$.   According to eqn. (<ref>), $\theta=\pi$ if and only if $\delta=0$.  In the case of a Riemann
surface $C$ without boundary, the meaning of $\theta=\pi$ is that the path integral with a line bundle $\L$ should be weighted by a factor $(-1)^{ c_1(\L)}$.
In the present case, because $C$ has a boundary and $c_1(\L)$ is a relative first Chern class, it is subtle to explain what is meant by $(-1)^{c_1(\L)}.$
A simple statement, however, is that when we multiply $Z_{{\rm xcap},\CT}$ by the trumpet path integral,
which for $\delta=0$ is odd under $\phi\to\phi+2\pi$, the two choices of $V$ will contribute with opposite signs.  This is a manifestation of the fact that $\theta=\pi$ if $\delta=0$.
If $\delta=1/2$, the trumpet is even under $\phi\to \phi+2\pi$, so the two choices of $V$ will contribute with the same sign, in accord with the fact that $\theta=0$.

However, $Z_{{\rm xcap},\CT}(b,\phi)-Z_{{\rm xcap},\CT}(b,\phi+2\pi)$ is imaginary, so if the two choices of $V$ contribute with opposite signs, then
it seems that the path integral will be imaginary.  Euclidean path integrals can be complex-valued in general, but an imaginary result here will not agree with expectations
from the boundary matrix ensemble.     In fact, a rather subtle additional  factor of
$\pm\i$ appears precisely when $\delta=0$.    A $\CT$-invariant (or $\tT$-invariant) theory with $n$ Majorana fermions all transforming in the same way under $\CT$ (or $\tT$)
has an anomaly that depends on the value of $n$ mod 8.   The $\N=2$ Schwarzian theory has two Majorana fermions, which transform oppositely under $\tT$, leading to no anomaly,
but transform the same way under $\CT$, so that the $\N=2$ Schwarzian theory with $\CT$ symmetry
has an anomaly coefficient[In $\N=1$ JT supergravity, the Schwarzian theory has a single
Majorana fermion, and is anomalous even in the absence of time-reversal symmetry.  This was important in [14].  The $\eta$-invariant entered in that analysis
for reasons similar to what we explain here.]  of 2.   If $\delta=1/2$, the boundary theory is anomalous in the same
way and the anomalous Schwarzian theory correctly reproduces the anomaly in the microscopic boundary description,[We can realize $\delta=1/2$ with an $\N=2$ SYK model
with an odd number $N=2r+1$ of complex fermions.  This is equivalent to $4r+2$ Majorana fermions, so the anomaly coefficient  is $4r+2$, or $\pm 2$ mod 8.   Thus the boundary
anomaly either equals the anomaly of the Schwarzian theory or  differs from it by 4.
If the two coefficients are equal, the Schwarzian theory simply reproduces the anomaly of the boundary theory.   If they differ by 4, we can compensate
with a bulk factor $(-1)^\chi$, which carries an anomaly coefficient of 4.  Including this factor
is a familiar option and does not affect the fact that the path integral is real.]    but if $\delta=0$, the boundary theory
is anomaly-free and we need to modify the bulk theory to compensate for the anomaly of the $\N=2$ Schwarzian theory.  In appendix <ref>, a factor in computing
the partition function of the $\N=2$ Schwarzian theory was ${\mathrm{Pf}}(D)$, where ${\mathrm{Pf}}$ is the Pfaffian (the square root of the determinant) and $D$, the rotation generator, can be viewed
as the 1-dimensional Dirac operator
of the two Majorana fermions that are described by the $\N=2$ Schwarzian theory.  This Dirac operator is defined on $S$, the boundary of the spacetime $C'$ obtained
by gluing the crosscap to the trumpet.   A general recipe[This recipe is suggested by
the Dai-Freed theorem [63] and was most fully justified from a physical point of view in [64].]  to define a potentially anomalous
fermion path integral on a manifold $S=\partial C'$ is to replace ${\mathrm{Pf}}(D)$ with $|{\mathrm{Pf}}(D)|\exp(-\i \pi \eta(D_{C'})/2)$, where $\eta(D_{C'})$ is the Atiyah-Patodi-Singer
eta-invariant of a bulk Dirac operator
$D_{C'}$ that extends $D$.  In the present case, we can define $D_{C'}$ as the Dirac operator (for some ${\rm pin}^-$ structure)
coupled to a flat gauge field on $C'$ defined by the monodromy $V$ of eqn. (<ref>) with $\phi$ replaced by $\alpha$.     Any other choice of $D_{C'}$ is equivalent topologically  to this, possibly with a different ${\rm pin}^-$ structure (or equivalently with $\phi=\alpha+2\pi$).   To see the consequences, first note that if $\alpha=0$, then
$D_{C'}$ is the direct sum of two copies of a Dirac operator coupled to a ${\rm pin}^-$ bundle.  Its $\eta$-invariant is then $-1$ or
$1$, depending on the ${\rm pin}^-$ structure, so $\exp(-\i\pi\eta/2)=\pm\i$.  When one varies $\alpha$, because $\eta$ is a topological invariant on a closed manifold of even dimension,
$\eta(D_{C'})$ is constant except for jumps that precisely ensure that $|{\rm Pf}(D)|e^{-\i\pi \eta(D_{C'})/2}$ varies smoothly with $\alpha$.   So $|{\rm Pf}(D)|e^{-\i\pi \eta(D_{C'})/2}=
\pm\i \,{\rm Pf}(D)$ (with a sign that does not depend on $\alpha$), and the effect of including the  eta-invariant and canceling the anomaly is just to multiply the path integral of the Schwarzian modes, naively
${\rm Pf}(D)$,  by an overall factor $\pm \i$.   The choice of
sign leads  to two bulk theories that differ by a factor of $(-1)^\chi$, dual as usual to two options in the boundary theory.

§.§ The Matrix Model Prediction

We will now compare to matrix model predictions in the two cases of $\tT$ and $\CT$ symmetry.

(A) We begin with  ${\sf T}$ symmetry, and consider statistically independent multiplets with $(\upalpha_0,\upbeta)=(0,1)$ or $(3,4)$. The spectral curve of a supermultiplet with charges $(q-\hq/2,q+\hq/2)$ is given by eqn. (<ref>). We remind the reader that in gravity it is more convenient to label a multiplet by the bulk charge $q$, which is also equal to the average microscopic $R$-charge of the multiplet. Let us first write down the prediction from $\cN=2$ JT gravity for the partition function of the crosscap more explicitly
Z_xcap,T(β,α)    =    ∫_0^∞ḅ   Z_tr(b,0) π(b/4)   =    ∑_q∈ℤ+δ-1/2 ( e^ıα(q-/2)  + e^ıα(q+/2)) ∫_0^∞ḅ   Z_tr^bos.(β,b) V^(q)_xcap,T(b),
where we introduced the “fixed charge” volume for the crosscap

V^(q)_xcap,T(b)=(b/4)/2.Case A
A factor of $1/(2\pi)$ comes from the normalization of the $\N=2$ trumpet in terms of its bosonic counterpart $Z_{\rm tr}^{\rm bos}$. This definition of the “fixed charge” crosscap volume in terms of the partition function is consistent with eqns. (<ref>) and (<ref>) specialized to $g=1/2$ and $n=1$. We used a subscript ${\sf T}$ in $V^{(q)}_{{\rm xcap},{\sf T}}$
to distinguish this crosscap volume from the one in case (B) below. The goal of this section is to reproduce eqn. (<ref>) from the matrix model.

The genus $g=1/2$ correction $R^{(q)}_{\rm xcap}(x)$ to the one boundary resolvent $R(x)$ of a supermultiplet $q$ can be derived from the matrix model loop equation, which we review in appendix <ref>, and is given by

R^(q)_xcap(x) = ∫x̣'/2πı 1/x'-x √(σ(x'))/√(σ(x)) ( (1-2/)y'(x')/2y(x') + _0-1/2x'),
and we can use $\sigma(x)=(x-\frac{q^2}{4\hq^{\,2}})$ (this function is introduced in the appendix), since there is a single edge in the double-scaling limit. The integration contour for $x'$ starts as a small circle around $x$ and can be deformed to compute the discontinuity along the cut. The answer for an $(\upalpha_0,\upbeta)=(0,1)$ AZ ensemble with spectral curve (<ref>) is given by
R^(q)_xcap(x) = 1/2√(-x+q^2/4^ 2) ∫_q^2/4^ 2^∞ √(x'-q^2/4^ 2)   x̣'/x'-x [-(2π√(x'-q^2/4^ 2))/√(x'-q^2/4^ 2)  ]
Shifting the integrand $x'\to \frac{q^2}{4\hq^{\,2}} + x'$ and writing $x$ in terms of a new variable $z$ as $x= \frac{q^2}{4\hq^{\,2}} - z^2$, we can put the matrix model prediction for the crosscap in the form

R^(q)_xcap(x= q^2/4^ 2-z^2) =- 1/2z ∫_0^∞ √(x') x̣'/x'+z^2 (2π√(x'))/√(x')
The matrix model prediction of the crosscap volume is given in terms of the resolvent by the integral transform

V^(q)_xcap,M(b) = ∫ẓ/2πı e^b z(-2 z) R^(q)_xcap(x= q^2/4 ^ 2-z^2 ),
where $z$ is integrated in the imaginary direction. To derive this relation, at the level of an individual multiplet, one can combine $Z_{\rm xcap}(\beta) = \int_0^\infty \d b Z_{\rm tr}^{\rm bos.}(\beta,b) e^{-\beta E_0} V_{\rm xcap,M}(b)$ with $R(x) = -  \int_0^\infty \d\beta\, e^{\beta x} Z(\beta)$. Apply first the integral transform to go from the partition function to the resolvent and use that $-\int_0^\infty \d \beta \, e^{\beta x} Z_{\rm tr}^{\rm bos} = e^{-b z}/(-2 z) $, with $x= E_0 - z^2$, to write the resolvent in terms of the volume. Applying an inverse Laplace transform, and using that $E_0= q^2/4 \hq^2$, one arrives at the expression (<ref>) given above.

We can insert (<ref>) into the right hand side of (<ref>), to derive $V^{(q)}_{{\rm xcap,M}}(b)$. Even though the integral appearing in (<ref>) diverges for large $x'$, we can obtain a finite answer for the volume by first integrating over $z$ and second over $x'$. We get

V^(q)_xcap,M(b)    =    ∫_0^∞x̣'  sin(b√(x'))(2π√(x'))/√(x') = (b/4)/2=V^(q)_xcap,T(b) .
This matches precisely the prediction from gravity in (<ref>), as indicated in the right hand side of eqn. (<ref>). The result for the $(\upalpha_0,\upbeta)=(3,4)$ ensemble is the same up to an overall minus sign, which can be reproduced in gravity by including a factor of $(-1)^{n_c}$ in the path integral, where $n_c$ is the number of crosscaps.

(B) We now analyze  ${\sf CT}$ symmetry. The gravity calculation performed in the previous section gives the following answer for the crosscap partition function
Z_xcap,CT(β,α)    =    ∫_0^∞ḅ   ∫_0^4π ϕ̣  Z_tr(b,ϕ) ( coshb/2+ cosq ϕ)b/4 /8(cosqϕ/2-ısinhb/4)^2   ,
The integrand is precisely (<ref>) and the range over the $\uU(1)$ holonomy $\phi$ is from $0$ to $4\pi$ since we need to integrate over all possible holonomies around the crosscap, given in eqn. (<ref>).
The integral is not absolutely convergent because of singularities when $b=0$ and $\hq \phi$ is an odd multiple of $\pi$; we  will integrate over $\phi$ before integrating over $b$, a procedure that seems to give sensible results.
Comparing with (<ref>), the crosscap volume at “fixed charge” is given by
V_xcap,CT^(q)(b) =∫_0^4π ϕ̣/2π   e^-ıϕq  ( coshb/2+ cosq ϕ)b/4 /8(cosqϕ/2-ısinhb/4)^2
A factor of $1/(2\pi)$ comes, again, from the normalization of the $\N=2$ trumpet in terms of $Z_{\rm tr}^{\rm bos.}$. This integral seems complicated but can be computed using residues in a straightforward way. In order to do this, change variables from $\phi$ to $w = e^{\i \frac{\phi}{2}}$ and write the integrand as a meromorphic function in $w$ implementing the following replacements $2\cos \hq \phi \to w^{2\hq}+w^{-2\hq}$, $2\cos \frac{\hq \phi}{2} \to w^{\hq}+w^{-\hq}$ and $e^{-\i \phi q} \to w^{- 2q}$. The corresponding integration contour in the complex $w$-plane is over the unit circle $|w|^2=1$. We can evaluate the integral as the sum over the residues at the poles inside the unit disk in the complex $w$-plane. The integrand has potentially a pole at $w=0$ and $\hq$ poles at $w^{\hq} = - \i e^{-b/4}$ (the poles at $w^{\hq} = \i e^{+b/4}$ are outside of the unit disk and do not contribute to this sum). The final answer is
V_xcap,CT^(q)(b) =
b/4/2 δ_q,0 - e^-ıπ|q|/ e^-|q|/2 b,     if $2q = 0 ~{\rm mod}~ \hq$,

0,     otherwise

As a preliminary observation, charge conjugation acts as $q\to -q$ and we see indeed that the crosscap volume depends only on $|q|$. We remind the reader that a ${\sf CT}$ symmetry is only possible if $\delta = 0$ or $1/2$. We separately analyze each case below.

$\delta = 1/2$: Let us consider first the anomalous theory. In this case the spectrum of $R$-charges $k$ is half-integral and therefore the bulk charge of a supermultiplet $q=k\pm \hq/2$ is an integer. The random matrix ensemble analysis implies that for all $q>0$ we should study an AZ ensemble with $(\upalpha_0,\upbeta)=(1,2)$ while for $q=0$ we should study the $(\upalpha,\upbeta)=(1,1)$ or $(1,4)$ ensembles, depending on the sign of ${\sf CT}^2$. Let us therefore begin with $q=0$, corresponding to a supermultiplet with $R$-charges $(-\hq/2,\hq/2)$. In this case the crosscap volume given in eqn. (<ref>) becomes
V_xcap,CT^(0)(b) = b/4/2 - 1
Next compare this with the matrix model crosscap in the $(\upalpha,\upbeta)=(1,1)$ AZ ensemble. The answer for $(1,4)$ is the same up to an overall minus sign that in gravity can be interpreted as adding a factor $(-1)^{n_c}$ in the path integral, where $n_c$ is the number of crosscaps. The matrix model prediction to the crosscap contribution to the resolvent, using the spectral curve given in eqn. (<ref>) and using the loop equation (<ref>), is given by
R^(0)_xcap(x=-z^2)    =     ∫x̣'/2πı 1/x'-x √(σ(x'))/√(σ(x)) ( (1-2/)y'(x')/2y(x') + _0-1/2x')

=   1/2z ∫_0^∞ √(x') x̣'/x'+z^2 [-(2π√(x'))/√(x')  + 1/πx'],
where $\sigma(x)= x$ in this case. We remind the reader in deriving this equation that for the $q=0$ supermultiplet, the gap vanishes $E_0(q=0)=0$ and there are no BPS states with charges $\pm \hq/2$. Since there is no pole in the spectral curve for this multiplet at $x=0$ we also do not need to add any logarithmic correction to the matrix potential. We can use the relation between the matrix model prediction for the volume and the matrix model resolvent to verify
V^(0)_xcap,M (b)    =    ∫ẓ/2πı e^b z (-2z) R^(0)_xcap(-z^2)
=    ∫_0^∞x̣'  sin(b√(x'))[(2π√(x'))/√(x')- 1/πx' ]

=   b/4/2  -1=V_xcap,CT^(0)(b) .
We therefore obtain a match between gravity and the matrix ensemble proposed at the beginning of this section for the $q=0$ sector.

Next we analyze the other multiplets with $q\neq 0$. An $(\upalpha_0,\upbeta)=(1,2)$ AZ ensemble predicts a vanishing crosscap contribution to the resolvent. In gravity we obtained that the crosscap volume vanishes for most charges except when $2q=0~{\rm mod}~\hq$, which in the $\delta=1/2$ theory reduces to $q = 0 ~{\rm mod}~\hq$. The crosscap volume in this case is

V_xcap,CT^(q)(b) = - (-1)^q e^-|q|/2 b.
How can we make sense of this non-zero result? It is customarily assumed that, when taking the large $L$ limit of a matrix integral, we keep fixed the matrix potential or equivalently the spectral curve $y(x)$. Nevertheless, one is free to include small corrections to the spectral curve (or the matrix potential) of order $1/L$.  This will contribute to the resolvent
at the same order in $e^{-S_0}$ as a crosscap.   This possibility has
not been necessary in previous discussions of two-dimensional gravity and matrix models, but it does seem to be necessary in $\N=2$ JT supergravity with $\CT$ symmetry.

Let us write $y(x) = y_0(x) + e^{-S_0} y_{1/2}(x) + \ldots $ and compute the correction $y_{1/2}(x)$ needed to reproduce (<ref>). We can use the crosscap volume from gravity to evaluate the change in the partition function
Z^(q)_xcap (β) =  ∫_0^∞ḅ  e^-b^2/4β/2√(πβ) e^-βq^2/4 ^2   (-1)^q+1 e^-|q|b/2 .
The correction to the resolvent is given by

R_xcap^(q)(x) = - ∫_0^∞β̣  e^βx   Z^(q)_xcap (β) =  (-1)^q/2z(z+|q|/2),   where  x=q^2/4^2-z^2.
The difference between $R(x)$ and $y(x)$ is $V'(x)/2$, which can be an arbitrary analytic function of $x$ up to the possible presence of a simple pole at $x=0$. But
$y$ is odd under the hyperelliptic involution $z\leftrightarrow -z$.   Therefore we can read off the change in
$y$ from the odd piece of (<ref>):

y_1/2(x) =(-1)^q |q|/2  1/2 x z.
The first check of our proposal is that  $y_{1/2}(x)$ has the right analytic behavior to be interpreted as a correction to the spectral curve. Following [58] and [59], the pole in $z$ at $z=0$ indicates this is the first term in a series that resums into a shift of the ground state energy of the supermultiplet

E_0(q) →E_0(q) + e^-S_0(-1)^q 2π|q|/ + 𝒪(e^-2S_0).
This statement, and the precise prefactors, can be verified easily by expanding the spectral curve (<ref>) under a small variation of $E_0$ near $z=0$. In addition,
the correction to the spectral curve (<ref>) has a pole at $x=0$ indicating the need for the correction to the matrix potential  to have a $\log x$ term as well.
A singularity of $y$ other than the
$1/z$ and $1/x$ singularities that we have actually found would not be consistent with interpreting $y_{1/2}(x)$ as a correction to the spectral curve.

This interpretation of $y_{1/2}$  has non-trivial implications for higher order corrections to the resolvent, and we will not attempt a full analysis here.  But we will verify our proposal in  the first nontrivial case. The leading order, connected, two-boundary partition function of the matrix integral of an AZ ensemble in the double scaling limit is universal and depends solely on $E_0$, the ground state energy,
Z_0,2=√(β_1β_2)/2π(β_1+β_2) e^-(β_1+β_2)E_0.
Therefore a change in $y$ will influence $Z_{0,2}$ only through the shift in $E_0$.
With this in mind, consider the topological expansion of the two-boundary partition function as $Z_2= Z_{0,2}+e^{-S_0}Z_{1/2,2}+{\mathcal O}(e^{-2S_0})$.  If the $e^{-S_0}Z_{1/2,2}$ contribution
reflects a correction to $y$, then we can predict $Z_{1/2,2}$ in terms of
the shift of the ground state energy computed in (<ref>).   In terms of the fixed charge partition function, the prediction is

Z^(q)_1/2,2 (β_1,β_2) = -(-1)^q 2π|q|/√(β_1β_2)/2π e^-(β_1 + β_2) E_0(q).
We can reproduce this prediction directly from gravity. For this purpose, we consider the path integral on a hyperbolic cylinder with a crosscap in the interior, see fig. <ref>. We first introduce $V_{\frac{1}{2},2}({\sf b}_1,{\sf b}_2)$ as the path integral of $\N=2$ JT gravity on the sphere with two geodesic boundaries and one crosscap. This quantity is easy to compute combining the rules given in section <ref> and section <ref>, since it does not involve a non-trivial action of the mapping class group. The answer is
V_1/2,2(b_1,b_2)    =    ∫_0^4π ϕ̣_3  ∫_0^∞ ḅ_3 V_0,3(b_1, b_2, b_3)( coshb_3/2+ cosq ϕ_3 )b_3/4 /8(cosqϕ_3/2-ısinhb_3/4)^2

=   ∑_q 1/2πq^2/4^2  e^ıq ϕ_1 + ıq ϕ_2 ∫_0^∞ ḅ_3 ∫_0^4π ϕ̣_3/2π   e^-ıϕ_3 q  ( coshb_3/2+ cosq ϕ_3 )b_3/4 /8(cosqϕ_3/2-ısinhb_3/4)^2

=    - ∑_q  (-1)^q/2π q^2/4^2  e^ıq ϕ_1 + ıq ϕ_2 ∫_0^∞ ḅ_3   e^-|q|/2 b

=    - ∑_q  (-1)^q/2π |q|/2  e^ıq ϕ_1 + ıq ϕ_2
In the first line, we start with a three-holed sphere and glue a crosscap to one hole, labeled $3$. We included an orientation reversal of  the crosscap $R$-symmetry holonomy with respect to the three-holed sphere as indicated by $\overline{{\sf b}}_3 = (b_3, -\phi_3)$. In the second line we used (<ref>) and wrote the delta function on holonomies as a sum over charges $q$
-δ”(ϕ_1 + ϕ_2 - ϕ_3) = 1/2π ∑_q∈ℤ q^2 e^ıq ϕ_1 + ıq ϕ_2 - ıq ϕ_3.
which allowed us to combine a factor of $e^{-\i \phi_3 q}$ with the crosscap contribution into the same integral we already computed above in (<ref>). Finally we performed the $\phi_3$ integral. In the last two lines we did not include the contribution of (<ref>) with $q=0$ since, even though the $b_3$ integral is divergent, it is multiplied by a factor of $q^2$ and therefore automatically vanishes. (This is consistent with the fact that the $q=0$ multiplet does not require any correction to the spectral curve.) Finally, we can glue this volume to the two external trumpets and obtain the correction to the two-boundary partition function in a sector of “fixed charge” $q$
Z^(q)_1/2,2(β_1,β_2)    =   - (-1)^q ∫_0^∞b_1 ḅ_1 Z_tr^bos.(b_1,β_1)∫_0^∞b_2 ḅ_2 Z_tr^bos.(b_2,β_2) (2π) |q|/2

=    -(-1)^q √(β_1β_2)/π (2π) |q|/2=- (-1)^q |q|/√(β_1β_2)
This answer obtained from the gravitational path integral, in the second line, matches precisely the matrix model prediction (<ref>) in the presence of a correction to the spectral curve, verifying our proposal.

[scale=0.8, baseline=([yshift=-0.1cm]current bounding box.center)]
[thick] (0,-1) ellipse (.5 and .15);
[thick, rotate around=-30:(-3,2.5)] (-3,2.5) ellipse (.15 and 1);
[thick, rotate around=30:(3,2.5)] (3,2.5) ellipse (.15 and 1);
[thick, bend right=30] (-1.8,2.45) to (1.8,2.45);
[thick, bend left=20] (-1.8,2.45) to (-2.5,3.35);
[thick, bend right=20] (1.8,2.45) to (2.5,3.35);
[thick, bend right=20] (-2.2,1.55) to (-3.5,1.65);
[thick, bend left=20] (2.2,1.55) to (3.5,1.65);
[thick, bend left=30] (-2.2,1.55) to (-0.5,-1);
[thick, bend right=30] (2.2,1.55) to (0.5,-1);
[thick] (-.3,-1-.12) to (.3,-1+.12);
[thick] (.3,-1-.12) to (-.3,-1+.12);

[thick, dashed, rotate around=-30:(-2,2)] (-2,2) ellipse (.15 and .5);
[thick, dashed, rotate around=30:(2,2)] (2,2) ellipse (.15 and .5);
at (-4,3) $\beta_1,\alpha_1$;
at (4,3) $\beta_2,\alpha_2$;

Contribution to the connected two-boundary path integral with a crosscap in the interior. The two NAdS boundaries are labeled by their inverse temperature and $R$-charge chemical potential, while the crosscap is indicated with a cross. The geodesic dividing the NAdS boundaries from the internal surface are denoted by dashed lines.

Even though we have found a matrix integral that reproduces the gravitational path integral to this order in the topological expansion, we should stress that we do not have an understanding of why in the presence of $\CT$ symmetry, and not otherwise, such a modification of the spectral curve is necessary.

$\delta = 0$: Finally we analyze the non-anomalous theory with $\delta=0$. Since the spectrum of $R$-charges $k$ is integral, the spectrum of bulk charges $q=k\pm\hq/2$ will be half-integral. Again we find from (<ref>) that most multiplets have a vanishing contribution to the crosscap except when $2 q = 0~{\rm mod}~\hq$. In those cases, the crosscap volume has the same $b$-dependence as considered above and we can again interpret it as a shift of the spectral curve. The check for the cylinder with a crosscap works as well.  The only new feature in this case is that the “fixed charge” crosscap volume is imaginary since it is proportional to $e^{-\frac{\i \pi |q|}{\hq}} = \i (-1)^{q+\frac{\hq}{2}} $ with half-integer $q$.  To get a real answer, we
need the factor of $\pm \i$, coming from the eta-invariant, that was explained at the end of section <ref>.

There is a subtlety here concerning the supermultiplets $(-\hq,0)$ and $(0,\hq\,)$ that contain states of $R$-charge zero.   While a generic supermultiplet in the presence of
$\CT$ symmetry is described by an AZ
multiplet with $(\upalpha_0,\upbeta)=(1,2)$, the $(-\hq,0)$ and $(0,\hq\,)$ multiplets are different because they both contain states of vanishing charge, and states of vanishing charge
are mapped to themselves
by $\CT$.   An analysis similar to what we have described before for in other examples shows that these multiplets are naturally described by ensembles with
$(\upalpha_0,\upbeta)=(-1,2)$. With an analytic potential, such an ensemble would produce a non-vanishing crosscap correction. As discussed in appendix <ref>, we
can add a $\log x$ term to the matrix potential that shifts the effective value of $\upalpha_0$.  After such a shift, we can make the measure for the singular values of $Q_{0}$ and $Q_{-\hq}$ take the same form as an $(\upalpha_0,\upbeta)=(+1,2)$ ensemble, therefore removing the contribution to the crosscap coming from the loop equations. After doing this, we can again add a small correction to the spectral curve in a way that reproduces the non-vanishing crosscap volume computed from gravity.

Finally, we note that in both the $\tT$-invariant case and the $\CT$-invariant case, the $b$ integral is divergent at $b=0$, which means that it is probably difficult to
extend the comparison between the matrix model and the crosscap to higher orders[In the bosonic case, the divergence can be regulated in gravity by working with the $(2,p)$ minimal string which becomes JT gravity only in the large $p$ limit [6, 14, 17]. It is reasonable to expect that if a similar duality between matrix integrals and the non-critical $\N=2$ string exists, the crosscap would again be finite.].   In the case of $\CT$ symmetry, the divergence is only at $q=0$ (assuming that one
integrates over $\phi$ before integrating over $b$, as we have done), so for other values
of $q$, there appears to be no difficulty in computing in higher orders.   However, this is precisely the case in which we have found that the matrix potential has higher order
corrections.  We do not know if further corrections are needed in going to higher orders, so we do not have clear predictions about the higher order amplitudes.

§ $\MATHCAL{N}=4$ JT GRAVITY BLACK HOLE

In this section we present some partial results regarding the study of pure $\cN=4$ JT gravity including a sum over topologies.

§.§ The $\cN=4$ JT Black Hole Spectrum

The general
structure of $\N=4$ JT supergravity is similar to what we have already discussed for $\N=2$.
First, the action has, besides a topological term proportional to $S_0$, a bulk action that can be written as a topological $BF$ theory with gauge supergroup $G=\PSU(1,1|2)$.  A
maximal bosonic subgroup of $G$ is  ${\rm SL}(2,\mathbb{R}) \times \SU(2)$. The first factor describes gravity while the second  is the gauge
group of  an $\SU(2)$ $BF$ theory. We denote generators of this group as $J^1, J^2, J^3$. For asymptotically NAdS boundary conditions, boundary terms are required; these are simpler to write in a second order formulation.   The bosonic fields in bulk are a graviton and
dilaton along with the $\SU(2)$ gauge fields.   The gravitino is now in the fundamental of $\SU(2)$.
Because the group $\PSU(1,1|2)$ has an $\SU(2)$ group of outer automorphisms, the $BF$ theory of $\PSU(1,1|2)$ has $\SU(2)$ as a global symmetry group.[If one gauges
this global $\SU(2)$, one gets a $BF$ theory of the supergroup $D(2,1;\alpha)$, potentially related to a near extremal limit of black holes that have AdS$_3$ throats with large $\cN=4$
superconformal symmetry.]  To distinguish the two $\SU(2)$'s, we refer to the one that is gauged as the $R$-symmetry group $\SU(2)_R$, and the other one we call

The disk partition function for $\N=4$ again reduces to the path integral of a boundary reparametrization mode, the $\N=4$ Schwarzian mode. We work at inverse temperature $\beta$ and we turn on a chemical potential for the Cartan of $\SU(2)$,  parametrizing the fugacity by $e^{\i \alpha J^3}$. This problem was studied in [24]. The boundary theory governs reparametrizations of the $\cN=4$ supercircle $(t,\theta,\bar{\theta})$ where $\theta$ is a complex $\SU(2)$ doublet and $\bar{\theta}$ its complex conjugate. The grand canonical partition function is found by imposing the following twisted conditions
(t,θ, θ̅) →(t+β, - e^ıαJ^3  θ, - e^- ıαJ^3 θ̅).
Since $\theta$ is in the fundamental representation, we can write $J^3=Z/2$ where $Z={\rm diag}(1,-1)$ is a Pauli matrix.  Thus $\alpha$  has  periodicity $\alpha\cong\alpha+4\pi$
as well as the $\SU(2)$ Weyl invariance, which acts by $\alpha\to -\alpha$.   The final result for the partition function is

Z_disk =e^S_0 ∑_n∈ℤ β (α/4)(α+4πn)/π^4(1-1/4π^2(α+4πn)^2)^2   e^π^2 /β(1-1/4π^2(α+4πn)^2) .
The derivation is outlined in appendix <ref>.

With NAdS boundary conditions, the boundary quantum mechanics has four supercharges $Q_{A\dot A}$, $A,\dot A=1,2$ transforming as a bifundamental
of $\SU(2)_R$ (acting on the $A$ index) and $\SU(2)_\gl$ (acting on $\dot A$) and satisfying
{Q_AȦ,Q_BḂ}=ε_ABε_Ȧ Ḃ H, with $H$ the Hamiltonian. Here $\varepsilon_{AB}$ and $\varepsilon_{\dot{A}\dot{B}}$ are antisymmetric tensors with $\varepsilon_{12}=\varepsilon_{\dot 1\dot 2}=1$.
Alternatively, since $(\SU(2)_R\times \SU(2)_\gl)/\Z_2\cong \SO(4)$ (where here $\Z_2$ is the product of the centers of the two $\SU(2)$'s), and a bifundamental
of $\SU(2)_R\times \SU(2)_\gl$ is a vector of $\SO(4)$, the supersymmetry algebra contains four hermitian supercharges $Q_I$, $I=1,\cdots,4$ transforming as a vector of $\SO(4)$,
and satisfying
{Q_I,Q_J}=2δ_IJ H .

In quantum mechanics with $\N=4$ supersymmetry, a  BPS multiplet  consists  of a  single  irreducible representation of $\SU(2)_R$ of any spin $J$, while  an irreducible
non-BPS multiplets decomposes under $\SU(2)_R$ as $J \oplus (J-1/2) \oplus (J-1/2) \oplus (J-1)$ with some $J\geq 1/2$
(for $J=1/2$, the multiplet instead reduces to  $0\oplus 0\oplus 1/2$). The partition function (<ref>) can be expanded in such multiplets as

Z_disk = e^S_0 +∑_J≥1/2 ( χ_J(α) + 2 χ_J-1/2(α) + χ_J-1(α) )  ∫_J^2^∞Ẹ e^-βE e^S_0J  sinh( 2π√(E-J^2))/π^2 E^2 .
where $\chi_J(\alpha) = \frac{\sin{((2J+1)\alpha/2)}}{\sin (\alpha/2)}$ is the character of a spin $J$ representation of $\SU(2)$, and the gap in the spectrum is given by $E_0(J)=J^2$ for a supermultiplet with maximum spin $J$.
(For $J=1/2$, since formally $\chi_{-1/2}(\alpha)=0$, the expression $ \chi_J(\alpha) + 2 \chi_{J-\frac{1}{2}}(\alpha) + \chi_{J-1}(\alpha)$ for the character of a non-BPS multiplet
reduces to $\chi_{1/2}(\alpha)+2\chi_0(\alpha)$, which is the character of the exceptional multiplet $J=1/2$.)   Since $J\geq 1/2$ in all cases,
the energy gap $J^2$  is  non-vanishing for all non-BPS multiplets.
In contrast to the case of $\N=2$, there is no multiplet with a vanishing gap.
The BPS states are described by the term $e^{S_0}$ in eqn. (<ref>), and since this term is independent of $\alpha$, they all have spin 0.

The spectrum extracted from this partition function is very simple.
In addition to $e^{S_0}$ BPS states, all of spin 0, there is for each $J\geq 1/2$ a continuous spectrum of non-BPS supermultiplets with highest spin $J$.
The density of states is
ρ_J(E) = e^S_0 J sinh(2π√(E-J^2))/π^2 E^2    E≥J0     E<J.       To obtain this answer from a matrix model, the spectral curve should be
y_J(x) = Jsin( 2π√(-x+J^2))/πx^2.
The behavior of the spectrum near threshold is the typical square root behavior $y_J(J^2 + \varepsilon) \sim \sqrt{\varepsilon}$ of a Dyson ensemble in random matrix theory,
or an AZ ensemble in which the threshold energy is positive. The behavior at $x=0$ merits discussion. Naively $ y_J(x)$  has a double pole at $x=0$:
y_J(x) = J sin(2πJ)/πx^2 - cos(2πJ)/ x +(analytic in x).
The double-pole would be problematic since it would imply a highly divergent term in the matrix potential at $x=0$ that would spoil the derivation of the loop equations in their usual form. This is explained in appendix <ref>. Fortunately, since $2J$ is an integer the coefficient of the double-pole vanishes: $\sin(2\pi J)=0$. The coefficient of the simple pole is non-zero and equal to $(-1)^{2J+1}$.   Thus the residue of the simple pole is positive or negative for half-integer or integer $J$.
A simple pole  is consistent with the loop equations in the presence of a logarithmic term in the potential, as discussed in the appendix.

$\N=4$ JT supergravity does not have some of the generalizations that exist for $\N=2$.  There is no analog of the parameter $\hq$,
since the $R$-symmetry group $\SU(2)_R$ is simply-connected. For essentially the same reason, there is no analog of the parameter $\delta$.

§.§ The Two-Boundary Wormhole

Having obtained the spectrum and the multiplet structure, the next question is whether different multiplets are statistically independent or not.
The most basic test of this question comes from the amplitude for a cylinder, the leading connected amplitude with two NAdS boundaries.
The path integral on the cylinder reduces to an integral over the moduli space of flat $\PSU(1,1|2)$ connections, modulo gauge transformations. These are labeled by the supergroup holonomy around the non-contractible circle. In order to restrict to gauge connections that have an interpretation in hyperbolic geometry we restrict to holonomies with the property that the $\SU(1,1)$ block is in the hyperbolic conjugacy class. We assume no other restriction. This holonomy is parametrized by a geodesic length $b$, describing the conjugacy class of the $\SU(1,1)$ block, and an $\SU(2)$ holonomy  conjugate to $e^{\i \phi J^3}$, with parameter $\phi$,
where in the fundamental representation $J^3=Z/2$ with $Z$ the diagonal Pauli matrix. Like $\alpha$, $\phi$ has a periodicity $\phi\to\phi+4\pi$ and a Weyl symmetry
$\phi\to -\phi$.   For concreteness we take $\phi>0$. There is also a Schwarzian reparametrization mode on each boundary.  The path integral of the reparametrization mode along each boundary is given by

Z_tr(β,α; b,ϕ) = ∑_n∈ℤ  (α/4)/4πβ e^-1/4βb^2 ( e^-1/4β(α-ϕ+4πn)^2-e^-1/4β(α+ϕ+4πn)^2)  .
We review the derivation in appendix <ref>. Using Poisson resummation, this expression can be written as a sum over representations:

Z_tr(β,α; b,ϕ) = ∑_J≥1/2 ( χ_J(α) + 2 χ_J-1/2(α) + χ_J-1(α) ) 2 sin(Jϕ)     1/4πe^-βJ^2-b^2/4β/2√(πβ).
The $\beta$ dependence is the same as that of the trumpet in bosonic JT ($Z_{\rm tr}^{\rm bos.}(\beta,b) =e^{-\frac{b^2}{4\beta}}/(2\sqrt{\pi \beta} )$) up to a shift of the group state energy to $E_0(J)=J^2$.

We can now compute the cylinder partition function by gluing two of these trumpets along a common geodesic boundary. The final answer is
Z_0,2(β_1, α_1; β_2,α_2) = 2π∫_0^∞bḅ ∫_0^4π ϕ̣  Z_tr (β_1,α_1; b,ϕ) Z_tr (β_2,α_2;b,-ϕ).
The factor of $2\pi$ comes from the integral over the twist parametrizing the maximal torus of $\SU(2)$, including a factor of $1/2$ because of the Weyl symmetry. (In any case, we should emphasize we do not claim to properly understand the absolute normalization of the trumpet.) After gluing the two trumpets we obtain
Z_0,2(β_1, α_1; β_2,α_2)    =    ∑_J_1,J_2≥1/2 ( χ_J_1(α_1) +2 χ_J_1-1/2(α_1) + χ_J_1-1(α_1) )

×( χ_J_2(α_2) +2 χ_J_2-1/2(α_2) + χ_J_2-1(α_2) )∫_0^4π ϕ̣/8π(2 sinJ_1ϕ)(2sinJ_2ϕ)       ×e^-β_1 J_1^2 - β_2 J_2^2 ∫_0^∞bḅ   e^-b^2/4β_1-b^2/4β_2/4 π√( β_1β_2).
The integral over the geodesic length $b$ gives the same as before, since the trumpet is again proportional to the bosonic one. Integration over $\phi$ imposes that $J_1=J_2$. Therefore again we can only have the same multiplet propagating on each boundary to get a non-zero answer. The final result is then
Z_0,2(β_1, α_1; β_2,α_2)  = ∑_J≥1/2 [ ∏_i=1,2 ( χ_J(α_i) +2χ_J-1/2(α_i) + χ_J-1(α_i) ) ] √(β_1 β_2)   e^-(β_1+β_2) J^2/2π(β_1+β_2).
The multiplets are statistically independent, and the temperature dependence matches the expectations from a double-scaled matrix integral with a gap  $E_0(J)= J^2$.   This is
encouraging for a possible random matrix description.

§.§ $\N=4$ Random Matrix Theory

Finally, in this section we will explore aspects  of $\N=4$ random matrix theory.   We will consider the case that only $J=1/2$ multiplets (multiplets that decompose under
$\SU(2)_R$ as $0\oplus 0\oplus 1/2$) are present.
By similar logic to that of section <ref>, we will argue that it is natural  for random $\N=4$ supersymmetry with all multiplets of $J=1/2$
to be described by an AZ ensemble with $(\upalpha,\upbeta)=(1,2)$, the familiar ensemble that is important in $\N=1$ and $\N=2$ JT supergravity.  We expect the same to be true
for multiplets of any $J$, but it would be technically more complicated to demonstrate this.

The $\N=4$ algebra in the $\SO(4)$ language of eqn. (<ref>) has an obvious resemblance  to a Clifford algebra.
That fact enables an easy construction of the $J=1/2$ representation, from which all other representations are easily constructed.   Consider four gamma matrices
$\gamma_I$, $I=1,\cdots ,4$, satisfying $\{\gamma_I,\gamma_J\}=2\delta_{IJ}$, transforming as a vector of $\SO(4)$ (or equivalently a
bifundamental of $\SU(2)_R\times \SU(2)_\gl$) and acting in the usual way on a four-dimensional
Hilbert space $\H_0$. As usual we let $\gamma_5=\gamma_1\gamma_2\gamma_3\gamma_4$ be the product of the four $\gamma_I$.
Then, we get an irreducible representation of $\N=4$ global supersymmetry by taking
Q_I=λγ_I,    H=λ^2,with a nonzero constant $\lambda$.   To make the $Q_I$ hermitian, $\lambda$ must be real, and by possibly conjugating with $\gamma_5$, we can reduce to $\lambda>0$.
A familiar fact about spinors is that $\H_0$ transforms under $\SU(2)_R\times \SU(2)_\gl$ as $(1/2,0)\oplus (0,1/2)$.   So we can describe $\H_0$ as $\S_{1/2}\oplus \S'_{1/2}$,
where $\S_{1/2}$ is a two-dimensional vector space in the spin 1/2 representation of $\SU(2)_R$, and $\S'_{1/2}$ is a two-dimensional vector space in the spin 1/2 representation
of $\SU(2)_\gl$.    However, since $\SU(2)_\gl$ is not part of the supersymmetry
algebra, we often want to describe $\H_0$ just as a representation of $\SU(2)_R$.  From that point of view, $\S'_{1/2}\cong \C^2$ is just a two-dimensional vector space on which
$\SU(2)_R$ acts trivially, so $\H_0=\C^2\oplus \S_{1/2}$.

We can now also easily describe an arbitrary irreducible representation of the $\N=4$ algebra.  For $J$ a positive integer or half-integer, let $\H_J$ be a Hilbert space of dimension
$2J+1$ on which $\SU(2)_R$ acts as spin $J$.   Let $\H=\H_0\otimes \H_{J-1/2}$.  Setting $Q_I=\lambda \gamma_I\otimes 1$, we get an irreducible representation of the $\N=4$
algebra.  $\H$  transforms under $(2)_R$ as $(J-1)⊕(J-1/2)⊕(J-1/2)⊕J$, and we label this representation by the highest spin it contains,
namely $J$.     We note that $(2)_$ actually acts on
this representation via its action on $_̋0$, transforming the supersymmetry generators in the usual way.  So even though it is not part of the
supersymmetry algebra, $(2)_$ acts on every irreducible representation of this algebra, and therefore on every representation.\footnote{In general, an outer automorphism of
a group $G$ will either act in a given representation $W$, or conjugate $W$ to some inequivalent representation.   In the present case, since the group $\SU(2)_\gl$ is connected,
there is no way for it to act nontrivially on the discrete label $J\in \{1/2,1,3/2,\cdots\}$ of a representation and instead it
acts in each representation.}  This fact turns out to be helpful in understanding the random matrix theory.

Here, however, as already noted,
we will restrict ourselves to considering a random realization of the $=4$ algebra that is isomorphic to the direct sum of some number $L$ of copies of the basic $J=1/2$
representation.    For this, we consider a bosonic Hilbert space $_̋b≅^2L$ and a fermionic Hilbert space $_̋f=_1/2⊗_̋f'$ with $_̋f'≅^L$.
Thus, $(2)_R$ acts trivially on $_̋b$, and $_̋f$ transforms as
the direct sum of $L$ copies of $_1/2$.   $(2)_R$ also has an expected action on $Q_I$ coming from
the fact that they transform as $(1/2,1/2)$ of $(2)_R×(2)_$.
Then we look for a set of four operators $Q_I:_̋b↔_̋f$ that obey the algebra (\ref{fouralg}) and intertwine with the action
of $(2)_R$, acting on the four $Q_I$ and on the Hilbert space $=̋_̋b⊕_̋f$ in the expected fashion.

The symmetry group of this problem is simply the group of unitary transformations of $$̋ that commute with $\SU(2)_R$.   This is $\uU(2L)\times \uU(L)$, where the first factor
acts on $\H_b$ and the second on $\H_f'$.

Given our knowledge of the classification of representations of the $\N=4$ algebra, we can use the group $\uU(2L)\times \uU(L)$ to put
any set of $Q_I$ that generate such a representation in a canonical form.   A representation of the $\N=4$ algebra
on $\H_b\oplus \H_f=\C^{2L}\oplus S_{1/2}\otimes \C^L$ is
generically[The reason that this is only generically the case is that a $J=1/2$ representation with $H=\lambda^2$
degenerates to a collection of BPS states if $\lambda=0$.]  the direct sum of $L$ copies of the $J=1/2$ representation.   In a single copy of that representation, we have as before
and $\H_b=\S'_{1/2}$, and we can assume that  $Q_I=\lambda\gamma_I$, with the $\gamma_I$ being generators of a Clifford
algebra on $\H_b\oplus\H_f$.  To get $L$ copies of this representation, we introduce an $L$-dimensional vector space $V\cong \C^L$ with trivial action of $\SU(2)_R$, and then take
$\H_b=S'_{1/2}\otimes V$, $\H_f=S_{1/2}\otimes V$, and
Q_I=γ_I⊗M, where $M$ is a hermitian matrix acting on $V$.  By a unitary transformation, we can put $M$ in a diagonal form $M={\rm diag}(\lambda_1,\lambda_2,\cdots,\lambda_L)$, and,
since it is possible to make a $\gamma_5$ transformation separately for each “eigenvalue” $\lambda_i$, we can assume that $\lambda_i\geq 0$ for all $i$.

We have already determined that the symmetry group in this problem, taking into account the $\SU(2)_R$ symmetry,
is $K=\uU(2L)\times \uU(L)$, with real dimension $5L^2$.   We will now show that, give the constraint of
$\SU(2)_R$-invariance, to specify $Q_I$ requires $8L^2$ real parameters.   And we will show that the supersymmetry algebra imposes $3L^2$ real constraints on the $Q_I$.
One might therefore expect that the moduli space of representations of the supersymmetry algebra with the assumed dimension and action of $\SU(2)_R$ would have
dimension $8L^2-5L^2-3L^2=0$.   Instead, by explicit construction in eqn. (<ref>), we see that this moduli space has dimension $L$, parametrized by $\lambda_1,\lambda_2,\cdots,
\lambda_L$.   What is happening is that the $5L^2$ generators of $K$ are not all effective at reducing the moduli space dimension, because an $L$-dimensional subalgebra,
consisting of the matrices on $V$ that commute with $M$, leaves fixed $Q_I$ in eqn. (<ref>).   Hence the dimension of the moduli space
is actually $8L^2-(5L^2-L)-3L^2 = L$.

This counting shows that the $3L^2 $ constraints that come from the supersymmetry algebra are all effective at reducing the dimension of the moduli space.
So they are all  independent; there are no relations between these constraints.
In section <ref>, we found that in $\N=2$ random matrix theory, there are relations between the constraints as soon as
at least  four consecutive values of the $R$-charge are present.
For $\N=4$, it turns out that there are relations between the constraints
for all non-BPS representations with $J>1/2$, even if only one type of representation is present.[In fact, there are not only relations between the constraints;
there are relations between the relations, and higher order relations between those relations, and so on, with a number of steps of order $J$.]   That is the main reason
that here we will analyze the random matrix theory only for $J=1/2$; the case of higher $J$ is technically more complicated.

It remains to verify  that after taking into account the action of $\SU(2)_R$, there are $8L^2$ parameters in the $Q_I$, and the supersymmetry algebra places
$3L^2$ relations among those parameters.

To count parameters in $Q_I$, we note first that we can write $Q_I=\Q_I+\Q_I^\dagger$,  where $\Q_I$ maps $\H_f$ to $\H_b$ and $\Q_I^\dagger$ is the adjoint map
from $\H_b$ to $\H_f$.   $\Q_I$ is constrained by $\SU(2)_R$ invariance, but it does not satisfy any condition of hermiticity (and there is no constraint associated to $\SU(2)_\gl$,
since $\SU(2)_\gl$ invariance is not assumed).    At this point, it is convenient to think of $\Q_I$ as a bifundamental of $\SU(2)_R\times \SU(2)_\gl$ rather than a vector
of $\SO(4)$.   Four operators $\Q_{A\dot A}:\S_{1/2}\otimes \C^L\to \C^{2L}$ that transform with spin 1/2 under $\SU(2)_R$ are equivalent to a pair
of operators $\X_{\dot A}:\C^L\to \C^{2L}$ that obey no particular constraint.
The two operators $\X_{\dot A}$ depend on a total of $2 \cdot L \cdot 2L=4L^2$ complex parameters, or $8L^2$ real parameters.   This is the number of real parameters
involved in the choice of the $Q_I$, after imposing the action of $\SU(2)_R$, but without imposing the constraint that the $Q_I$ satisfy the $\N=4$ algebra.

The number of parameters involved in imposing the $\N=4$ algebra can be determined similarly.
A priori, $\{Q_I,Q_J\}$ transforms as $(0,0)\oplus (1,1)$ under $\SU(2)_R\times \SU(2)_\gl$.  The $(0,0)$ part is the Hamiltonian, and the supersymmetry constraint says
that the $(1,1)$ part vanishes.    Of course, $\{Q_I,Q_J\}$ maps $\H_b$ to $\H_b$ and $\H_f$ to $\H_f$.   An operator that transforms as spin 1 under $\SU(2)_R$ cannot
map $\H_b$ nontrivially to itself, since $\H_b$ transforms with spin 0 under $\SU(2)_R$.   Since $\H_f$ transforms under $\SU(2)_R$ with spin 1/2, and spin 1/2 arises in
the decomposition $1/2\otimes 1 =1/2\oplus 3/2$, it is possible for a spin 1 operator
to map $\H_f$ to itself.  A set of three hermitian operators mapping $\H_f=\S_{1/2}\otimes \C^L$ to itself are determined by a single hermitian operator
on $\C^L$, and thus depends on $L^2$ real parameters.   The $(1,1)$ part of $\{Q_I,Q_J\}$ consists of 9 hermitian operators on $\H_f$, making up three vectors of $\SU(2)_R$.
This is equivalent to three hermitian operators on $\C^L$, so the number of real parameters is $3L^2$, as claimed earlier.

We still have  to determine the measure for integration over the “eigenvalues” $\lambda_i$.   By definition, when we do this, we are trying to compute the measure on the
space of $Q_I$'s that do satisfy the constraints of $\N=4$ supersymmetry  (modulo equivalence generated by the group $K$).    As noted earlier, every solution of the constraints
is invariant under $\SU(2)_\gl$, acting via a suitable subgroup of $K$.   In computing the measure, we have to consider (in first order) perturbations of the $Q_I$ that do
not necessarily preserve the $\SU(2)_\gl$ symmetry, but $\SU(2)_\gl$ is still very useful in organizing and constraining the calculation.

As in eqn. (<ref>), when the constraints are satisfied, $\H_b$ and $\H_f$ have decompositions $\H_b=\S'_{1/2}\otimes \C^L$, $\H_f=\S_{1/2}\otimes \t \C^L$,
where $\SU(2)_\gl$ acts on $\S'_{1/2}$ (in the spin 1/2 representation) and not on anything else.   Here $\C^L$ and $\t \C^L$ are two different vector spaces, both
of dimension $L$; to emphasize that they are not the same, we now denote the second one as $\t\C^L$.   With this decomposition, it is now clear that the subgroup of
$K=\uU(2L)\times \uU(L)$ that commutes with $\SU(2)_\gl$ is $K'=\uU(L)\times \uU(L)$, where the two factors act respectively on $\C^L$ and $\t\C^L$.
The group $K'$ has dimension $2L^2$, while $K$ has dimension $5L^2$.   The part of the Lie algebra of $K$ that is orthogonal to the Lie algebra of $K'$
consists of $L^2$ copies of the spin 1 representation of $\SU(2)_\gl$.   (These constraints come from operators transforming as $(1,1)$ under $\SU(2)_R\times \SU(2)_\gl$
and acting on an $\SU(2)_\gl$-invariant Hilbert space $\H_f$, so they all transform with spin 1 under $\SU(2)_\gl$.)

It turns out that the effect on the measure of $3L^2$ symmetry generators transforming at $L^2$ copies of spin 1 of $\SU(2)_\gl$ cancels the effect on the
measure of $3L^2$ constraints transforming in the same way.  Before demonstrating this cancellation, we will first argue that if it does occur,
then a random realization of the $\N=4$ algebra with the representation content that we have assumed is described by an AZ ensemble with $(\upalpha,\upbeta)=(1,2)$.

We simply write down the general form of the four hermitian operators $Q_I$, assuming that they intertwine with an action of $\SU(2)_R\times \SU(2)_\gl$, not
just $\SU(2)_R$ as required by the supersymmetry algebra:
Q_I=γ_I 1+γ_5/2 M +γ_I 1-γ_5/2 M^†. Here $M$ is a completely unconstrained linear operator mapping $\C^L$ to $\t\C^L$, and $M^\dagger $ is its adjoint acting in the opposite direction.
In other words, $M$ is a bifundamental of $K'=\uU(L)\times \uU(L)$.   As we explained in section <ref>, a bifundamental of $\uU(L)\times \uU(L)$ is
naturally associated in random matrix theory to an AZ ensemble with $(\upalpha,\upbeta)=(1,2)$.

So all that remains is to show that there is no net effect on the measure of the symmetry generators and constraints that transform as spin 1 of $\SU(2)_\gl$.
The analysis can be simplified by using the residual symmetries.   After putting $Q_I$ in the canonical form (<ref>) with a diagonal matrix
$M={\rm diag}(\lambda_1,\cdots,\lambda_L)$,  there is  a residual $\uU(1)^L$ symmetry of diagonal matrices that commute with $M$.
A cancellation between symmetries and constraints occurs separately for each $\uU(1)^L$ representation.   Symmetries and perturbations of $L$ that are $\uU(1)$-invariant
preserve the diagonal structure of $Q_I$, and do not “mix” with the off-diagonal symmetries and perturbations.   For each pair of distinct eigenvalues $\lambda_i,\lambda_j$, the $ij$ matrix
elements of a symmetry generator or a perturbation of the $Q_I$ transform in a $\uU(1)^L$ representation that is different for each pair $i,j$.   This implies that one can
analyze the $ij$ matrix elements for given $i,j$ independently of the others.

To understand the cancellation that occurs for diagonal matrix elements, we do not really need to do a calculation.   Diagonal symmetry generators and perturbations of the $Q_I$
preserve the diagonal structure, and as long as this diagonal structure is preserved, the different diagonal matrix elements do not influence each other.
Therefore, to study a diagonal matrix element, it suffices to take $L=1$.
To understand why a cancellation must occur in this case, all we need to know is that for $L=1$,  the $Q_I$ are proportional to $\lambda$, the canonical form being
$Q_I=\lambda\gamma_I$.   This means that the vector fields corresponding to the generators of $K$ that have spin 1 under $\SU(2)_\gl$ are all proportional to $\lambda$,
so each of these generators produces a factor $\lambda$ in the measure.   On the other hand, for any first order variation $\delta Q_I$ of the $Q_I$, the constraints
give delta functions that assert the vanishing of the $(1,1)$ part of $\{\delta Q_I,Q_J\}+\{Q_I,\delta Q_J\}$.   This quantity is linear in the $Q_I$, so it is linear in $\lambda$.
So the constraints give delta functions of the general form $\delta(\lambda f)$, where $f$ is independent of $\lambda$.   As $\delta(\lambda f)=\frac{1}{\lambda}\delta(f)$,
each constraint gives a factor $1/\lambda$, canceling the factors that come from the symmetry generators.

To study off-diagonal symmetries and perturbations, it suffices to consider the case of $L=2$ with $M={\rm diag}(\lambda_1,\lambda_2)$.
The generators of $\SO(4)$ acting on $\H_0=\S'_{1/2}\otimes \S_{1/2}$ are the antihermitian matrices $\sigma_{IJ}=\frac{1}{2}[\gamma_I,\gamma_J]$.   With an appropriate
orientation convention, the matrices on $\H_0$ that transform as spin 1 of $\SU(2)_\gl$ – that is, the $\SU(2)_\gl$ generators – are $\zeta^{IJ}\sigma_{IJ}$, where $\zeta_{IJ}=\frac{1}{2}
\epsilon_{IJKL}\zeta^{KL}$ is self-dual ($\epsilon_{IJKL}$ is the antisymmetric tensor with $\epsilon_{1234}=1$).    Thus, with
Q_I=γ_I⊗ λ_1     0 0     λ_2, we can consider the off-diagonal symmetry generators
r(ζ) = ζ^IJσ_IJ ⊗0     10     0-h.c. The induced variation of $Q_I$ is
δQ_I = [r(ζ),Q_I]= -2ζ_IJγ^J  0     λ_1+λ_20     0  +2ζ_IJγ^Jγ_5
0     λ_1-λ_20     0 +h.c. In the norm squared of $\delta Q_I$ (namely $\Tr\,\delta Q_I^2$), there is no cross term between the parts of $\delta Q_I$ with and without a factor of $\gamma_5$.
So the norm squared is a multiple of
$$(\zeta^{IJ}\bar\zeta_{IJ})\left( (\lambda_1-\lambda_2)^2+(\lambda_1+\lambda_2)^2\right)=2\zeta^{IJ}\bar\zeta_{IJ} (\lambda_1^2+\lambda_2^2).$$
each of the six independent real components of $\zeta_{IJ}$ contributes a factor $(\lambda_1^2+\lambda_2^2)^{1/2}$ to the measure, giving an overall factor $(\lambda_1^2+\lambda_2^2)^3$.

However, the corresponding matrix elements of the constraints cancel this factor.   A normalized off-diagonal perturbation $\delta' Q_I$ in the same $2\times 2$ block that
is orthogonal to $\delta Q_I$ is
δ' Q_I=1/√(λ_1^2+λ_2^2) (ζ_IJγ^J 0    λ_1-λ_20   0 +ζ_IJγ^Jγ_5  0     λ_1+λ_20   0 )
+ h.c.. We then find that
{Q_I,δ' Q_J}+{Q_J,δ' Q_I}=  2(σ_IKζ_JK+σ_JKζ_IK) 0     √(λ_1^2+λ_2^2)0     0 +h.c. For each independent real component of $\zeta_{IJ}$, we impose the vanishing of $\{Q_I,\delta' Q_J\}+\{Q_J,\delta' Q_I\}$ as a constraint.   In each case, the delta
function that imposes this constraint has an argument that is a multiple of $\sqrt{\lambda_1^2+\lambda_2^2}$.   The resulting factor in the measure
is $1/(\lambda_1^2+\lambda_2^2)^3$, canceling the measure factor that came from the corresponding symmetry generators.

We thank M. Heydeman and D. Stanford for discussion. GJT is supported by the Institute for Advanced Study and the NSF under Grant  PHY-2207584, and by the Sivian Fund. Research of EW supported in part by NSF Grant PHY-2207584.

§ THE DISK AND THE TRUMPET

In this appendix we explain the derivation of the disk and trumpet partition functions for $\cN=2$ and $\cN=4$ JT supergravity.

In both cases, there are subtleties concerning the overall normalization, which depend on the normalization that one assumes for zero-mode wavefunctions.

In the case of the disk, the overall normalization is inessential, in the sense that a change in this normalization is equivalent to a shift in the ground state
entropy $S_0$.   However, if one does change this normalization, one must compensate by renormalizing the path integral of a three-holed sphere by an inverse factor.

In the case of the trumpet, since the trumpet has an inner boundary and is always glued to something else,  it is not meaningful to specify the normalization
of the partition function without also specifying the measure that will be used in gluing.   In the body of this paper, we have normalized the gluing measure as $\d b\d \varrho \d\phi\d\varphi$.

In practice, we will use normalizations that have been justified in related problems by comparing to the symplectic form of the bulk moduli space in bosonic JT gravity [6]
or by comparing a localization procedure to canonical quantization in some other cases, for example in [44].    However, the most conclusive justification for the
normalizations that we will assume is that with these normalizations, $\cN=2$ JT supergravity agrees with matrix model predictions, as found in the body of the
article.[It would presumably be possible to get a more complete understanding of the normalizations
by computing the precise relationship between the symplectic measure and the torsion for $\N=2$ super Riemann surfaces.  It is expected
that they are equivalent, in the sense that the symplectic measure and the torsion differ by a factor $e^{c\chi}$, where $\chi$ is the Euler characteristic of a surface, and $c$ is
a constant.  If this relative normalization were known, some considerations of [6] could be adapted to the $\N=2$ case.]

In all cases, the  path integrals that we will evaluate are one-loop exact, meaning that they can be evaluated by summing over fixed points of a rotation symmetry,
with each fixed point weighted by a Gaussian integral.   A conventional approach to the one-loop calculation around a critical point
is to expand the action, in the present context a Schwarzian action,
to get the appropriate quadratic action, find the measure – a symplectic measure if the localization is justified using the symplectic structure of the phase space [34] –
and then  do a Gaussian integral.   However, the localization procedure implies a shortcut which
we will use repeatedly: the output of the Gaussian integral is $1/\sqrt{\det' D}$ (or the inverse of this for fermions), where $D$ is the operator that generates the rotation symmetry
of the fixed point, and the notation $\det'$ means that modes of $D$ that can be generated by symmetries of the disk or trumpet should be discarded.   We note that the modes
that are discarded are zero-modes in the sense that they do not appear in the action or in the symplectic form, but they are in general not zero-modes of $D$.  So in general a few
eigenvalues of  $D$ have to be omitted by hand.  We call the eigenvalues of $D$ “rotation eigenvalues.”

§.§ $\cN=2$ JT supergravity

The trumpet We follow the notation of section <ref>. We consider first  the $n=0$ fixed point
y(t,θ) = 2π/β t,    ψ(t,θ)= e^ı(ϕ-α)t/β θ.
The general answer can be obtained by a shift $\alpha \to \alpha+2\pi n$. The action of this fixed point is easy to evaluate. Since the fermions are turned off, we can restrict the $\cN=2$ Schwarzian to its bosonic subsector. It was found in [35, 34, 39, 40] that a reparametrization $(t,\theta) \to (y(t), e^{\i \sigma(t)} \theta)$ on the trumpet has an action
S = - 1/2∫_0^β ṭ [ {tanh(b  y/4π) ,t}- 2 (∂_t σ)^2 ] = b^2/4 β + ^  2/β(α- ϕ)^2.
The operator that generates the rotation symmetry of this fixed point is
D = ∂/∂t - 2π/β ∂/∂y - (ϕ-α)/β  ı(ψ∂_ψ- ψ̅ ∂_ψ̅).
We first consider the reparametrization mode and expand $y(t,\theta) = 2\pi t/\beta + \delta y(t)$ with $\delta y(t+\beta)=\delta y(t)$. The eigenvalue equation $\i D \delta y = \lambda \delta y $ reduces to $\i \partial_t \delta y = \lambda \delta y$; the eigenvalues are $\lambda_m=2\pi m /\beta$ with $m\in \mathbb{Z}$. The trumpet has a bosonic $\uU(1)$ isometry  $y \to y +{\rm const}$ that corresponds to $m=0$ and has to be removed from the spectrum of fluctuations. The contribution of the reparametrization mode to the one-loop determinant of the trumpet is
Z_1-Loop, T^Rep. =∏_m≥14π/λ_m =  ∏_m≥1 2β/m = 1/2√(πβ).
To extract $\sqrt{\det' D}$, we took the product of eigenvalues over $m>0$ only, after discarding the zero-mode at $m=0$.
The final result has been obtained using zeta-function regularization.  In zeta-function regularization, replacing $\lambda_m$ by $\lambda_m/4\pi$ has the effect of
multiplying the determinant by a constant factor.    The normalization we chose is the same as in [6, 14].

Next we consider the rotation eigenvalues coming from the phase mode $\psi \to e^{ \i \delta \sigma(t) } \psi $. On such modes the eigenvalue equation for $D$ becomes $\i \partial_t \delta \sigma = \lambda \delta \sigma  $. The zero-mode of $\delta \sigma$ comes from a $\uU(1)$ symmetry of the trumpet and should be discarded.
The final result is proportional to $ \prod_{m\geq 1} 2\beta/m$. The normalization is a bit subtle. One can work with either an “ultralocal” measure, which gives a result consistent with the canonical quantization of the phase mode, or a symplectic measure appropriate to the BF theory origin of this mode. It was shown in appendix A of [44] that the partition function with the symplectic measure is equal to the partition function with the ultralocal measure divided by the volume of the group, in this case $2\pi$. This leads to the answer
Z_1-Loop, T^(1) =2∏_m≥1 2β/ m = /√(πβ).
The one-loop determinant corresponding to the ultralocal measure is then $\hq\sqrt{4\pi/\beta}$, and we can check this is consistent with canonical quantization. The partition function of the phase mode alone would be, taking $\phi=0$ for simplicity, $\sum_{n\in\mathbb{Z}}  \frac{\hq \sqrt{4\pi}}{\sqrt{\beta}} e^{-\frac{\hq^2(\alpha + 2\pi n)^2}{\beta}}= \sum_{k\in \mathbb{Z}} e^{\i \alpha k} e^{-\beta\frac{ k^2}{4\hq^{\, 2}}}$, where we used Poisson resummation to write the right hand side as a sum over states with different charges and energies.

Next we analyze the rotation eigenvalues of the fermionic fields. We write a generic fermionic perturbation around the fixed point as $\psi \to \psi + \delta \psi$ with $\delta\psi (t) = e^{\i \hq \phi t/\beta} \chi(t)$, and $\chi(t+\beta)=-\chi(t)$. The eigenvalue equation $\i D\delta \psi = \lambda \delta \psi$ then reduces to
( ı∂/∂t -   α/β ) χ= λχand therefore the eigenvalues are $ (2\pi m+\hq\,\alpha)/\beta$ with $m \in \mathbb{Z}+1/2$. Importantly, the answer is independent of the holonomy around the geodesic and depends only on the chemical potential at the NAdS boundary. The regularized version of this determinant is

Z_1-Loop, T^ferm. = ∏_m ∈ℤ+ 1/2 m+ α/2π/2β = 2 cos( α/2 ),
which makes sense  since this is the partition function of a complex fermion of charges $\pm \hq/2$ in quantum mechanics.   In this example, there are no zero-modes,
so there is no subtlety with the normalization.

We can put the three contributions together with the classical action and write the result for the $n=0$ fixed point:
cos(α/2)/πβ e^-b^2/4β-^  2/β(α-ϕ)^2.
Shifting $\alpha \to \alpha + 2\pi n $ gives the final answer presented in (<ref>). Deriving (<ref>) from (<ref>) is straightforward. To do this first rewrite (<ref>) as
Z_tr(β,α; b,ϕ)   =    Z_tr^bos.(β,b) 2cos(α/2) ∑_n∈ℤ    e^2πın(δ+1/2) /√(πβ)e^-^  2/β(α-ϕ+2πn)^2 .
Since $\hq$ is an odd integer, we have substituted $\cos\big(\pi \hq (\frac{\alpha}{2\pi}+n)\big)=e^{\i \pi n} \cos(\frac{\hq\alpha}{2})$, to simplify the $n$-dependence in the sum. Now we can apply Poisson resummation on the final term
∑_n∈ℤ  e^2πın(δ+1/2)   /√(πβ) e^-^  2/β(α-ϕ+2πn)^2 = 1/2π∑_q ∈ℤ + δ-1/2  e^ıαq e^-ıϕq e^-βq^2/4^ 2
This immediately leads to (<ref>) after multiplication by  $2\cos(\frac{\hq\alpha}{2})$.

The disk Next we consider the evaluation of the disk partition function. The calculation is very similar to the trumpet and we only point out the differences. The $n=0$ fixed point is given by $y=2\pi t/\beta$ and $\psi = e^{- \i \hq \alpha t/\beta} \theta$ and the bosonic part of the action is now instead
S = - 1/2∫_0^β ṭ [ {tan(y/2) ,t}+ 2 (∂_t σ)^2 ] = - π^2/β - ^ 2/βα^2.
We analyze next the rotation eigenvalues and start with the reparametrization mode. In the case of the trumpet we restricted to $m\geq1$ since $m=0$ corresponds to the $\uU(1)$ isometry of the geometry. The hyperbolic disk has a full ${\rm SL}(2,\mathbb{R})$ isometry, generated by the modes with $m=0,\pm 1$. This leads to a modified one-loop determinant
Z_1-Loop, D^Rep. = ∏_m≥2 2β/m = 1/4√(π)β^3/2.
The one-loop determinant of the phase mode is unchanged and gives $Z_{\rm 1-Loop,~D}^{\uU(1)} = \hq /\sqrt{\pi \beta}$. Finally the $\N=2$ hyperbolic disk has four real fermionic isometries, corresponding to complex modes with rotation eigenvalues labeled by $m=\pm 1/2$. These are the fermionic generators of $\SU(1,1|1)$. The fermionic one-loop determinant is
Z_1-Loop, D^Ferm.= ∏_m≠±1/2 m+α/2π/2β = 16 β^2/1-^  2 α^2/π^2  2 cos(α/2).
(In this case, we extracted the square root by considering just one of the two fermions, and instead allowed both signs of $m$.)
We can now put the ingredients together. We also multiply the disk partition function by $1/(4\pi)^2$ which we can think of as a shift in the action by a term proportional to the Euler characteristic. The normalization of the disk partition function correlates with the normalization used in section <ref> for the torsion of a three-holed sphere.
With these choices, the final answer is
cos(α/2)/2π^3(1-^  2 α^2/π^2) e^π^2/β(1-^  2 α^2/π^2),
and the disk partition function is obtained by summing over shifts $\alpha \to \alpha + 2\pi n $, reproducing the result in section <ref>. This result was first derived in [34].

We now explain how to obtain the spectrum (<ref>) from the partition function. First, use $2\cos \pi \hq (n+\frac{\alpha}{2\pi})=(-1)^{n}2 \cos(\frac{\hq\alpha}{2})$ to write the partition function as
Z_disk(β,α) = 2cos(α/2) ∑_n∈ℤ   e^2πın(δ+1/2) /4π^3 (1-^  2 (α+2πn)^2/π^2) e^π^2/β(1-^  2 (α+2πn )^2/π^2),
Let us begin by finding the BPS states. These are the states with zero energy and therefore they are the only contribution that survives a zero temperature $\beta \to \infty$ limit. The classical action is proportional to the temperature so in this limit we can evaluate it to zero
Z_disk(β→∞,α) = 2cos(α/2) ∑_n∈ℤ    e^2πın(δ+1/2) /4π^3 (1-^  2 (α+2πn)^2/π^2),
This is not in a useful form, in particular we would like to extract the number of BPS states for each charge $k$. To do this we apply Poisson resummation,

∑_n∈ℤ e^     2πın(δ+1/2)2  cos(α/2) /4π^3 (1-^  2 (α+2πn)^2/π^2) = ∑_k ∈ℤ+δ, |k|</2   e^ıαk cos(πk/)/4π^2,
by taking the Fourier transform of the summand in the left hand side with respect to $n$, using the poles coming from the denominator. We now derive the density  of non-BPS states.  In order to do this it is convenient to take a derivative with respect to $\beta$
-∂_βZ_disk    =     2cos(α/2) ∑_n∈ℤ e^ 2πın(δ+1/2) /4πβ^2 e^π^2/β (1-^  2 (α+2πn )^2/π^2),

=       2cos(α/2) ∑_q∈ℤ+δ-1/2 e^ıαq/2π e^π^2 /β/4 √(π)β^3/2 e^-βE_0(q) ,

=      ∑_q∈ℤ+δ-1/2 e^ıα(q-/2)  + e^ıα(q+/2)/2π  ∫_0^∞ṣ 2 s sinh2πs/4π^2 e^- β(E_0(q)+s^2) ,

=      ∑_q∈ℤ+δ-1/2 e^ıα(q-/2)  + e^ıα(q+/2)/2π  ∫_E_0(q)^∞Ẹ   sinh2π√(E-E_0(q))/4π^2 e^- βE .
In the second line, we apply Poisson resummation to rewrite the sum over fixed points as a sum over charges. Since the derivative with respect to $\beta$ removed the denominator that came from the fermion zero-modes, this Poisson resummation is straightforward.  Motivated by the fact that the temperature dependence in the second line is that of
bosonic JT gravity, in the third line we rewrote this expression as an integral over a parameter $s$.
In the fourth line, we changed integration variables from $s$ to $E=E_0+s^2$.

It is straightforward to integrate the final line in the formula above with respect to $\beta$. The integration produces simply a factor of $1/E$ inside the integral over energies. We conclude
Z_disk = Z_disk(β=∞) + ∑_q∈ℤ+δ-1/2 (e^ıα(q-/2)  + e^ıα(q+/2))∫_E_0(q)^∞Ẹ  sinh2π√(E-E_0(q))/8π^3 e^- βE .
The integral on the right vanishes for $\beta\to\infty$ in all cases; it vanishes exponentially fast if $E_0(q)>0$, and as $1/\sqrt \beta$ if $E_0(q)=0$.
We already computed the integration constant $Z_{\rm disk} (\beta=\infty)$ in (<ref>), since it corresponds to the BPS  contribution. This leads to (<ref>).

Schwarzian Theory from Virasoro We briefly show how to reproduce these results from a canonical quantization approach. We can view the Schwarzian path integral arising from the disk as computing a classical partition function of a system whose phase space is infinite-dimensional and parametrized by $\cN=2$ superconformal transformations modulo $\SU(1,1|1)$. This is a super Virasoro coadjoint orbit, and an extension of [65, 66], see also [41], would imply that the quantization of this phase space is given by the vacuum representation of the $\cN=2$ Virasoro algebra with central charge $c=3 \widehat{c}$. The quantum partition function is the character of this representation. The classical limit of the quantum result, meaning the large $\widehat{c}$ limit of the Virasoro character, reproduces the integral over the classical phase space, i.e. the Schwarzian path integral.

The $\N=2$ Virasoro vacuum character on a torus with modulus $q=e^{2\pi \i \tau}$ and $\uU(1)$ fugacity $y=e^{2\pi \i z}$ is given by [67]
Tr_vac [ q^L_0 y^J_0] = q^-c-1/8 1-q/(1+yq^1/2)(1+y^-1q^1/2) θ_3 (τ, z) /η(τ)^3,    Z_vac(q,y) = e^ıπc /τ(z-1/2)^2 Tr [ q^L_0 y^J_0]
The trace is over the vacuum representation, but to compare with the Schwarzian partition function we need to include an elementary exponential factor  to go from a sum over states to a path integral $Z_{\rm vac}$; see section 5 of [68] for a general discussion on the origin of this factor. Finally, both in integrating over the coadjoint orbit
(to perform the Schwarzian path integral) and in quantizing it, one has to sum over the same fixed points.   In quantizing the orbit, the different fixed points give representations of
the $\N=2$ super Virasoro algebra that
differ by integer spectral flow [67], and summing over them gives a representation of the $\N=2$ super Virasoro group, not just the corresponding Lie algebra.

The torus is a product of two circles. As explained in [41], to relate the coadjoint orbit quantization on the torus  in the large $\widehat{c}$ limit to the Schwarzian path integral, we need to send the size of one of the circles to zero so that the modular parameter $\tau$ vanishes as $\tau \sim 1/\widehat{c}$. In this limit, the Schwarzian mode lives on the circle that remains of finite size. More precisely, the relation between the 2d CFT parameters and the Schwarzian ones is
τ=ı2π/β 2/c-1,  and  z= 1/2+ α/2π τ,     c →∞.
The half-integer shift in $z$ appears so that, for $\alpha=0$, the fermions are periodic around the circle which disappears in the Schwarzian limit. Otherwise, there would be no fermions in the remaining large circle where the Schwarzian mode lives, and one would reproduce instead the partition function of a bosonic theory.

The  large $\widehat{c}$ limit of the following quantities will be useful:
1-q →2 πıτ,  1+y^±1 q^1/2 →-ıπτ( 1± α/π),  θ_3(τ,z)/η(τ)^3 →(-ıτ) 2 cos( α/2).
Using these relations, the semiclassical limit of the $\N=2$ Virasoro vacuum character is
e^ıπc /τ (z-1/2)^2q^-c-1/8 1-q/(1+yq^1/2)(1+y^-1q^1/2) θ_3 (τ, z) /η(τ)^3 →4/π cos( α/2)/1-^ 2 α^2/π^2  e^π^2/β(1-^  2α^2/π^2).
This reproduces the disk partition function for the $n=0$ fixed point. The sum over $n$ in the Schwarzian limit corresponds to the integral spectral flow of [67] ensuring that the $\uU(1)$ symmetry is compact.

We can repeat this analysis for the trumpet. This case corresponds to a different coadjoint orbit of the $\N=2$ Virasoro group, where we only mod out by $\uU(1)\times \uU(1)$ generated by the $R$-symmetry and time translations. The Hilbert space that quantizes this phase space is identified with a general non-degenerate representation of the Virasoro group labeled by scaling dimension $h$ and $\uU(1)$ charge ${\sf q}$. These parameters correspond to the length of the geodesic and the $R$-charge holonomy of the Schwarzian trumpet, through the relations
h = c-1/8 + c-1/8 b^2/4π^2+q^2/2(c-1) ,   q = c -1/2π   ϕ.
These identifications are the same one obtains when considering the bosonic Virasoro times Kac-Moody group. The relevant super Virasoro character is
Tr_(h,q) [ q^L_0 y^J_0] = q^h-c-1/8y^q   θ_3 (τ, z) /η(τ)^3
The Schwarzian limit of the path integral version gives
e^ıπc/τ (z-1/2)^2q^h-c-1/8y^q   θ_3 (τ, z) /η(τ)^3 → 2 cos(α/2)/β e^-b^2/4β -^  2/β(α- ϕ)^2.
This again reproduces precisely the answer coming from localization, after summing over integral spectral flow.

Supersymmetric Defect We finish by considering the path integral of the boundary graviton on the hyperbolic disk with a defect. If the deficit angle is bigger than $\pi$, we can obtain the partition function by analytically continuing the trumpet by $(b,\phi) \to (\i \gamma, \phi)$. The opening angle around the defect is $\gamma$ in these conventions, and
$e^{\i \hq\,\phi}$ is the holonomy around the defect. When $\gamma$ and $\phi$ are arbitrary, the answer is the same as the trumpet.

When $\gamma = 2\pi\pm 2\hq \phi$ the defect is supersymmetric. In terms of the connection with Virasoro, these defects correspond to chiral primary fields with scaling dimension related to the charge as $h=|{\sf q}|/2$. Picking ${\sf q}$ positive leads to $\gamma = 2\pi -2 \hq \phi$ while negative ${\sf q}$ leads to $\gamma = 2\pi + 2\hq \phi$.

The classical action is a smooth function of $\gamma,\phi$, and nothing special happens when the defect becomes supersymmetric.  The same is true for the rotation eigenvalues,
but one fermionic eigenvalue, with  $m=+1/2$ or $-1/2$ depending on the sign appearing in $\gamma$, is now associated to a symmetry of the defect geometry and should be
discarded. Something similar was analyzed for $\N=4$ JT gravity defects in [69]. Taking this into account,  the final answer for the partition function is
Z_susy-defect(β,α)= w∑_n∈ℤ e^2πi n(δ+1/2) 2  cos(α/2) /4π^3 (1∓(α+2πn)/π) e^1/4β(2π±ϕ)^2- ^  2/β(α-ϕ+ 2πn)^2.
We added a weighting factor $w$ that comes with each defect insertion.

§.§ $\cN=4$ JT supergravity

The previous analysis can be repeated for $\N=4$ JT gravity, with some modifications due to the $\SU(2)$ $R$-symmetry. We begin again with the trumpet. We refer to the $\SU(2)$ holonomy around the geodesic as $U$. In a $2\times 2$ representation, this $\SU(2)$ matrix can be diagonalized to $e^{\i \phi Z/2}$, where $Z={\rm diag}(1,-1)$. We can parametrize the $\N=4$ cylinder with coordinates $(x,\psi,\bar{\psi})$, with $\psi=(\psi^1,\psi^2)$ in the fundamental of $\SU(2)$.  The choice of holonomy translates into the identification
(x,ψ,ψ̅ ) ≅(x+2π, - U ψ, - U^-1 ψ̅).
Similar to the conventions of section <ref>, we take the geodesic boundary at ${\rm Im}\,x=0$ and the NAdS boundary to be at $x=\i T + y$ with real $y$. The geometry of the trumpet then induces the following identification on the boundary:
(y,ψ,ψ̅ ) ≅(y+2π, - U ψ, - U^-1 ψ̅).
The boundary quantum mechanics is formulated on a supercircle with coordinates $(t,\theta,\bar{\theta})$, where again $\theta$ is in the fundamental representation. The equivalence relation that corresponds to computing the partition function with fugacity $V\in \SU(2)$ is
(t, θ,θ̅) ≅(t+β, -V θ, - V^-1 θ̅).
Following the notation in the main text, we  label $V$ by its eigenvalues $e^{\i \alpha Z/2}$. Fixed points take the general form
y(t,θ)= 2π/β t,    ψ(t,θ) = g(t) θ,    g:S^1 →(2).
For this to satisfy the appropriate periodicity conditions, $g(t)$ must satisfy
g(t+β) = U g(t) V^-1.
$U$ and $V$ can be diagonalized by separate unitary transformations of $\psi$ and $\theta$.   After doing so, the fixed point can be written, in analogy to the $\N=2$ case, as

ψ^1(t,θ) = e^ı(ϕ-α) t/2 β θ^1,   ψ^2(t,θ) = e^-ı(ϕ-α) t/2 β θ^2,
where we wrote $\psi$ and $\theta$ in components to be more explicit.

As in $\N=2$ we have a family of fixed points obtained by a shift $\alpha \to \alpha + 4 \pi n$ with $n\in \mathbb{Z}$.   Because the diagonalization of $U$ or $V$ is unique
only up to the action of the Weyl group, which exchanges the eigenvalues, there is also a second family of fixed points obtained by $\alpha\to-\alpha$.
For concreteness we focus first on the choice in (<ref>).

The classical action is easy to obtain. In the basis that diagonalizes $U$ and $V$, the bosonic sector of the Schwarzian action is essentially the same as for $\N=2$.  Taking account of the normalization used in defining $\alpha$ and $\phi$, the action is
S = b^2/4β + 1/4β(α-ϕ)^2.
Next, we need to compute the  eigenvalues of the generator
D = ∂/∂t - 2π/β ∂/∂y - ı(ϕ-α)/β (ψ^1 ∂_ψ^1 -ψ^2 ∂_ψ^2 -ψ̅^1 ∂_ψ̅^1 + ψ̅^2 ∂_ψ̅^2)
of the rotation symmetry of the fixed point.
The eigenvalues for  the bosonic reparametrization mode are the same as $\N=2$ or $\N=0$, so  again $Z_{\rm 1-Loop,~T}^{\rm Rep.} = \frac{1}{2\sqrt{\pi \beta}}$. The fermionic rotation eigenvalues are also straightforward: we essentially have two copies of the fermions that we had for $\N=2$, leading to
Z_1-Loop, T^Ferm. = 4 cos^2(α/4).
What remain are the modes associated to time-dependent $\SU(2)$ transformations. The fluctuations in the Cartan of $\SU(2)$ are again precisely the same as the phase mode of $\N=2$. The rotation eigenvalues for off-diagonal generators are
λ_m,± =±2πm+α/β,1cm m∈.This leads, with zeta-function regularization, to $\sqrt{\det D}=\sin\frac{\alpha}{2}$.
The overall normalization can be fixed
using the relation, described in appendix A of  [44], between the one-loop determinant derived from the symplectic measure, appropriate for this calculation, and the one-loop determinant derived from the ultralocal measure which is equivalent to canonical quantization. The final result is
Z_1-Loop, T^(2)
=1/8 √(πβ) 1/sin(α/2).
The path integral around one fixed point is therefore
(α/4)/4πβ e^-b^2/4β - (α-ϕ)^2/4β.
To obtain the final answer given in (<ref>) we need to sum over $\alpha \to \alpha + 4\pi \mathbb{Z}$ and add another contribution with $\alpha \to -\alpha$. Since $\cot(-\frac{\alpha}{4}) = -\cot(\frac{\alpha}{4})$ there is a relative minus sign between the two.

We now explain how to re-express the  result  (<ref>)  in terms of a sum over $\N=4$ supermultiplets. We begin with an identity derived from Poisson resummation:

∑_n∈ℤ  √(π)/2√(β)sin(α/2) ( e^-1/4β(α-ϕ+4πn)^2-e^-1/4β(α+ϕ+4πn)^2) = ∑_J≥1/2 χ_J-1/2(α) sin(Jϕ) e^-βJ^2 .
Here $\chi_J(\alpha)=\frac{\sin((J+1/2)\alpha)}{\sin(\alpha/2)}$ is the character of the spin $J$ representation of $\SU(2)$.
To get the $\N=4$ trumpet path integral, the expression in eqn. (<ref>) has to be multiplied by the bosonic JT gravity trumpet, and also by a factor of $4 \cos^2 \alpha/2$. The latter turns the $\SU(2)$ character into a sum over $\N=4$ multiplets, since
4cos^2(α/4)  χ_J-1/2(α) = χ_J(α) + 2 χ_J-1/2(α) + χ_J-1(α).
Putting all this together, we arrive at (<ref>).

Next we consider the disk path integral and point out the main differences with the trumpet. The fixed points and action are the same as the trumpet but with $\phi=0$. A new feature is that we do not need to sum over the action of the Weyl group and there is a unique family of saddles related by $\alpha \to \alpha + 4\pi \mathbb{Z}$. The bosonic reparametrization one-loop determinant is the same as, for example, the $\N=2$ disk computed earlier in this section. The fermionic one-loop determinant is essentially two copies of the $\N=2$ disk one, so it is proportional to
4 cos^2 (α/4)/(1-α^2/4π^2)^2.
The denominator comes from the eight fermionic isometries of the $\N=4$ disk which make up the supergroup $\PSU(1,1|2)$. We have two complex fermions each with frequency $+1/2$ and $-1/2$ in time. Finally, for the $\SU(2)$ mode, the rotation eigenvalues are the same as before, but the $m=0$ modes in eqn. (<ref>) are now generated by a symmetry of the bulk
and so should be omitted.  This multiplies $\sqrt{\det'\,D}$ by $\beta/\alpha$, so $1/\sqrt{\det' D}$ is now  $\alpha/\beta^{3/2}\sin\frac{\alpha}{2}$, where the overall coefficient
can be determined from [70].  Putting the pieces together, we obtain
β(α/4) α/π^4(1-α^2/4π^2)^2 e^π^2/β(1-α^2/4π^2)
A sum over shifts of $\alpha$ gives the result (<ref>) for the disk partition function. We also choose the normalization of $S_0$ to match the normalization in (<ref>).

The evaluation of the density of states works similarly to the procedure we outlined for $\N=2$ JT gravity. First we determine the BPS states. To do this, we take $\beta \to \infty$. The new feature is the factor of $\beta$ remaining in the one-loop determinant. This appears to give a divergent answer, but the coefficient vanishes
∑_n∈ℤ β(α+ 4πn)/(1-1/4π^2(α+4πn)^2)^2 = 0.
This implies we should not set $1/\beta=0$ in the action and instead Taylor expand in $1/\beta$. The linear term cancels the factor of $\beta$ in the one-loop determinant and now the prefactor is
Z(β→∞)  →e^S_0 ∑_n∈ℤ (α/4)(α+4πn)/π^2 (1-(α+4πn)^2/4π^2) = e^S_0.
Since this is independent of $\alpha$, there are BPS states only at $J=0$.
Once we have the number of BPS states, we can use the same trick as before. We take two derivatives $\partial^2_\beta  Z_{\rm disk}$. The two derivatives acting on the classical term will cancel the denominator from the fermion zero-modes. After that we can easily apply Poisson resummation, as  for the trumpet. Finally we integrate over $\beta$ twice to get back $Z_{\rm disk}$ and use the BPS states as boundary condition at $1/\beta=0$.

One can reproduce these results again from taking a large $c$ limit of $\N=4$ Virasoro characters (the vacuum for the disk, generic for the trumpet). In this case, the relevant torus partition functions were computed by Eguchi and Taormina in [71, 72, 73, 74].

§ REVISED LOOP EQUATIONS AND TOPOLOGICAL RECURSION

In this section we analyze the loop equations of the AZ $(\upalpha,\upbeta)$ ensembles, following the presentation in [14]. We focus on the new features that appear relative
to bosonic JT gravity and $\N=1$ JT supergravity.

A first new feature is that in $\N=2$ and $\N=4$ JT supergravity, the number of BPS states, extracted from the disk partition function, is proportional to $e^{S_0}$. In contrast, in  $\cN=1$ JT supergravity, the number of supersymmetric  states does not scale with $e^{S_0}$ and vanishes in
the absence  of Ramond punctures. A second and related new feature is that we need to allow a logarithmic singularity in the matrix potential. We will see that even though the two effects are physically distinct, they can be analyzed together.

We have already seen in sections <ref> and <ref> that different multiplets are statistically independent. Therefore we can focus on any one of them separately.
Consider a single supermultiplet of charges $(k,k+\hq\,)$ with $L$ nonzero matrix eigenvalues $\lambda_i$ and  $\nu$ BPS states.   In the case of an $(\upalpha,\upbeta)$ AZ ensemble, the
measure in integration over the $\lambda_i$ is

∏_iλ̣_i∏_i<j |λ_i^2 - λ_j^2|^   ∏_i=1^L λ_i^  e^- L /2 V(λ_i^2).
To maintain the standard structure of topological recursion in matrix integrals, $V$ should be independent of[We have actually had to relax this condition
in section <ref> in the discussion of $\CT$-invariant models, but it is a useful condition in the rest of the article.]   $L$.   From   eqn. (<ref>), we see immediately that a shift in $\upalpha$ by a multiple
of $L$ can be absorbed in an $L$-independent logarithmic correction to $V$.   In sections <ref> and <ref>, we have found in all cases that
$\upalpha=\upalpha_0+\upbeta\nu$, where $\upalpha_0$ is the value of $\upalpha$ in an ensemble of  some given type in the absence of BPS states, and $\nu$ is the number of
BPS states.   Moreover, in $\N=2$ and $\N=4$ JT supergravity, $\nu$ is always a multiple of $e^{S_0}$ and thus, in the matrix model, it is a multiple of $L$.
So the $\upbeta\nu$ term in $\upalpha=\upalpha_0+\upbeta\nu$ can be eliminated by adding to $V(x)$ an $L$-independent term $\upgamma \log x$, $\upgamma=\nu/L$.
The upshot is that we can reduce to the case of only considering
$(\upalpha_0,\upbeta)$ ensembles  if we allow the matrix potential to have a logarithmic singularity at $x=0$:
V(x) = logx + Ṽ(x),where $\tilde{V}(x)$ is an analytic function.[For  discussion of matrix models with such logarithmic potentials, see [75, 76, 77, 78]. ]
One of the reasons that this makes sense in our application is that in all cases except the multiplet of charges $(-\h q/2,\h q/2)$, the threshold energy $E_0$ is positive and (at least
to all orders in the topological expansion) the support of
the matrix eigenvalues does not reach the logarithmic singularity at $x=0$; on the other hand, the $(-\h q/2,\h q/2)$ multiplet has no BPS states and can be described by an analytic

As usual, we define the resolvent
R(x) = ∑_j=1^L 1/x-λ_i^2,    R(x_1,…,x_n) = R(x_1)…R(x_n).
The sum runs over the generically nonzero matrix eigenvalues, not the BPS states.
The loop equations, introduced in [79] and put in their optimal form in [80], are derived from the identity
( (1-2/)∂_x + _0-1/x - L V'(x) ) ⟨R(x, I )⟩_c + ⟨R(x,x,I) ⟩_c+ ∑_J  ⟨R(x,J) ⟩_c ⟨R(x,I/J) ⟩_c       + 2/x ∑_k=1^n ∂_x_k x⟨R(x,I/x_k)⟩_c - x_k ⟨R(I)⟩_c/x-x_k =-L/x⟨P(x;I) ⟩.
See section 4 of [14] for a derivation. We follow here the notation of that reference. We denote $I=\{ x_1,\ldots,x_n\}$. The quantity appearing on the right hand side is
P(x;I) = ∑_a=1^L xV'(x)-λ_a^2 V'(λ_a^2)/x-λ_a^2  R(I).
The connected resolvent has an 't Hooft expansion at large $L$ given by
⟨R(I) ⟩_c ≅∑_g=0,1/2,1,…  R_g,n(I)/L^2g+n-2,
and inserting this expansion in (<ref>) gives a recursion relation for the genus $g$ correlators $R_{g,n}(I)$. The derivation of the loop equations relies on the fact that the function $P(x;I)$ is analytic in $x$. This is obvious if the potential is analytic but  remains true in the presence of a logarithmic term. To see this, notice that $x \partial_x( \log x) = 1$ and therefore the two terms in the numerator coming from the logarithm cancel.

We start with the order $L^2$ term in the loop equation. It is given by
R_0,1(x)^2 - V'(x) R_0,1(x) = 1/x( analytic in x).
Introducing $y(x) = R_{0,1}(x) - V'(x)/2$, this equation can be rewritten as
y(x)^2 = ^2/4x^2 + 1/x(analytic in x),
an equation that defines the spectral curve – a double cover of the $x$-plane.
Thus  a logarithmic term in the potential (or  a number of BPS states proportional to $L$) leads to a pole in $y(x)$ at $x=0$.  The sign of the residue of the pole is fixed by demanding that the resolvent $R_{0,1}(x)$ does not have a pole, since this would imply an unprotected exact zero-energy supermultiplet, which would be unnatural. Therefore we always have opposite poles in $V'(x)$ and $y(x)$.

In a single cut AZ matrix model
with a possible logarithmic term in the potential, apart from a possible pole at $x=0$, $y$ has a cut on an interval $(a_-,a_+)$, $0\leq a_-<a_+$.
In our applications to[In $\N=4$ JT supergravity, we have not attempted to incorporate $\tT$ or $\CT$ symmetry, so we  have only encountered $(1,2)$ ensembles, always
with $a_->0$ and with BPS states only at $J=0$.] $\N=2$ JT supergravity,   $(a_-,a_+)=(E_0(q),\infty)$, with $E_0(q)$ the threshold energy of the given multiplet with $R$-charges $q\pm \hq/2$.
The only case with $E_0(q)=0$ is the multiplet with charges $\pm \hq/2$.   In this case, there are no BPS states and no pole in $y$, so this multiplet is described by  the standard $(1,2)$ ensemble considered in [14] with square matrices and a regular potential.
In the rest of this discussion we consider the other cases, which all have  $a_-=E_0(q) > 0$.
When $|q|< \hq/2$, the pole in $y_q(x)$ matches with the number of BPS states at charge $q-\hq/2$ or $q+\hq/2$, consistent with an interpretation in terms of rectangular matrices and no
logarithmic term in the potential.
When $|q|>\hq/2$ there are no BPS states and therefore the supercharge is a square matrix, but the pole remains in $y_q(x)$.
It is interpreted in terms of a logarithmic term in the matrix potential for that multiplet.

We consider now $I=\{x_1\}$ and $g=0$. The loop equation to order $L$ becomes
2 y(x) R_0,2(x,x_1) + 2/ R_0,1(x)/(x-x_1)^2 = 1/x(analytic in x).
On the right hand side, we can replace $R_{0,1}(x)$ by $y(x)$, since the difference, proportional to $V'(x)$ times a function analytic in the neighborhood of the cut, has the same form as the right hand side. Now we can multiply this equation by $\sqrt{\sigma(x)}/(x y(x))$, where $\sigma(x)=(x-a_-)(x-a_+)$. This ratio is analytic in $x$ in a neighborhood of the cut,
since the  branch cuts in $y(x)$ at $a_\pm$ are removed by the numerator.
It has
no pole at $x=0$, since $y$ has a pole there. Following the derivation in section 4 of [14], this leads to
R_0,2(x_1,x_2) = 1/(x_1-x_2)^2 ( x_1 x_2 - a_+ + a_-/2 (x_1+x_2) + a_+ a_-/√(σ(x_1) ) √(σ(x_2)) -1).
This is the same as the answer for a Dyson $\upbeta$-ensemble, or for an AZ ensemble with either an analytic potential or $\nu$ of order one.  A possible
logarithmic term in the potential has not affected $R_{0,2}(x_1,x_2)$.

We can now move on and look at the loop equation with $I=\emptyset$ to order $L$. In this case we get
( (1-2/) ∂_x + _0-1/x ) R_0,1 (x) + 2 y(x) R_1/2,1(x) = 1/x(analytic in x).
In section <ref> we referred to $R_{\frac{1}{2},1}(x)$ as $R_{\rm xcap}(x)$. So far the answer is the same as in [14]. It is convenient, especially in the double scaling limit, to rewrite the loop equations in terms of $y(x)$ only. At this point we get new terms
( (1-2/) ∂_x + _0-1/x ) y (x) + 2 y(x) R_1/2,1(x) =- ( (1-2/) ∂_x + _0-1/x ) V' (x)/2+ 1/x(analytic in x).
For an analytic potential, the  terms on the right hand side can be combined as an analytic function divided by $x$, leading to the derivation in [14].
Allowing for a logarithmic term in the potential, the general result is
( (1-2/) ∂_x + _0-1/x ) y (x) + 2 y(x) R_1/2,1(x) =(-_0-1)/2 x^2+ 1/x(analytic in x).
For generic $\upalpha_0,\upbeta$, and $\upgamma$, the $1/x^2$ term on the right hand side of this equation will modify the analysis.   However,
in all cases encountered in the present article, either $\upgamma=0$ or $\upbeta-\upalpha_0-1=0$.
Specifically, we encounter $(1,2)$, $(0,1)$, and $(3,4)$ ensembles with various values of $\upgamma$, and $(1,1)$ and $(1,4)$ ensembles with $\upgamma=0$.
Given this, the double pole drops out and  a simple exercise in complex analysis and integration similar to the analysis in [14] gives the crosscap prediction
R_1/2,1(x) = ∫_𝒞 x̣'/2πı 1/x'-x √(σ(x'))/√(σ(x)) ( (1-2/)y'(x)/2 y(x') + _0-1/2x').
The contour $\mathcal{C}$ surrounds the cut of the spectral curve counterclockwise in the $x$ plane. We use this expression in section <ref> to match with the gravity result for the crosscap.

We are ready to consider the most general case for the loop equations of the AZ ensemble with either $\nu$ proportional to $L$ or a logarithmic term in the potential. We expand (<ref>) to order $L^{2-2g-n}$. The result is
2 x y(x) R_g,n+1(x,I) + x F_g,n(x,I) = (analytic   in   x),
F_g,n(x,I)    =    ((1-2/)∂_x + _0-1/x )R_g-1/2,n+1(x,I)        + R_g-1,n+2(x,x,I) + ∑_stable R_h,|J|+1 (x,J) R_g-h,|I/J|+1(x,I/J)      +2 ∑_k=1^n ( R_0,2(x,x_k) + 1/ 1/(x-x_k)^2 ) R_g,n(x,I/x_k).
The recursion relation one derives from this is
R_g,n+1(x,I) = ∫_𝒞 x̣'/2πı 1/x'-x √(σ(x'))/√(σ(x)) F_g,n(x',I)/2 y(x').
This is exactly the same as the one derived in [14]. The only new feature is that now $y(x)$ is allowed to have a pole at $x=0$.   For $(\upalpha_0,\upbeta)=(1,2)$, the recursion relation becomes identical to the one of the Dyson $\upbeta=2$ ensemble. We used this fact in section <ref>.    When $V'(x)$ has a pole at $x=0$ (or $\nu\propto L$), getting a recursion relation of the
same form as if $V'(x)$ is regular and there are no BPS states has
depended on the condition
$\upbeta-\upalpha_0-1=0$, which made it possible at an initial stage of the expansion to replace $R_{0,1}(x)$ with $y(x)$ without disturbing the structure of the topological recursion.
Conveniently this condition is satisfied in our applications.

In conclusion,
allowing for a simple pole in $V'(x)$ or a number of BPS states of order $e^{S_0}$ has very little effect  on the structure of the topological recursion,
at least if $\upbeta-\upalpha_0-1=0$.  What happens if we have instead a double pole in $V'(x)$? In this case the analysis is modified. To begin with, the function $P(x;I)$ would
not be analytic in $x$ anymore: it would itself have a pole. This would potentially have an effect at all orders in the genus expansion. It is therefore a nice feature that a potential
double pole in the spectral curve of $\cN=4$ JT gravity, which
appears if the formula for $y_J(x)$ is analytically continued away from integer values of  $2J$, actually vanishes when $2J$ is an integer. This is of course the only
relevant case to consider since $J$ is the spin of a representation of $\SU(2)$.

§ RELATION TO MIRZAKHANI'S TOPOLOGICAL RECURSION

Finally, in the rest of this appendix we show that the loop equations (<ref>) imply the topological recursion for the matrix model volumes written in (<ref>). This has been done before for the specific spectral curve associated to bosonic JT gravity [57] and also $\N=1$ JT supergravity [14]. Here we generalize this derivation to an arbitrary spectral curve. This helps streamline the proof, given in section <ref>, that the $\N=2$ volumes introduced in this paper satisfy the matrix model loop equations, but might be also of use in other applications.

Let us begin by rewriting the loop equations in the following way. It is convenient, instead of working with the resolvent argument $x$, to introduce a variable $z$ such that $x=E_0-z^2$, where $E_0$ is the end-point of the density of states. Then we can rewrite (<ref>) as an integral over $z$
R_g,n+1(E_0-z^2,I) =- ∫_ε+ ıℝ ẓ'/2πı z'^2/z'^2-z^2  F_g,n(E_0-z'^2,I)/ z y(E_0-z'^2).
We remind the reader that the $x'$ contour of integration was around the branch-cut of $y(x)$ in the counterclockwise direction. This corresponds, in the $z$ plane, to a contour along the imaginary axis with a small positive real part $\varepsilon>0$ as indicated in eqn. (<ref>). In the case of random matrix ensembles without time reversal the kernel $F_{g,n}(x,I)$ simplifies

(_0,)=(1,2):   F_g,n(x,I)    =    R_g-1,n+2(x,x,I) + ∑_stable R_h,|J|+1 (x,J) R_g-h,|I/J|+1(x,I/J)      +2 ∑_k=1^n ( R_0,2(x,x_k) + 1/2 1/(x-x_k)^2 ) R_g,n(x,I/x_k).
The volumes computed through the matrix model observables are defined by the integral transform

V_g,n,M(B) = ∫_ε+ ıℝ R_g,n(E_0-z_1^2,…,E_0-z_n^2) ∏_j=1^n ẓ_j/2πı -2z_j/b_je^b_j z_j.
This relation is the same as (<ref>), but defined in terms of resolvents instead. The inverse relation will also be needed

R_g,n(E_0-z_1^2,…, E_0-z_n^2) = ∫_0^∞V_g,n,M(B) ∏_j=1^n b_j ḅ_j e^-b_j z_j/-2z_j.
We are now ready to derive (<ref>). The procedure is the following. Begin by writing the volume $V_{g,n+1,M}(b,B)$ as an integral transform of $R_{g,n+1}(x,I)$ using eqn. (<ref>). Then use the loop equations to replace $R_{g,n+1}(x,I)$ in the right hand side in terms of other resolvents using (<ref>) and (<ref>). Next, we can write the resolvents on the right hand side of the loop equations as integrals over the volumes using eqn. (<ref>). We focus on the first term in eqn. (<ref>) first. We obtain
bV_g,n+1,M(b,B)    ⊃   b ∫_ε'+ıℝ ẓ/2πı 2z/b e^b z ∫_ε+ıℝ ẓ'/2πı z'^2/z'^2-z^2 1/z y(E_0-z'^2)

×∫_0^∞b' ḅ' ∫_0^∞b” ḅ” V_g-1,n+2,M(b',b”,B) e^-b'z'/2z' e^-b” z'/2z'

⊃    1/2 ∫_0^∞b' ḅ' ∫_0^∞b” ḅ” V_g-1,n+2,M(b',b”,B)   ∫_ε+ıℝ ẓ'/2πı  e^-(b'+b”)z'/y(E_0-z'^2)∫_ε'+ıℝ ẓ/2πıe^b z/z'^2-z^2

⊃    1/2 ∫_0^∞b' ḅ' ∫_0^∞b” ḅ” V_g-1,n+2,M(b',b”,B)  ∫_ε+ıℝ ẓ'/2πı  e^-(b'+b”)z'/y(E_0-z'^2) ( - sinh(b z')/z')

⊃    1/2 ∫_0^∞b' ḅ' ∫_0^∞b” ḅ” V_g-1,n+2,M(b',b”,B) D(b'+b”,b)
where $\varepsilon' > \varepsilon>0$. In the first step, first two lines, we replaced explicitly the first term of (<ref>). The Laplace and inverse Laplace transforms required in order to go from resolvent to volume cancel each other for all arguments inside the set $B$, related to the set $I$ in the resolvent. We also multiply by a factor of $b$ for later convenience. In the second step, third line, we change the order of integration and leave the $b'$ and $b''$ integrals to be done last. The answer already takes the same form as the first term on the right hand side of (<ref>), with a kernel obtained from evaluating the $z$ and $z'$ integrals. We can change the order of integration and perform the $z$ integral first, picking up the residues at $z=\pm z'$. The answer is shown in parenthesis in the fourth line. In the fifth line we write the term in its final form introducing

D(x,y) = - ∫_ε+ıℝ ẓ/2πı e^-x z sinh(y z)/z y(E_0-z^2).
This is precisely the result quoted in eqn. (<ref>). The exact same manipulations apply to the second term in the right hand side of (<ref>). We are left to consider the final term in the second line of (<ref>), which has a slightly different structure
bV_g,n+1,M(b,B)    ⊃    ∑_k=1^n ∫ẓ/2πı  e^b z ∫ẓ'/2πı z'/z'^2-z^2 ∫_0^∞b' ḅ' e^-b' z'/y(E_0-z'^2)

×V_g,n,M(b',B/b_k) ∫ẓ_k/2πı  2 z_ke^b_k z_k/b_k ( 1/2 z' z_k(z'+z_k)^2+1/ (z'^2-z_k^2)^2)_=2 cosh(b_k z')/2z'

⊃    ∑_k=1^n ∫_0^∞b' ḅ' V_g,n,M(b',B/b_k)∑_± ∫ẓ'/2πı e^-(b' ±b_k)z'/2y(E_0-z'^2) ∫ẓ/2πı   e^b z/z'^2-z^2_= -sinh(b z')/z'

⊃   1/2 ∑_k=1^n ∫_0^∞b' ḅ' V_g,n,M(b',B/b_k)(D(b'+b_k,b)+D(b'-b_k,b))
In the first step, first and second line, we perform the integral transformations to turn the loop equation into a statement for the volumes. The main difference now is that since the term in the kernel $F_{g,n}$ we are considering involves $R_{g,n}(x,I/x_k)$ we need to keep explicitly the integration over $z_k$. The integral over $z_k$ can be done via residues and gives a sum of two terms proportional to $\cosh(b_k z') = \sum_{\pm} e^{\pm b_k z'}$. In the third line we integrate first over $z$ and in the fourth line we recognize the sum of the same kernel $D(x,y)$ as before evaluated at $x=b'+b_k$ and $x=b'-b_k$.

Putting (<ref>) (and a similar contribution from the second term of (<ref>)) and (<ref>) together, we reproduce the full answer (<ref>) with the kernel (<ref>). The result is valid for any spectral curve. Moreover, since the loop equations for the Dyson ensemble $\upbeta=2$ and the AZ ensemble $(\upalpha_0,\upbeta)=(1,2)$ are the same, it is valid in both cases (the only difference being that for the AZ ensemble $E_0$ is strictly non-negative and if $E_0=0$ then $y\sim 1/z$ near $z=0$ while for the Dyson ensemble $y \sim z$ near $z=0$ always). For example, for $E_0=0$ and $y(x) = \frac{\sin(2\pi \sqrt{-x})}{4\pi}$ we reproduce the bosonic version of Mirzakhani's topological recursion.

§ HOLONOMY CONSTRAINT FOR THREE-HOLED SPHERE

In this appendix, we fill in details that were omitted in section
<ref>, by computing the fermionic contributions to the $\uU(1)$ holonomy constraints for the three-holed sphere and  to the parameter $\kappa$, as a function of other moduli.
We assume a three-holed sphere with two boundary holonomies conjugate to $U_0$ and $V_0$ defined in (<ref>).  The third holonomy is therefore conjugate
to $W_0=V_0^{-1}U_0^{-1}$.

We need to compute the three eigenvalues of $W_0$.
Because $W_0$ is unimodular, its eigenvalues can be expressed in terms of two parameters
$c$ and $\phi_c$, as in eqn. (<ref>).
To determine these parameters, it is enough to find two equations that they satisfy. Such
equations are  provided by  conjugation invariants, such as the supertraces ${\rm Str}\,W_0$ and ${\rm Str}\,W_0^{-1}$, which can be compared directly from what
one computes from the diagonal form (<ref>) of $W_0$.
To simplify some expressions, we introduce $H_1 = - e^{\i \hq\phi_a}$, $H_2 = -e^{\i \hq\phi_b}$ and $H_3 = - e^{\i \hq\phi_c}$. The supertrace of $W$ is given by
Str  W = 2 cosha-b/2/H_1 H_2 +κ/H_1 H_2- 1/H_1 ^2 H_2^2 +  ψ_2 (ψ_1 -e^b/2 ψ_3) e^-b/2 ψ_1 ψ_4 /H_1^2 H_2 .
The supertrace of $W^{-1} = U V$ is
Str  W^-1 = H_1H_2 2 cosha-b/2 +H_1H_2 κ- H_1^2H_2^2 +  H_1H_2^2[ ψ_3 (e^a/2 ψ_2 +κψ_4) +e^-a/2 ψ_1 ψ_4].
In terms of $c$ and $\phi_c$, the supertraces should be
Str  W    =   H_3 2 cosh(c/2) - H_3^2.

Str   W^-1    =   H_3 2 cosh( c/2) -H_3^2.
We now have enough equations to determine $H_3$ and $\kappa$. Due to the fermionic nature of $\psi$, these parameters can be written in the form

\begin{equation}
\kappa = \kappa_0 + \sum_{i<j} \kappa_{ij} \psi_i \psi_j + \kappa_4 \psi_1 \psi_2 \psi_3 \psi_4,~~~\phi_a +\phi_b + \phi_c=\sum_{i<j} f_{ij} \psi_i \psi_j + f_4 \psi_1\psi_2\psi_3\psi_4,
\eeq
where $\kappa_0\equiv - 2 \cosh \frac{c}{2} - 2 \cosh \frac{a-b}{2} $. The goal here is to compute the coefficients $\kappa_{0},\kappa_{ij},\kappa_{4}$ and $f_{0},f_{ij},f_{4}$ explicitly. They are functions of $\phi_{a,b}$ and the geodesic lengths only. When the fermionic moduli vanish, the $H_i$ obey the bosonic constraint $H_1 H_2 H_3  = 1$.

To begin the derivation, we retain $\kappa$ as an independent variable, though eventually we will want to express it in terms of lengths and holonomies.  From the equations for the supertraces of $W$ and $W^{-1}$, we can deduce
\beq
H_1^2 H_2^2 H_3^2 = \frac{2H_1H_2 \cosh \frac{a-b}{2} + H_1 H_2 \kappa - H_1^2 H_2^2 -1 + H_2 \psi_1 \psi_2 + e^{-\frac{b}{2}} H_2 (-\psi_1 \psi_4 + e^{b} \psi_2 \psi_3)}{2H_1H_2 \cosh \frac{a-b}{2} + H_1 H_2 \kappa - H_1^2 H_2^2 -1 -  H_1 H_2^2  (\kappa \psi_3 \psi_4 + e^{-\frac{a}{2}}\psi_1 \psi_4 - e^{\frac{a}{2}} \psi_2 \psi_3 )}
\eeq
Let us define now the combination $X\equiv 2H_1H_2 \cosh \frac{a-b}{2} + H_1 H_2 \kappa - H_1^2 H_2^2 -1$ to shorten some equations that will appear next, such that
\beq
H_1^2 H_2^2 H_3^2 = \frac{X + H_2 \psi_1 \psi_2 + e^{-\frac{b}{2}} H_2 (-\psi_1 \psi_4 + e^{b} \psi_2 \psi_3)}{X -  H_1 H_2^2  (\kappa \psi_3 \psi_4 + e^{-\frac{a}{2}}\psi_1 \psi_4 - e^{\frac{a}{2}} \psi_2 \psi_3 )}
\eeq
We denote by $(H_1^2 H_2^2 H_3^2)_n$ the term with $n$ fermions (when the formula is written in terms of $\kappa$). Then
\beq
(H_1^2 H_2^2 H_3^2)_0=1
\eeq
\beq
(H_1^2 H_2^2 H_3^2)_2= \frac{H_2}{X} \psi_1 \psi_2  +\frac{H_2^2 H_1 e^{-\frac{a}{2}} -H_2 e^{-\frac{b}{2}}}{X} \psi_1 \psi_4 -\frac{H_2^2 H_1 e^{\frac{a}{2}} -H_2 e^{\frac{b}{2}}}{X} \psi_2 \psi_3  +\frac{H_2^2 H_1 \kappa }{X} \psi_3 \psi_4
\eeq
\beq
(H_1^2 H_2^2 H_3^2)_4=\frac{H_2^3 H_1 (\kappa + 2 \cosh\frac{a-b}{2}- 2 H_1H_2) }{X^2}\psi_1 \psi_2 \psi_3 \psi_4
\eeq
Consider the fact that we know the bosonic solution for $\kappa= - 2 \cosh \frac{c}{2} - 2 \cosh \frac{a-b}{2} + \mathcal{O}(\psi^2) + \mathcal{O}(\psi^4) $. We can see what happens when $\kappa$ is expanded in fermionic moduli. First, the term with no fermions is always $H_1^2H_2^2H_3^2=1$. To get the terms with two fermions we can simply replace $\kappa= - 2 \cosh \frac{c}{2} - 2 \cosh \frac{a-b}{2} $ in $(H_1^2 H_2^2 H_3^2)_2$. To get the term with four fermions we do the same replacement in $(H_1^2 H_2^2 H_3^2)_4$, but we also need to include $\psi^2$ corrections to $(H_1^2 H_2^2 H_3^2)_2$ which contribute to quartic order in fermions. This last step is the main thing left to do.

To find $\kappa$ to order $\psi^2$ can be done in the following way. Consider one of the supertrace equations $H_3 2 \cosh \frac{c}{2} - H_3^2 = \frac{2 \cosh \frac{a-b}{2}}{H_1 H_2}-\frac{1}{H_1^2 H_2^2} + \frac{\kappa}{H_1 H_2} + \mathcal{O}(\psi^2)$, where the precise form of the $\psi^2$ terms can be found above. We just argued that we know $H_3$ to order $\psi^2$. Then we can replace it in the left hand side of this formula to find  $\kappa$ to order $\psi^2$. Once we know $\kappa$ to order $\psi^2$, we can compute $H_3$ to order $\psi^4$, as explained in the previous paragraph. Knowing $H_3$ to order $\psi^4$ we can replace it again in the same supertrace equation, and matching terms with four fermions obtain $\kappa_4$. Following this procedure we find
\bea
&& \kappa_{12} =-\frac{H_2(H_1H_2+\cosh \frac{c}{2})}{H_1H_2 2\cosh \frac{c}{2} + H_1^2H_2^2+1 },\\
&&\kappa_{14}= \frac{H_2(e^{-\frac{b}{2}}(H_1H_2+\cosh \frac{c}{2})+e^{-\frac{a}{2}}(1+H_1H_2\cosh \frac{c}{2}))}{H_1H_2 2\cosh \frac{c}{2} + H_1^2H_2^2+1 }\\
&& \kappa_{23} =-\frac{H_2(e^{\frac{b}{2}}(H_1H_2+\cosh \frac{c}{2})+e^{\frac{a}{2}}(1+H_1H_2\cosh \frac{c}{2}))}{H_1H_2 2\cosh \frac{c}{2} + H_1^2H_2^2+1 } , \\
&& \kappa_{34} =- \frac{2H_2 (1+ H_1 H_2 \cosh \frac{c}{2})(\cosh \frac{c}{2} + \cosh \frac{a-b}{2})}{H_1H_2 2\cosh \frac{c}{2} + H_1^2H_2^2+1} .
\ea
The final $\kappa_4$ is for now undetermined but we will be able to compute it below, after using this result to fully determine $H_3$. Define for now
\beq
H_1^2H_2^2H_3^2=1 + \sum_{i<j} \hat{f}_{ij} \psi_i \psi_j + \hat{f}_4 \psi_1\psi_2\psi_3\psi_4,
\eeq
Then as we explained, to get $ \hat{f}_{ij}$ we can simply replace $\kappa \to \kappa_0$ while to get $ \hat{f}_4$ we need to include the dependence with the $\kappa_{ij}$ terms. The final answer is
\bea
&& \hat{f}_{12} = -\frac{H_2}{H_1H_2 2\cosh \frac{c}{2} + H_1^2H_2^2+1},~~~ \hat{f}_{14}=\frac{H_2(e^{-\frac{b}{2}}-e^{-\frac{a}{2}} H_1 H_2)}{H_1H_2 2\cosh \frac{c}{2} + H_1^2H_2^2+1},\\
&&  \hat{f}_{23} =\frac{H_2(-e^{\frac{b}{2}}+e^{\frac{a}{2}} H_1 H_2)}{H_1H_2 2\cosh \frac{c}{2} + H_1^2H_2^2+1} ,~~~ \hat{f}_{34}=\frac{2H_1H_2^2(\cosh \frac{c}{2} + \cosh \frac{a-b}{2})}{H_1H_2 2\cosh \frac{c}{2} + H_1^2H_2^2+1},\\
&&  \hat{f}_4=- \frac{H_1 H_2^3(\cosh \frac{c}{2} + H_1 H_2)}{(H_1H_2 2\cosh \frac{c}{2} + H_1^2H_2^2+1)^2}.\label{eq:Fexpapp}
\ea
For the reasons explained above, knowing $ \hat{f}_4$ allows us to compute $\kappa_4$. The final answer is
\beq
\kappa_4 = \frac{H_2^2(\cosh \frac{c}{2} + 2 H_1 H_2 + H_1^2 H_2^2 \cosh \frac{c}{2})}{2(H_1H_2 2\cosh \frac{c}{2} + H_1^2H_2^2+1)^2}
\eeq
This completes the derivation of the fermionic contribution to $\kappa$. After writing it in the original variables it becomes the eqn. \eqref{eq:Kexp} quoted in the main text. In terms of the holonomy constraint, the final step is to rewrite it in terms of $\phi$ instead of $H$. A simple Taylor expansion gives
\bea
\hq(\phi_a+\phi_b+\phi_c) &=& \frac{1}{2}\log\left( 1 + \sum_{i<j} \hat{f}_{ij} \psi_i \psi_j +  \hat{f}_4 \psi_1\psi_2\psi_3\psi_4 \right) \\
&=&  \sum_{i<j}\hq f_{ij} \psi_i \psi_j + \hq f_4 \psi_1\psi_2\psi_3\psi_4.
\ea
where $f_{ij}={\scriptstyle \frac{1}{2\i \hq}}\hat{f}_{ij}$ and $f_4 = {\scriptstyle \frac{1}{2\i \hq}}(\hat{f}_4 - \hat{f}_{12}\hat{f}_{34} - \hat{f}_{14} \hat{f}_{23})$. After using \eqref{eq:Fexpapp} and simplifying we obtain \eqref{eq:Fexp}.

\section{JT gravity with $\uU(1)$ gauge field}\label{app:JTU1bos}

In analyzing the torsion  of a three-holed sphere in section \ref{torthree}, we encountered a subtlety in the treatment of the $\uU(1))$ $R$-symmetry.
We encountered in eqn. (\ref{answer}) an uncanceled factor $1/[\d r_z]$, which we interpreted as an instruction to divide by the volume of the $R$-symmetry group,
with the volume form $[\d r_z]$.

Here we will verify that  this prescription gives the correct answer in the much simpler case of  bosonic JT gravity coupled to a $\uU(1)$ gauge field, related to $BF$ theory of the
group $G={\rm PSL}(2,\mathbb{R}) \times \uU(1)$.  The partition function in this case can
be obtained by other means. Consider a flat $G$-bundle on a three-holed sphere, with two holonomies being $U_0 =\pm \Big(\hspace{-1mm}\begin{array}{cc}
e^{a/2} & \kappa \\
0& e^{-a/2}
\end{array}\hspace{-1mm}\Big) \otimes e^{\i \phi_a} $ and $V_0 = \pm \Big(\hspace{-1mm}\begin{array}{cc}
e^{-b/2} &0 \\
1& e^{b/2}
\end{array}\hspace{-1mm}\Big)\otimes e^{\i \phi_b}$. The torsion of the three-holed sphere, according to our prescription, is given by
\bea
\tau = \frac{1}{2\pi} \, 8 \sinh \frac{a}{2} \sinh \frac{b}{2}\sinh \frac{c}{2}\, \delta(\phi_a+\phi_b+\phi_c) \, \d a\, \d b \, \d c \, \d \phi_a \, \d \phi_b \, \d \phi_c,
\ea
where, following our prescription, we have replaced $1/[\d r_z]$ with $1/2\pi$, where $2\pi$ is the volume of $\uU(1)$ assuming that the measure is $\d r_z$.The third holonomy $\phi_c$ can be  integrated in using the constraint  $1=\int \d \phi_c\, \delta(\phi_a + \phi_b+\phi_c)$. The torsion of a circle is
\bea
\tau_S = 4 \sinh^2 \frac{a}{2} \, \d a \, (\d \varrho)^{-1} \, \d \phi \, (\d \varphi)^{-1}.
\ea
When we glue along a circle, the measure is $\d a \, \d \phi \, \d \varrho \, \d \varphi$. The $\uU(1)$ twist parameter $\varphi$ is a modulus of a flat $\UU(1)$ connection,
though as $\uU(1)$ is abelian, nothing in the computation is being twisted.
Integrating over the twist parameter gives a measure  $ a \d a \, 2\pi \d \phi$. Normalizing by the torsion of the circle and fixing the boundary moduli, the measure of the three-holed sphere is
\bea
\mu_{0,3} = \frac{1}{2\pi} \delta(\phi_a+\phi_b+\phi_c).
\ea
We derive a recursion relation for the volumes of hyperbolic surfaces with a flat $\uU(1)$ connection, adapting the procedure we used for $\N=2$ JT supergravity. We obtain
\bea
b V_{g,n+1} ({\sf b}, {\sf B}) &=& \frac{1}{2}(2\pi) \int_0^\infty  b'\text{d}b' \, \int_0^{2\pi} \d \phi' \, \int_0^\infty b''\text{d}b'' \int_0^{2\pi} \d \phi''\, \delta(\phi+\phi'+\phi'')\mathsf{D}_2 (b,b',b'')\nn
&&\hspace{-5mm} \times \left(V_{g-1,n+2}(\bar{{\sf b}}',\bar{{\sf b}}'',{\sf B}) + \sum_{\rm stable} V_{h_1,|B_1|+1} (\bar{{\sf b}}',{\sf B}_1) V_{h_2,|B_2|+1} (\bar{{\sf b}}'',{\sf B}_2) \right)\nn
&&\hspace{-5mm}+\sum_{k=1}^{n} \int_0^{\infty} b'\text{d}b' \int_0^{2\pi} \d \phi' \delta(\phi+\phi'+\phi_k)\Big(b -\mathsf{T}_2(b,b',b_k) \Big) V_{g,n} (\bar{{\sf b}}',{\sf B} /{\sf b}_k) .
\ea
Integration over the $\uU(1)$ twist parameters produces a factor of $(2\pi)^2 = \int \d \varphi' \d \varphi''$ in the first line and a factor of $(2\pi)= \int \d \varphi'$ in the third line. One of those factors of each line cancels with a similar one in the normalization of $\mu_{0,3}$ which comes from our treatment of $\d r_z$. The initial conditions for this recursion are straightforward to obtain using the volumes for a three-holed sphere and for a torus with one boundary. The solution to this recursion is
\beq
V_{g,n}({\sf B}) = (2\pi)^{2g-1} \delta( \phi_1 + \ldots + \phi_{n}) V_g^{\rm bos.}(B),
\eeq
where $2\pi = {\rm Vol}(U(1))$ and $V_{g,n}^{\rm bos.}(B)$ is the bosonic volumes obtained by the recursion relation of Mirzakhani. We can compute the ``fixed charge'' volumes defined through the Fourier transform $V_{g,n}^{(q)}(B) = \int \frac{\d^n \phi}{(2\pi)^n} e^{\i q \sum_i \phi_i} V_{g,n}({\sf B})$. The explicit expression is given by
\bea
V_{g,n}^{(q)}(B) = (2\pi)^{2g-2} \, V_{g,n}^{\rm bos.}(B).
\ea
The contribution $(2\pi)^{2g-2}$ is consistent with eqn. (21) of [44] derived independently using $BF$ theory techniques.

\bibliographystyle{JHEP}

\providecommand{\href}[2]{#2}\begingroup\raggedright\begin{thebibliography}{10}

[1]
A.~Almheiri, T.~Hartman, J.~Maldacena, E.~Shaghoulian and A.~Tajdini,
\emph{{The entropy of Hawking radiation}},
\href{https://doi.org/10.1103/RevModPhys.93.035002}{\emph{Rev. Mod. Phys.}
{\bfseries 93} (2021) 035002}
[\href{https://arxiv.org/abs/2006.06872}{{\ttfamily 2006.06872}}].

[2]
G.~W. Gibbons and S.~W. Hawking, \emph{{Action Integrals and Partition
Functions in Quantum Gravity}},
\href{https://doi.org/10.1103/PhysRevD.15.2752}{\emph{Phys. Rev. D}
{\bfseries 15} (1977) 2752}.

[3]
J.~M. Maldacena, \emph{{Eternal black holes in anti-de Sitter}},
\href{https://doi.org/10.1088/1126-6708/2003/04/021}{\emph{JHEP} {\bfseries
04} (2003) 021} [\href{https://arxiv.org/abs/hep-th/0106112}{{\ttfamily

[4]
J.~S. Cotler, G.~Gur-Ari, M.~Hanada, J.~Polchinski, P.~Saad, S.~H. Shenker
et~al., \emph{{Black Holes and Random Matrices}},
\href{https://doi.org/10.1007/JHEP05(2017)118}{\emph{JHEP} {\bfseries 05}
(2017) 118} [\href{https://arxiv.org/abs/1611.04650}{{\ttfamily

[5]
P.~Saad, S.~H. Shenker and D.~Stanford, \emph{{A semiclassical ramp in SYK and
in gravity}},  \href{https://arxiv.org/abs/1806.06840}{{\ttfamily

[6]
P.~Saad, S.~H. Shenker and D.~Stanford, \emph{{JT gravity as a matrix
integral}},  \href{https://arxiv.org/abs/1903.11115}{{\ttfamily 1903.11115}}.

[7]
R.~Jackiw, \emph{{Lower Dimensional Gravity}},
\href{https://doi.org/10.1016/0550-3213(85)90448-1}{\emph{Nucl. Phys.}
{\bfseries B252} (1985) 343}.

[8]
C.~Teitelboim, \emph{Gravitation and hamiltonian structure in two spacetime
dimensions}, {\emph{Physics Letters B} {\bfseries 126} (1983) 41}.

[9]
A.~Almheiri and J.~Polchinski, \emph{{Models of AdS$_{2}$ backreaction and
holography}}, \href{https://doi.org/10.1007/JHEP11(2015)014}{\emph{JHEP}
{\bfseries 11} (2015) 014} [\href{https://arxiv.org/abs/1402.6334}{{\ttfamily

[10]
K.~Jensen, \emph{{Chaos in AdS$_2$ Holography}},
\href{https://doi.org/10.1103/PhysRevLett.117.111601}{\emph{Phys. Rev. Lett.}
{\bfseries 117} (2016) 111601}
[\href{https://arxiv.org/abs/1605.06098}{{\ttfamily 1605.06098}}].

[11]
J.~Maldacena, D.~Stanford and Z.~Yang, \emph{{Conformal symmetry and its
breaking in two dimensional Nearly Anti-de-Sitter space}},
\href{https://doi.org/10.1093/ptep/ptw124}{\emph{PTEP} {\bfseries 2016}
(2016) 12C104} [\href{https://arxiv.org/abs/1606.01857}{{\ttfamily

[12]
J.~Engelsöy, T.~G. Mertens and H.~Verlinde, \emph{{An investigation of
AdS$_{2}$ backreaction and holography}},
\href{https://doi.org/10.1007/JHEP07(2016)139}{\emph{JHEP} {\bfseries 07}
(2016) 139} [\href{https://arxiv.org/abs/1606.03438}{{\ttfamily

[13]
T.~G. Mertens and G.~J. Turiaci, \emph{{Solvable Models of Quantum Black Holes:
A Review on Jackiw-Teitelboim Gravity}},
\href{https://arxiv.org/abs/2210.10846}{{\ttfamily 2210.10846}}.

[14]
D.~Stanford and E.~Witten, \emph{{JT Gravity and the Ensembles of Random Matrix
Theory}},  \href{https://arxiv.org/abs/1907.03363}{{\ttfamily 1907.03363}}.

[15]
A.~Altland and M.~R. Zirnbauer, \emph{{Nonstandard symmetry classes in
mesoscopic normal-superconducting hybrid structures}},
\href{https://doi.org/10.1103/PhysRevB.55.1142}{\emph{Phys. Rev. B}
{\bfseries 55} (1997) 1142}

[16]
C.~V. Johnson, \emph{{Nonperturbative Jackiw-Teitelboim gravity}},
\href{https://doi.org/10.1103/PhysRevD.101.106023}{\emph{Phys. Rev. D}
{\bfseries 101} (2020) 106023}
[\href{https://arxiv.org/abs/1912.03637}{{\ttfamily 1912.03637}}].

[17]
T.~G. Mertens and G.~J. Turiaci, \emph{{Liouville quantum gravity --
holography, JT and matrices}},
\href{https://doi.org/10.1007/JHEP01(2021)073}{\emph{JHEP} {\bfseries 01}
(2021) 073} [\href{https://arxiv.org/abs/2006.07072}{{\ttfamily

[18]
J.~Maldacena, \emph{{A simple quantum system that describes a black hole}},
\href{https://arxiv.org/abs/2303.11534}{{\ttfamily 2303.11534}}.

[19]
T.~Mohaupt, \emph{{Black hole entropy, special geometry and strings}},
\href{https://doi.org/10.1002/1521-3978(200102)49:1/3<3::AID-PROP3>3.0.CO;2-#}{\emph{Fortsch.
Phys.} {\bfseries 49} (2001) 3}
[\href{https://arxiv.org/abs/hep-th/0007195}{{\ttfamily hep-th/0007195}}].

[20]
A.~Cabo-Bizet, D.~Cassani, D.~Martelli and S.~Murthy, \emph{{Microscopic origin
of the Bekenstein-Hawking entropy of supersymmetric AdS$_{5}$ black holes}},
\href{https://doi.org/10.1007/JHEP10(2019)062}{\emph{JHEP} {\bfseries 10}
(2019) 062} [\href{https://arxiv.org/abs/1810.11442}{{\ttfamily

[21]
S.~Choi, J.~Kim, S.~Kim and J.~Nahmgoong, \emph{{Large AdS black holes from
QFT}},  \href{https://arxiv.org/abs/1810.12067}{{\ttfamily 1810.12067}}.

[22]
F.~Benini and E.~Milan, \emph{{Black Holes in 4D $\mathcal{N}$=4
Super-Yang-Mills Field Theory}},
\href{https://doi.org/10.1103/PhysRevX.10.021037}{\emph{Phys. Rev. X}
{\bfseries 10} (2020) 021037}
[\href{https://arxiv.org/abs/1812.09613}{{\ttfamily 1812.09613}}].

[23]
J.~Boruch, M.~T. Heydeman, L.~V. Iliesiu and G.~J. Turiaci, \emph{{BPS and
near-BPS black holes in $AdS_5$ and their spectrum in $\mathcal{N}=4$ SYM}},
\href{https://arxiv.org/abs/2203.01331}{{\ttfamily 2203.01331}}.

[24]
M.~Heydeman, L.~V. Iliesiu, G.~J. Turiaci and W.~Zhao, \emph{{The statistical
mechanics of near-BPS black holes}},
\href{https://arxiv.org/abs/2011.01953}{{\ttfamily 2011.01953}}.

[25]
H.~W. Lin, J.~Maldacena, L.~Rozenberg and J.~Shan, \emph{{Looking at
supersymmetric black holes for a very long time}},
\href{https://arxiv.org/abs/2207.00408}{{\ttfamily 2207.00408}}.

[26]
H.~W. Lin, J.~Maldacena, L.~Rozenberg and J.~Shan, \emph{{Holography for people
with no time}},  \href{https://arxiv.org/abs/2207.00407}{{\ttfamily

[27]
I.~C.~H. Ip, R.~C. Penner and A.~M. Zeitlin, \emph{{$\mathcal{N}=2$
super-Teichm\"uller theory}},
\href{https://doi.org/10.1016/j.aim.2018.08.001}{\emph{Adv. Math.} {\bfseries
336} (2018) 409} [\href{https://arxiv.org/abs/1605.08094}{{\ttfamily

[28]
T.~Kanazawa and T.~Wettig, \emph{{Complete random matrix classification of SYK
models with $\mathcal{N}=0$, $1$ and $2$ supersymmetry}},
\href{https://doi.org/10.1007/JHEP09(2017)050}{\emph{JHEP} {\bfseries 09}
(2017) 050} [\href{https://arxiv.org/abs/1706.03044}{{\ttfamily

[29]
J.~J.~M. Verbaarschot and I.~Zahed, \emph{{Spectral density of the QCD Dirac
operator near zero virtuality}},
\href{https://doi.org/10.1103/PhysRevLett.70.3852}{\emph{Phys. Rev. Lett.}
{\bfseries 70} (1993) 3852}
[\href{https://arxiv.org/abs/hep-th/9303012}{{\ttfamily hep-th/9303012}}].

[30]
A.~Anderson, R.~C. Myers and V.~Periwal, \emph{{Complex random surfaces}},
\href{https://doi.org/10.1016/0370-2693(91)90401-B}{\emph{Phys. Lett. B}
{\bfseries 254} (1991) 89}.

[31]
A.~Anderson, R.~C. Myers and V.~Periwal, \emph{{Branched polymers from a double
scaling limit of matrix models}},
\href{https://doi.org/10.1016/0550-3213(91)90411-P}{\emph{Nucl. Phys. B}
{\bfseries 360} (1991) 463}.

[32]
S.~Dalley, C.~V. Johnson and T.~R. Morris, \emph{{Multicritical complex matrix
models and nonperturbative 2-D quantum gravity}},
\href{https://doi.org/10.1016/0550-3213(92)90217-Y}{\emph{Nucl. Phys. B}
{\bfseries 368} (1992) 625}.

[33]
S.~Dalley, C.~V. Johnson and T.~R. Morris, \emph{{Nonperturbative
two-dimensional quantum gravity}},
\href{https://doi.org/10.1016/0550-3213(92)90218-Z}{\emph{Nucl. Phys. B}
{\bfseries 368} (1992) 655}.

[34]
D.~Stanford and E.~Witten, \emph{{Fermionic Localization of the Schwarzian
Theory}}, \href{https://doi.org/10.1007/JHEP10(2017)008}{\emph{JHEP}
{\bfseries 10} (2017) 008}
[\href{https://arxiv.org/abs/1703.04612}{{\ttfamily 1703.04612}}].

[35]
W.~Fu, D.~Gaiotto, J.~Maldacena and S.~Sachdev, \emph{{Supersymmetric
Sachdev-Ye-Kitaev models}},
\href{https://doi.org/10.1103/PhysRevD.95.026009}{\emph{Phys. Rev. D}
{\bfseries 95} (2017) 026009}
[\href{https://arxiv.org/abs/1610.08917}{{\ttfamily 1610.08917}}].

[36]
M.~Heydeman, G.~J. Turiaci and W.~Zhao, \emph{{Phases of $ \mathcal{N} $ = 2
Sachdev-Ye-Kitaev models}},
\href{https://doi.org/10.1007/JHEP01(2023)098}{\emph{JHEP} {\bfseries 01}
(2023) 098} [\href{https://arxiv.org/abs/2206.14900}{{\ttfamily

[37]
V.~Oganesyan and D.~A. Huse, \emph{Localization of interacting fermions at high
temperature}, \href{https://doi.org/10.1103/PhysRevB.75.155111}{\emph{Phys.
Rev. B} {\bfseries 75} (2007) 155111}.

[38]
Y.~Y. Atas, E.~Bogomolny, O.~Giraud and G.~Roux, \emph{Distribution of the
ratio of consecutive level spacings in random matrix ensembles},
\href{https://doi.org/10.1103/PhysRevLett.110.084101}{\emph{Phys. Rev. Lett.}
{\bfseries 110} (2013) 084101}.

[39]
S.~Forste and I.~Golla, \emph{{Nearly AdS$_2$ sugra and the super-Schwarzian}},
\href{https://doi.org/10.1016/j.physletb.2017.05.039}{\emph{Phys. Lett. B}
{\bfseries 771} (2017) 157}
[\href{https://arxiv.org/abs/1703.10969}{{\ttfamily 1703.10969}}].

[40]
S.~F\"orste, J.~Kames-King and M.~Wiesner, \emph{{Towards the Holographic Dual
of N = 2 SYK}}, \href{https://doi.org/10.1007/JHEP03(2018)028}{\emph{JHEP}
{\bfseries 03} (2018) 028}
[\href{https://arxiv.org/abs/1712.07398}{{\ttfamily 1712.07398}}].

[41]
T.~G. Mertens, G.~J. Turiaci and H.~L. Verlinde, \emph{{Solving the Schwarzian
via the Conformal Bootstrap}},
\href{https://doi.org/10.1007/JHEP08(2017)136}{\emph{JHEP} {\bfseries 08}
(2017) 136} [\href{https://arxiv.org/abs/1705.08408}{{\ttfamily

[42]
M.~C\'ardenas, O.~Fuentealba, H.~A. Gonz\'alez, D.~Grumiller, C.~Valc\'arcel
and D.~Vassilevich, \emph{{Boundary theories for dilaton supergravity in
2D}}, \href{https://doi.org/10.1007/JHEP11(2018)077}{\emph{JHEP} {\bfseries
11} (2018) 077} [\href{https://arxiv.org/abs/1809.07208}{{\ttfamily

[43]
S.~Forste, A.~Gerhardus and J.~Kames-King, \emph{{Supersymmetric Black Holes
and the SJT/nSCFT$_1\!$ Correspondence}},
\href{https://arxiv.org/abs/2007.12393}{{\ttfamily 2007.12393}}.

[44]
D.~Kapec, R.~Mahajan and D.~Stanford, \emph{{Matrix ensembles with global
symmetries and \textquoteright{}t Hooft anomalies from 2d gauge theory}},
\href{https://doi.org/10.1007/JHEP04(2020)186}{\emph{JHEP} {\bfseries 04}
(2020) 186} [\href{https://arxiv.org/abs/1912.12285}{{\ttfamily

[45]
D.~Ray and I.~Singer, \emph{{R Torsion and the Laplacian on Riemannian
manifolds}}, \href{https://doi.org/10.1016/0001-8708(71)90045-4}{\emph{Adv.
Math.} {\bfseries 7} (1971) 145}.

[46]
A.~S. Schwarz, \emph{{The Partition Function of Degenerate Quadratic Functional
and Ray-Singer Invariants}},
\href{https://doi.org/10.1007/BF00406412}{\emph{Lett. Math. Phys.} {\bfseries
2} (1978) 247}.

[47]
K.~Reidemeister, \emph{{Homotopieringe und Linsenraume}},
\href{https://doi.org/10.1007/BF02940717}{\emph{Abhandlungen aus dem
Mathematischen Seminar der Universitat Hamburg} {\bfseries 11} (1935) }.

[48]
J.~Cheeger, \emph{Analytic torsion and the heat equation}, {\emph{Annals of
Mathematics} {\bfseries 109} (1979) 259}.

[49]
W.~Muller, \emph{{Analytic torsion and R-torsion of riemannian manifolds}},
{\emph{Advances in Mathematics} {\bfseries 28} 233}.

[50]
W.~Muller, \emph{{Analytic torsion and R-torsion for unimodular
representations}}, {\emph{Journal of the American Mathematical Society}
{\bfseries 6} 721}.

[51]
J.-M. Bismut and W.~Zhang, \emph{{An extension of a theorem by Cheeger and
Muller}}, {\emph{Asterisque} {\bfseries 205} 235}.

[52]
E.~Witten, \emph{{Notes On Supermanifolds and Integration}},
\href{https://doi.org/10.4310/PAMQ.2019.v15.n1.a1}{\emph{Pure Appl. Math.
Quart.} {\bfseries 15} (2019) 3}
[\href{https://arxiv.org/abs/1209.2199}{{\ttfamily 1209.2199}}].

[53]
E.~Witten, \emph{{On quantum gauge theories in two-dimensions}},
\href{https://doi.org/10.1007/BF02100009}{\emph{Commun. Math. Phys.}
{\bfseries 141} (1991) 153}.

[54]
M.~Mirzakhani, \emph{Simple geodesics and weil-petersson volumes of moduli
spaces of bordered riemann surfaces}, {\emph{Inventiones mathematicae}
{\bfseries 167} (2007) 179}.

[55]
M.~Mirzakhani, \emph{{Weil-Petersson volumes and intersection theory on the
moduli space of curves}},
\href{https://doi.org/10.1090/S0894-0347-06-00526-1}{\emph{J. Am. Math. Soc.}
{\bfseries 20} (2007) 1}.

[56]
E.~Witten, \emph{{Notes On Super Riemann Surfaces And Their Moduli}},
\href{https://doi.org/10.4310/PAMQ.2019.v15.n1.a2}{\emph{Pure Appl. Math.
Quart.} {\bfseries 15} (2019) 57}
[\href{https://arxiv.org/abs/1209.2459}{{\ttfamily 1209.2459}}].

[57]
B.~Eynard and N.~Orantin, \emph{{Weil-Petersson volume of moduli spaces,
Mirzakhani's recursion and matrix models}},
\href{https://arxiv.org/abs/0705.3600}{{\ttfamily 0705.3600}}.

[58]
H.~Maxfield and G.~J. Turiaci, \emph{{The path integral of 3D gravity near
extremality; or, JT gravity with defects as a matrix integral}},
\href{https://arxiv.org/abs/2006.11317}{{\ttfamily 2006.11317}}.

[59]
E.~Witten, \emph{{Matrix Models and Deformations of JT Gravity}},
\href{https://arxiv.org/abs/2006.13414}{{\ttfamily 2006.13414}}.

[60]
F.~Rosso and G.~J. Turiaci, \emph{{Phase transitions for deformations of JT
supergravity and matrix models}},
\href{https://doi.org/10.1007/JHEP02(2022)187}{\emph{JHEP} {\bfseries 02}
(2022) 187} [\href{https://arxiv.org/abs/2111.09330}{{\ttfamily

[61]
G.~J. Turiaci, M.~Usatyuk and W.~W. Weng, \emph{{2D dilaton-gravity,
deformations of the minimal string, and matrix models}},
\href{https://doi.org/10.1088/1361-6382/ac25df}{\emph{Class. Quant. Grav.}
{\bfseries 38} (2021) 204001}
[\href{https://arxiv.org/abs/2011.06038}{{\ttfamily 2011.06038}}].

[62]
L.~Eberhardt and G.~J. Turiaci, \emph{{2D dilaton gravity and the
Weil-Petersson volumes with conical defects}},
\href{https://arxiv.org/abs/2304.14948}{{\ttfamily 2304.14948}}.

[63]
X.-z. Dai and D.~S. Freed, \emph{{eta invariants and determinant lines}},
\href{https://doi.org/10.1063/1.530747}{\emph{J. Math. Phys.} {\bfseries 35}
(1994) 5155} [\href{https://arxiv.org/abs/hep-th/9405012}{{\ttfamily

[64]
E.~Witten and K.~Yonekura, \emph{{Anomaly Inflow and the $\eta$-Invariant}},
in \emph{{The Shoucheng Zhang Memorial Workshop}}, 9, 2019,
\href{https://arxiv.org/abs/1909.08775}{{\ttfamily 1909.08775}}.

[65]
E.~Witten, \emph{{Coadjoint Orbits of the Virasoro Group}},
\href{https://doi.org/10.1007/BF01218287}{\emph{Commun. Math. Phys.}
{\bfseries 114} (1988) 1}.

[66]
A.~Alekseev and S.~L. Shatashvili, \emph{{Path Integral Quantization of the
Coadjoint Orbits of the Virasoro Group and 2D Gravity}},
\href{https://doi.org/10.1016/0550-3213(89)90130-2}{\emph{Nucl. Phys. B}
{\bfseries 323} (1989) 719}.

[67]
T.~Eguchi and Y.~Sugawara, \emph{{Modular bootstrap for boundary N = 2
Liouville theory}},
\href{https://doi.org/10.1088/1126-6708/2004/01/025}{\emph{JHEP} {\bfseries
01} (2004) 025} [\href{https://arxiv.org/abs/hep-th/0311141}{{\ttfamily

[68]
P.~Kraus, \emph{{Lectures on black holes and the AdS(3) / CFT(2)
correspondence}}, {\emph{Lect. Notes Phys.} {\bfseries 755} (2008) 193}
[\href{https://arxiv.org/abs/hep-th/0609074}{{\ttfamily hep-th/0609074}}].

[69]
L.~V. Iliesiu, M.~Kologlu and G.~J. Turiaci, \emph{{Supersymmetric indices
factorize}},  \href{https://arxiv.org/abs/2107.09062}{{\ttfamily

[70]
R.~Picken, \emph{{The propagator for quantum mechanics on a group manifold from
an infinite-dimensional analogue of the Duistermaat-Heckman integration
formula }}, {\emph{Journal of Physics A: Mathematical and General} {\bfseries
22} (1989) 2285}.

[71]
T.~Eguchi and A.~Taormina, \emph{{Unitary Representations of $N=4$
Superconformal Algebra}},
\href{https://doi.org/10.1016/0370-2693(87)91679-0}{\emph{Phys. Lett.}
{\bfseries B196} (1987) 75}.

[72]
T.~Eguchi and A.~Taormina, \emph{{Character Formulas for the $N=4$
Superconformal Algebra}},
\href{https://doi.org/10.1016/0370-2693(88)90778-2}{\emph{Phys. Lett.}
{\bfseries B200} (1988) 315}.

[73]
T.~Eguchi and A.~Taormina, \emph{{On the Unitary Representations of $N=2$ and
$N=4$ Superconformal Algebras}},
\href{https://doi.org/10.1016/0370-2693(88)90360-7}{\emph{Phys. Lett.}
{\bfseries B210} (1988) 125}.

[74]
T.~Eguchi, Y.~Sugawara and A.~Taormina, \emph{{Modular Forms and Elliptic
Genera for ALE Spaces}},  in \emph{{Workshop on Exploration of New Structures
and Natural Constructions in Mathematical Physics: On the Occasion of
Professor Akhiro Tsuchiya's Retirement}}, 3, 2008,
\href{https://arxiv.org/abs/0803.0377}{{\ttfamily 0803.0377}}.

[75]
M.~Srednicki, \emph{{A New construction of the Penner model}},
\href{https://doi.org/10.1142/S0217732392004237}{\emph{Mod. Phys. Lett. A}
{\bfseries 7} (1992) 2857}
[\href{https://arxiv.org/abs/hep-th/9206085}{{\ttfamily hep-th/9206085}}].

[76]
R.~C. Penner, \emph{{The moduli space of a punctured surface and perturbative
series}}, {\emph{Bull. Am. Math. Soc.} {\bfseries 15} (1986) 73}.

[77]
R.~C. Penner, \emph{{Perturbative series and the moduli space of Riemann
surfaces}}, {\emph{J. Diff. Geom.} {\bfseries 27} (1988) 35}.

[78]
J.~Ambjorn, C.~F. Kristjansen and Y.~Makeenko, \emph{{Generalized Penner models
to all genera}}, \href{https://doi.org/10.1103/PhysRevD.50.5193}{\emph{Phys.
Rev. D} {\bfseries 50} (1994) 5193}
[\href{https://arxiv.org/abs/hep-th/9403024}{{\ttfamily hep-th/9403024}}].

[79]
A.~A. Migdal, \emph{{Recursion Equations in Gauge Theories}}, {\emph{Sov. Phys.
JETP} {\bfseries 42} (1975) 413}.

[80]