††institutetext: 1 Stanford Institute for Theoretical Physics, Stanford, CA
USA††institutetext: 2 Physics Department, Princeton University, Princeton, NJ
USA††institutetext: 3 Physics Department, University of Washington, Seattle,
WA USA††institutetext: 4 Institute for Advanced Study, Princeton, NJ USA

# Spin-Statistics for Black Hole Microstates

Yiming Chen1,2 and Gustavo J. Turiaci3,4

###### Abstract

The gravitational path integral can be used to compute the number of black
hole states for a given energy window, or the free energy in a thermal
ensemble. In this article we explain how to use the gravitational path
integral to compute the separate number of bosonic and fermionic black hole
microstates. We do this by comparing the partition function with and without
the insertion of $(-1)^{\sf F}$. In particular we introduce a universal
rotating black hole that contributes to the partition function in the presence
of $(-1)^{\sf F}$. We study this problem for black holes in asymptotically
flat space and in AdS, putting constraints on the high energy spectrum of
holographic CFTs (not necessarily supersymmetric). Finally, we analyze
wormhole contributions to related quantities.

##  1 Introduction

The gravitational path integral Gibbons:1976ue  is normally used to compute
the total number of black hole microstates with some constraints such as a
fixed energy window or temperature. The result is usually phrased in terms of
the free energy $F(\beta)$ as a function of temperature or the microcanonical
entropy $S(E)$ as a function of the energy

$\displaystyle Z(\beta)={\rm Tr}\,\,e^{-\beta H}=e^{-\beta
F(\beta)},~{}~{}~{}~{}{\rm Tr}_{E}\,\,1=e^{S(E)}.$ (1)

The purpose of this article is to apply this formalism to derive statistical
properties of how these black hole microstates are distributed according to
whether they are bosonic or fermionic. In principle, for Lorentz invariant
theories the quantum statistics of a state is correlated with the angular
momentum: states with half-integer angular momentum are fermions while states
with integer angular momentum are bosons. This would correctly suggest we can
estimate the quantum statistics of a black hole microstate by looking at its
angular momentum. The problem with this approach is that in the limit of small
$G_{N}$, where we can rely on semiclassical gravity, charges including the
angular momentum are large and of order $1/G_{N}$. The distinction of whether
the angular momentum is integer or half-integer is therefore beyond the
classical approximation. The determination of the quantum statistics from the
gravitational path integral in the way described in this paragraph therefore
requires incorporating quantum effects.

To extract the distribution of fermionic and bosonic black hole microstates in
a simpler way, we focus on the following quantities which generalize (1):

$Z_{\mathrm{spin}}(\beta)={\rm Tr}\,(-1)^{\sf F}e^{-\beta H}=e^{-\beta
F_{\mathrm{spin}}(\beta)},~{}~{}~{}~{}~{}{\rm Tr}_{E}\,(-1)^{\sf
F}=e^{S_{\mathrm{spin}}(E)}.$ (2)

These are the free energy $F_{\mathrm{spin}}(\beta)$ as a function of
temperature, and the entropy $S_{\mathrm{spin}}(E)$ as a function of energy,
both computed with an insertion of $(-1)^{\sf F}$ the operator that assigns a
value of $+1$ to bosonic states and $-1$ to fermionic states. The quantities
in (2) together with (1) are enough to extract the number of bosonic against
fermionic black hole microstates. The reason to study (2) is that, as we
explain in this paper, this quantities can be computed using the gravitational
path integral in the classical approximation.

The idea to evaluate (2) is the following. For concreteness we frame the
discussion in the context of ${\rm AdS}_{d+1}/{\rm CFT}_{d}$ and use the
gravitational path integral to compute the partition function of the boundary
CFT on $S^{1}\times S^{d-1}$ both with and without an insertion $(-1)^{\sf
F}$. Let us consider the fixed temperature ensemble since similar
considerations apply when fixing energy. When evaluating the gravitational
path integral, since the thermal partition function $Z(\beta)$ is an Euclidean
observable it gets a contribution from the Euclidean section of a non-rotating
black hole as well as from thermal AdS. In principle the only difference
between the partition function with or without an insertion of $(-1)^{\sf F}$
is whether fermionic fields are periodic or anti-periodic around the time
cycle, so the same black hole and thermal AdS saddles should contribute to
both quantities. Nevertheless, in the presence of the $(-1)^{\sf F}$
insertion, fluctuations of fermionic fields around the black hole saddle are
singular since the choice of spin structure is incompatible with the time
circle contracting at the horizon.

So what saddles do contribute to $F_{\mathrm{spin}}$ or $S_{\mathrm{spin}}$?
The first obvious option is thermal AdS. Since the time circle does not
contract anywhere inside the thermal AdS geometry we can evaluate
$F_{\mathrm{spin}}$ or $S_{\mathrm{spin}}$ by counting bosonic vs fermionic
low energy exitations around vacuum AdS. To leading order in the small $G_{N}$
limit this predicts a small value for
$F_{\mathrm{spin}},S_{\mathrm{spin}}\sim\mathcal{O}(1)$. This is much smaller
than the results without the $(-1)^{\sf F}$ insertion since
$F,S\sim\mathcal{O}(1/G_{N})$, dominated by the black hole saddle. If this was
true, it implies most states are evenly distributed between bosons and
fermions and there is a large cancellation in $F_{\mathrm{spin}}$ or
$S_{\mathrm{spin}}$. Some evidence for a huge cancellation between bosonic and
fermionic states in a theory without supersymmetry was presented from the
boundary side in Cherman:2018mya  (and references therein).

The purpose of this article is to introduce another universal saddle that
always contributes to both $F_{\mathrm{spin}}$ and $S_{\mathrm{spin}}$,
analogous to the black hole contribution to $F$ or $S$. For concreteness we
focus first on the free energy. Choose an arbitrary direction and define $J$
as the angular momentum generating rotations around this direction. One can
generalize (1) and compute ${\rm Tr}\,\left(e^{-\beta H}e^{\beta\Omega
J}\right)$, where $\Omega$ is the angular velocity of the ensemble. On the
boundary side this is implemented by imposing a twist in the boundary
conditions: as fields go around the time cycle they get multiplied by
$+e^{\beta\Omega J}$ (bosons) or $-e^{\beta\Omega J}$ (fermions). The black
hole that contributes to this quantity in the gravitational path integral has
now rotation. This implies that the contractible cycle is not time anymore,
but a combination of time and the angle around the $J$ direction. This
partition function becomes $F_{\mathrm{spin}}$ when
$\beta\Omega=2\pi{\mathrm{i}}$, since $(-1)^{\sf F}=e^{2\pi{\mathrm{i}}J}$.
Therefore there is a universal rotating black hole saddle that contributes to
$F_{\mathrm{spin}}$ such that periodic fermions around the time cycle produces
a smooth spin structure.

The way described in the previous paragraph of generating black hole solutions
with periodic fermions was considered exclusively for supersymmetric solutions
in Cabo-Bizet:2018ehj ; Bobev:2020pjk ; Iliesiu:2021are . The point of the
present paper is to apply this same construction to general theories that are
either not supersymmetric, or supersymmetric but the partition function with a
$(-1)^{\sf F}$ insertion is not protected.

In the next sections we study the rotating black hole geometries that
contribute to $F_{\mathrm{spin}}$ and $S_{\mathrm{spin}}$ and analyze their
consequences. For the spacetime dimensions we analyzed, we always find that
the rotating black hole saddle that contribute to $F_{\mathrm{spin}}$ has a
higher free energy than thermal AdS with periodic fermions making it always
subleading. The free energy $F$ has a phase transition as a function of
temperature when thermal AdS and the black hole change dominance, which on the
boundary side is interpreted as a confinement-deconfinement phase transition
Witten:1998zw . Instead, even after including the rotating black hole saddle,
the free energy $F_{\mathrm{spin}}$ in the presence of $(-1)^{\sf F}$ has no
phase transition and depends smoothly on the temperature being always
controlled by thermal AdS. In the presence of bulk gauge fields, it is easy to
remedy this and make our rotating black hole dominant: one can consider an
ensemble where some charge is fixed such that thermal AdS is discarded.

A different way to make the rotating black hole be dominant is to work in the
fixed energy ensemble and compute the entropy $S_{\mathrm{spin}}$. The
rotating black hole does contribute a positive and large amount to
$S_{\mathrm{spin}}$. For illustration take ${\rm AdS}_{4}$. The entropy
without and with the insertion of $(-1)^{\sf F}$ are given, at large energies,
by (see Section 2.2.1 for conventions)

$S(E)\sim E^{2/3},~{}~{}~{}~{}~{}S_{\mathrm{spin}}(E)\sim E^{1/2}.$ (3)

The black hole result $S(E)\sim E^{2/3}$ is consistent with a local 3d theory
on the boundary. The rotating black hole contributing to $S_{\mathrm{spin}}$
gives a positive contribution that grows with energy, albeit slower than the
total number of states. The results for ${\rm AdS}_{3}$ are similar
qualitatively. Instead, in ${\rm AdS}_{5}$ we find that both
$F_{\mathrm{spin}}$ and $S_{\mathrm{spin}}$ are dominated by thermal AdS.

We also study the contribution from wormholes to the partition function in the
presence of a $(-1)^{\sf F}$ insertion. More precisely, we study a
generalization of the spectral form factor Cotler:2016fpe ; Saad:2018bqo
where $(-1)^{\sf F}$ is inserted. We discuss a construction similar to the
double cone of Saad:2018bqo  and evaluate its contribution to the
gravitational path integral. We argue that at late time the spectral form
factor is not sensitive to the $(-1)^{\sf F}$ insertion.

We finish this introduction with some general comments and relation to other
work. A universal formula was derived in Pal:2020wwd ; Harlow:2021trr  for the
density of states for theories with finite-group symmetry. If the finite-group
is $\mathbb{Z}_{2}$ the formula predicts that high energy states are equally
distributed between even and odd states. In our context the finite-group is
generated by $(-1)^{\sf F}$ and their result would apply if no saddle would
contribute an amount to $F_{\mathrm{spin}}$ (or $S_{\mathrm{spin}}$)
comparable to $F$ (or $S$), such that to leading order there are as many
bosonic as fermionic states. In this context, our rotating black hole provides
a universal geometric configuration that gives a subleading correction to the
result of Pal:2020wwd ; Harlow:2021trr . Of course there can be other
contributions such as defects which provide other corrections to Pal:2020wwd ;
Harlow:2021trr  (see the discussion section for further comments), but such a
contribution would not be universal and would depend on the details of the
defect and the theory in which they are embedded.

Another interesting feature of our rotating black hole solution is that it
does not respect ensemble equivalence: the saddle that dominates at fixed
temperature (thermal AdS) is completely different than the saddle that
dominates the microcanonical ensemble (the rotating black hole). In Section
2.2 we give an interpretation of why this is so. In the context of ${\rm
AdS}_{3}/{\rm CFT}_{2}$ Tauberian theorems that incorporate rotation
Mukhametzhanov:2020swe ; Pal:2022vqc ; Pal:2023cgk  might be powerful enough
to prove when different ensembles are equivalent or not and therefore studying
$Z_{\mathrm{spin}}$ in that context could be interesting.

The rest of the paper is organized as follows. In Section 2 we find new black
hole geometries that contribute to the partition function with an insertion of
$(-1)^{\sf F}$. In Section 2.1 we study this problem for black holes in flat
space, with and without charge, and in Section 2.2 we generalize this to ${\rm
AdS}_{4}$, ${\rm AdS}_{5}$ and finally ${\rm AdS}_{3}$. In Section 3 we study
the contribution from wormholes to the partition function in the presence of a
$(-1)^{\sf F}$ insertion. We conclude in Section 4 with discussions of our
results and future directions, leaving technical details for appendices.

##  2 Black hole solutions in the presence of $(-1)^{\sf F}$

###  2.1 Flat Space Solutions

In this section we explain the general principle to compute the distribution
of fermionic and bosonic black hole microstates using the gravitational path
integral. We begin by studying black holes in asymptotically four dimensional
flat spacetime. As we will see, the calculation in flat space is a bit
singular. However we decide to include it since it is a good starting point to
illustrate the main ideas. The action, in Euclidean signature, is given by

$I=-\frac{1}{16\pi G_{N}}\int\sqrt{g}R-\frac{1}{8\pi
G_{N}}\oint\sqrt{h}K+({\rm matter}).$ (4)

The details of the matter sector will not be important other than the
assumption that part of it is fermionic. This assumption does not show up
explicitly in the following discussion, while its importance will be discussed
in Section 4. This theory has classical black hole solutions described by the
Kerr metric, which in Boyer-Lindquist coordinates is given by

${\mathrm{d}}s^{2}=-\frac{\rho^{2}\Delta}{\Xi}{\mathrm{d}}t^{2}+\frac{\rho^{2}}{\Delta}{\mathrm{d}}r^{2}+\rho^{2}{\mathrm{d}}\theta^{2}+\frac{\Xi}{\rho^{2}}\sin^{2}\theta\left({\mathrm{d}}\varphi-\frac{2Ear}{\Xi}{\mathrm{d}}t\right)^{2}$
(5)

with the functions

$\displaystyle\rho^{2}=r^{2}+a^{2}\cos^{2}\theta,$
$\displaystyle\Delta=r^{2}-2Er+a^{2},$ (6)
$\displaystyle\Xi=(r^{2}+a^{2})^{2}-a^{2}\Delta\,\sin^{2}\theta.$

$E$ and $a$ are parameters of the solution, and their interpretation depends
on the ensemble we choose to work with. In a microcanonical ensemble, $E$ is
the ADM mass while $J=aE$ is the angular momentum. In the classical limit both
of these quantities are large, of order $1/G_{N}$ and one is _a priori_ not
sensitive to questions such as whether $J$ is integer (bosonic) or half-
integer (fermionic). Below we work in units such that $G_{N}=1$.

In the grandcanonical ensemble, we want instead to fix the inverse temperature
$\beta$, the length of the thermal circle, and the angular velocity $\Omega$
at infinity. For the metric given in equation (5) this is achieved by imposing
the following identification

$(t_{\rm E},\varphi)\sim(t_{\rm
E}+\beta,\varphi+{\mathrm{i}}\beta\Omega)\sim(t_{\rm E},\varphi+2\pi),$ (7)

where $t_{\rm E}=-{\mathrm{i}}t$ is the Euclidean time. In this ensemble, the
parameters $a$ and $E$ should be seen as being fixed by the temperature and
angular velocity in the following way. The Kerr metric has an outer event
horizon at $r_{+}=E+\sqrt{E^{2}-a^{2}}$. It will be useful sometimes below to
trade the parameter $E$ by $r_{+}$ since $E=\frac{r_{+}^{2}+a^{2}}{2r_{+}}$.
Demanding that the solution (5) is smooth at the horizon, given the
identification (7), determines the parameters $a$ and $E$ (or $r_{+}$) as
functions of $\beta$ and $\Omega$ by

$\beta=\frac{4\pi
r_{+}(a^{2}+r_{+}^{2})}{(r_{+}^{2}-a^{2})},~{}~{}~{}~{}\Omega=\frac{a}{r_{+}^{2}+a^{2}}.$
(8)

It is for this choice of ensemble that the action quoted in equation (4) with
the Gibbons-Hawking-York boundary term poses a well-defined variational
problem.

Let us begin by computing the partition function in the grandcanonical
ensemble with $\Omega=0$, where we sum over states with arbitrary mass and
angular momentum. This is of course a very well known result Gibbons:1976ue .
Classically, $\Omega=0$ implies that $a=0$ and then the metric (5) reduces to
the Schwarszchild black hole. For any value of $\beta$ and $\Omega$, the
action (4) evaluated on this solution is equal to $I=\beta E-A/4-\beta\Omega
J$, where $A=4\pi(r_{+}^{2}+a^{2})$ is the area of the event horizon. For
$\Omega=0$ the free energy then is

$-\beta F=\log Z=-\frac{\beta^{2}}{16\pi},~{}~{}~{}~{}F=\frac{\beta}{16\pi}.$
(9)

In a quantum mechanical description of the black hole microstates, this
quantity corresponds to $Z(\beta)={\rm Tr}\,e^{-\beta H}$, where the trace is
taken over states of any mass or angular momentum. We can also work in an
ensemble of fixed mass, and again restrict to $\Omega=0$. We refer to this
partition function as $e^{S(E)}$, and can be interpreted as the number of
states of energy $E$ regardless of the spin.111In the semiclassical
approximation it makes more sense to specify the energy $E$ to be in a small
window $(E-\delta E,E+\delta E)$ with small enough $\delta E$. Whenever we
specify the energy from now on we always mean up to $\delta E$. On the black
hole side the energy corresponds to the mass $E$. This quantity can be
obtained either by Legendre transform of the result at finite temperature, or
by finding the appropriate boundary terms in the action. Either way the answer
is

$S(E)=4\pi E^{2}.$ (10)

This is the standard Bekenstein-Hawking area law for a Schwarszchild black
hole.

The result above gives a prediction for the total number of black hole
microstates of a given energy, ${\rm exp}\\{S(E)\\}$. The question we want to
address here is what fraction of these states are fermionic states and what
fraction are bosonic. On general grounds, we expect that both bosonic and
fermionic microstates constitute about half of the total number of states.
This is because we can turn a bosonic black hole into a fermionic one by
throwing in a fermion, which barely changes its energy. However, we do not
expect the ratio to be precisely one half. We might have slightly more bosonic
states or fermionic states depending on the energy window we are looking at.
It is this subtle difference that we are after.

To approach this question, we analyze a quantity whose evaluation is easy to
formulate using the gravitational path integral. In terms of the quantum
system describing the black hole microstates we would like to compute the
partition function with a fermion parity operator, an insertion of $(-1)^{\sf
F}$.222As we mentioned in the introduction, the quantity
$\textrm{Tr}\,(-1)^{\sf F}e^{-\beta H}$ is commonly studied for supersymmetric
theories, and is called the Witten index Witten:1982df . Here we are _not_
considering supersymmetric theories. This is

${\rm Tr}\,(-1)^{\sf F}e^{-\beta H}=e^{-\beta
F_{{\mathrm{spin}}}},~{}~{}~{}~{}~{}{\rm Tr}_{E}\,(-1)^{\sf
F}=e^{S_{{\mathrm{spin}}}(E)}$ (11)

in an ensemble with fixed temperature or energy, respectively. We will use the
gravitational path integral to compute the free energy $F_{\mathrm{spin}}$ and
entropy $S_{\mathrm{spin}}$ in the presence of a $(-1)^{\sf F}$ insertion, to
leading order in the classical limit.

$\beta$$t_{\rm E}$$2\pi$$\varphi$${\mathrm{i}}\beta\Omega$

$\beta$$t_{\rm E}$$2\pi$$\varphi$${\mathrm{i}}\beta\Omega=2\pi$

Figure 1: $(t_{\rm E},\varphi)$-plane at fixed $r$ and $\theta\neq 0,\pi$.
Working in the grandcanonical ensemble implies two identifications given in
equation (7), the fundamental domain is bounded by the dashed lines. We show
the thermal cycle $(t_{E},\varphi)\sim(t_{E}+\beta,\varphi)$ (red arrow) and
the cycle that contracts at the horizon (blue arrow). Left: General value of
$\Omega$. Fermions are always antiperiodic around the blue cycle and around
$\varphi\sim\varphi+2\pi$ since they are both contractible in the black hole
geometry. Fields get multiplied by $e^{\beta\Omega J}$ (bosons) or
$-e^{\beta\Omega J}$ (fermions) as they go around the red cycle. Right: Choice
of angular velocity that computes the partition function with a $(-1)^{\sf F}$
insertion. $\Omega$ is such that fermions are antiperiodic along the blue
cycle but now become periodic around the red one since
$e^{2\pi{\mathrm{i}}J}=1$ (bosons) or $-1$ (fermions).

In a naive approach to this question, one would conclude that with the
insertion of $(-1)^{\sf F}$ we should study the same Schwarszchild black hole
geometry, but impose that fermions should be periodic around the thermal
circle. Since this circle contracts at the horizon, this is not a smooth
choice of spin structure. This approach is naive since we can generate
periodic boundary conditions for fermions starting from (7) and setting
instead $\beta\Omega=\pm 2\pi{\mathrm{i}}$. Now the solution is the Kerr black
hole and the rotation implies the time cycle does not contract at the horizon
anymore. Instead it is precisely the one along the first identification in
equation (7) that contracts and fermions are antiperiodic around this cycle
Iliesiu:2021are . We illustrate this argument in figure 1. Another way to
think about this solution is to write

$(-1)^{\sf F}=e^{\pm 2\pi{\mathrm{i}}J}.$ (12)

Using this relation, the insertion of $(-1)^{\sf F}$ corresponds to a specific
choice of the angular velocity.

Let us make a few clarifications. The boundary conditions corresponding to the
grandcanonical ensemble given in (7) break the full rotational isometries of
flat space down to the one generated by $\partial_{\varphi}$. This is true for
generic values of $\Omega$. An exception is when $\Omega=0$: the full $SU(2)$
rotational symmetry is restored in the boundary and the bulk geometry, the
Schwarszchild metric, preserves these isometries everywhere in the interior of
the spacetime. Another exception is when $\beta\Omega=\pm 2\pi{\mathrm{i}}$:
the relation (12) is independent of the choice of axis for $J$ in the right
hand side and therefore the boundary conditions preserve the full rotational
invariance. Nevertheless, the new feature is that now the bulk geometry, the
Kerr metric, does break the rotational isometries down to $\partial_{\varphi}$
since it requires singling out an axis of rotation. This implies the presence
of a moduli space of Kerr black holes with arbitrary axis of rotation, and the
rotational invariance arises from integrating over this moduli space. The
presence of these continuum bosonic zero-modes is not problematic since the
moduli space is compact333This was pointed out in Iliesiu:2021are  and
analyzed in detail in H:2023qko  in the context of concrete supergravity
examples., and in any case this issue would only arise when evaluating quantum
corrections which we will not attempt here. Given the choice of axis we made
here, the two values $\beta\Omega=+2\pi{\mathrm{i}}$ or $-2\pi{\mathrm{i}}$
are connected inside the moduli space, and therefore we can restrict from now
on to $\beta\Omega=+2\pi{\mathrm{i}}$ for concreteness.

After these generalities, let us compute $F_{\mathrm{spin}}$ using the
rotating black hole solution. We first need to impose
$\beta\Omega=2\pi{\mathrm{i}}$ and solve for $r_{+}$, or equivalently $E$, and
$a$. More generally, we have the following relation between $r_{+}$ and
$a={\mathrm{i}}{\sf a}$

$\frac{2{\sf a}r_{+}}{{\sf
a}^{2}+r_{+}^{2}}=\frac{\beta\Omega}{2\pi{\mathrm{i}}}\equiv\omega.$ (13)

We want to set $\omega=1$. Assuming $r_{+}$ is finite, this leads to ${\sf
a}=\pm r_{+}$ which in turn implies $\beta=0$. This is an issue since in
principle we should be able to compute the free energy for any temperature
using the gravitational path integral. A way out is to consider the fixed
energy ensemble instead, which we do below. Another option is to work with
$\omega=1-\varepsilon$ and take $\varepsilon\to 0$ at the end of the
calculation. After eliminating $a$, the inverse temperature is given by

$\beta=4\pi
r_{+}\sqrt{\varepsilon(2-\varepsilon)},~{}~{}\Rightarrow~{}~{}r_{+}=\frac{\beta}{4\pi\sqrt{\varepsilon(2-\varepsilon)}}.$
(14)

This expression is valid for any $\varepsilon$, but in the $\varepsilon\to 0$
limit we get $r_{+}\sim 1/\sqrt{\varepsilon}\to\infty$. It might be worrisome
that computing $F_{\mathrm{spin}}$ involves a geometry with $r_{+}\to\infty$.
Nevertheless one can check that in this limit the energy, the regularized
action, and the curvature tensor, all remain finite as explained in Appendix
B. (This issue will not arise for black holes in AdS.) Thus we will accept
this solution and continue our analysis. We can estimate the contribution of
this saddle to the difference between fermionic and bosonic states by
computing the on-shell action. We obtain

$I=\beta E-\frac{A}{4}-\beta\Omega
J=\frac{\beta^{2}}{8\pi(1+\sqrt{\varepsilon(2-\varepsilon)})},$ (15)

for an arbitrary value of $\varepsilon$. Taking the limit $\varepsilon\to 0$
gives $I=\beta^{2}/8\pi$ and therefore

$-\beta
F_{\mathrm{spin}}=-\frac{\beta^{2}}{8\pi},~{}~{}~{}~{}F_{\mathrm{spin}}=\frac{\beta}{8\pi}.$
(16)

To interpret this result, we repeat this analysis in the fixed energy
ensemble. In this case, instead of determining $r_{+}$ from $\beta$, $r_{+}$
is determined by the fixed mass $E$. We obtain $r_{+}\sim
E/\sqrt{4\varepsilon}$ in the small $\varepsilon$ limit. In the fixed energy
ensemble the classical action has an extra boundary term $-\beta E$ and the
total answer for the action is

$I_{\rm fixed~{}E}=-\frac{A}{4}-\beta\Omega J\to 2\pi E^{2}.$ (17)

Therefore the entropy in the presence of $(-1)^{\sf F}$, counting the
difference between fermionic and bosonic black hole microstates of mass $E$ in
the semiclassical limit, is given by

$S_{\mathrm{spin}}(E)=2\pi E^{2}=\frac{1}{2}S(E).$ (18)

This implies that if the total number of black hole microstates of energy $E$
grows as $N_{\rm tot}$, then the absolute value of the difference in number of
bosonic minus fermionic states grows as $\sqrt{N_{\rm tot}}$. Notice at the
classical level we cannot determine which statistic has more states. To
determine the overall sign would require evaluating the one-loop determinant.

A possible interpretation for the singular behavior of the flat space
calculation could be the following. The flat space black hole is coupled to
radiation extending to infinity. In flat space, we have the possibility of
introducing very light particles far away from the black hole that carry large
angular momentum. This suggests that in this case it might be subtle to define
a closed system involving the black hole and its environment, for which the
trace $\textrm{Tr}\,(-1)^{\sf F}$ can be defined. Such a problem does not
exist in AdS space, and as we will see in Section 2.2 the calculation is
indeed non-singular.

There is a final point we want to address before moving on. Since we assume
that the theory includes fermionic and bosonic fields, a background can be
probed by operators of half-integer or integer angular momentum. Therefore the
boundary condition in equation (7) only depends on
${\mathrm{i}}\beta\Omega~{}{\rm mod}~{}4\pi\mathbb{Z}$, while the black hole
solution depends on its real valued lift ${\mathrm{i}}\beta\Omega$. This is
resolved by summing over saddles obtained by integer shifts
${\mathrm{i}}\beta\Omega\to{\mathrm{i}}\beta\Omega+4\pi\mathbb{Z}$. What
subset of these saddles should be included is an open question, and affects
the entropy both with and without the insertion of $(-1)^{\sf F}$. We leave
this for future work.

#### Charged black hole

We can extend the previous calculation to black holes in the presence of a
Maxwell field, in a background of fixed electric charge $Q$, which we take to
be positive. The classical solution of Einstein-Maxwell theory we would need
in this case is the Kerr-Newman black hole. We normalize the charge $Q$ such
that the thermodynamic potentials derived from the Kerr-Newman solution are

$\beta=\frac{4\pi
r_{+}(r_{+}^{2}+a^{2})}{r_{+}^{2}-a^{2}-Q^{2}},~{}~{}~{}\Omega=\frac{a}{r_{+}^{2}+a^{2}}$
(19)

while the mass is given in terms of $r_{+}$ as
$E=\frac{r_{+}^{2}+a^{2}+Q^{2}}{2r_{+}}$ and $J=aE$. The saddle contributing
to the grandcanonical partition function, with $\Omega=0$ and therefore no
$(-1)^{\sf F}$ insertion, has $a=0$. This leads to the Euclidean section of
the Reissner-Nordstrom black hole. At fixed temperature, the on-shell action
has an explicit but complicated expression at fixed temperature. The result
simplifies when written in the fixed energy ensemble and results in an entropy

$S(E,Q)=\pi\big{(}E+\sqrt{E^{2}-Q^{2}}\big{)}^{2},$ (20)

the standard Bekenstein-Hawking area law for the Reissner-Nordstrom black
hole. We only consider the black hole solutions with $E\geq Q$.

We can use the gravitational path integral to compute the difference between
the number of bosonic and fermionic black hole microstates. To do this, for
the same reasons as outlined before, we need to set
$\beta\Omega=2\pi{\mathrm{i}}$. This implies, using equation (19) and writing
$a={\mathrm{i}}{\sf a}$, that $2{\sf a}r_{+}=r_{+}^{2}+{\sf a}^{2}-Q^{2}$ and
therefore ${\sf a}=r_{+}-Q$. The horizon radius is fixed in terms of the
inverse temperature as $r_{+}=\frac{Q(\beta-2\pi Q)}{\beta-4\pi Q}.$444The
radius of the horizon is negative for $2\pi Q<\beta<4\pi Q$. One option to
deal with temperatures in this range is to define the geometry along a complex
contour in the $r$-plane that approaches $r_{+}$ from infinity without going
close to the singularity at $r=0$. This type of complex geometry would violate
the criterion put forth in Witten:2021nzp . For $\beta<2\pi Q$, $r_{+}$ is
positive again but a new horizon appears with a larger value of $r$, which
again has to be avoided along the complex plane. If we consider the classical
limit of $\mathcal{N}=2$ supergravity in 4d, these complex geometries seem
necessary to ensure that the index is temperature independent. In this case
the on-shell action appropriate to the fixed temperature ensemble has a very
simple form $I=-\pi Q^{2}+\beta Q$, implying that

$-\beta F_{\mathrm{spin}}=\pi Q^{2}-\beta Q,$ (21)

which is the result found in Iliesiu:2021are .

This solution has two issues. The first is that it does not have a smooth
$Q\to 0$ limit since $r_{+}\to 0$. The second is that for this solution the
energy is $E=Q$ independently of $\beta$. Instead we should be able to compute
the difference between bosonic and fermionic microstates in a fixed energy
ensemble with any $E\neq Q$. This requires finding a different saddle in the
fixed temperature ensemble. To do this set
$\beta\Omega=2\pi{\mathrm{i}}(1-\varepsilon)$ and take the limit
$\varepsilon\to 0$ more carefully. This way we discover a new solution such
that the horizon radius is

$r_{+}\sim\frac{\sqrt{\beta^{2}-(4\pi
Q)^{2}}}{4\pi\sqrt{2\varepsilon}},~{}~{}~{}\varepsilon\ll 1.$ (22)

This solution is similar to the one we found in the uncharged case. Now the
average ADM energy is given by $E=\beta/4\pi$ and becomes a free parameter set
by the temperature (and happens to be independent of the charge). The action
in the fixed temperature ensemble is now $I=\pi Q^{2}+\beta^{2}/8\pi$, or
equivalently

$-\beta F^{\prime}_{\mathrm{spin}}=-\pi Q^{2}-\frac{\beta^{2}}{8\pi}.$ (23)

Notice the sign change of the temperature independent term between
$F_{\mathrm{spin}}$ and $F_{\mathrm{spin}}^{\prime}$. The true free energy
should be the minimum between $F_{\mathrm{spin}}$ and
$F^{\prime}_{\mathrm{spin}}$, which happens to be $F_{\mathrm{spin}}$ for any
temperature.

The goal of Iliesiu:2021are  was to use this type of solution to compute the
index of the black hole in $\mathcal{N}=2$ supergravity. In this case we
expect the solution with $E=Q$ to be the only contribution. The boundary
conditions in this context are supersymmetric and any solution that breaks
supersymmetry (if $E\neq Q$ there is no Killing spinor) will have a fermionic
zero-mode that makes this contribution vanish.555Even if $E=Q$ some
supersymmetries are broken since the Reissner-Nordstrom preserves only four
supercharges, and the contribution to the index also vanishes. One can fix
this considering the helicity supertrace instead of the index, which can be
also computed with the gravitational path integral. The second solution above
with $r_{+}\to\infty$ therefore makes a vanishing contribution to the index in
supergravity due to this fermion zero-modes. If we are working with a non-
supersymmetric theory of gravity, then both solutions should be in principle
included. For any temperature, the black hole with $r_{+}$ finite has always a
lower free energy and is dominant. Nevertheless, when working at fixed energy
$E>Q$ the solution with finite $r_{+}$ disappears and only the solution with
$r_{+}\to\infty$ remains.

Figure 2: Plot of $S_{\mathrm{spin}}/S$ for a charged black hole of energy $E$
as a function of $q=Q/E$. It is always smaller than one and for a given energy
$E$, it interpolates between $1/2$ for $Q=0$ and $1$ for $Q=E$.

Let us then consider entropy in the fixed energy ensemble with the insertion
of $(-1)^{\sf F}$, using the relation between energy and temperature derived
above, $\beta=4\pi E$. The action, including the boundary terms necessary to
change ensemble, is given by $\pi(2E^{2}-Q^{2})$ and therefore the entropy
counting the difference between bosonic and fermionic microstates is

$S_{\mathrm{spin}}(E,Q)=\pi(2E^{2}-Q^{2}).$ (24)

We do not expect any black hole microstate with $E<Q$ regardless of its
statistics, so this result should only be considered for $E>Q$. The ultimate
reason why a complex saddle with $E<Q$ is not allowed has to be tracked back
to the integration contour of the gravitational path integral.

In the uncharged case we found $S_{\mathrm{spin}}(E,Q=0)=\frac{1}{2}\cdot
S(E,Q=0)$. We can now evaluate the ratio between the entropy with and without
a $(-1)^{\sf F}$ insertion and we find the following charge dependent result

$\frac{S_{\mathrm{spin}}(E,Q)}{S(E,Q)}=\frac{2-q^{2}}{(1+\sqrt{1-q^{2}})^{2}}\leq
1,~{}~{}~{}q\equiv\frac{Q}{E},$ (25)

where $0\leq q\leq 1$. We plot this function in figure 2. When $q=0$ we
recover the $1/2$ we found for the uncharged black hole. When $q=1$ we
reproduce the expectation that number of states counted with a $(-1)^{\sf F}$
is the same as the total number of states, and therefore to leading order all
states have the same statistics. (This interpretation is not correct after
including quantum effects which imply the zero-temperature entropy at $E=Q$
does not correspond to a real degeneracy Ghosh:2019rcj ; Iliesiu:2020qvm ;
Heydeman:2020hhw ; Turiaci:2023wrh .) As indicated above, this ratio is always
smaller than one. This is consistent with unitarity since the total number of
states cannot be smaller than the number of bosonic minus fermionic states.

###  2.2 AdS Solutions

####  2.2.1 AdS4

In this section we analyze the quantum statistics of black hole microstates in
asymptotically AdS4 spaces. Through AdS/CFT this puts a constraint on the
quantum statistics of high energy states of holographic CFT3’s.

As explained for example in Chamblin:1999tk  it is possible to construct
backgrounds in string theory where the low energy theory is described by
Einstein-Maxwell in asymptotically AdS4 coupled to matter

$I=-\frac{1}{16\pi}\int{\mathrm{d}}^{4}x\sqrt{g}[R-2\Lambda-F^{2}]+I_{\rm
bdy}.$ (26)

We work in units with $G_{N}=1$. We parametrize the cosmological constant by
$\Lambda=-3/\ell^{2}$ and $I_{\rm bdy}$ are the boundary terms necessary
depending on the ensemble. The black hole solutions are given by the following
generalization of the Kerr-Newman black hole

${\mathrm{d}}s^{2}=-\frac{\Delta_{r}}{\rho^{2}}\Big{[}{\mathrm{d}}t-\frac{a\sin^{2}\theta}{\Xi}{\mathrm{d}}\phi\Big{]}^{2}+\frac{\rho^{2}}{\Delta_{r}}{\mathrm{d}}r^{2}+\frac{\rho^{2}}{\Delta_{\theta}}{\mathrm{d}}\theta^{2}+\frac{\Delta_{\theta}\sin^{2}\theta}{\rho^{2}}\left[a{\mathrm{d}}t-\frac{r^{2}+a^{2}}{\Xi}{\mathrm{d}}\phi\right]^{2}$
(27)

where

$\displaystyle\rho^{2}=r^{2}+a^{2}\cos^{2}\theta,~{}~{}~{}\Xi=1-a^{2}/\ell^{2},$
(28)
$\displaystyle\Delta_{r}=(r^{2}+a^{2})(1+\frac{r^{2}}{\ell^{2}})-2mr+q^{2},~{}~{}~{}\Delta_{\theta}=1-\frac{a^{2}}{\ell^{2}}\cos^{2}\theta.$
(29)

We take the charge to be electric and the gauge field is
$A=-\frac{qr}{\rho^{2}}({\mathrm{d}}t-\frac{a\sin^{2}\theta}{\Xi}{\mathrm{d}}\phi)+\alpha{\mathrm{d}}t$,
where $\alpha$ is a constant chosen to make the gauge connection smooth at the
horizon. The solution is parametrized by $m$, $a$, and $q$, which roughly
correspond to the mass/energy (or temperature), the angular momentum (or
angular velocity), and the charge (or chemical potential).

In the Euclidean section, one can find the inverse temperature and angular
velocity by demanding smoothness at the horizon, located at
$\Delta_{r}(r_{+})=0$, which gives

$\beta=\frac{4\pi(r_{+}^{2}+a^{2})}{r_{+}(1+\frac{a^{2}}{\ell^{2}}+3\frac{r_{+}^{2}}{\ell^{2}}-\frac{a^{2}+q^{2}}{r_{+}^{2}})},~{}~{}~{}\Omega=\frac{a(1+r_{+}^{2}/\ell^{2})}{r_{+}^{2}+a^{2}}.$
(30)

We can trade the parameter $m$ by $r_{+}$ using the relation
$m=\frac{(r_{+}^{2}+a^{2})(1+\frac{r_{+}^{2}}{\ell^{2}})+q^{2}}{2r_{+}}$. The
ADM charges, the energy $E$, angular momentum $J$, and $\text{U}(1)$ charge
$Q$ are given by

$E=\frac{m}{\Xi^{2}},~{}~{}~{}~{}J=aE,~{}~{}~{}~{}Q=\frac{q}{\Xi}.$ (31)

Finally, the area of the even horizon is given by
$A=4\pi(r_{+}^{2}+a^{2})/\Xi$.

Below we consider in detail the uncharged case $q=0$ (the generalization to
$q\neq 0$ is striaghtfoward). The analysis is very similar to the one
presented in the previous section so we will be brief, emphasizing mainly the
differences.

As before, we begin by reminding the reader the results for the free energy
and entropy without an insertion of the fermion parity operator. In the fixed
temperature ensemble we can sum over all states by taking $\Omega=0$ and
therefore the black hole solution has $a=0$. The size of the horizon is
determined through the (inverse) temperature by

$\beta=\frac{4\pi
r_{+}}{1+\frac{3r_{+}^{2}}{\ell^{2}}},~{}~{}~{}r_{+}=\frac{2\pi\ell^{2}}{3\beta}\left(1+\sqrt{1-\frac{3\beta^{2}}{4\pi^{2}\ell^{2}}}\right).$
(32)

This solution only exists for $\beta\leq 2\pi\ell/\sqrt{3}$. The free energy,
computed from the on-shell action $I=\beta E-A/4-\beta\Omega J$ is given by

$\displaystyle-\beta F$ $\displaystyle=$
$\displaystyle\frac{(2\pi\ell^{2}+\sqrt{4\pi^{2}\ell^{4}-3\beta^{2}\ell^{2}})(-3\beta^{2}+\pi(2\pi\ell^{2}+\sqrt{4\pi^{2}\ell^{4}-3\beta^{2}\ell^{2}}))}{27\beta^{2}}$
(33) $\displaystyle\sim$
$\displaystyle\frac{16\pi^{3}\ell^{4}}{27\beta^{2}},~{}~{}~{}~{}\beta\to 0.$
(34)

From the first line we can derive that the black hole dominates over thermal
AdS for temperatures higher than $\beta<\pi\ell$. The second line shows the
result in the high temperature limit. There is a second solution for $r_{+}$
which has higher free energy and therefore never dominates the ensemble.

In the fixed energy ensemble, the black hole radius is constrained by
$E=\frac{r_{+}(r_{+}^{2}+\ell^{2})}{2\ell^{2}}$. There are three solutions of
this equation, one real and two complex conjugate ones. The two complex
solutions have negative real part of the entropy. The real solution has an
entropy given by

$S(E)=\frac{\pi(3^{2/3}\ell^{2}-3^{1/3}\ell^{4/3}(\sqrt{3\ell^{2}+81E^{2}}-9E)^{2/3})^{2}}{9\ell^{4/3}(\sqrt{3\ell^{2}+81E^{2}}-9E)^{2/3}}\sim
2^{2/3}\ell^{4/3}\pi E^{2/3},~{}~{}~{}~{}E\to\infty,$ (35)

where we also indicated the behavior at large energies. In this limit the
radius of the black hole is $r_{+}\sim 2^{1/3}\ell^{2/3}E^{1/3}$.

We are now ready to consider the distribution between fermionic and bosonic
black hole microstates. We begin by computing the free energy
$F_{\mathrm{spin}}$ in the presence of a fermion parity operator insertion
$(-1)^{\sf F}$. Again, we implement this by imposing
$\beta\Omega=2\pi{\mathrm{i}}$. The first observation we can make is that
nothing interesting happens that requires regulating the limit $\beta\Omega\to
2\pi{\mathrm{i}}$, as opposed to the flat space analysis. We find two
solutions for $a={\mathrm{i}}{\sf a}$ given by

${\sf a}=r_{+},~{}~{}~{}~{}{\sf
a}=\frac{r_{+}(\ell^{2}+3r_{+}^{2})}{\ell^{2}-r_{+}^{2}}.$ (36)

The first solution has $\beta=0$ and $E=0$ so it is the same as ${\sf
a}=r_{+}$ in flat space. The second solution has temperature and energy

$\beta=\frac{16\pi\ell^{2}r_{+}^{3}}{3r_{+}^{4}-2\ell^{2}r_{+}^{2}-\ell^{4}},~{}~{}~{}E=-\frac{4r_{+}^{3}\ell^{2}(\ell^{2}-r_{+}^{2})^{2}}{(9r_{+}^{4}-2\ell^{2}r_{+}^{2}+\ell^{4})^{2}}$
(37)

To compute $F_{\mathrm{spin}}$ we focus on the first relation and solve for
$r_{+}(\beta)$. For general $\beta$ and $\ell$, we find four solutions for
$r_{+}$. Two are real and two are complex conjugated, and they are all finite.
In the flat space limit, one of the solution for the black hole radius becomes
the one found in the previous section with $r_{+}\to\infty$. We see therefore
that the background cosmological constant regulates the large $r_{+}$ limit of
the flat space black hole, without the need to go away from
$\beta\Omega=2\pi{\mathrm{i}}$. We find that all these four solutions satisfy,
for any $\beta$ and $\ell$,

${\rm Re}\left[-\beta F_{\mathrm{spin}}\right]<0,$ (38)

and therefore thermal AdS always dominates the ensemble computing
$F_{\mathrm{spin}}$. The possible presence of other saddles which are neither
thermal AdS nor black holes and support periodic fermion boundary conditions
is not ruled out. Assuming this is not the case and the free energy is indeed
of order one, controlled by thermal AdS, it suggests a large cancellation in
$F_{\mathrm{spin}}$ compared with $F$ since there are actually a large number
of black hole states in a given energy window.

Next we consider the entropy $S_{\mathrm{spin}}$, counting black hole states
in the presence of a $(-1)^{\sf F}$ insertion. Using equation (37) we solve
for $r_{+}(E)$. It’s clear by looking at the equation for $E$ that it can be
rewritten as the roots of an eight order polynomial. There are therefore eight
solutions (that cannot be written analytically). We find for positive values
of the energy that the solutions always come in four pairs of complex
conjugated values of $r_{+}$. Out of these four pairs, two pairs have negative
real part of the action while two pairs have positive real part of the action.
We denote the action of these solutions by $I_{1}$ and $I_{2}$, such that
${\rm Re}(I_{1,2})<0$ and ${\rm Im}(I_{1,2})>0$, and their complex conjugate.
Out of the two pairs with negative real part of the action, one pair has the
smallest real part which we choose to be $I_{2}$.

In general, in the presence of two competing saddles with the same real part
we should add their contributions in the total partition function. The one-
loop determinants arising from each saddle are important in writing the total
contribution. In our case the two saddles are related by
${\mathrm{i}}\to-{\mathrm{i}}$ and therefore the one-loop determinants are
related by complex conjugation. The total answer is

$Z\approx Z_{\rm 1-loop}\,e^{-I}+\overline{Z_{\rm
1-loop}}\,e^{-\overline{I}}\approx|Z_{\rm 1-loop}|e^{-{\rm
Re}(I)}\,2\cos\left({\rm Im}(I)+{\rm Arg}\,Z_{\rm 1-loop}\right).$ (39)

This applies whether $Z$ computes the partition function in the fixed energy
or temperature ensemble. As a function of either, the real part of the action
and the absolute value of the one-loop determinant determine the envelope of
the partition function while the imaginary part of the action leads to rapid
oscillations around the envelope. The overall phase of these oscillations
requires knowledge of the one-loop determinants which we do not evaluate here,
but the frequency to leading order is determined by ${\rm Im}(I)$. A similar
feature takes place in the context of the WKB approximation in quantum
mechanics. This conclusion applies to both $I_{1}$ and $I_{2}$.

The pair of solutions with minimal ${\rm Re}(I)$, with an action $I_{2}$,
would naively be the one that dominates the ensemble. Nevertheless, we find it
to have strange properties. For example, it satisfies ${\rm Re}(r_{+})<0$
666One can check that they do not satisfy the Kontsevich-Segal-Witten
criterion Kontsevich:2021dmb ; Witten:2021nzp , see Appendix A for more
discussion. and moreover if we continue this solution smoothly from
$\beta\Omega=2\pi{\mathrm{i}}$ down to $\beta\Omega=0$ it does not become the
familiar black hole that computes the partition function and instead remains
complex. On the other hand the other solution with action $I_{1}$ is less
dominant but has a more reasonable behavior. For example, ${\rm Re}(r_{+})>0$
and becomes the black hole computing the partition function when continued to
$\beta\Omega=0$. Moreover, only this solution becomes the one studied in
Section 2.1 in the flat space limit. For these reasons we conjecture that the
integration contour in the gravitational path integral is such that the
solutions $I_{2}$ do not contribute and instead the pair of complex conjugate
solutions with action $I_{1}$ and ${\rm Re}(r_{+})>0$ dominate the ensemble.

Figure 3: Left: Plot of the envelope of $S_{\mathrm{spin}}/S$ for a black hole
in AdS4 with $\ell=10$ as a function of energy $E$, in blue. It starts at
$1/2$ (flat space result), grows slightly at small energy and then decay to
zero at large energy. Right: Sketch - not to scale - of the rapid fluctuations
as a function of energy in ${\rm Tr}_{E}\,(-1)^{\sf F}$ in blue and the
envelope (black) growing exponentially with $\sqrt{E}$. Even though we can
trust the frequency of oscillations, the overall phase is sensitive to quantum
corrections.

To get some insight on these results we can find the black hole solution
analytically in the large energy limit. Looking at equation (37) the solution
for large $E$ is such that the denominator vanishes
$9r_{\star}^{4}-2\ell^{2}r_{\star}^{2}+\ell^{4}=0$. (As opposed to the
partition function without a $(-1)^{\sf F}$ insertion, large $r_{+}$ leads to
small energies.) This equation has four solutions
$r_{\star}=\pm\frac{\ell}{3}\sqrt{1\pm{\mathrm{i}}2^{3/2}}$ for each choice of
signs. Then we expand $r_{+}=r_{\star}+\delta r$ and solve for $\delta r$ as a
function of $E$. For large $E$, $\delta r$ will be small, so we can solve for
$\delta r$ perturbatively. The solution we propose to keep is the one that
arises from a small fluctuation around
$r_{\star}=\frac{\ell}{3}\sqrt{1\pm{\mathrm{i}}2^{3/2}}\approx
0.47\ell\pm{\mathrm{i}}0.33\ell$. The other solutions have negative real part
for $r_{\star}$. In this limit the action is given by

$I_{1}=c\sqrt{\ell^{3}E}+2\pi{\mathrm{i}}\ell E+\ldots,~{}~{}~{}~{}c\approx
3.81,$ (40)

where the dots denote terms subleading in the large energy limit. The other
saddle in the pair has action
$\overline{I}_{1}=c\sqrt{\ell^{3}E}-2\pi{\mathrm{i}}\ell E$. Since these two
solutions have the same real part of the action they both contribute and
should be added up in the path integral. The result for the entropy in the
presence of $(-1)^{\sf F}$ is therefore

$S_{\mathrm{spin}}(E)\approx c\sqrt{\ell^{3}E}+\log\cos(2\pi\ell E)$ (41)

in the large energy limit. As explained above, knowing the actual phase of the
oscillatory term requires including quantum effects, a problem we leave for
future work. This result explains the reason why the black hole states do not
contribute to the canonical ensemble with the insertion of $(-1)^{\sf F}$. We
find within each energy window the difference between bosonic and fermionic
states is large in magnitude, of order $e^{c\sqrt{\ell^{3}E}}$.777This
expression looks like the Cardy formula in 2d and would be interesting if
there was a two dimensional interpretation of this result from the CFT side.
It might very well be a coincidence. Nevertheless whether most states are
bosonic or fermionic depends also on the energy due to the oscillatory factor
coming from the imaginary part of the action, making the sum over energies
(involved in the fixed temperature ensemble) smaller than each individual
term.

The contribution to the difference between fermionic and bosonic states,
coming from the black hole studied here, is qualitatively consistent with
Cherman:2018mya . The authors of Cherman:2018mya  propose that without
supersymmetry, large $N$ adjoint QCD in four dimensions has a large
cancellation between bosonic and fermionic states. The partition function with
an insertion of $(-1)^{\sf F}$ is found to grow with energy, with a rate
slower than the partition function. This is similar to the prediction we find
here for three dimensional conformal field theories.

####  2.2.2 AdS5

In this section we make some comments regarding the generalization to five
dimensional AdS, relevant to four dimensional conformal field theories. We
consider a low energy theory with an action given by Einstein-Hilbert with
cosmological constant $\Lambda=-6/\ell^{2}$, with $\ell$ being the five-
dimensional AdS radius. For simplicity we consider the case without charge.
The black hole metric with mass and angular momentum is given by

$\displaystyle{\mathrm{d}}s^{2}$ $\displaystyle=$
$\displaystyle-\frac{\Delta_{\theta}(1+\frac{r^{2}}{\ell^{2}})}{\Xi_{a}\Xi_{b}}{\mathrm{d}}t^{2}+\frac{2m}{\rho^{2}}\left(\frac{\Delta_{\theta}{\mathrm{d}}t}{\Xi_{a}\Xi_{b}}-\omega\right)^{2}+\frac{\rho^{2}{\mathrm{d}}r^{2}}{\Delta_{r}}+\frac{\rho^{2}{\mathrm{d}}\theta^{2}}{\Delta_{\theta}}$
(42)
$\displaystyle+\frac{r^{2}+a^{2}}{\Xi_{a}}\sin^{2}\theta{\mathrm{d}}\phi^{2}+\frac{r^{2}+b^{2}}{\Xi_{b}}\cos^{2}\theta{\mathrm{d}}\psi^{2}.$

where we define $\rho^{2}=r^{2}+a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta$ and
the functions

$\displaystyle\Xi_{a}=1-\frac{a^{2}}{\ell^{2}},~{}~{}~{}~{}\Xi_{b}=1-\frac{b^{2}}{\ell^{2}},~{}~{}~{}~{}~{}\omega=a\sin^{2}\theta\frac{{\mathrm{d}}\phi}{\Xi_{a}}+b\cos^{2}\theta\frac{{\mathrm{d}}\psi}{\Xi_{b}},$
$\displaystyle\Delta_{r}=\frac{(r^{2}+a^{2})(r^{2}+b^{2})(1+\frac{r^{2}}{\ell^{2}})}{r^{2}}-2m,~{}~{}\Delta_{\theta}=1-\frac{a^{2}}{\ell^{2}}\cos^{2}\theta-\frac{b^{2}}{\ell^{2}}\sin^{2}\theta.$
(43)

The boundary has the topology of $S^{1}\times S^{3}$ with $S^{1}$ being
thermal time. The spatial three-sphere is parametrized by $\phi$ and $\psi$
which are $2\pi$ periodic and $\theta\in[0,\pi/2]$. When considering a
solution with a Lorenzian interpretation one has to restrict $a^{2}<\ell^{2}$
and $b^{2}<\ell^{2}$. In the Euclidean path integral involved in computing the
free energy or entropy, we allow these parameters to be complex. The two
angular momenta $J_{1}$ and $J_{2}$ generate rotations around $\phi$ and
$\psi$ respectively. The energy and angular momenta for this black hole are
given by

$E=\frac{\pi
m(2\Xi_{a}+2\Xi_{b}-\Xi_{a}\Xi_{b})}{4\Xi_{a}^{2}\Xi_{b}^{2}},~{}~{}J_{1}=\frac{\pi
2am}{4\Xi_{a}^{2}\Xi_{b}},~{}~{}J_{2}=\frac{\pi 2bm}{4\Xi_{a}\Xi_{b}^{2}}.$
(44)

The thermodynamic potentials, the temperature and the two angular velocities
conjugated to the two angular momenta $J_{1}$ and $J_{2}$, are derived from
imposing smoothness at the Euclidean horizon and are given by

$\beta=\frac{2\pi
r_{+}(r_{+}^{2}+a^{2})(r_{+}^{2}+b^{2})}{r_{+}^{4}[1+\ell^{-2}(2r_{+}^{2}+a^{2}+b^{2})]},~{}~{}\Omega_{1}=\frac{a(1+\frac{r_{+}^{2}}{\ell^{2}})}{(r_{+}^{2}+a^{2})},~{}~{}~{}\Omega_{2}=\frac{b(1+\frac{r_{+}^{2}}{\ell^{2}})}{(r_{+}^{2}+b^{2})}.$
(45)

The area of the event horizon that appears in the Bekenstein-Hawking entropy
is given by
$A=\frac{\pi^{2}(r_{+}^{2}+a^{2})(r_{+}^{2}+b^{2})}{2r_{+}\Xi_{a}\Xi_{b}}$.
The on-shell action in the fixed temperature and angular velocity ensemble is
$I=\beta E-A/4-\beta\Omega_{1}J_{1}-\beta\Omega_{2}J_{2}$. In the fixed energy
and angular velocity ensemble, the on-shell action is instead
$I=-A/4-\beta\Omega_{1}J_{1}-\beta\Omega_{2}J_{2}$.

In the fixed temperature ensemble we can compute the free energy with no
insertion of $(-1)^{\sf F}$. The saddle point in this case contributing to the
partition function is a black hole with $a=b=0$ and $r_{+}$ is computed as a
function of inverse temperature $\beta$. The solution is only real for
$\beta<\pi\ell/2$. Moreover the black hole solution dominates the ensemble
only for even higher temperatures $\beta<2\pi\ell/3$. In the range
$2\pi\ell/3<\beta$ thermal AdS dominate the ensemble and therefore $\beta_{\rm
HP}=2\pi\ell/3$ is the Hawking-Page transition corresponding to the
confinement-deconfinement transition of the dual gauge theory Witten:1998zw .
At very high temperatures, the size of the black hole is
$r_{+}\approx\ell^{2}\pi/\beta$ and the free energy is large and negative

$-\beta F\approx\frac{\ell^{6}\pi^{5}}{8\beta^{3}}$ (46)

and in a fixed energy ensemble the entropy grows as $S(E)\sim E^{3/4}$.

We now move on to the computation of $F_{\mathrm{spin}}$ and
$S_{\mathrm{spin}}$. In this case (and really for black holes in any number of
dimensions bigger than four) we have more freedom to decide how to implement
the $(-1)^{\sf F}$ since there are more than one angular momenta. In our
definition $J_{1}$ and $J_{2}$ are two generators that each rotate along an
$\mathbb{R}^{2}\subset\mathbb{R}^{4}$. From the point of view of the field
theory living in the four dimensional boundary of AdS5, Lorentz invariance on
$\mathbb{R}^{4}$ implies that $J_{1}$ and $J_{2}$ are either both half-integer
(corresponding to fermionic states) or both integer (corresponding to bosonic
states). In other words $2J_{1}=2J_{2}~{}{\rm mod}~{}2\mathbb{Z}$. Therefore
we could implement the fermion parity operator either by $(-1)^{\sf
F}=e^{2\pi{\mathrm{i}}J_{1}}$ or $(-1)^{\sf F}=e^{2\pi{\mathrm{i}}J_{2}}$. For
concreteness, we choose to use $J_{1}$ and therefore we can set
$\Omega_{2}=J_{2}=0$ on our black hole.

There are other notions of fermion parity operator in five dimensions. The
group of rotations of $S^{3}$ can be written as ${\rm SO}(4)\approx{\rm
SU}(2)_{L}\times{\rm SU}(2)_{R}$ and the two ${\rm SU}(2)$ factors are
generated by $J_{L}=\frac{J_{1}+J_{2}}{2}$ and $J_{R}=\frac{J_{1}-J_{2}}{2}$.
Therefore one could also define other parity operators such as $(-1)^{{\sf
F}_{\sf L}}=e^{2\pi{\mathrm{i}}J_{L}}$ and $(-1)^{{\sf F}_{\sf
R}}=e^{2\pi{\mathrm{i}}J_{R}}$. The free energy and entropy in the presence of
such insertions can be computed in a similar way, but they would not be
probing the quantum statistics of the black hole microstate which requires the
$(-1)^{\sf F}$ insertion of the previous paragraph.

After these clarifications we implement the insertion of $(-1)^{\sf F}$ by
setting $\beta\Omega_{1}=2\pi{\mathrm{i}}$ and $\beta\Omega_{2}=0$ for
concreteness. The second condition implies $b=0$ and the solution for
$a={\mathrm{i}}{\sf a}$ is

$\frac{\beta\Omega_{1}}{2\pi{\mathrm{i}}}=\frac{{\sf
a}(r_{+}^{2}+\ell^{2})}{2r_{+}^{3}-{\sf
a}^{2}r_{+}+\ell^{2}r_{+}}=1,~{}~{}\Rightarrow~{}~{}{\sf
a}=-\frac{\ell^{2}}{r_{+}}-2r_{+}.$ (47)

(There is also another solution with ${\sf a}=r_{+}$ and with $\beta=E=0$
which we ignore.) With (47), the temperature and energy are given by

$\beta=\frac{2\pi\ell^{2}(3r_{+}^{2}+\ell^{2})}{2r_{+}^{3}+\ell^{2}r_{+}},~{}~{}~{}~{}E=-\frac{\pi(3r_{+}^{2}+\ell^{2})(4r_{+}^{4}+7\ell^{2}r_{+}^{2}+\ell^{4})}{8(4r_{+}^{2}+\ell^{2})^{2}}.$
(48)

In the fixed temperature ensemble we can solve $r_{+}(\beta)$ and compute the
free energy $-\beta F_{\mathrm{spin}}=-\beta E+A/4+\beta\Omega_{1}J_{1}$. In a
fixed energy ensemble we solve instead $r_{+}(E)$ and
$S_{\mathrm{spin}}=A/4+\beta\Omega_{1}J_{1}$.

In the fixed temperature ensemble, we find numerically that all solutions for
$r_{+}$ lead to a free energy ${\rm Re}[-\beta F_{\mathrm{spin}}]<0$ and
therefore thermal AdS5 dominates the ensemble for $F_{\mathrm{spin}}$ at all
temperatures. There is no Hawking-Page transition. This is similar to the
results for AdS4, and it is also qualitatively consistent with Cherman:2018mya
. From the boundary side the Hawking-Page transtition present in the free
energy is interpreted as a confinement-deconfinement phase transtition. On the
other hand the free energy in the presence of $(-1)^{\sf F}$ has no phase
transition and is always in the confined phase regardless of temperature.

In the fixed energy ensemble we find numerically that ${\rm
Re}[S_{\mathrm{spin}}]=0$ for all the solutions for $r_{+}$. Therefore, as
opposed to the result in four dimensions, $S_{\mathrm{spin}}$ is controlled by
the difference between bosonic and fermionic low energy excitations around
thermal AdS, and not from a black hole solution. In this case we expect any
type of bulk spin defect that supports periodic fermions to dominate as long
as it has large enough entropy. We will come back to this later in the
discussion section.

####  2.2.3 AdS3

We now consider the case of black holes in asymptotically AdS3 spaces. We
focus on theories with a gravity sector described by the Einstein-Hilbert
action with a negative cosmological constant, with the AdS3 length given by
$\ell$. The black hole solution in this theory is given by the BTZ metric

${\mathrm{d}}s^{2}=-f{\mathrm{d}}t^{2}+\ell^{2}\frac{{\mathrm{d}}r^{2}}{f}+r^{2}({\mathrm{d}}\phi-\frac{r_{-}r_{+}}{r^{2}}{\mathrm{d}}t)^{2},~{}~{}~{}f=\frac{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}{r^{2}}.$
(49)

This black hole carries energy and angular momentum, given by

$E=\frac{\ell}{8}+\frac{r_{+}^{2}+r_{-}^{2}}{8\ell},~{}~{}~{}J=\frac{r_{+}r_{-}}{4\ell}.$
(50)

To be consistent with the other sections, we define the energy such that
vacuum AdS has $E=0$, which is an unconventional choice in the context of 2d
CFTs. The corresponding inverse temperature and angular velocity are given by

$\beta=\frac{2\pi\ell
r_{+}}{r_{+}^{2}-r_{-}^{2}},~{}~{}~{}~{}~{}\Omega=\frac{r_{-}}{r_{+}}.$ (51)

The area of the horizon is $A=2\pi r_{+}$. It is convenient to work with the
left and right-moving temperatures which are $\beta_{L}=\beta+\beta\Omega$ and
$\beta_{L}=\beta-\beta\Omega$. The on-shell action of the BTZ black hole is
given, in an ensemble of fixed left- and right-moving temperatures, by
$I=\beta E-A/4-\beta\Omega J$, while the action of vacuum AdS is just zero.
The contribution of the BTZ black hole in a fixed temperature and angular
velocity ensemble to the free energy is

$-\beta
F=-\frac{\beta_{L}c}{24}+\frac{c}{24}\frac{4\pi^{2}}{\beta_{L}}-\frac{\beta_{R}c}{24}+\frac{c}{24}\frac{4\pi^{2}}{\beta_{R}},$
(52)

where $c=3\ell/2$. We are working with units where $G_{N}=1$ and therefore the
semiclassical limit corresponds to large AdS radius in Planck units, or $c\gg
1$.

We shall begin by considering the free energy without and with the $(-1)^{\sf
F}$ insertion. If $\beta\Omega=0$ or $2\pi{\mathrm{i}}$ we obtain respectively

$-\beta F=-\frac{\beta
c}{12}+\frac{c}{12}\frac{4\pi^{2}}{\beta},~{}~{}~{}~{}-\beta
F_{\mathrm{spin}}=-\frac{\beta
c}{12}+\frac{c}{12}\frac{4\pi^{2}\beta}{\beta^{2}+4\pi^{2}}.$ (53)

From the expression for $F$ we see that the black hole only contributes for
$\beta<2\pi$, when compared with thermal ${\rm AdS}_{3}$. Instead, $-\beta
F_{\mathrm{spin}}<0$ for any value of the temperature. This means that thermal
AdS with periodic fermions always dominates over the rotating black hole
introduced here, in the canonical ensemble. This situation is similar to AdS
in other dimensions.

We now analyze this problem in the fixed energy ensemble. The result for the
total black hole entropy is the well-known Cardy formula
$S(E)=2\pi\sqrt{\frac{c}{3}(E-\frac{c}{12})}$ for $E\geq\frac{c}{12}$. (Only
for $E\geq c/6$ the BTZ black hole also dominates the fixed temperature
ensemble Hartman:2014oaa .). What is the gravity prediction for
$S_{\mathrm{spin}}$? The procedure is very similar to the one implemented
already so we will be brief. Setting $\beta\Omega=2\pi{\mathrm{i}}$, we find
eight solutions for $r_{+}$ given $E$: four pairs of complex conjugated ones.
Two pairs have ${\rm Re}[S_{\mathrm{spin}}]\leq 0$, and one pair has ${\rm
Re}[S_{\mathrm{spin}}]>0$. The pair with positive entropy gives the leading
contribution but they come with an imaginary part of opposite sign, leading to
a similar oscillatory term as in AdS4. The result is

${\rm
Re}\,[S_{\mathrm{spin}}(E)]=\sqrt{2}\pi\sqrt{\frac{c}{3}E}=\frac{1}{\sqrt{2}}S(E),~{}~{}~{}~{}E\gg
c.$ (54)

and ${\rm Im}\,[S_{\mathrm{spin}}]=\pm 2\pi E$. Therefore even though we
expect the entropy in the presence of $(-1)^{\sf F}$ to be large, the
cancellation found in the fixed temperature ensemble is due to the rapid
oscillations as a function of energy. This situation is completely analogous
to ${\rm AdS}_{4}$.

We would like to emphasize that $(-1)^{\sf F}$ is defined as
$e^{2\pi{\mathrm{i}}J}$ and therefore counts the difference between black hole
microstates with integer or half-integer scaling dimensions. This is different
than the elliptic genus usually computed in two dimensional supersymmetric
theories. The elliptic genus counts the difference between states with even or
odd charges with respect to a ${\rm U}(1)$ gauge field (which could be the
Cartan of a bigger non-Abelian group). When AdS3 arises form string theory,
this charge comes from angular momentum on a compact direction and therefore
can also be interpreted as a different version of a fermion parity operator
similar to the discussion about ${\rm AdS}_{5}$.

##  3 Contribution from wormholes

One might wonder whether wormholes contribute to the quantity
$\rho_{\textrm{spin}}(E)\equiv\textrm{Tr}_{E}(-1)^{\sf F}$. To be more
precise, what we mean is that the semiclassical geometry we discussed in the
previous sections only give a coarse-grained estimation to the quantity
$\rho_{\textrm{spin}}(E)$, which varies smoothly with the energy $E$, in spite
of the oscillation at the scale of $1/\ell$. However, in a microscopic theory,
one expects that on top of the smoothly varying average,
$\rho_{\textrm{spin}}(E)$ also contains a fluctuating piece that depends
sensitively to the precise energy window one chooses. We can quantify the
typical size of the fluctuating piece by studying the connected correlator

$\langle\rho_{\textrm{spin}}(E)\rho_{\textrm{spin}}(E^{\prime})\rangle_{\textrm{c}}\equiv\langle\rho_{\textrm{spin}}(E)\rho_{\textrm{spin}}(E^{\prime})\rangle-\langle\rho_{\textrm{spin}}(E)\rangle\langle\rho_{\textrm{spin}}(E^{\prime})\rangle.$
(55)

Here the bracket $\langle\cdot\rangle$, in a fixed microscopic theory, can be
simply viewed as averaging over a small range of energy $E$ while keeping
$E-E^{\prime}$ fixed. In a theory with only bosons, the correlator (55) is
nothing but the correlator of the full density of state $\rho(E)$. Here we are
interested in theories that have fermions, where the bosonic and fermionic
states contributes to $\rho_{\textrm{spin}}(E)$ with different signs.

Following the insight in Penington:2019kki ; Saad:2018bqo ; Stanford:2020wkf ,
the connected contribution to the average (55) can be quantified by looking at
the contribution from wormholes in the gravitational path integral. In
supersymmetric theories, however, it has been argued in Iliesiu:2021are  that
wormholes do not contribute to the square of the index. However, there is no
argument in non-supersymmetric theories, which we are focusing on here.

In this section, we discuss a universal wormhole contribution to a closely
related quantity

$\langle|Y_{E,\textrm{spin}}(T)|^{2}\rangle_{\textrm{c}}\equiv\langle|\textrm{Tr}_{E}\,(-1)^{\sf
F}e^{-{\mathrm{i}}HT}|^{2}\rangle_{\textrm{c}}$ (56)

where we introduced a factor of unitary time evolution $e^{-{\mathrm{i}}HT}$
into the trace other than the original $(-1)^{\sf F}$ factor. Evidently, this
is the analogue of the spectral form factor Cotler:2016fpe ; Saad:2018bqo
when we don not have the $(-1)^{\sf F}$ insertion and we will therefore call
it the spin spectral form factor.

Figure 4: An illustration of the double cone wormhole geometry, where we are
only drawing the radial and time directions.

The wormhole contribution to the ordinary spectral form factor

$\langle|Y_{E}(T)|^{2}\rangle_{\textrm{c}}\equiv\langle|\textrm{Tr}_{E}\,e^{-{\mathrm{i}}HT}|^{2}\rangle_{\textrm{c}}$
(57)

was discussed in Saad:2018bqo . In particular, in the microcanonical ensemble
where one focuses on states around energy $E$, the gravity solution is the
double cone wormhole which exists universally in any theories containing black
holes. To construct it, one starts with a two-sided black hole geometry with
energy $E$ and perform a quotient in time by period $T$, resulting in a
geometry of the form in figure 4. The resulting geometry has a zero mode
corresponding to the relative time shift between the two sides, which leads to
a linear $T$ dependence in the spectral form factor. The geometry is naively
singular since the time circle shrinks to zero size at the bifurcation
surface, while Saad:2018bqo  proposed a simple prescription to regulate the
geometry by deforming the radial coordinate slightly into the complex plane.
The time circle is non-shrinking everywhere in the geometry. Another feature
of the double cone is that its classical action is zero due to the
cancellation between the left and right sides. Since the action vanishes,
black holes carrying different angular momentum and possible gauge charges
contribute equally (the coefficient multiplying $T$ in the one-loop
determinant is charge independent as well), and one should sum over them. In
the case without any symmetry and the statistics of eigenvalues follow
Gaussian unitary ensemble (GUE) universality, we have888Any unitary Lorentz
invariant theory comes with an additional symmetry that modifies the statement
here. See footnote 10 for more discussion.

$\langle|Y_{E}(T)|^{2}\rangle_{\textrm{c}}\approx\int dE\,\frac{T}{2\pi},$
(58)

where the integral over $E$ runs over the energy window involved in defining
the spectral form factor.

To understand the computation of (56), we first discuss an analogous situation
where we have a $\text{U}(1)$ gauge symmetry which is also interesting in its
own right.

###  3.1 Analogue in the case with a $\text{U}(1)$ gauge symmetry

In a gravity theory with a $\text{U}(1)$ gauge symmetry, we can consider an
analogue of the quantity (56) as

$\langle|Y_{E,\mu}(T)|^{2}\rangle_{\textrm{c}}\equiv\langle|\textrm{Tr}_{E}\,e^{-{\mathrm{i}}(H-\mu
Q)T}|^{2}\rangle_{\textrm{c}},\quad\mu\in\mathbb{R}$ (59)

namely we weight the states carrying charge $Q$ by a pure phase factor
$e^{{\mathrm{i}}\mu QT}$. In gravity, the way to implement this phase factor
is to impose a non-zero boundary condition for the gauge field at infinity,
such that the holonomy of the gauge field satisfies

$e^{{\mathrm{i}}\int dt\,A_{t}}|_{\textrm{bdry}}=e^{{\mathrm{i}}\mu T}.$ (60)

In the gauge that $\partial_{t}A_{t}=0$, (60) determines the boundary value of
the gauge field up to constant shifts

$A_{t}|_{\textrm{bdry}}=\mu+\frac{2\pi n}{T},\quad n\in\mathbb{Z}.$ (61)

Let us focus on the $n=0$ case in the following. The sum over the shifts by
$n$ is important for ensuring the quantization condition for the $\text{U}(1)$
charge, but it is not the main focus here (see Appendix. C for an explicit
example of how it works). We have $A_{t}|_{\textrm{bdry}}=\mu$ and we look for
the analogous double cone geometry satisfying the boundary condition. The
immediate guess will be to take the Lorentzian two-sided charged black hole
solution, with energy given by $E$ and charge $q$ set by chemical potential
$\mu$ through the usual relation in black hole thermodynamics. For example, in
four flat spacetime dimensions, we would be tempting to take

$q=4\pi\mu r_{+}$ (62)

where $r_{+}$ is the radius of the outer horizon. However, even though what is
described above will be a valid double cone geometry that contributes to (59),
it is only one out of an entire family of solutions. In fact, for the double
cone geometry, even after we fixed the boundary value for the gauge field, the
charge $q$ can still freely vary instead of being fixed by (62) or analogous
formulae in other dimensions. To understand this point, we can first ask how
(62) could be derived if one were to consider the standard black hole
thermodynamics. There the geometry in question is a Euclidean black hole, with
the classical solution of gauge field being

$A_{t_{\rm E}}=-\mu+\frac{q}{4\pi r}$ (63)

and the condition (62) comes from requiring that $A_{t_{\rm E}}|_{r=r_{+}}=0$,
namely the gauge field configuration is smooth at the horizon, where the time
circle shrinks to a point. However, for the double cone geometry, the crucial
difference is that the time circle never shrinks in the geometry, so we no
longer have the constraint $A_{t}|_{r=r_{+}}=0$. Saying it in a different way,
the coefficient of the $1/r$ piece in the gauge field - the physical charge
$q$ \- is not determined by $\mu$ and can freely vary.

Physically, the fact we no longer have a map between $q$ and $\mu$ for the
double cone geometry is reasonable based on the expectation that no particular
value of charge $q$ would dominate (59). The same phenomena takes place in the
original double cone with respect to the mass of the black hole: the time
cycle is not contractible when periodically identifying time on the Lorentzian
geometry. Therefore the mass is not fixed by the asymptotic boundary length
and the double cone contributes equally at all energies. Indeed, the double
cone geometry has zero classical action regardless of the value of $q$, so
black holes with all possible charges $q$ contribute equally to (59) at the
classical level. In other words, we have

$\langle|Y_{E,\mu}(T)|^{2}\rangle_{\textrm{c}}\approx\int dE\int
dq\,\frac{T}{2\pi}.$ (64)

For the specific case of JT gravity coupled to a $\text{U}(1)$ gauge field,
the wormhole contribution has been studied in Kapec:2019ecr . In Appendix C we
review the calculation in JT gravity. Above we are simply restating the main
features they found in a general setting. See also Cotler:2022rud  for
discussion on closely related wormhole solutions with a $\text{U}(1)$ gauge
field.

###  3.2 Wormhole contribution to the spin spectral form factor

The lesson in the $\text{U}(1)$ symmetry case can be generalized
straightforwardly to the spin spectral form factor (56). To implement the
$(-1)^{\sf F}$, similar to what we discussed in Section 2, we turn on an
angular potential $\Omega=2\pi/T$ at infinity. In the discussion of Section 2
where we were studying black hole solutions with a single boundary, it was
important that the black hole has particular values of angular momentum $J$
determined by $\Omega$ such that the spin structure is smooth at the horizon.
However, in the case of a wormhole geometry, since the time circle is never
shrinking, we do not have any constraints on the angular momentum $J$ of the
black hole. This is the analogue of the phenomenon that the charge is not
fixed by the chemical potential in the previous section.

As a conclusion, the wormhole contributions to the spin spectral form factor
are simple. One simply consider all the possible black hole geometries with
energy $E$, including those with different angular momentum $J$. The only
difference from the ordinary double cone is that we have a periodic boundary
condition in time for fermionic fields. This change in boundary condition is
invisible at the classical level, but will affect the one-loop fluctuations
around the geometry. However, given that the one-loop determinant of the
matter fields around the double cone geometries goes to one at sufficiently
late time Saad:2018bqo ; CIM ,999On the flip side, the matter fluctuations are
important at early time, so what we said in this section does not generalized
easily to $\langle|\textrm{Tr}_{E}(-1)^{\sf F}|^{2}\rangle_{\rm c}$. we expect
that at large enough $T$ we have

$\langle|\textrm{Tr}_{E}\,(-1)^{\sf
F}e^{-{\mathrm{i}}HT}|^{2}\rangle_{\textrm{c}}\approx\langle|\textrm{Tr}_{E}\,e^{-{\mathrm{i}}HT}|^{2}\rangle_{\textrm{c}}.$
(65)

This is the main result of this section.

###  3.3 Interpretation in terms of random matrix universality

In this section we give an interpretation of the fact that at late times the
spectral form factor is equal with or without the $(-1)^{\sf F}$ insertion,
see equation (65), in terms of late time quantum chaos and random matrix
universality.

To simplify the discussion let us assume that the only symmetry of the random
matrix ensemble is the presence of a $\mathbb{Z}_{2}$ symmetry generated by
$(-1)^{\sf F}$. This implies that the Hilbert space can be decomposed into a
bosonic and fermionic sector
${\mathcal{H}}={\mathcal{H}}_{b}\oplus{\mathcal{H}}_{f}$ such that both
$(-1)^{\sf F}$ and the Hamiltonian can be written as $2\times 2$ blocks:

$(-1)^{\sf F}=\left(\begin{array}[]{c|c}1&0\cr\hline\cr
0&-1\end{array}\right),~{}~{}~{}~{}~{}~{}H=\left(\begin{array}[]{c|c}H_{b}&0\cr\hline\cr
0&H_{f}\end{array}\right).$ (66)

The appropriate random matrix ensemble therefore is to take $H_{b}$ and
$H_{f}$ to be statistically independent random matrices, assuming the absence
of time reversal symmetry. This situation was studied in the context of JT
gravity and in particular the connection with wormholes in Stanford:2019vob ;
Kapec:2019ecr . The fact that the two sectors are statistically independent
imply in particular that

$\left\langle{\rm Tr}_{{\mathcal{H}}_{b}}e^{-\beta_{1}H}\,\,{\rm
Tr}_{{\mathcal{H}}_{f}}e^{-\beta_{2}H}\right\rangle_{\rm c}=0,$ (67)

which we will use below. Finally, we did not specify yet which ensemble
$H_{b}$ and $H_{f}$ are drawn from. As explained in section 4 of Yan:2023rjh ,
the ${\sf CPT}$ theorem implies that the bosonic sector is in the orthogonal
ensemble (GOE) while fermionic sector is in the symplectic ensemble
(GSE).101010Following the notation of Witten:2023snr , the ${\sf CPT}$ theorem
says that the operator ${\sf R}{\sf T}$ is a symmetry of any unitary Lorentz
invariant theory. ${\sf R}$ generates a transformation that reverses the sign
of one coordinate (any) and ${\sf T}$ is the time reversal operator defined
such that it anticommutes with conserved charges. In Euclidean signature ${\sf
RT}$ generates a rotation of $180^{o}$ in a plane determined by ${\sf R}$ and
the time direction. The fact that for a full rotation $({\sf
RT})^{2}=(-1)^{\sf F}$ implies the appropriate ensemble for the bosonic and
fermionic sectors.  This will not affect much the results below.

Let us first evaluate the partition function. Decomposing the Hilbert space
into a bosonic and fermionic sector implies that for each realization of the
Hamiltonian we can write

${\rm Tr}_{{\mathcal{H}}}\,e^{-\beta H}={\rm
Tr}_{{\mathcal{H}}_{b}}\,e^{-\beta H}+{\rm Tr}_{{\mathcal{H}}_{f}}\,e^{-\beta
H}.$ (68)

The connected contribution to the product of two partition functions is
therefore given by

$\left\langle{\rm Tr}_{{\mathcal{H}}}e^{-\beta_{1}H}\,\,{\rm
Tr}_{{\mathcal{H}}}e^{-\beta_{2}H}\right\rangle_{\rm c}=\left\langle{\rm
Tr}_{{\mathcal{H}}_{b}}e^{-\beta_{1}H}\,\,{\rm
Tr}_{{\mathcal{H}}_{b}}e^{-\beta_{2}H}\right\rangle_{\rm c}+\left\langle{\rm
Tr}_{{\mathcal{H}}_{f}}e^{-\beta_{1}H}\,\,{\rm
Tr}_{{\mathcal{H}}_{f}}e^{-\beta_{2}H}\right\rangle_{\rm c}$ (69)

where we use that the mixed terms vanish due to the statistically independence
of the sectors. The right hand side is a universal quantity in the large rank
limit of random matrices but we will not need its details below.

Next we evaluate the partition function with a $(-1)^{\sf F}$. In terms of the
decomposition into bosonic and fermionic states the answer for each
realization is now

${\rm Tr}_{{\mathcal{H}}}\,(-1)^{\sf F}e^{-\beta H}={\rm
Tr}_{{\mathcal{H}}_{b}}\,e^{-\beta H}-{\rm Tr}_{{\mathcal{H}}_{f}}\,e^{-\beta
H}.$ (70)

When computing the connected contribution to the product of two partition
functions, the minus sign that appears above is irrelevant since the sectors
are statistically independent

$\displaystyle\left\langle{\rm Tr}_{{\mathcal{H}}}(-1)^{\sf
F}e^{-\beta_{1}H}\,\,{\rm Tr}_{{\mathcal{H}}}(-1)^{\sf
F}e^{-\beta_{2}H}\right\rangle_{\rm c}$ $\displaystyle=$
$\displaystyle\left\langle{\rm Tr}_{{\mathcal{H}}_{b}}e^{-\beta_{1}H}\,{\rm
Tr}_{{\mathcal{H}}_{b}}e^{-\beta_{2}H}\right\rangle_{\rm c}$ (71)
$\displaystyle+\left\langle{\rm Tr}_{{\mathcal{H}}_{f}}e^{-\beta_{1}H}\,{\rm
Tr}_{{\mathcal{H}}_{f}}e^{-\beta_{2}H}\right\rangle_{\rm c}.$

Therefore regardless of the details of the right hand side, we find that the
answer is identical in both cases with or without the $(-1)^{\sf F}$
insertion:

$\left\langle{\rm Tr}_{{\mathcal{H}}}(-1)^{\sf F}e^{-\beta_{1}H}\,\,{\rm
Tr}_{{\mathcal{H}}}(-1)^{\sf F}e^{-\beta_{2}H}\right\rangle_{\rm
c}=\left\langle{\rm Tr}_{{\mathcal{H}}}e^{-\beta_{1}H}\,\,{\rm
Tr}_{{\mathcal{H}}}e^{-\beta_{2}H}\right\rangle_{\rm c}$ (72)

After analytically continuing in $\beta$ and going to the microcanonical
ensemble, this results is equivalent to equation (65) derived as a consequence
of the evaluation of the double cone contribution to the gravitational path
integral. The late time behavior of either term on the right hand side of (69)
has the same behavior as the double cone. First, it is a quantity that does
not grow with the rank of the matrix and correspondingly the double cone on-
shell action vanishes. Second, it has a late time ramp with a linear growth in
$T$.

So far we incorporated the $(-1)^{\sf F}$ as a generator of a $\mathbb{Z}_{2}$
symmetry but did not take into account the fact that it is embedded in a
bigger group of rotations in higher dimensions. At late times when the double
cone becomes reliable, this approximation for the random matrix ensemble was
good enough to reproduce the relation (65). Nevertheless the model here so far
does not reproduce the correct prefactor of the linear-in-$T$ term on either
side of equation (65). Equation (69) implies the coefficient of the linear-
in-$T$ term for a theory with a $\mathbb{Z}_{2}$ symmetry is twice the result
for a theory without such a symmetry. According to the discussion in the
previous section, the spectral form factor gets multiplied not by $2$ but by a
divergent factor given by the sum over the possible angular momenta of the
black hole used to generate the double cone. This factor is divergent but can
be regulated by working in an ensemble where the angular momentum is specified
in a finite window (which would be the natural way to generalize fixing an
energy window that regulates the integral over energies in (58)).

A more accurate ensemble incorporates the rotation group $G$ (for example
$G={\mathrm{SU}}(2)$ in four dimensions or
$G={\mathrm{SU}}(2)\times{\mathrm{SU}}(2)$ in five dimensions). The Hilbert
space is decomposed into irreducible representations of $G$. For example the
case relevant to four dimensions is
${\mathcal{H}}=\oplus_{J=0,\frac{1}{2},1,\ldots}{\mathcal{H}}_{J}$, where each
${\mathcal{H}}_{J}$ represents an irreducible $(2J+1)$-dimensional
representation of $G={\mathrm{SU}}(2)$.111111Here $J$ labels the eigenvalue of
$\vec{J}^{2}$ which is fixed within each representation. In the previous
sections we used $J$ to denote the eigenvalue of $\vec{J}\cdot\vec{n}$
instead, the eigenvalue of the angular momentum itself along a given
direction. Since the Hamiltonian commutes with the generators of $G$, it is
block diagonal according to the decomposition of the Hilbert space into
sectors transforming in fixed representations, and consists of statistically
independent random matrices in each sector. We can embed the $\mathbb{Z}_{2}$
group generated by $(-1)^{\sf F}$ inside $G$. Moreover, now the ${\sf CPT}$
theorem implies the existence of an ${\sf RT}$ symmetry that squares to
$(-1)^{\sf F}$ and in even dimensions (after possibly combining with a
rotation) anticommutes with the generator of rotations $\vec{J}$, so even-spin
sectors are GOE and odd-spin are GSE.121212This is true for ${\mathrm{SU}}(2)$
since all representations are real. If the group $G$ has complex
representations then such sectors would remain GUE.

An insertion of $(-1)^{\sf F}$ introduces a minus sign in the partition
function for all representations of $G$ that are odd under $(-1)^{\sf F}$ (for
example half-integer $J$’s for $G={\mathrm{SU}}(2)$). Since different
representations are statistically independent, this sign disappears when
evaluating the spectral form factor for essentially the same reason as the
$\mathbb{Z}_{2}$ case studied earlier in this section. The main difference is
that now the right hand side of (69) becomes not a sum over an even and odd
sector, but a sum over a contribution from all angular momenta appearing in
all representations of $G$. Working this out in more detail (following e.g.
Kapec:2019ecr ) gives an extra factor of

$2\sum_{J=0,\frac{1}{2},1,\ldots}(2J+1)^{2},$

multiplying the value of the ramp for a theory without any symmetry. The
factor of $2$ comes from the ensemble being GOE/GSE while the $(2J+1)^{2}$
factor comes from the ${\mathrm{SU}}(2)$ structure. This sum over spins gives
a divergent factor (which can be regulated working in an ensemble where the
angular momentum is restricted to a window $\delta J$ around an average value
$J$) that is now at least qualitatively the same that we identified in the
gravity calculation in Section 3.1 and 3.2. We leave a more precise match of
this normalization (involving a more careful analysis of quantum effects
around the double cone in the presence of rotation) for future work.

##  4 Discussion

In this article we explained how to use the gravitational path integral to
estimate the difference between the number of bosonic and fermionic black hole
microstates. In particular, we focused on a contribution coming from the
complex rotating black hole saddle point. This is a universal contribution
that does not depend on the specific matter content of the theories, similar
to the black hole that contributes to the ordinary partition function. We
discussed its contribution for black holes in various dimensions, as well as
the case of charged black holes in four dimensional flat space. We also
describe wormhole contributions to the quantities we are computing similar to
the double cone of Saad:2018bqo . In this section we will make some further
comments and point out some interesting directions.

##### Purely bosonic theories

We stated at the beginning of Section 2 that we are considering theories that
contain fermions. However, one might be puzzled about where this assumption
came into our discussion. Indeed, we never used the explicit details about the
matter sector and only used the black hole geometries. On the other hand, our
calculation must fail in a purely bosonic theory of quantum gravity (if it
exists), since we are finding that the entropy with $(-1)^{\sf F}$ insertion
is different from the entropy, meaning that there exists at least some
fermionic states.

In fact, the resolution to this puzzle is simple and instructive. In a theory
with only bosons, turning on an angular potential
$\Omega=2\pi{\mathrm{i}}/\beta$ should be viewed as trivial, since all the
states are invariant under a rotation by angle $2\pi$. However, this
invariance is not explicit in terms of individual bulk saddle points. Instead,
it is enforced by summing over saddles with shifts of $\Omega$ by
$2\pi{\mathrm{i}}n/\beta,\,n\in\mathbb{Z}$. As a consequence, in both the
calculation of $Z$ and $Z_{\textrm{spin}}$ we are summing over saddles with
$\Omega=2\pi{\mathrm{i}}\mathbb{Z}/\beta$, so we trivially have
$Z=Z_{\textrm{spin}}$, consistent with the fact that the theory does not have
fermions. In such theories, the rotating black holes analyzed in this paper
are simply subleading contributions to the partition function.

So we see that our calculation relies on the assumption about the existence of
fermions. In particular, it cannot be used to argue that a quantum gravity
theory that contains black holes must have fermions. It would be nice to know
whether this is true, and if so whether one can find some other argument for
this statement.

##### Other contributions to $\textrm{Tr}\,(-1)^{\sf F}$

In this paper, we have only considered universal contributions to the quantity
$\textrm{Tr}\,(-1)^{\sf F}$, either in the canonical or microcanonical
ensemble. Our result should really be thought of as giving a _lower bound_ to
the size of this quantity, since in general there could be other more dominant
contributions that depends on the details of the theory. (An upper bound,
automatic from the boundary but not obvious from the bulk, is the partition
function without the $(-1)^{\sf F}$.) One particular possibility is that a
theory might contain codimension two defects which implements $(-1)^{\sf F}$
when one brings an operator around it. Such a defect can be placed at the tip
of a Euclidean black hole such that it implements $(-1)^{\sf F}$ when going
around the thermal circle, and it will give rise to a solution that computes
$\textrm{Tr}\,(-1)^{\sf F}$.131313We thank Miguel Montero for discussing this
point. This is a special case of the general story for black holes carrying
discrete gauge charges Coleman:1991ku .

The detailed property of such a defect is a theory dependent question.
However, there is an interesting construction of such a defect that only
requires the knowledge of the low energy spectrum Arkani-Hamed:2007ryu
.141414We thank Juan Maldacena for this suggestion. One imagines that instead
of having a Euclidean time circle which shrinks to zero at the horizon, it is
stabilized to a finite radius by the Casimir energy coming from the light
fields, in a similar way as described in Arkani-Hamed:2007ryu . Given that the
circle remains finite size, we are then free to choose the periodic boundary
condition for the fermions, without the need to worry about the smoothness of
the spin structure. Of course, this choice of boundary condition itself
affects the Casimir energy one uses to find the solution.

The intriguing aspect of such a solution is that it connects the properties of
low energy spectrum to some properties of very high energy spectrum. We hope
to return to the details of these solutions in the future.

##### Expectation from field theories

We studied a universal contribution from gravity to the difference between
bosonic and fermionic states, using AdS/CFT. This raises the question of what
expectations do we have from field theory for this quantity.

The first obvious one is that the difference between bosonic and fermionic
states cannot be large than the sum of them. This is obvious from field theory
but becomes a non-trivial constraint from gravity. In particular, all the
universal rotating black hole solutions we found satisfy this property. We
found that $Z_{\mathrm{spin}}$, the partition function with a $(-1)^{\sf F}$
insertion, is exponentially subleading in the large $N$ limit compared to $Z$,
the partition function without the insertion. This phenomenon was discussed in
Cherman:2018mya  (see also references therein) for large $N$ adjoint QCD,
where the authors find a large cancellation between bosonic and fermionic
states. Our results were compared with theirs in the last paragraph of Section
2.2.1.

A more interesting constraint comes from the thermal effective field theory
put forth in Benjamin:2023qsc . This predicts a specific dependence of the
free energy with temperature and angular velocity. Their result was derived
without the insertion of $(-1)^{\sf F}$. In the presence of such insertion one
would expect their thermal effective theory still applies although the
specific value of the effective theory parameters can change. Benjamin:2023qsc
also uses this free energy to extract the microcanonical density of states.
Using the rotating black hole for the we find $F_{\mathrm{spin}}$ is
controlled by thermal AdS, but in some cases like AdS4 the microcanonical
density of states $S_{\mathrm{spin}}$ is controlled by the black hole.
Furthermore, $S_{{\mathrm{spin}}}$ grows only as $\sqrt{E}$ and is
subextensive in the volume ($S_{\textrm{spin}}\propto V^{\frac{3}{4}}$). It
would be interesting to explore to what extent this is a violation of the
thermal effective field theory.

Acknowledgements We thank Tom Hartman, Shota Komatsu, Henry Lin, Juan
Maldacena, Miguel Montero, Sridip Pal, Douglas Stanford, Edward Witten for
discussion. We specially thank Juan Maldacena for initial collaboration. YC is
supported by a Procter Fellowship from Princeton University. GJTs work was
supported by the Institute for Advanced Study and the NSF under Grant No.
PHY-2207584, and by the Sivian Fund, and currently by the University of
Washington and the DOE award DE-SC0024363.

##  Appendix A The Kontsevich-Segal-Witten criterion for complex Kerr saddles

In this appendix, we apply the Kontsevich-Segal-Witten (KSW) criterion for
complex metrics Kontsevich:2021dmb ; Witten:2021nzp  to the complex Kerr black
hole solutions that are relevant for the discussion in this paper. The KSW
criterion selects reasonable gravitational saddle points to be included in the
gravitational path integral, which is derived by demanding that the
fluctuations of various quantum fields on the background are suppressed. To
what extent it is a strict rule that one should follow is still an open
question.151515For example, in the derivation it was assumed that the
fluctuations of the quantum fields are integrated along real coutours. If one
allows for deformations of the integration contours for the matter fields,
then the criterion will be weakened. See also Maldacena:2019cbz ; Bah:2022uyz
for some recently discussed geometries that violate the criterion but
nonetheless lead to physically sensible results.

Concretely, the criterion states that for a complex metric $g$ on a $D$
dimensional manifold, if one picks a real basis such that the metric is
diagonal

$g_{ij}=\lambda_{i}\delta_{ij},\quad i,j=1,...,D,$ (73)

then the KSW criterion demands that at every point of the manifold we have

$\sum_{i=1}^{D}|\textrm{Arg}(\lambda_{i})|<\pi.$ (74)

In this appendix we discuss how the analysis can be done for either the Kerr
metric in 4d flat space or Anti-de Sitter space. The discussion can be
generalized other dimensions straightforwardly.

The 4d Kerr black hole metric in flat space is

$\displaystyle{\mathrm{d}}s^{2}=\frac{\rho^{2}\Delta}{\Xi}{\mathrm{d}}t_{E}^{2}+\frac{\rho^{2}}{\Delta}{\mathrm{d}}r^{2}+\rho^{2}{\mathrm{d}}\theta^{2}+\frac{\Xi}{\rho^{2}}\sin^{2}\theta\left({\mathrm{d}}\widetilde{\varphi}+{\mathrm{i}}\left(\frac{2Ear}{\Xi}-\Omega\right){\mathrm{d}}t_{E}\right)^{2}$
(75)

where the definition of various functions involved are given in (2.1). Note
that we’ve introduced coordinate $\widetilde{\varphi}$ such that the
coordinates are identified as
$(t_{E},\widetilde{\varphi})\sim(t_{E}+\beta,\widetilde{\varphi})\sim(t_{E},\widetilde{\varphi}+2\pi)$.
The metric is characterized by two parameters $(a,r_{+})$. The case of
$r_{+},a\in\mathbb{R}$ was already analyzed in Witten:2021nzp , with the
conclusion being that the metric is disallowed under the criterion. The
problem comes from that the imaginary piece in (75), which leads to a negative
$g_{t_{E}t_{E}}$ component of the metric at large radius. The physical
interpretation is simple - in flat space the fluctuations corresponding to
adding particles far away from the black hole carrying large angular momenta
are not suppressed. The situation becomes more complicated when we allow both
$r_{+}$ and $a$ to be complex. In this case, it is not only the far away
region we need to worry about, the near horizon region could also become
dangerous. Even though the KSW criterion demands that (74) should be satisfied
everywhere on the manifold, the full analysis appears complicated and here we
will only focus on the near horizon region as well as the asymptotic region,
where the analysis can be simplified. We will see that simply from these two
limits we can already get interesting constraints on $r_{+},a$.

We first focus on the near horizon part of the geometry. The analysis here is
similar to the one for complexified non-rotating black holes in Chen:2022hbi .
By introducing new coordinates

${\mathrm{d}}\varrho=\frac{{\mathrm{d}}r}{\sqrt{\Delta}},\quad{\mathrm{d}}u=\frac{2\pi{\mathrm{d}}t_{E}}{\beta},\quad
u\sim u+2\pi,$ (76)

the near horizon metric can be put into the following form

${\mathrm{d}}s^{2}=(r_{+}^{2}+a^{2}\cos^{2}\theta)\left({\mathrm{d}}\varrho^{2}+\varrho^{2}{\mathrm{d}}u^{2}\right)+(r_{+}^{2}+a^{2}\cos^{2}\theta){\mathrm{d}}\theta^{2}+\frac{r_{+}^{2}+a^{2}}{r_{+}^{2}+a^{2}\cos^{2}\theta}\sin^{2}\theta\,{\mathrm{d}}\widetilde{\varphi}^{2}.$
(77)

Note that the off-diagonal term ${\mathrm{d}}u{\mathrm{d}}\widetilde{\varphi}$
will be of order $\varrho^{2}$ so can be ignored in the near horizon limit. We
can absorb the factor $(r_{+}^{2}+a^{2}\cos^{2}\theta)$ into
${\mathrm{d}}\varrho^{2}$ and therefore the radial plus time part of the
metric can take a form that is completely real. As explained in Chen:2022hbi ,
this is always achievable due to the smoothness of the geometry at the
horizon. Therefore, the only nontrivial phases that will enter (74) comes from
${\mathrm{d}}\theta^{2}$ and ${\mathrm{d}}\widetilde{\varphi}^{2}$ parts of
the metric. Concretely, (74) requires

$\left|\textrm{Arg}\left(r_{+}^{2}+a^{2}\cos^{2}\theta\right)\right|+\left|\textrm{Arg}\left(\frac{r_{+}^{2}+a^{2}}{r_{+}^{2}+a^{2}\cos^{2}\theta}\right)\right|<\pi$
(78)

for any $\theta\in[0,\pi)$. We could check that this is satisfied by the
solutions we discussed in Section 2.1, since there we have
$a={\mathrm{i}}\zeta r_{+}$, with $r_{+}\in\mathbb{R}$ and $\zeta$ taken to
one from below.

We can also verify that our metric is allowable in the asymptotic region. The
leading behavior of the metric at $r\rightarrow\infty$ takes the following
form

${\mathrm{d}}s^{2}=-\frac{\beta^{2}}{4\pi^{2}}\Omega^{2}r^{2}\sin^{2}\theta{\mathrm{d}}u^{2}-2{\mathrm{i}}r^{2}\frac{\beta}{2\pi}\Omega\sin^{2}\theta{\mathrm{d}}u{\mathrm{d}}\widetilde{\varphi}+{\mathrm{d}}r^{2}+r^{2}{\mathrm{d}}\theta^{2}+r^{2}\sin^{2}\theta{\mathrm{d}}\widetilde{\varphi}^{2}.$
(79)

For the solution we considered in Section 2.1, we have
$\beta\Omega=2\pi{\mathrm{i}}$, which leads to a real metric in (79) with
positive coefficients in front of ${\mathrm{d}}u^{2}$ and
${\mathrm{d}}\widetilde{\varphi}^{2}$. From the general discussion in
Witten:2021nzp  we know that such a metric is allowable.

For the AdS4 case discussed in Section 2.2.1, the analysis is similar to the
flat space case. The near horizon region imposes a family of constraints that
says for any $\theta$ we should have

$\left|\textrm{Arg}\left(\frac{r_{+}^{2}+a^{2}\cos^{2}\theta}{\ell^{2}-a^{2}\cos^{2}\theta}\right)\right|+\left|\textrm{Arg}\left(\frac{(r_{+}^{2}+a^{2})(\ell^{2}-a^{2}\cos^{2}\theta)}{(r_{+}^{2}+a^{2}\cos^{2}\theta)(\ell^{2}-a^{2})^{2}}\right)\right|<\pi.$
(80)

We’ve checked that the complex solution we considered in Section 2.2.1 satisfy
(80), while the solutions we dropped do not. Asymptotically, the metric (27)
behaves as

${\mathrm{d}}s^{2}\approx\frac{\ell^{2}}{r^{2}}{\mathrm{d}}r^{2}+\frac{r^{2}}{\ell^{2}}\frac{\beta^{2}}{4\pi^{2}}{\mathrm{d}}u^{2}+{\mathrm{i}}\frac{\beta}{2\pi}\frac{r^{2}}{\ell^{2}}\frac{2a\sin^{2}\theta}{1-a^{2}/\ell^{2}}{\mathrm{d}}u{\mathrm{d}}\phi+\frac{r^{2}\sin^{2}\theta}{1-a^{2}/\ell^{2}}{\mathrm{d}}\phi^{2}+\frac{r^{2}}{1-\frac{a^{2}}{\ell^{2}}\cos^{2}\theta}{\mathrm{d}}\theta^{2}$
(81)

Since the metric contains off-diagonal terms, one has to first find a real
basis in which the metric is diagonal and then apply the criterion (74)
Witten:2021nzp . We performed this exercise and found that the solution we
considered in (2.2.1) is also allowable in the asymptotic region.

##  Appendix B Smoothness of the Kerr black hole

In this section we explain how the Kerr black hole found in Section 2.1 is
smooth even though it requires taking a limit $r_{+}\to\infty$. The curvature
squared of the Kerr metric is given by

$R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}=\frac{48E^{2}(r^{2}-a^{2}\cos^{2}\theta)[(r^{2}+a^{2}\cos^{2}\theta)^{2}-16r^{2}a^{2}\cos^{2}\theta)]}{(r^{2}+a^{2}\cos^{2}\theta)^{6}}.$
(82)

Since after setting $\beta\Omega$ to be pure imaginary the metric is real in
Euclidean signature, we use a criterion for smoothness that this quantity is
finite $|R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}|<\infty$. The calculation is
straightfoward so we will simply quote the result. As we take the
$\varepsilon\to 0$ limit of the solution in Section 2.1 we find that for
generic spacetime points the curvature squared vanishes
$|R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}|\sim\mathcal{O}(\varepsilon^{6})$.
The only exception is at the north and south pole $\theta=0,\pi$ at the
horizon $r=r_{+}$, where we find
$|R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}|=24/E^{4}+\mathcal{O}(\varepsilon^{1})$,
which is still finite.

##  Appendix C Review of spectral form factor in JT gravity with a
$\text{U}(1)$ gauge field

Here we review the calculation of the spectral form factor in a random matrix
model with $\text{U}(1)$ global symmetry, dual to JT gravity coupled to a
$\text{U}(1)$ Maxwell theory. The partition functions for general Riemann
surfaces for this theory was discussed in Kapec:2019ecr . Here we review their
story for the two boundary wormhole, and in particular, the computation where
we focus on a microcanonical window around energy $E$, with width $\Delta$. We
will work in the unit that the length scale $\phi_{r}$ governing the
Schwarzian fluctuations in JT is one.

We start with the expression for the double trumpet in this theory
Kapec:2019ecr

$Z(\beta_{L},\mu_{L};\beta_{R},\mu_{R})=2\pi\int_{0}^{2\pi}{\mathrm{d}}\phi\int_{0}^{\infty}{\mathrm{d}}b\,b\,Z_{tr}(\beta_{L},\mu_{L};b,\phi)Z_{tr}(\beta_{R},\mu_{R};b,-\phi),$
(83)

where $Z_{tr}(\beta,\mu;b,\phi)=Z_{tr}^{gauge}Z_{tr}^{grav}$, with
$Z_{tr}^{grav}$ being the ordinary JT trumpet partition function

$Z_{tr}^{grav}(\beta)=\frac{1}{\sqrt{2\pi\beta}}\exp\left[-\frac{b^{2}}{2\beta}\right]$
(84)

and

$\displaystyle Z_{tr}^{gauge}(\beta,\mu,\phi)$
$\displaystyle=\frac{1}{\sqrt{4\pi\beta}}\sum_{n\in\mathbb{Z}}\exp\left[-\frac{1}{2\beta}(2\pi
n-{\mathrm{i}}\mu\beta-\phi)^{2}\right]$ (85)
$\displaystyle=\frac{1}{\sqrt{8\pi^{2}}}\sum_{n\in\mathbb{Z}}\int{\mathrm{d}}q\,\exp\left[q(\beta\mu+2\pi{\mathrm{i}}n-{\mathrm{i}}\phi)\right]\exp\left[-\beta\frac{q^{2}}{2}\right].$

The variable $\phi$ in (83) and (85) comes from the holonomy of the gauge
field at the throat of the wormhole. Note that in the second line of (85), we
could further sum over $n$ which sets $q$ to be integers, but we will defer
this step such that the procedure is more analogues to the general story
described in Section 3.1. Taking (85) into (83), we get

$\displaystyle Z(\beta_{L},\mu_{L};\beta_{R},\mu_{R})=$
$\displaystyle\frac{1}{8\pi^{2}}\sum_{n_{L},n_{R}}\int{\mathrm{d}}q_{L}\int{\mathrm{d}}q_{R}\int_{0}^{2\pi}{\mathrm{d}}\phi\int_{0}^{\infty}{\mathrm{d}}b\,b\frac{1}{\sqrt{\beta_{L}\beta_{R}}}\exp\left[-\frac{b^{2}}{2\beta_{L}}-\frac{b^{2}}{2\beta_{R}}\right]$
(86)
$\displaystyle\exp\left[\beta_{L}\mu_{L}q_{L}+\beta_{R}\mu_{R}q_{R}\right]\exp\left[-\beta_{L}\frac{q_{L}^{2}}{2}-\beta_{R}\frac{q_{R}^{2}}{2}\right]$
$\displaystyle\exp\left[q_{L}(2\pi{\mathrm{i}}n_{L}-{\mathrm{i}}\phi)+q_{R}(2\pi{\mathrm{i}}n_{R}+{\mathrm{i}}\phi)\right]$

Redefine $n=n_{R},m=n_{L}+n_{R}$, we can rewrite the exponent on the last line
of (86) as

$q_{L}\left(2\pi{\mathrm{i}}(m-n)-{\mathrm{i}}\phi\right)+q_{R}\left(2\pi{\mathrm{i}}n+{\mathrm{i}}\phi\right)=2\pi{\mathrm{i}}mq_{L}+(2\pi{\mathrm{i}}(n+\phi))(q_{R}-q_{L}).$
(87)

We can combine the sum over $n$ and the integral over $\phi$ into a single
integral of $n+\phi$ over the real axis, which imposes that $q_{L}=q_{R}$.
Then we have

$\displaystyle Z(\beta_{L},\mu_{L};\beta_{R},\mu_{R})=$
$\displaystyle\frac{1}{8\pi^{2}}\sum_{m}\int{\mathrm{d}}q\int_{0}^{\infty}{\mathrm{d}}b\,b\frac{1}{\sqrt{\beta_{L}\beta_{R}}}\exp\left[-\frac{b^{2}}{2\beta_{L}}-\frac{b^{2}}{2\beta_{R}}\right]$
(88)
$\displaystyle\exp\left[q(\beta_{L}\mu_{L}+\beta_{R}\mu_{R}+2\pi{\mathrm{i}}m)\right]\exp\left[-\beta_{L}\frac{q^{2}}{2}-\beta_{R}\frac{q^{2}}{2}\right]$

Again, we defer the sum over $m$ which sets the charge to be integers.

We are interested in the quantity $\langle|Y_{E,\mu}(T)|^{2}\rangle_{c}$
defined in (59). It can be computed by applying a transformation of (83)

$\displaystyle\langle|Y_{E,\mu}(T)|^{2}\rangle_{c}$
$\displaystyle\propto\int{\mathrm{d}}\beta_{L}{\mathrm{d}}\beta_{R}\,e^{\beta_{L}E+\beta_{R}E+\frac{1}{2}(\beta_{L}^{2}+\beta_{R}^{2})\Delta^{2}}Z(\beta_{L}-{\mathrm{i}}T,\mu;\beta_{R}+{\mathrm{i}}T,\mu)$
(89)
$\displaystyle\propto\int{\mathrm{d}}\beta_{L}{\mathrm{d}}\beta_{R}\,e^{\beta_{L}E+\beta_{R}E+\frac{1}{2}(\beta_{L}^{2}+\beta_{R}^{2})\Delta^{2}}\times$
$\displaystyle\quad\sum_{m}\int{\mathrm{d}}q\int_{0}^{\infty}{\mathrm{d}}b\,b\frac{1}{\sqrt{(\beta_{L}-{\mathrm{i}}T)(\beta_{R}+{\mathrm{i}}T)}}\exp\left[-\frac{b^{2}}{2(\beta_{L}-{\mathrm{i}}T)}-\frac{b^{2}}{2(\beta_{R}+{\mathrm{i}}T)}\right]$
$\displaystyle\quad\exp\left[q((\beta_{L}+\beta_{R})\mu+2\pi{\mathrm{i}}m)\right]\exp\left[-(\beta_{L}-{\mathrm{i}}T)\frac{q^{2}}{2}-(\beta_{R}+{\mathrm{i}}T)\frac{q^{2}}{2}\right]$

To understand this seemingly complicated expression, we can look for the
saddle points for the integrals for $\beta_{L},\beta_{R}$ and $b$. Similar to
the ordinary double cone Saad:2018bqo , the saddle point value of $\beta_{L}$
and $\beta_{R}$ are located at zero, while the saddle point for $b$ is located
at

$b_{*}=\sqrt{2\left(E+\mu q-\frac{q^{2}}{2}\right)}T.$ (90)

We can then expand (89) around the saddle points and evaluate the one loop
determinant. The final result of the calculation is very simple

$\displaystyle\langle|Y_{E,\mu}(T)|^{2}\rangle_{c}$
$\displaystyle\propto\sum_{m}\int{\mathrm{d}}q\,e^{2\pi{\mathrm{i}}mq}\,T$
(91) $\displaystyle\propto\sum_{q}T.$

We kept the sum over $m$ in the first line because in the discussion of 3.1 we
did not discuss the effect of summing over shifts (we had $m=0$ there), so we
simply have a continuous integral over the $\text{U}(1)$ charge. Of course,
the effect of summing over $m$ is to enforce charge quantization. The final
result (91) is easy to interpret. We simply have an independent random matrix
in each of the charge sectors, which contributes one copy of the linear $T$
growth. The final result is the sum over all the charge sectors. In
particular, the final result is independent of the chemical potential $\mu$.

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