#  BPS and near-BPS black holes in AdS5
and their spectrum in $\mathcal{N}=4$ SYM

###### Abstract

We study quantum corrections in the gravitational path integral around nearly
$1/16$-BPS black holes in asymptotically $AdS_{5}\times S^{5}$ space, dual to
heavy states in 4D $\mathcal{N}=4$ super Yang-Mills. The analysis provides a
gravitational explanation of why $1/16$-BPS black holes exhibit an exact
degeneracy at large $N$ and why all such states have the same charges,
confirming the belief that the superconformal index precisely counts the
entropy of extremal black holes. We show the presence of a gap of order
$N^{-2}$ between the $1/16$-BPS black holes and the lightest near-BPS black
holes within the same charge sector. This is the first example of such a gap
for black holes states within the context of AdS5 holography. We also derive
the spectrum of near-BPS states that lie above this gap. Our computation
relies on finding the correct version of the $\mathcal{N}=2$ super-Schwarzian
theory which captures the breaking of the $SU(1,1|1)$ symmetry when the black
hole has finite temperature and non-zero chemical potential. Finally, we
comment on possible stringy and non-perturbative corrections that can affect
the black hole spectrum.

###### Contents

  1. 1 Introduction
  2. 2 The $\mathcal{N}=2$ super-Schwarzian
    1. 2.1 From the Schwarzian to near-extremal black holes
    2. 2.2 The model
    3. 2.3 Exact partition function and its spectrum
  3. 3 BPS and near-$1/16$ BPS black holes in AdS5
    1. 3.1 General black hole solutions in AdS5
    2. 3.2 The $1/16$-BPS black hole solution
    3. 3.3 What we want to compute
    4. 3.4 Near horizon geometry and supersymmetries
    5. 3.5 The low-temperature expansion of the action
    6. 3.6 The black hole spectrum including quantum corrections
    7. 3.7 Type IIB string corrections to the Schwarzian Result
  4. 4 Further examples: black holes in $\mathcal{N}=(2,2)$ supergravity in AdS3
    1. 4.1 States and representations of 2D $\mathcal{N}=(2,2)$ CFT
    2. 4.2 Naive Spectrum
    3. 4.3 Corrected Spectrum
  5. 5 Discussion and future work
  6. A Details on the low temperature expansion
  7. B Details on Killing spinors

##  1 Introduction

$1/16$-BPS states have recently played a crucial role in analyzing the duality
between Type-IIB string theory in AdS${}_{5}\times S_{5}$ and 4D
$\mathcal{N}=4$ super-Yang Mills [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. Such states can be accurately counted
by computing the superconformal index, a grand canonical partition function
with multiple chemical potentials for the black hole angular momenta and
$R$-charges turned on. On the boundary side the superconformal index can be
obtained exactly since its independent of the coupling [24, 25]. Expanding the
exact answer in the large $N$ limit, most contributions to the superconformal
index were matched with a corresponding Euclidean gravity saddles, with the
dominant contributions given by well-known supersymmetric black hole solutions
[23]. The computation of the superconformal index, on both the boundary and
the bulk side, has provided a detailed check of holography and a detailed
counting of black hole micro-states.

A lot less is known about black hole states that are not protected by
supersymmetry. In particular, an important issue in classical black hole
thermodynamics is the breakdown of the statistical description of black holes
[26, 27], occurring at low temperatures whose scale is power-law suppressed in
the number of degrees of freedom describing the black hole. When quantum
effects are included, two resolutions to this thermodynamic breakdown were
found. First, for non-supersymmetric black holes in flatspace or AdS, a recent
computation [28] which accounted for quantum effects occurring in the near-
horizon region of near-extremal black holes showed an effective continuum of
states with a strongly modified spectrum at the scale identified in [26]. In
the second resolution, due to the addition of fermionic degrees of freedom in
the computation of quantum effects, the spectrum of nearly-supersymmetric
black holes in supergravity in 4D flatspace or $(4,4)$ supergravity in AdS3
was shown to be drastically different [29]: there is an exact degeneracy of
supersymmetric black holes at extremality followed by a gap precisely at the
energy scale at which a failure of black hole thermodynamics is predicted.
Thus, quantum effects in the near-horizon region proved to be important in
both cases. However, exact degeneracies at extremality and gaps between
extremal and near-extremal states which were predicted in stringy
constructions [30, 31, 32] were only found in supersymmetric theories.

In the case of supersymmetric black holes in AdS5, we want to ask whether
there is a gap (with an appropriate power-law suppression in $N$) in the
spectrum of masses between the $1/16$-BPS black holes and the lightest
unprotected black hole state (which we call near-BPS) in a sector with the
same angular momenta and R-charge quantum numbers. These states preserve less
supersymmetry than their flatspace counterparts, and a rigorous understanding
of whether a gap is truly present has, up to the point of this paper, not been
achieved.111In the supersymmetric flatspace and $(4,4)$ AdS3 cases the
presence of a gap in stringy examples was explained in [31, 32]. We could not
find a string theory argument about the existence of the gap in the
literature. Even when it comes to the $1/16$-BPS states themselves, not all of
their properties are known. Firstly, the superconformal index cannot
distinguish whether such states are purely bosonic or a combination of bosonic
and fermionic states, which would yield cancellations in the index. In other
words, it is unclear whether the entropy associated to the index precisely
matches the actual entropy of BPS black holes. Secondly, from the
gravitational perspective, the degeneracy of the extremal $1/16$-BPS states
has not been rigorously understood due to the presence of an infinite number
of zero modes observed when computing the one-loop determinant for black holes
at extremality.222As we will explain shortly, these zero-modes get lifted when
studying the partition function at small but non-zero temperature. This
regularizes the one-loop determinant in the finite temperature partition
function and this regularization will be responsible for the observed
degeneracy among supersymmetric black holes.

In this paper, we address all these questions by computing the low-temperature
expansion of the partition function of near-BPS black holes. At zero
temperature, the leading contribution to the partition function comes from
$1/16$ BPS black holes that exhibit an $SU(1,1|1)$ isometry in their near-
horizon region [33]. As the temperature is turned on, the $SU(1,1|1)$ isometry
is broken and we explain how the gravitational modes associated to this
breaking are effectively captured by the $\mathcal{N}=2$ super-Schwarzian
theory [34, 35, 36, 37, 38, 29].333The relation between these black holes and
the $\mathcal{N}=2$ Schwarzian theory was explored at the classical level in
[38, 39]. At small temperatures, the super-Schwarzian theory and the
associated modes in supergravity become strongly coupled. Luckily the
partition function of the super-Schwarzian can be found exactly and this
consequently allows us to reliably compute the low-temperature corrections to
the free energy of such black holes, appearing at linear order in $T$ (from
classically evaluating the super-Schwarzian action and the black hole action)
and logarithmic order in $T$ (from evaluating quantum corrections in the
super-Schwarzian or, equivalently, specific set of modes of the graviton,
gravitino and gauge fields in the black hole background). This calculation is
valid up to temperatures that are much smaller than the one identified in
[26].

Studying these quantum corrections to the black hole spectrum leads to the
main results of our paper. We find that there is indeed a gap between the
$1/16$-BPS state and the lightest near-BPS black hole states precisely at the
energy scale identified in [26] (once again, in contrast to the non-
supersymmetric case of black holes in AdS5). Above this gap there is a
continuum of states whose density we can predict (a precise understanding of
the discreteness of the spectrum in this sector would require a better non-
perturbative understanding of type IIB string theory). Additionally, we find
that the $1/16$ BPS black hole states all have the same charges, to leading
order in $N$, confirming that there is no cancellation in the superconformal
index. This improves a previous argument for black holes with an emergent
$SU(1,1|1)$ symmetry described in [40], that did not take into account the
physics from the Schwarzian mode. This is important since there are also
theories with an emergent $SU(1,1|1)$ symmetry and vanishing index, and we
show this does not happen for the $1/16$-BPS states of $\mathcal{N}=4$ Yang
Mills.

Before diving into a more quantitative description of our results, it is
useful to first describe some of the properties of the black hole solutions
whose spectrum we determine in this paper. When viewed from the 10D
perspective, such black hole solutions have five angular momenta: two angular
momenta parametrizing rotations in AdS5 as well as three angular momenta
parametrizing rotations on the $S^{5}$. On the boundary side, the former are
the angular momenta of the dual state within the conformal group, which we
will denote by $J_{1,2}$ while the latter are three Cartans of the $SO(6)$
R-symmetry group, which for simplicity we will set to all be equal and denote
by $R$. The mass of the black hole can be fixed in terms of the temperature in
addition to these five charges. For extremal BPS black holes, their mass can
be determined in terms of four (of the five) angular momenta since there is an
additional non-linear relation between these charges necessary in order for
Killing spinor solutions to exist in the geometry while at zero-temperature.
For instance, the BPS value of $R$ as well as the mass of the black hole, are
uniquely determined at the BPS value by $J_{1,2}$.

With the quantum numbers of such black holes in mind, our computation of the
partition function predicts the density of states of the near-BPS black holes
as a function of the scaling dimension $\Delta$ of the corresponding states in
$\mathcal{N}=4$ Yang Mills. The answer is given by

$\displaystyle\rho_{R,J_{1},J_{2}}(\Delta)$ $\displaystyle\sim
e^{S^{*}}\bigg{(}\underbrace{\delta_{R,R^{*}}\delta\left(\Delta-\Delta_{BPS}\right)}_{\begin{subarray}{c}\text{Degenerate}\\\
1/16-\text{BPS black holes}\end{subarray}}$
$\displaystyle+\,\underbrace{\frac{\sinh\left(\frac{\pi}{2}\sqrt{\frac{1}{\Delta_{\text{gap}}}\left(\Delta-\Delta_{\text{extremal}}\right)}\right)}{2\pi(\Delta-\Delta_{\text{BPS}})}}_{\begin{subarray}{c}\text{Continuum
density of states}\\\ \text{above the
gap}\end{subarray}}\,\,\,\underbrace{\Theta\left(\Delta-\Delta_{\text{extremal}}\right)}_{\begin{subarray}{c}\text{Gap
resulting}\\\ \text{from quantum
corrections}\end{subarray}}+\underbrace{(R\rightarrow
R+1)}_{\begin{subarray}{c}\text{Degeneracy within}\\\
\text{supermultiplet}\end{subarray}}\bigg{)}\,.$ (1.1)

The first line represents the degenerate contribution of extremal states when
the R-charge is fixed to its BPS value $R^{*}$ by the angular momenta
$J_{1,2}$. The degeneracy is precisely found to be $e^{S^{*}}$, where $S^{*}$
is the Bekenstein-Hawking entropy of the extremal supersymmetric black
hole.444This degeneracy is up to corrections logarithmic in $S^{*}$ or,
equivalently, up to logarithmic area corrections in the free energy.  The
second line represents the contribution of the continuum of states (with non-
perturbative gaps expected to be exponentially small in $N$) starting at the
scaling dimension $\Delta_{\text{extremal}}$ which then represents the
extremal black hole state. This gives a gap above the BPS scaling dimension
$\Delta_{\text{BPS}}$ (defined for $R=R^{\star}$), that scales as

$\displaystyle\Delta_{\text{extremal}}(R)-\Delta_{\rm BPS}$
$\displaystyle=\frac{1}{2}(R-R^{\star})+\Delta_{\text{gap}}\left(2R-2R_{*}-{1}\right)^{2}\,,$
$\displaystyle\Delta_{\text{gap}}$
$\displaystyle=\frac{\tilde{\Delta}(\mathcal{J}_{1},\,\mathcal{J}_{2})}{N^{2}}+O\left(\frac{1}{N^{3}}\right)\,,$
(1.2)

where $N$ is related to the 5D Newton constant by
$N^{2}=\frac{\pi\ell_{AdS_{5}}^{3}}{2G_{5}}$, and where we rescale the two
angular momenta $J_{1,2}=N^{2}\mathcal{J}_{1,2}$,555Since $\mathcal{J}_{1,2}$
do not need to scale with any parameter in the theory for the black holes
discussed in this paper. and where
$\tilde{\Delta}(\mathcal{J}_{1},\mathcal{J}_{2})$ is a function that we
determine exactly. Since the states that are part of the continuum are
unprotected, they come in super-multiplets with charges $R$ and $R+1$. Now we
can see the meaning of the gap scale $\Delta_{\rm gap}$. For any charge $R\neq
R^{\star}$, the spectrum starts at $\Delta_{\text{extremal}}(R)$ with no gap.
But for $R=R^{\star}$, we have the BPS states at scaling dimension
$\Delta_{\text{BPS}}$ and the first excited state above them in this charge
sector start at $\Delta_{\text{BPS}}+\Delta_{\text{gap}}$ (the continuum
sector at $R=R^{\star}$ comes from supermultiplets with highest R-charge
$R^{\star}$ and $R^{\star}+1$). This is a large-$N$ analysis and therefore we
cannot rule out the possibility of an order $O(1)$ number of states between
$\Delta_{\text{BPS}}$ and $\Delta_{\text{BPS}}+\Delta_{\text{gap}}$. We
conjecture that $\Delta_{\rm gap}$ is the true gap of the theory at large $N$.

When the chemical potential of the $R$-symmetry is fixed appropriately, the
grand canonical partition function can also be used to compute the
superconformal index of such black holes and the only contribution comes from
the first line of (1.1). This result can be obtained by directly computing the
corresponding supersymmetric index in the $\mathcal{N}=2$ super-Schwarzian
theory. This matches the result from the leading gravity saddle found in [8,
23] and, additionally, shows that the one-loop determinant from the
gravitational theory matches the computation of the index in the boundary
theory. An interesting feature that comes out of our analysis is that the IR
R-charge of the $\mathcal{N}=2$ super-Schwarzian theory is shifted by a known
value $R^{*}$ compared to the UV R-charge of $\mathcal{N}=4$ Yang Mills.

We would like to emphasize that since we work in a mostly canonical ensemble
where almost all charges are fixed, we avoid the subtleties encountered in the
grand canonical ensemble regarding which sheet the superconformal index is
computed [22]. In our formalism, these issues would appear from attempting to
sum over charges to construct the grand canonical answer.

Additionally, we compute the leading (in $\alpha^{\prime}$) non-zero stringy
correction to the black hole spectrum. While as expected the scaling dimension
and charges of BPS states remains unaffected, $\Delta_{\text{gap}}$ and the
scaling dimension of the extremal state $\Delta_{\text{extremal}}(R)$ are
affected, and are pushed towards lower energies. Nevertheless, because the
$\mathcal{N}=2$ super-Schwarzian theory remains a good effective description
even in the presence of stringy corrections, the overall dependence of the
density of states on $\Delta_{\text{gap}}$ and $\Delta_{\text{extremal}}(R)$
remains as in (1.1).

A technical difficulty in our computation comes from identifying the correct
parameters of the $\mathcal{N}=2$ super-Schwarzian, that capture the correct
breaking of the $SU(1,1|1)$ isometry as the temperature of the near-$1/16$ BPS
black holes is increased. In particular, there are multiple version of the
$\mathcal{N}=2$ super-Schwarzian since the R-symmetry group associted to the
breaking of the near-horizon isometry is $U(1)$. One can change the radius of
the associated $U(1)$ mode in the super-Schwarzian by choosing a different
value for the fundamental charge in the theory, and, additionally, one can add
a topological $\theta$-angle term associated to this $U(1)$ mode in the super-
Schwarzian action. Depending on this fundamental charge and on the value of
$\theta$, the $\mathcal{N}=2$ super-Schwarzian density of states can have
widely different behaviors:666This is closely related to how the spectrum of a
particle moving on a circle is also controlled by the radius of the circle and
by the $\theta$-angle term that one can also add to the action [41].  for
instance, in some cases there is gap and the BPS states are purely bosonic,
while in others the theory has no such gap and has degenerate bosonic and
fermionic BPS states (leading to possible cancellations in the superconformal
index). For black hole in AdS5, we explain how each parameter is fixed from
the perspective of the $\mathcal{N}=4$ SYM boundary theory and of the bulk
supergravity theory. In particular, we find that the fundamental $R$-charge in
the super-Schwarzian is fixed to be $1$ from the quantization of the $SO(6)$
R-symmetry charges and two angular momenta along $S^{3}$ and find that
$\theta=0$ by carefully evaluating the 10D supergravity action. Fixing these
two parameters determines the value of the gap and density of states that we
described above. Moreover, we also determine the Schwarzian energy and
R-charge in the IR, in terms of the $\mathcal{N}=4$ SYM scaling dimensions
$\Delta$ and R-charge $R$.

One might however wonder whether there are near-BPS black holes in AdS/CFT
whose density of states is also controlled by the $\mathcal{N}=2$ super-
Schwarzian but with different values for the fundamental charge and
$\theta$-angle. For instance, one might ponder whether there are situations
when there is no thermodynamic mass gap above the BPS state. We construct such
near-BPS black holes in AdS3 for $\mathcal{N}=(2,2)$ and $\mathcal{N}=(2,0)$
supergravity. We explain how, in such a case, the density of states predicted
by the super-Schwarzian can also be obtained from modular invariance in the
boundary $2D$ $\mathcal{N}=(2,2)$ superconformal field theory by making mild
assumptions about the boundary theory (i.e. that it has large central charge
and a non-vanishing twist gap).777As noticed in [36, 42] in theories with
other amounts of supersymmetry, this connection originates from the equality
between the partition function of the (super)Schwarzian and the semiclassical
limit (large central charge) of the vacuum (super)Virasoro character.

The rest of this paper, is organized as follows. In section 2 we first
describe the connection between the Schwarzian theory and the spectrum of
near-extremal black holes more broadly. We then describe the various features
of the $\mathcal{N}=2$ super-Schwarzian theories which will be needed to
extract the spectrum of near-BPS black holes. In section 3, we give a detailed
description of the BPS and near-BPS black hole solutions in AdS5. We work in a
mixed ensemble with some charges fixed, and argue that in this ensemble the
index is always in a deconfined phase. We follow this by an analysis focused
on the steps necessary to determine the coupling, fundamental charge and
$\theta$-angle in the corresponding $\mathcal{N}=2$ super-Schwarzian theory by
analyzing the low-temperature expansion of the action and the super-algebra of
the near-horizon isometry present for BPS black holes. Putting these results
together, we give a detailed discussion of the spectrum of near-BPS black
holes. We conclude with a discussion about the leading stringy corrections to
the spectrum of such black holes. We discuss further appearance of the
$\mathcal{N}=2$ super-Schwarzian within holography in section 4 for black
holes in $(2,2)$ supergravity in AdS3, dual to heavy states in 2D $(2,2)$
SCFTs. Finally, in section 5 we discuss possible non-perturbative corrections
to the spectrum of $\mathcal{N}=4$ SYM and 2D $(2,2)$ SCFTs and conjecture
that the gap persists in the spectrum even when including states that exhibit
string excitations in the black hole background.

##  2 The $\mathcal{N}=2$ super-Schwarzian

###  2.1 From the Schwarzian to near-extremal black holes

The connection between the dynamics of near-extremal black holes and the one-
dimensional Schwarzian theory has been extensively studied [43, 44, 45, 46,
47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 28, 29]. Through this connection,
a better understanding of the spectrum of near-extremal black holes has been
achieved and the previously open question about the existence of a
thermodynamic mass gap has been resolved for a variety of near-extremal black
holes [28, 29]. Nevertheless, there are still near-extremal black holes whose
connection to the Schwarzian has not yet been completely understood and
consequently, their spectrum near-extremality remains unknown. One such
example, are the near-BPS black holes in AdS5 whose connection to the
Schwarzian will be extensively discussed in section 3. It is useful to first
review the general mechanism through which the spectrum of near-extremal black
holes is related to that of the Schwarzian with various amounts of
supersymmetry.

The clear-cut way to understand the relation between the two spectra is by
performing a dimensional reduction of the full gravitational theory in the
near-horizon region to AdS2 and isolate the low-energy modes whose one-loop
determinant solely is temperature $T$ dependent at leading order. In the limit
in which the inverse-temperature $\beta$ is much larger than the horizon size
of the black hole at extremality $r_{0}$ ($\beta\gg r_{0}$) and under the
assumption that the extremal entropy $S_{0}$ is the largest dimensionless
parameter in the problem, the theory that results from the dimensional
reduction is a two-dimensional theory of gravity coupled to a dilaton field
(which parametrizes the area of the sphere on which the reduction is
performed). In addition, the two-dimensional theory includes gauge fields that
result either from the s-wave reduction of gauge fields (whose gauge group we
will denote by $G$) in the original theory or as Kaluza-Klein modes that
capture the isometry group of the space on which the dimensional reduction was
performed (whose gauge group we will denote by $G_{\text{iso}}$). In theories
of supergravity which include fermions in higher dimensions, an analogous
procedure leads to a coupling between a dilatino field and the gravitino in
the 2D theory. Expanding this action in $S_{0}$, one finds that the theory can
be approximated by JT gravity plus a BF theory whose gauge group is
$G_{BF}=G\times G_{\text{iso}}$. For higher dimensional theories of
supergravity, the dilaton and metric as well as the gauge field entering in
the BF theory all couple to the gravitino and dilatino. Fluctuations in the
region outside of the near-horizon region can be captured by a boundary term
for the JT gravity action and for the BF action computed along the curve that
separates the near-horizon region from the asymptotic region.

For concreteness, we will briefly review the example of large near-extremal
Reissner-Nordstrom black holes in (bosonic) Einstein-Maxwell theory in AdS5
(i.e. whose horizon size at extremality $r_{0}$ is much larger than the AdS5
size $r_{0}\gg\ell_{\text{AdS}_{5}}$ and whose inverse-temperature $\beta\gg
r_{0}$). The spectrum of such black holes was extensively studied in [28].
Following the steps outlined above, the canonical partition function of such
black holes can be expressed as

$\displaystyle Z(\beta$
$\displaystyle,\,Q,\,J_{1,2}=0)=\mathrm{Tr}_{Q}~{}e^{-\beta H}$
$\displaystyle=S_{0}(Q)^{\\#}~{}e^{S_{0}(Q)-\beta E_{0}(Q)}\int D\phi
Dg_{\mu\nu}e^{-\int_{\text{AdS}_{2}}\phi(R+\Lambda_{\text{AdS}_{2}})-\frac{\phi_{b}(Q)}{\epsilon}\int_{\partial\text{AdS}_{2}}(K-1)}\bigg{[}1+O\left({1}/{S_{0}}\right)\bigg{]}$
$\displaystyle=S_{0}(Q)^{\\#}~{}e^{S_{0}(Q)-\beta
E_{0}(Q)}\int\frac{Df(\tau)}{SL(2,\mathbb{R})}\,e^{\phi_{b}(Q)\int_{0}^{\beta}d\tau\text{Sch}(f,\tau)}\bigg{[}1+O\left({1}/{S_{0}}\right)\bigg{]}$
$\displaystyle=\underbrace{S_{0}(Q)^{\\#}}_{\begin{subarray}{c}\text{Sen's
logarithmic}\\\
\text{corrections}\end{subarray}}\underbrace{\left(\frac{1}{M_{SL(2)}\beta}\right)^{3/2}}_{\begin{subarray}{c}\text{Schwarzian}\\\
\text{all-loop
correction}\end{subarray}}\,\,\,\exp\underbrace{\left({S_{0}(Q)-\beta
M_{0}(Q)+\frac{2\pi^{2}}{M_{SL(2)}\beta}}\right)}_{\begin{subarray}{c}\text{Extremal
entropy,}\\\ \text{extremal energy,}\\\ \text{ and the Schwarzian }\sim
1/\beta\text{ correction}\\\ \text{to the
action}\end{subarray}}\bigg{[}1+O\left(\frac{1}{S_{0}}\right)\bigg{]}\,,$
(2.1)

where the extremal mass $M_{0}(Q)$, the “extremal entropy” $S_{0}(Q)$ and the
Schwarzian coupling denoted by $\phi_{b}(Q)$ (or by the more physically
meaningful notation $M_{SL(2)}^{-1}$ which we shall explain shortly) are given
by888Here, $Q$ can be viewed as the charge associated to the supergravity
gauge field $A$, whose conventions we set in (3.1) and thereafter. For reader
convenience, in this section we leave the radius of AdS5 to be arbitrary and
set to $\ell_{\text{AdS}_{5}}$.

$\displaystyle
M_{0}(Q)\sim\frac{G_{5}^{1/3}(|Q|)^{4/3}}{\ell_{\text{AdS}_{5}}^{2}}\,,\qquad
S_{0}(Q)\sim|Q|\,,\qquad\phi_{b}(Q)=M_{SL(2)}^{-1}\sim\frac{\ell_{\text{AdS}_{5}}^{2}|Q|^{\frac{2}{3}}}{G_{5}^{\frac{1}{3}}}\,,$
(2.2)

where $\ell_{\text{AdS}_{5}}$ is the radius of $AdS_{5}$ and $G_{5}$ is the
Newton constant. To obtain the first line of (2.1), we peform a dimensional
reduction on $S^{3}$ in the $AdS_{2}\times S^{3}$ near-horizon region and then
expand the resulting action at large $S_{0}$. For large black holes, the
radius of $AdS_{2}$ is the same as the radius of $AdS_{5}$, which fixes
$\Lambda_{{\rm AdS}_{2}}=6/(\ell_{\text{AdS}_{5}}^{2})$. The leading order
result yields the action of JT gravity, whose degrees of freedom are the 2D
metric of the near-horizon region and the dilaton $\phi$ which parametrizes
the size of $S^{3}$. Above, the BF terms associated to the gauge group
$G_{BF}=U(1)\times SO(4)$ do not give a non-trivial contribution since we are
fixing both the $U(1)$ and $SO(4)$ fluxes when fixing the charge of the black
hole to $Q$ and its angular momenta $J_{1}=J_{2}=0$ (this will be contrasted
with the case of the grand-canonical partition function below) [58]. There is
additionally a Gibbons-Hawking-York (GHY) boundary term at the edge of the
near-horizon region where the dilaton (fixed to $\phi|_{\partial
AdS_{2}}=\phi_{b}/\epsilon$) as well as the induced boundary metric are fixed
(to have a proper boundary length $\beta/\epsilon$).

In addition to these these modes there are also KK modes obtained from the
dimensional reduction on $S^{3}$. The one-loop determinant of most KK modes
corresponding to massless fields in the original theory yield logarithmic
corrections to the extremal entropy, given by $S_{0}^{\\#}$ which can in
principle be computed using the methods in [59, 60, 61, 62, 28].999For fields
that are not massless in the original gravitational theory the one-loop
determinant yields an answer that, to leading order, is independent of the
extremal entropy $S_{0}$. Here, the exact value of $\\#$ depends on the
massless field content in the original gravitational theory and is unimportant
in the analysis of this paper since it does not affect the energy dependence
of the density of states but only its overall scaling. The remaining modes
which are not taken into account are precisely the JT gravity modes which we
have separated in the first line of (2.1). At zero temperature, these are the
zero-modes that are ubiquitous when computing one-loop determinants in black
hole backgrounds and are given by the set of large diffeomorphisms which do
not vanish close to the boundary of the near-horizon region.101010See for
instance [61] for a treatment of these modes. However, when working at finite
temperature, these zero-modes are lifted and become the boundary modes of the
near-horizon region weighed by the Schwarzian action. This can be seen when
going from the first to the second line of (2.1) by integrating out the
dilaton and rewriting the GHY term in terms of a field $f(\tau)$ which
parametrizes the set of possible large diffeomorphisms. Equivalently,
$f(\tau)$ parametrizes the shape of the boundary of the near-horizon region.
The Schwarzian theory can be seen to be weekly-coupled when $T\gg M_{SL(2)}$
and becomes strongly coupled when $T\sim M_{SL(2)}$. Finally, to go from the
second to the third line and compute the partition function for any coupling,
one can use the fact that the path integral of the Schwarzian theory is
exactly solvable and is in fact one-loop exact. This one-loop determinant can
be obtained by accounting for the three remaining bosonic zero-modes that
survive even at finite temperature and are due to the near-horizon
$SL(2,\mathbb{R})$ isometry.

The low-temperature expansion of the action is sufficient to obtain the full
partition function in (2.1). This is because the Schwarzian theory can be
viewed as the effective theory for the breaking of the near-horizon
$SL(2,\mathbb{R})$ isometry, as the temperature is turned-on. This motivates
denoting the Schwarzian coupling by $M_{SL(2)}$.

The result in (2.1) implies that the density of states obtained by Laplace
transforming (2.1),

$\displaystyle\rho(E,Q)=S_{0}(Q)^{\\#}~{}e^{S_{0}(Q)}\sinh\left(2\pi\sqrt{\frac{E}{M_{SL(2)}}}\right)\Theta(E-M_{0})\,.$
(2.3)

Before we proceed, it is worth pondering the meaning of the continuum of
states in (2.3). The partition function (2.1) is obtained in a $1/S_{0}$
expansion and we have only kept the leading order term. Consequently, the
density of states (2.3) also receives perturbative and non-perturbative
corrections in $S_{0}$. Such corrections, which we do not have control over,
could in principle lead to a discretum of states as one would expect from the
perspective of the boundary dual CFT. Nevertheless, (2.3) is still useful:
when integrating the density of states within some energy interval we should
obtain the number of black hole microstates within that energy interval, once
again, up to $1/S_{0}$ corrections.

For the purposes of this paper, it is also useful to study the partition
function in the grand-canonical ensemble, imposing that the holonomy of the
$U(1)$ gauge field is fixed to $e^{\oint A}=e^{-\beta\mu}$ at the boundary of
the spacetime, along the thermal circle. The partition function can be
instructively rewritten as,

$\displaystyle Z(\beta,\mu)$ $\displaystyle=\mathrm{Tr}\,e^{-\beta H-\beta\mu
Q}$
$\displaystyle=\frac{S_{0}^{\\#}}{\left(M_{SL(2)}\beta\right)^{3/2}}e^{S_{0}-\beta\mu
Q_{*}-\beta
M_{0}+\frac{2\pi^{2}}{M_{SL(2)}\beta}}\sum_{q\in\mathbb{Z}}e^{2\pi\xi q-\beta
M_{U(1)}\frac{q^{2}}{4}}$ $\displaystyle={S_{0}^{\\#}}e^{S_{0}-\beta\mu
Q_{*}-\beta
M_{0}+\frac{2\pi^{2}}{M_{SL(2)}\beta}}\sum_{n\in\mathbb{Z}}\frac{1}{M_{SL(2)}^{3/2}M_{U(1)}^{1/2}\beta^{2}}e^{\frac{4\pi^{2}(\xi+in)^{2}}{M_{U(1)}\beta}}\,,$
(2.4)

The grand-canonical partition function can be obtained from (2.1) by summing
over fixed charges. This sum is dominated by the charge $Q_{*}$, which in
terms of the chemical potential $\mu$ is given by the solution of
$\mu=\partial_{Q}M_{0}(Q)|_{Q=Q_{*}}$.111111$S_{0}$, $M_{0}$, $M_{SL(2)}$ and
$M_{U(1)}$ in (2.1) depend on the charge $Q=Q_{*}$ as in (2.2) and (2.5).
Above, $q$ can be viewed as the quadratic fluctuation around the charge
$Q_{*}$. Similarly, $\xi/\beta$ can be viewed as the effective $U(1)$ chemical
potential in the near-horizon region, while $M_{U(1)}$ can be viewed as the
the coupling of the $U(1)$ mode associated to the s-wave gauge field in the
near-horizon region. For concreteness, these are given by the thermodynamic
relations

$\displaystyle 2\pi\xi=\left(\frac{\partial S_{0}}{\partial
Q}\right)_{T=0}\,,\qquad M_{U(1)}=\frac{2}{\left({\partial
Q}/{\partial\mu}\right)_{T=0}}\,.$ (2.5)

After Poisson resummation in $q$, one can obtain the sum over $n$ in (2.1).
The sum over $n$ represents a sum over classical saddles for the solutions of
the gauge field. The partition function need only depend on the holonomy
associated to the chemical potential $\xi$ imposed in the near-horizon region.
Thus, the partition function should be periodic under shift of $\xi\to\xi+in$
with $n\in\mathbb{Z}$, which is precisely what the sum over classical saddles
implements. Similar to the Schwarzian, excitations of the gauge field in the
black hole background are captured by large $U(1)$ gauge transformations that
do not vanish at the boundary of the near-horizon region. Such transformations
can be parametrized by a $U(1)$ mode that captures the breaking of the $U(1)$
symmetry as the chemical potential is turned on. Like the Schwarzian, this
theory is one-loop exact, and its on-shell action and one-loop determinant are
captured in the sum over $n$ in (2.1). Once again, the full partition function
of this effective theory can be read off from the low-temperature expansion of
the on-shell action.

We can additionally fix the angular velocities of the black hole instead of
its angular momenta. This can be done by setting the holonomy of the $SO(4)$
gauge field that is obtained from the dimensional reduction to $e^{i\oint
B}=e^{-\beta\omega_{1}\widehat{J}_{1}-\beta\omega_{2}\widehat{J}_{2}}$ where
$\widehat{J}_{1}$ and $\widehat{J}_{2}$ are two Cartans of $SO(4)$ whose
eigenvalues are $J_{1}$ and $J_{2}$. Similar to the $U(1)$ mode, excitations
of the $SO(4)$ gauge field can be captured by an $SO(4)$ valued field which
parametrizes non-vanishing gauge transformations at the boundary of the near-
horizon region. As above, there are multiple saddles for this $SO(4)$-mode
which is once again due to the fact that the partition function does not
explicitly depend on $\omega_{1}$ or $\omega_{2}$, but rather on the holonomy
$e^{-\beta\omega_{1}\widehat{J}_{1}-\beta\omega_{2}\widehat{J}_{2}}$.

The black holes described above should also appear in the bosonic truncation
of a supergravity theory. Nevertheless, black holes which at extremality
preserve some amount of supersymmetry are instead described by different
versions of the super-Schwarzian theory. This is because the effective theory
which computes the low-temperature expansion of the partition function does
not only capture the breaking of the near-horizon bosonic isometry group
$SL(2,\mathbb{R})\times G_{BF}$; instead, it captures the breaking of a super-
group. As mentioned above, in such a case there are additional modes that
remain massless in the near-horizon region, which contribute in the low-
temperature expansion of the partition function - these are the dilatino and
the gravitino. These modes do not decouple from the bosonic modes and
consequently the sum over $n$ in (2.1) has a more complicated dependence with
$\beta$ and $n$. Nevertheless, as we will explain in section 2.3 the couplings
determining the low-temperature expansion of the partition function can still
be read-off from the on-shell action, as was the case in (2.1) or (2.1).

Thus, instead of performing the full-dimensional reduction, in this paper we
will take the simpler approach of reading off the effective action which
captures the low-temperature expansion of the partition function. As discussed
above, we will identify this effective theory by (a) the near-horizon isometry
that gets broken when the temperature, chemical potential and angular
velocities are turned on and (b) the low-temperature expansion of the on-shell
action. This will determine the correct version of the super-Schwarzian theory
needed to describe the near-BPS black holes discussed in this paper. Before
diving into that identification, it is useful to first discuss the properties
of the effective theory that will end up being important in our analysis, the
$\mathcal{N}=2$ super-Schwarzian theory.

###  2.2 The model

The $\mathcal{N}=2$ super-Schwarzian theory was described in detail in [34].
It is a theory that is described by $\mathcal{N}=2$ super-reparametrizations.
Just like in the bosonic case, the bulk origin of these super-
reparametrizations is given by the action of super-diffeomorphisms on the
metric, gravitino and gauge field at the boundary of the near-horizon region.
After imposing the appropriate chirality constrains, these super-
reparametrizations can be described in super-space coordinates
$(\tau,\theta,\overline{\theta})\to(\tau^{\prime},\theta^{\prime},\overline{\theta}^{\prime})$
in terms of two time-dependent bosonic fields $f(\tau)$ and
$e^{ir\sigma(\tau)}\in U(1)$, where $r$ is a normalization constant we will
discuss below, as well as two fermionic fields $\eta(\tau)$ and
$\overline{\eta}(\tau)$:

$\displaystyle\tau^{\prime}$ $\displaystyle=$ $\displaystyle f(\tau)+\ldots,$
(2.6) $\displaystyle\theta^{\prime}$ $\displaystyle=$ $\displaystyle
e^{ir\sigma(\tau)}\sqrt{f^{\prime}(\tau)}\theta+\eta(\tau)+\ldots$ (2.7)
$\displaystyle\overline{\theta}^{\prime}$ $\displaystyle=$ $\displaystyle
e^{-ir\sigma(\tau)}\sqrt{f^{\prime}(\tau)}\overline{\theta}+\overline{\eta}(\tau)+\ldots,$
(2.8)

where the dots can be obtained by explicitly solving the super-
reparametrizations constraints. The fields entering in these super-
reparametrizations become the degrees of freedom of the $\mathcal{N}=2$ super-
Schwarzian theory. The Schwarzian derivative is then given by

$S(f,\sigma,\eta,\overline{\eta})=\frac{\partial_{\tau}\overline{D}\overline{\theta}^{\prime}}{\overline{D}\overline{\theta}^{\prime}}-\frac{\partial_{\tau}D\theta^{\prime}}{D\theta^{\prime}}-2\frac{\partial_{\tau}\theta^{\prime}\partial_{\tau}\overline{\theta}^{\prime}}{(\overline{D}\overline{\theta}^{\prime})(D\theta^{\prime})}=\ldots+\theta\overline{\theta}S_{b}(f,\sigma,\eta,\overline{\eta})\,,$
(2.9)

from which one can write the $\mathcal{N}=2$ super-Schwarzian action in terms
of super-coordinates

$\displaystyle I_{\mathcal{N}=2\text{ Schw}}$ $\displaystyle=$
$\displaystyle-\frac{1}{M_{SU(1,1|1)}}\int d\tau d\theta
d\overline{\theta}S=-\frac{1}{M_{SU(1,1|1)}}\int d\tau S_{b}\,,$ (2.10)
$\displaystyle=$
$\displaystyle\frac{1}{M_{SU(1,1|1)}}\int\text{Sch}(f,\tau)+2(\partial_{\tau}\sigma)^{2}+({\rm
fermions})\,.$ (2.11)

Just like in (2.1), the action of the $\mathcal{N}=2$ super-Schwarzian can be
obtained from an $\mathcal{N}=2$ bulk JT theory which in addition to the
metric and dilton present in the bosonic theory described in section 2.1, also
contains fermionic super-partners, the gravitino and dilatino as well as a
$U(1)$ R-symmetry gauge field coupled to a zero-form Lagrange multiplier.
After integrating-out the dilaton, dilatino and $U(1)$ Lagrange multiplier
what remains are only the large super-diffeomorphisms describing the
transformations of the metric, gravitino and $U(1)$ gauge field at the
boundary of the near-horizon region. These can be again parametrized by the
fields $f(\tau)$, $\sigma(\tau)$, $\eta(\tau)$ and $\overline{\eta}(\tau)$
entering in (2.6). Once again, just like for the bosonic counter-part (2.1),
these super-diffeomorphisms enter in the boundary terms necessary for the
$\mathcal{N}=2$ JT theory to satisfy the variational principle when imposing
Dirichlet boundary conditions for all the aforementioned fields. Plugging in
the large super-diffeomorphisms parametrized by $f(\tau)$, $\sigma(\tau)$,
$\eta(\tau)$ and $\overline{\eta}(\tau)$, these boundary terms are then
precisely equal to the super-Schwarzian action (2.10). When performing the
path integral over the set of large super-diffeomorphisms which are weighed by
the boundary term, one should be careful to not over-count identical super-
geometries. For this reason, we should quotient the path integral over these
modes by whatever super-isometry is present in the bulk. In $\mathcal{N}=2$ JT
gravity this is $SU(1,1|1)$. These super-isometries can be explicitly
identified in (2.11) as a global $SU(1,1|1)$ acting on the fields
$f,\sigma,\eta,\overline{\eta}$ [34]. Consequently, we should quotient the
space of configurations of $f,\,\sigma,\,\eta,\,\overline{\eta}$ by such
transformations.

In addition, one can add a topological term in the bulk associated to the
$U(1)$ R-symmetry gauge field. On the boundary, one should add to the action
an associated topological term to the $U(1)$ mode, $\sigma$:

$I_{\text{topological}}=i\theta r\int d\tau\,(\partial_{\tau}\sigma)\,,$
(2.12)

Due to its topological nature this term is also invariant under the
$SU(1,1|1)$ transformations. Because $e^{ir\sigma(\tau)}\in U(1)$ and thus
$\sigma\sim\sigma+2\pi/r$, we can identify theories with
$\theta\sim\theta+2\pi$ and will therefore restrict to $\theta\in[0,2\pi)$.
From the bulk perspective, since the boundary holonomy of the $U(1)$
R-symmetry gauge field is given by $e^{i(\sigma(\beta+\tau)-\sigma(\tau))}$,
$r$ is determined by the smallest R-charge among any field that could possibly
be coupled to the $U(1)$ R-symmetry gauge field in $\mathcal{N}=2$ JT.121212In
the context of SYK, $r=1/\widehat{q}$ in [34] and represents the number of
fermionic fields that are needed to construct the $\mathcal{N}=2$ SYK
supercharges. However, in section 3 we will find that the relevant effective
theory has $r=1$ which is not realized by any known SYK model.

As in (2.1), we will next review the features of the resulting partition
function

$Z(\beta,\alpha)=\int\frac{\mathcal{D}f\mathcal{D}\sigma\mathcal{D}\eta\mathcal{D}\overline{\eta}}{SU(1,1|1)}e^{-I_{\mathcal{N}=2\text{
Schw}}-I_{\text{topological}}}\,,$ (2.13)

where the partition function depends on the inverse temperature $\beta$ which
sets the length of the thermal circle and on a $U(1)$ chemical potential
$\alpha$ associated to the mode $\sigma$, under which fermionic fields are
also charged. Thus, the periodicity conditions for the resulting path integral
become $f(\tau+\beta)=f(\tau)$, $e^{i\sigma(\tau+\beta)}=e^{2\pi
i\alpha}e^{i\sigma(\tau)}$, $\eta(\tau+\beta)=-e^{2\pi ir\alpha}\eta(\tau)$
and similarly for $\overline{\eta}$.

###  2.3 Exact partition function and its spectrum

The partition function of the $\mathcal{N}=2$ super-Schwarzian theory can be
computed exactly [35, 36]. We work in conventions where the $U(1)_{R}$ charge
of the complex supercharge is one and the minimal R-charge of fundamental
fields is fractional and given by $r$. Consistency of the spectrum with
$\mathcal{N}=2$ supersymmetry requires that $1/r$ is an integer, since states
related by applying a supercharge should both be in the spectrum. The
partition function is one-loop exact and given by [35, 36]

$Z_{\mathcal{N}=2\text{
Schw}}(\beta,\alpha)=\sum_{n\in\frac{1}{r}\cdot\mathbb{Z}}e^{i\,r\theta
n}~{}\frac{2\cos\left(\pi(\alpha+n)\right)}{\pi\left(1-4(\alpha+n)^{2}\right)}~{}e^{S_{0}+\frac{2\pi^{2}}{\beta
M_{SU(1,1|1)}}(1-4(\alpha+n)^{2})}.$ (2.14)

The sum over $n$ is a sum over saddles analogous to the previous result (2.1),
where the $U(1)_{R}$ mode $\sigma(\tau)$ has different windings around the
thermal circle, $\sigma(\tau+\beta)=\sigma(\tau)+2\pi n/r$. The exponential
terms all come from the evaluation of the classical action on the saddle point
configurations. Finally the prefactor of the exponentials is the one loop
determinant of the Schwarzian mode, $U(1)_{R}$ mode, and the fermions. The
non-trivial dependence on the chemical potential comes form the fermions,
while the possible $\beta$ dependence in the one loop determinant cancels
since $SU(1,1|1)$ has four bosonic and four fermionic generators. When the
theta angle in (2.12) vanishes the partition function is invariant under
$\alpha\sim\alpha+1/r$. Thus, for simplicity we can restrict to
$\alpha\in[0,1/r)$. The theory is also charge conjugation invariant only when
$\theta=0$ or $\theta=\pi$, since only then is the partition function real.

Supersymmetry in the on-shell action. The contribution of the
$SL(2,\mathbb{R})$ Schwarzian mode to the action is independent of the winding
mode $n$ and gives $\frac{2\pi^{2}}{\beta M_{SU(1,1|1)}}$, while the
contribution of the $U(1)_{R}$ mode is given by $-\frac{8\pi^{2}}{\beta
M_{SU(1,1|1)}}(\alpha+n)^{2}$. Thus, the equality of coupling seen in (2.11),
translates to a relation between the $\alpha$-dependent terms in the on-shell
and the $\alpha$ independent terms, in the convention in which
$\alpha\sim\alpha+1/r$.

Density of states within each supermultiplet. We can Fourier transform the
above result to obtain the decomposition of $Z_{\mathcal{N}=2\text{
Schw}}(\beta,\alpha)$ as a sum over $U(1)_{R}$ charges $Z$, with the smallest
charge equal to $r$. Because of supersymmetry the spectrum should organize
itself in supermultiplets $(Z)\oplus(Z-1)$ for states with $E\neq 0$ and
solely charge $Z$ for states with $E=0$. Based on symmetry principles the
partition function should thus be decomposed as

$Z_{\mathcal{N}=2\text{ Schw}}(\beta,\alpha)=\sum_{Z}e^{2\pi i\alpha
Z}\rho_{\rm ext}(Z)+\sum_{Z}\int dEe^{-\beta E}\left(e^{i2\pi\alpha
Z}+e^{i2\pi\alpha(Z-1)}\right)\rho_{\rm cont}(Z,E).$ (2.15)

In contrast to Schwarzian theories with smaller amounts of supersymmetry
($\mathcal{N}=0$ and $\mathcal{N}=1$), the density of states for
$\mathcal{N}=2$ splits into an extremal piece and a continuous piece. Since
the first term in (2.15) is temperature independent this implies that those
states are extremal and yield an exact Dirac delta-function in the density of
states at $E=0$. Rewriting (2.14) as in (2.15), gives [36]

$\displaystyle Z_{\mathcal{N}=2\text{
Schw}}(\alpha,\beta)=\sum_{\begin{subarray}{c}Z\in
r\cdot\mathbb{Z}-\frac{r\theta}{2\pi},\\\
|Z|<\frac{1}{2}\end{subarray}}e^{2\pi i\alpha Z}~{}e^{S_{0}}~{}r\cos\left(\pi
Z\right)$ (2.16) $\displaystyle+\\!\\!\\!\sum_{Z\in
r\cdot\mathbb{Z}-\frac{r\theta}{2\pi}}r\left(e^{2\pi i\alpha Z}+e^{2\pi
i\alpha(Z-1)}\right)\int_{E_{\text{gap}}(Z)}^{\infty}dEe^{-\beta
E}\frac{e^{S_{0}}\sinh{\left(2\pi\sqrt{\frac{2}{M_{SU(1,1|1)}}(E-E_{\text{gap}}(Z))}\right)}}{2\pi
E},$

where $E_{\text{gap}}(Z)\equiv\frac{M_{SU(1,1|1)}}{8}(Z-\frac{1}{2})^{2}$
denote the energy gap for a supermultiplet labeled by the charge $Z$. The
first line in (2.16) captures the contribution of BPS states that have $E=0$
and whose R-charges $Z$ can thus take the values $Z\in
r\cdot\mathbb{Z}-\frac{r\theta}{2\pi}$, with the constrain $|Z|<\frac{1}{2}$
which ensures the density of states is positive. The contribution of the
second line captures the non-BPS states which have a continuous density.

The BPS to non-BPS energy gap. When does such a theory have a gap between the
BPS states and lightest non-BPS state? If $Z=1/2$ is included in the sum over
charges in (2.16), then $E_{\text{gap}}(Z=1/2)=0$ which implies that there is
a continuum of states starting at $E=0$ and that there is no gap in the
spectrum. For $\theta=0$, there is no gap if $1/r$ is even and there is a gap
if $1/r$ is odd. For $\theta=\pi$, the opposite is true: there is no gap if
$1/r$ is even and the gap is present if $1/r$ is odd. If $\theta\neq\\{0\text{
or }\pi\\}$, then the gap is always present.

The supersymmetric index. For particular values of the $U(1)_{R}$ chemical
potential $\alpha$, the grand canonical partition function computes a
supersymmetric index. In particular, for any $\alpha$ such that
$e^{i2\pi\alpha}=-1$ the fermionic degrees of freedom $\eta$ and
$\overline{\eta}$ become periodic and consequently, turning on this fugacity
is equivalent to the insertion of $(-1)^{F}$ in the super-Schwarzian path
integral. This works for any $\alpha_{k}=k-1/2$ with $k=1,\ldots,r^{-1}$. For
$r=1$ there is only one index (with $\alpha=1/2$) and if we fix
$-\pi\leq\theta\leq\pi$ then it is given by

$\mathcal{I}=Z_{\mathcal{N}=2\text{
Schw}}(\beta,\alpha=1/2)=e^{S_{0}}e^{-i\frac{\theta}{2}}\cos\left(\frac{\theta}{2}\right).$
(2.17)

In such a case, note that $Z(\beta\to\infty,\alpha\to
0)=Z(\beta,\alpha\to\frac{1}{2})$ which implies that in such a case the
“ground state” of the $\mathcal{N}=2$ super-Schwarzian are purely bosonic.

The ’t Hooft anomaly. Finally, we will comment on the symmetries of the
particular theories with $\theta=0$ and $\theta=\pi$, for which charge
conjugation invariance appears, at least classically. For a related toy model
see Appendix D of [41]. When $\theta=0$ and $1/r\in\mathbb{Z}$ the theory
presents both charge conjugation and $U(1)_{R}$ symmetry. This is evident from
the fact the partition function (2.14) is real and that the charges are
integer multiples of $r$. Instead, the theory with $\theta=\pi$ has a ’t Hooft
anomaly between charge conjugation and $U(1)_{R}$ symmetries. For example, the
localization calculation leading to (2.14) is manifestly charge conjugation
invariant, since it is real. Nevertheless the spectrum in (2.16) is shifted by
a half-integer unit of charge, inconsistent with $U(1)_{R}$ symmetry. This
issue can be fixed by shifting the charge by a constant $Z\to Z+r/2$, making
the charge operator still commutes with the Hamiltonian; however, the price to
pay is to break charge conjugation invariance. For this reason the theory with
$\theta=\pi$ has a ’t Hooft anomaly, while $\theta=0$ preserves the symmetries
at the quantum level.

##  3 BPS and near-$1/16$ BPS black holes in AdS5

In the previous section, we have reviewed the role Jackiw-Teitelboim gravity
and the Schwarzian theory play in determining the form of the spectrum of near
extremal black holes in general. It can be thought of as the soft mode coming
from the broken symmetries that emerge in the near extremal limit. We also
consider in particular the case of a broken $SU(1,1|1)$ symmetry which is
described by $\mathcal{N}=2$ Schwarzian theory.

In this section we are going to apply this to $1/16$-BPS black holes in
AdS${}_{5}\times S^{5}$, which AdS/CFT predicts are dual to $1/16$ BPS states
in $\mathcal{N}=4$ Super Yang-Mills. We will begin by reviewing general
properties of these black holes in sections 3.1 and 3.2, taken from [6], but
see also [63, 4, 5, 64, 7]. Black holes in AdS5 have a-priori several related
but distinct limits:

  * •

The near-extremal limit in which $T\rightarrow 0$.

  * •

The extremal black holes in which $T=0$.

  * •

The supersymmetric limit in which the geometry has Killing spinors.

  * •

The BPS limit which has $T=0$ and an enhanced set of Killing spinors.

By adjusting the parameters of the solution, one may move independently off of
extremal and supersymmetric surfaces in parameter space [65, 8]. We adopt the
convention of these references to refer to the intersection of these surfaces
as the _BPS limit_.

Recently the Bekenstein-Hawking entropy of exactly $1/16$-BPS black holes was
reproduced from the superconformal index of $\mathcal{N}=4$ Yang Mills [24,
25, 8, 9, 10]. The goal of this section is to reproduce this result from the
point of view of the gravitational path integral, and in the process predict
the spectrum of excited black holes states above the BPS ones; these states
are _not_ protected by exact supersymmetry. In section 3.3 we specify a mixed
ensemble where some charges are fixed that will allow us to use the results of
section 3.1 without requiring the most general black hole solution. In section
3.4 we analyze the BPS limit and show the black holes have an $AdS_{2}\times
S^{3}\times S^{5}$ throat with an emergent $SU(1,1|1)$ symmetry. We identify
then the $\mathcal{N}=2$ Schwarzian theory controlling the near BPS spectrum
from the explicit breaking of this superconformal group. In section 3.5 we
verify this at the level of the classical action of the black hole saddle and
in section 3.6 we put everything together and give a picture of the spectrum
of nearly $1/16$-BPS states in $\mathcal{N}=4$ Yang Mills.

###  3.1 General black hole solutions in AdS5

The goal of this paper is to extract information about the spectrum of nearly
$1/16$-BPS states in $\mathcal{N}=4$ Yang Mills. According to AdS/CFT, these
states are dual to black holes and we will study their spectrum using the
gravitational path integral. Since we want to know the energy and charges of
these states, we will compute and derive the spectrum from the dual field
theory partition function on $S^{1}_{\beta}\times S^{3}$ with angular
velocities and chemical potentials conjugate to the $SO(6)$ R-charges turned
on. From the bulk perspective, the gravitational path integral instructs us to
sum over all geometries satisfying boundary conditions appropriate for our
choice of ensemble in asymptotically $AdS_{5}$. Therefore we need to know
black hole solutions carrying these charges, which we briefly review next,
mostly to establish our conventions.

The theory of gravity in asymptotically $AdS_{5}$ space dual to four
dimensional $\mathcal{N}=4$ Yang Mills arises from a dimensional reduction of
ten dimensional type IIB supergravity on $AdS_{5}\times S^{5}$. From this
perspective the $SO(6)$ gauge symmetry in the bulk is identified with the
isometries of $S^{5}$, with the $SO(6)$ charges associated to rotations along
the $S^{5}$. Since $SO(6)$ has rank three it leads to three gauge fields
$A_{1,2,3}$ and charges $R_{1,2,3}$. The resulting theory in $AdS_{5}$ is
quite complicated, but simplifies drastically when the rotation happens
symmetrically along the three Cartan directions on $S^{5}$, meaning $A\equiv
A_{1}=A_{2}=A_{3}$ and $R\equiv R_{1}=R_{2}=R_{3}$. We will restrict only to
that case in this paper, the reduction leads to a simple theory of minimal
gauged supergravity on $AdS_{5}$ with a single gauge field131313Note that, as
in almost all string constructions of AdS/CFT, the radius of the sphere is of
the same order as the AdS radius. The five-dimensional theory studied here is
a consistent truncation of the 10D theory, and all black hole solutions we
study may be uplifted to 10D type IIB, as we show in 3.4.. The bosonic part of
the action is given by (e.g. [66, 67]):

$\displaystyle I\supset\frac{1}{16\pi G_{5}}\int\left[(R+12)\ast
1-\frac{2}{3}F\wedge\ast F+\frac{8}{27}F\wedge F\wedge A\,\right].$ (3.1)

We work in units in which $\ell_{\textrm{AdS}_{5}}=1$ and the gauge coupling
$g$ has been set to $1$. When this effective action is obtained from string
theory, the dimensionful parameters will appear in the full action through the
5D Newton constant $G_{5}$. The Chern-Simons term affects both the definition
of asymptotic electric charges and the value of the on-shell action.

Before writing the solutions we should determine the boundary conditions. We
will parametrize the bulk by coordinates $(\tau,r,\theta,\phi,\psi)$ where
$(\phi,\psi)$ are $2\pi$ periodic and $\theta\in[0,\frac{\pi}{2}]$. The
coordinates $(\tau,\theta,\phi,\psi)$ also parametrize the boundary
$S^{1}\times S^{3}$ at large radius $r$. This means that the metric should
behave asymptotically, after a rescaling of $r$, as
$ds^{2}=\frac{dr^{2}}{r^{2}}+r^{2}\left(d\tau^{2}+d\Omega_{3}^{2}\right)+\mathcal{O}(r^{0})$,
with the unit $S^{3}$ element $d\Omega_{3}^{2}=d\theta^{2}+\sin^{2}\theta
d\phi^{2}+\cos^{2}\theta d\psi^{2}$. In order to fix the R-charge chemical
potential to be $\Phi$, we fix the gauge field holonomy at large $r$ through
$A=-\Phi d\tau+\mathcal{O}(r^{-1})$. The above electric potential is related
to the $R$-charge chemical potentials as
$\Phi_{1}=\Phi_{2}=\Phi_{3}=\frac{2}{3}\Phi$ [23]. Following [68] we fix the
angular velocities along the $S^{3}$ and the temperature through the
identification

$(\tau,\theta,\phi,\psi)\sim(\tau+\beta,\theta,\phi-i\Omega_{1}\beta,\psi-i\Omega_{2}\beta).$
(3.2)

A general solution with this asymptotic behavior is a charged rotating black
hole with unequal angular momenta. The solution, which we write in Lorentzian
signature for simplicity defining $t=-i\tau$, was found in [6], but we follow
the conventions of [23]. In terms of asymptotically static coordinates the
metric is given by

$\displaystyle ds^{2}=$ $\displaystyle-\frac{\Delta_{\theta}dt\left(2\nu
q+\rho^{2}\left(r^{2}+1\right)dt\right)}{\rho^{2}\Xi_{a}\Xi_{b}}+\frac{2\nu
qw}{\rho^{2}}+\frac{\rho^{2}dr^{2}}{\Delta_{r}}+\frac{f_{t}\left(\frac{\Delta_{\theta}dt}{\Xi_{a}\Xi_{b}}-w\right){}^{2}}{\rho^{4}}$
(3.3)
$\displaystyle+\frac{\rho^{2}d\theta^{2}}{\Delta_{\theta}}+\frac{\left(a^{2}+r^{2}\right)\sin^{2}\theta
d\phi^{2}}{\Xi_{a}}+\frac{\left(b^{2}+r^{2}\right)\cos^{2}\theta
d\psi^{2}}{\Xi_{b}},$ $\displaystyle A=$
$\displaystyle\frac{3q}{2\rho^{2}}\left(\frac{\Delta_{\theta}dt}{\Xi_{a}\Xi_{b}}-w\right)-\Phi
dt,$ (3.4)

where

$\displaystyle\nu$ $\displaystyle=a\cos^{2}\theta d\psi+b\sin^{2}\theta
d\phi,\qquad\rho=\sqrt{a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta+r^{2}},$ (3.5)
$\displaystyle\Delta_{r}$
$\displaystyle=\frac{\left(a^{2}+r^{2}\right)\left(b^{2}+r^{2}\right)\left(r^{2}+1\right)+2abq+q^{2}}{r^{2}}-2m,$
(3.6) $\displaystyle f_{t}$
$\displaystyle=2ab\rho^{2}q+2m\rho^{2}-q^{2},\qquad\Delta_{\theta}=1-a^{2}\cos^{2}\theta-b^{2}\sin^{2}\theta,$
(3.7) $\displaystyle\Xi_{a}$
$\displaystyle=1-a^{2},\qquad\Xi_{b}=1-b^{2},\qquad w=\frac{a\sin^{2}\theta
d\phi}{\Xi_{a}}+\frac{b\cos^{2}\theta d\psi}{\Xi_{b}}.$ (3.8)

The above metric gives a four parameter family of solutions for charged and
rotating black holes in five dimensions, parametrized by $(m,q,a,b)$, where we
will restrict to $0<a,b<1$. These parameters are related to
$(m,q,a,b)\to(\beta,\Phi,\Omega_{1},\Omega_{2})$ by imposing the solution is
smooth at the Euclidean horizon, located at the radius $r_{+}$ defined as the
largest positive root of $\Delta_{r}(r_{+})=0$. We trade from now on the
variable $m$ by $r_{+}$ through

$\displaystyle
m=\frac{\left(a^{2}+r_{+}^{2}\right)\left(b^{2}+r_{+}^{2}\right)\left(r_{+}^{2}+1\right)+2abq+q^{2}}{2r_{+}^{2}}.$
(3.9)

One can show the solution is smooth with appropriate choice of parameters
following the methods in [68]. The cycle becoming contractible at the horizon
is generated by the vector field $V=\frac{\partial}{\partial
t}+\Omega_{1}\frac{\partial}{\partial\phi}+\Omega_{2}\frac{\partial}{\partial\psi}$,
where

$\displaystyle\Omega_{1}=\frac{a\left(b^{2}+r_{+}^{2}\right)\left(r_{+}^{2}+1\right)+bq}{\left(a^{2}+r_{+}^{2}\right)\left(b^{2}+r_{+}^{2}\right)+abq},\qquad\Omega_{2}=\frac{b\left(a^{2}+r_{+}^{2}\right)\left(r_{+}^{2}+1\right)+aq}{\left(a^{2}+r_{+}^{2}\right)\left(b^{2}+r_{+}^{2}\right)+abq},$
(3.10)

denote angular velocities on the horizon. Smoothness also determines the
temperature to be

$\beta^{-1}=T=\frac{r_{+}^{4}\left(\left(a^{2}+b^{2}+2r_{+}^{2}\right)+1\right)-(ab+q)^{2}}{2\pi
r_{+}\left(\left(a^{2}+r_{+}^{2}\right)\left(b^{2}+r_{+}^{2}\right)+abq\right)}.$
(3.11)

The chemical potential at the boundary is fixed by the asymptotic value of
$A\to-\Phi dt$, when $r\to\infty$. The relation between $\Phi$ and $q$ is
determined by demanding the solution is regular at the Euclidean horizon.
Finally smoothness of the gauge potential $A$ at the horizon
$A_{\mu}V^{\mu}|_{r_{+}}=0$ gives the final relation

$\Phi=\frac{3qr_{+}^{2}}{2[\left(a^{2}+r_{+}^{2}\right)\left(b^{2}+r_{+}^{2}\right)+abq]}.$
(3.12)

Altogether, smoothness gave us the four relations needed to solve for
$(r_{+},q,a,b)$ in terms of $(\beta,\Phi,\Omega_{1},\Omega_{2})$.
Interestingly, in addition to the solution we have just discussed, there are
other distinct solutions with the same boundary conditions obtained by integer
shifts of the chemical potentials an angular velocities, first analyzed in the
context of $AdS_{5}$ black holes in [23]. We will comment on these solutions
in section 3.3, where they play an important role in our analysis.

Alternatively, these black holes can be identified through specifying the
values of their four conserved charges $(E,R,J_{1},J_{2})$ canonically
conjugate to $(\beta,\Phi,\Omega_{1},\Omega_{2})$. The charges can be defined
by the ADM procedure which gives

$\displaystyle E$
$\displaystyle=\frac{\pi\left(2abq\left(\Xi_{a}+\Xi_{b}\right)+m\left(-\Xi_{a}\Xi_{b}+2\Xi_{a}+2\Xi_{b}\right)\right)}{4G_{5}\Xi_{a}^{2}\Xi_{b}^{2}}+\frac{3\pi}{32G_{5}}\,,\,\,\,\,R=\frac{\pi
q}{2G_{5}\Xi_{a}\Xi_{b}},$ (3.13) $\displaystyle J_{1}$
$\displaystyle=\frac{\pi\left(bq\left(a^{2}+1\right)+2am\right)}{4G_{5}\Xi_{a}^{2}\Xi_{b}}\,,\,\,\,\,J_{2}=\frac{\pi\left(aq\left(b^{2}+1\right)+2bm\right)}{4G_{5}\Xi_{a}\Xi_{b}^{2}}.$
(3.14)

The energy of vacuum $AdS_{5}$ is given by $E_{0}=\frac{3\pi}{32G_{5}}$. We
will denote the black hole mass by $M\equiv E-E_{0}$. To reiterate, $R$ is
proportional to the angular momenta along $S^{5}$ distributed symmetrically
along the Cartan directions.

Having found the solution of the equations of motion filling the boundary
conditions, we can approximate the partition function in this grand-canonical
ensemble by the exponential of the classical action

$\displaystyle Z(\beta,\Omega_{1},\Omega_{2},\Phi_{1},\Phi_{2},\Phi_{3})\sim
e^{-I_{GCE}(\beta,\Omega_{1},\Omega_{2},\Phi_{1},\Phi_{2},\Phi_{3})}.$ (3.15)

As mentioned above, there are other saddles which are relevant in the near-
extremal limit which we discuss in section 3.3 but ignore for now. The action
$I_{GCE}(\beta,\Omega_{1,2},\Phi_{1,2,3})$, including now the GHY boundary
term and the holographic counterterm [69] appropriate to this ensemble where
we fix the metric and gauge potential at infinity, is given by

$I=\frac{1}{16\pi G_{5}}\int d^{5}x\sqrt{-g}\mathcal{L}+\frac{1}{8\pi
G_{5}}\int_{\partial}d^{4}x\sqrt{-h}K-\frac{3}{8\pi
G_{5}}\int_{\partial}d^{4}x\sqrt{-h}\left(1+\frac{R_{h}}{12}\right),$ (3.16)

where $h=g|_{\rm bdy}$ and $R_{h}$ is the Ricci curvature scalar of $h$.
Evaluating all contributions to the on-shell action (3.16), together with the
Chern-Simons term, one obtains [70, 71]

$\displaystyle
I_{GCE}=\frac{\pi\beta}{4G_{5}\Xi_{a}\Xi_{b}}\left(m-(r_{+}^{2}+a^{2})(r_{+}^{2}+b^{2})-\frac{q^{2}r_{+}^{2}}{(r_{+}^{2}+a^{2})(r_{+}^{2}+b^{2})+abq}\right)+\frac{3\pi\beta}{32G_{5}}.$
(3.17)

Defining the Bekenstein-Hawking entropy as the area of the horizon

$S=\frac{A}{4G_{5}}=\frac{\pi^{2}\left(\left(a^{2}+r_{+}^{2}\right)\left(b^{2}+r_{+}^{2}\right)+abq\right)}{2G_{5}r_{+}\Xi_{a}\Xi_{b}}\,,$
(3.18)

the on-shell action satisfies the so-called quantum statistical relation,

$\displaystyle I_{GCE}=\beta(E-TS-\Omega_{1}J_{1}-\Omega_{2}J_{2}-\Phi R),$
(3.19)

which can explicitly be checked using the expressions for the charges and
potentials. Finally, if we want to compute the partition function in a fixed
charge sector we need to not only write $(r_{+},q,a,b)$ in terms of the
charges but also add the apropriate boundary terms in the action to make the
variational principle well-defined [72]. For example, if we wanted to fix the
charge $R$ this amounts to adding an extra $I\to I+\beta\Phi R$ term in the
action, understanding that $\Phi$ should be written as a function of charges,
and similarly for angular momentum.

###  3.2 The $1/16$-BPS black hole solution

As discussed in the introduction of section 3, the general AdS5 black hole
solution has different limits corresponding to extremality ($T\rightarrow 0$)
and supersymmetry (existence of a Killing spinor). Written in terms of
charges, the supersymmetric (but not yet extremal) condition is [73]

$\displaystyle
M-\left(\frac{3}{2}R+J_{1}+J_{2}\right)\bigg{|}_{\text{SUSY}}=0,$ (3.20)

which after inserting the explicit expressions for $M$, $R$, and $J_{1,2}$
valid for both BPS and non-BPS black holes, takes the form

$\displaystyle q=\frac{m}{1+a+b}.$ (3.21)

We next want to impose extremality, meaning that the solution has zero
temperature. This can be achieved by imposing the further condition

$\displaystyle m=(a+b)(1+a)(1+b)(1+a+b).$ (3.22)

The size of the horizon for these BPS black holes is given by the simple
expression

$\displaystyle r_{+}=r^{*}\equiv\sqrt{(a+b+ab)}.$ (3.23)

In terms of $r^{*}$ the constraint (3.21) becomes $q=q^{*}=(a+b)(1+a)(1+b)$.
It is easy to check that (3.21) together with (3.22) implies that $T=0$. This
means that, while in general one could have supersymmetric non-extremal
($T\neq 0$) black holes, imposing extremality leads to the _BPS condition_ ,
which is the intersection of the supersymmetric and extremal surfaces. Note
that this also means that BPS black holes are now labeled by only two
parameters $(a,b)$, through the relations $(r^{*}(a,b),q^{*}(a,b))$. In the
above, following [8], we introduced the $(\phantom{r})^{*}$–notation, which
from now on will denote quantities evaluated after imposing supersymmetry and
extremality.

The black hole solutions that are both extremal and supersymmetric are the
ones dual the $1/16$-BPS states in $U(N)$ $\mathcal{N}=4$ Yang Mills at large
$N$. In terms of the parameters $(a,b)$ the BPS value of the Bekenstein-
Hawking entropy is

$\displaystyle S^{*}=\frac{\pi^{2}(a+b)\sqrt{ab+a+b}}{2G_{5}(1-a)(1-b)}.$
(3.24)

Importantly, we cannot yet determine from gravity whether this is the true
entropy of BPS states without including quantum effects from the gravity path
integral, as explained in section 2. The field theory calculation of the index
supports this interpretation of $S^{*}$ which we will verify from gravity as
well below. The BPS values for the energy and charges are

$\displaystyle M^{*}$ $\displaystyle=$
$\displaystyle-\frac{\pi\left(2a^{2}b^{2}+a^{3}(b+1)+a\left(b^{3}-3\right)+b\left(b^{2}-3\right)\right)}{4G_{5}(a-1)^{2}(b-1)^{2}},~{}~{}R^{*}=\frac{\pi(a+b)}{2G_{5}(1-a)(1-b)},$
(3.25) $\displaystyle J_{1}^{*}$ $\displaystyle=$
$\displaystyle\frac{\pi(a+b)(a(b+2)+b)}{4G_{5}(a-1)^{2}(1-b)},~{}~{}~{}~{}~{}J_{2}^{*}=\frac{\pi(a+b)(ab+a+2b)}{4G_{5}(1-a)(b-1)^{2}}.$
(3.26)

It is easy to verify the relation
$M^{*}-\frac{3}{2}R^{*}-J_{1}^{*}-J_{2}^{*}=0$. Thus, due to the BPS
constraints, given the angular momenta $J_{1}^{*}$ and $J_{2}^{*}$ we uniquely
determine the equal $U(1)^{3}\in SO(6)$ charge $R^{*}$ for the BPS black hole.
A field theory explanation of this constraint from the point of view of
$\mathcal{N}=4$ Yang Mills was suggested in [74, 75]. The chemical potentials
potentials associated to the BPS black holes are given by
$\Phi^{*}=\frac{3}{2}$, $\Omega_{1}^{*}=\Omega_{2}^{*}=1$.

This restriction on parameters by the BPS condition is not unfamiliar. In the
case of black holes in four dimensional ungauged supergravity, supersymmetry
only implies that the mass is equal to the charge, while BPS states satisfy
the further condition that the angular momentum vanishes. A motivation for
this is obvious in this example, black holes with real angular momentum that
are supersymmetric have a naked singularity in Lorentzian signature141414From
a Euclidean perspective, having supersymmetric solutions with imaginary
angular momentum and non-zero temperature is perfectly fine, giving a real
smooth metric, and just computes the index from gravity [76].. The issue with
supersymmetric non-extremal $AdS_{5}$ black holes is instead the presence of
closed timelike curves (CTC) in Lorentzian signature outside the horizon [6],
and imposing extremality together with supersymmetry removes this pathology.

###  3.3 What we want to compute

In this section we clarify two issues that arise from the discussion so far.
The first is that in the grand-canonical ensemble the solution reviewed in
section 3.1 is not the only one. Take for example the gauge potential $A$.
While we demanded that $A_{\mu}V^{\mu}|_{\textrm{horizon}}=0$, the gauge
invariant statement that the holonomy is trivial along a contractible cycle
allows for more general values of the potential at the horizon, which can be
removed by a large gauge transformation but lead to physically distinct
solutions. The other saddles are mostly subleading but become important at low
temperatures, which is the regime of interest of this paper. They were
considered in the near extremal limit in the context of a different black hole
solution in [28, 29], and we will discuss them in the context of AdS3 in
section 4.

The second issue is related to the fact that the supergravity theory
considered above and the corresponding electrically charged black holes arise
from the dimensional reduction of Type IIB supergravity solutions on $S^{5}$
in the sector in which the solutions have equal Kaluza-Klein momenta $R\equiv
R_{1}=R_{2}=R_{3}$ on the $S^{5}$ factor. In a grand canonical ensemble, even
if we fix the chemical potentials conjugate to the R-charges $R_{i}$ to be
equal, these new solutions discussed in the previous paragraph will inevitably
involve configurations that do not respect this symmetry. This would require
dealing with the much more complicated solution of type IIB with scalar fields
turned on [77], which we will not attempt in this paper.

We will address these issues in the following way. First, we will use
information about the UV completion of the theory given by $\mathcal{N}=4$
Yang Mills to determine the full space of saddle point configurations. Second,
we will choose an ensemble for which only configurations with
$R_{1}=R_{2}=R_{3}$ contribute. Our conventions in this subsection closely
follow those of [23].

We begin by identifying the new solutions of the equations of motion. The most
obvious way to generate new solutions is the following. Looking at the
boundary conditions specified in section 3.1 it is clear that any geometry
obtained by an integer shift $\Omega_{i}\to\Omega_{i}+\frac{2\pi
i\mathbb{Z}}{\beta}$ solves the same equations with the same boundary
condition, while being physically distinct. Therefore the gravitational path
integral instructs us to sum over all of them. Something similar is true for
the gauge potential. Two configurations related by $A\to A+\frac{2\pi
i\mathbb{Z}}{\beta}dt$ satisfy the same gauge invariant boundary condition at
infinity while continuing to be smooth since the extra integer flux can be
undone by a large gauge transformation. This is not surprising since the
R-charge chemical potential in $AdS_{5}$ corresponds to angular velocity along
$S^{5}$ direction in ten dimensions. The partition function with all chemical
potentials fixed is given in the large $N$ limit by a sum over saddles

$\displaystyle Z(\beta,\Omega_{1},\Omega_{2},\Phi_{i})$
$\displaystyle=\mathrm{Tr}[e^{-\beta
H+\beta\Omega_{1}J_{1}+\beta\Omega_{2}J_{2}+\frac{1}{2}\beta\Phi_{1}R_{1}+\frac{1}{2}\beta\Phi_{2}R_{2}+\frac{1}{2}\beta\Phi_{3}R_{3}}]$
$\displaystyle\to\sum_{n_{1},n_{2},m_{1},m_{2},m_{3}}Z_{\text{one-
loop}}\,e^{-I_{GCE}(\beta,\Omega_{1}+\frac{2\pi
in_{1}}{\beta},\Omega_{2}+\frac{2\pi in_{2}}{\beta},\Phi_{1}+\frac{2\pi
im_{1}}{\beta},\Phi_{2}+\frac{2\pi im_{2}}{\beta},\Phi_{3}+\frac{2\pi
im_{3}}{\beta})},$ (3.27)

where $I_{GCE}(\beta,\Omega_{1},\Omega_{2},\Phi_{1},\Phi_{2},\Phi_{3})$ is the
action of the black hole considered in [77] with Dirichlet boundary
conditions. When $\Phi_{1}=\Phi_{2}=\Phi_{3}=\frac{2}{3}\Phi$ (which along
with the factors of $\frac{1}{2}$ in (3.27) defines our normalization for
$\Phi_{i}$) this action above becomes equivalent to the action computed in
section 3.1. We will discuss the one-loop contribution later. At this point we
need to make a choice regarding the range of integers that are allowed in the
sum (3.27). One option is to consider more carefully the global nature of the
gauge groups in five dimensions from a ten dimensional perspective given by
Type IIB. A simpler option is to use AdS/CFT and analyze the properties of
this partition function given by $\mathcal{N}=4$ Yang Mills, which we turn to
next.

The $\mathcal{N}=4$ Yang Mills theory has the superconformal symmetry
$PSU(2,2|4)$, and parallel to the gravity analysis we decompose the Lorentz
and R-symmetry factors to the Cartan subalgebra $U(1)_{J_{1}}\times
U(1)_{J_{2}}\times U(1)^{3}_{R_{i}}$, with $i=1,2,3$. In $\mathcal{N}=1$
language, this theory is a particular instance of a vector multiplet coupled
to a triplet of adjoint chiral multiplets $X_{i}$, where we normalize the
R-charges such that $R_{i}$ assigns charge 2 to the corresponding chiral
multiplet. This means that scalars have even integer R-charge and fermions
have odd integer R-charge. States of the theory are also labeled by the two
angular momenta $J_{1}$, $J_{2}$ which as usual are integer or half-integer
and satisfy spin statistics. With these conventions, the fermion number
operator, is given by $F=2J_{1,2}=R_{i}$ mod 2. This constraint implies a
particular periodicity of the grand canonical partition function under shifts
of the chemical potential [23], which we can use to deduce the allowed saddles
we need to include. Comparing with the gravitational answer (3.27) the
constraint implies the following restriction on the solutions:

$\displaystyle m_{1}+m_{2}+m_{3}+n_{1}+n_{2}=2\mathbb{Z}.$ (3.28)

This determines the saddles that should in principle be included based on
smoothness and global properties of the gauge groups151515Actually there
should be more restrictions on these integers since otherwise the sum strictly
diverges. A rationale for which saddles should be included or not is currently
under investigation [78]. Additional restrictions besides (3.28) also were
observed in [23] when computing the superconformal index. These issues are not
relevant in the limit considered in this paper.. The need to sum over saddles
was pointed out in this context in [23] for supersymmetric configurations
only, but is true more broadly.

This brings us to our second issue. We see clearly now that even if we
restrict to $\Phi_{1}=\Phi_{2}=\Phi_{3}=\frac{2}{3}\Phi$, the sum over
integers will necessarily involve configurations outside the scope of section
3.1. In order to resolve this we will go to a mixed ensemble where some
charges will be fixed. First of all we will introduce some notation, writing
the grand-canonical partition function as

$Z(\beta,\Omega_{1},\Omega_{2},\Phi_{i})=\mathrm{Tr}[e^{-\beta\\{\mathcal{Q},\mathcal{Q}^{\dagger}\\}+\beta(\Omega_{1}-1)J_{1}+\beta(\Omega_{2}-1)J_{2}+\frac{1}{2}\beta(\Phi_{1}-1)R_{1}+\frac{1}{2}\beta(\Phi_{2}-1)R_{2}+\frac{1}{2}\beta(\Phi_{3}-1)R_{3}}],$
(3.29)

where
$\\{\mathcal{Q},\mathcal{Q}^{\dagger}\\}=H-J_{1}-J_{2}-\frac{1}{2}(R_{1}+R_{2}+R_{3})$
and $\mathcal{Q},\mathcal{Q}^{\dagger}$ is the supercharge preserved by the
$1/16$-BPS states at zero temperature. This will allow us later to make a more
direct connection with the superconformal index [25]. The definition of
$\\{\cal{Q},\cal{Q}^{\dagger}\\}$ is chosen to match the supersymmetry
condition, (3.20), when all $R_{i}$ are equal. We redefine the chemical
potentials to track their temperature scaled deviations from their BPS values
[8], as well as new flavor charges

$\displaystyle\mathfrak{j}_{1,2}\equiv
J_{1,2}+\frac{R_{3}}{2},~{}~{}~{}\mathfrak{q}_{1,2}\equiv\frac{R_{1,2}-R_{3}}{2}\,,$
(3.30)

which commute with the $\mathcal{N}=1$ subalgebra, as well as

$\displaystyle\Delta_{1,2}=\frac{\beta}{2\pi
i}(\Phi_{1,2}-1),\quad\omega_{1,2}=\frac{\beta}{2\pi
i}(\Omega_{1,2}-1),\quad\alpha=\frac{\beta}{4\pi i}\varphi,$ (3.31)

where

$\displaystyle\varphi=\Phi_{1}+\Phi_{2}+\Phi_{3}-\Omega_{1}-\Omega_{2}-1\,.$
(3.32)

In terms of these, the grand canonical partition function becomes

$\displaystyle
Z(\beta,\omega_{1},\omega_{2},\Delta_{1},\Delta_{2},\varphi)=\mathrm{Tr}[e^{-\beta\\{\mathcal{Q},\mathcal{Q}^{\dagger}\\}+2\pi
i\omega_{1}\mathfrak{j}_{1}+2\pi i\omega_{2}\mathfrak{j}_{2}+2\pi
i\Delta_{1}\mathfrak{q}_{1}+2\pi i\Delta_{2}\mathfrak{q}_{2}+2\pi i\alpha
R_{3}}].$ (3.33)

These variables have the advantage of removing the constraints between the
integer shifts appearing in the different solutions. Indeed, the partition
function (3.27) with a constraint (3.28) can be equivalently written as

$\displaystyle
Z(\beta,\omega_{1},\omega_{2},\Delta_{1},\Delta_{2},\alpha)=\sum_{n_{1},n_{2},m,m_{1},m_{2}}Z_{\text{one-
loop}}\,e^{-I_{\rm
GCE}(\beta,\omega_{1}+n_{1},\omega_{2}+n_{2},\Delta_{1}+m_{1},\Delta_{2}+m_{2},\alpha+m)},$
(3.34)

with unconstrained integers $(m,m_{1},m_{2},n_{1},n_{2})$. This in part
motivates these redefinitions. If we expect to obtain an emergent
$\mathcal{N}=2$ Schwarzian mode in the nearly $1/16$-BPS limit, it should come
with a sum over saddles involving an unconstrained integer.

Avoiding black hole solutions with unequal R-charges suggests we use a mixed
ensemble in which we fix the inverse temperature $\beta$ and the BPS chemical
potential $\alpha$, but Laplace transform with respect to the other potentials
to obtain a trace over charges. For simplicity we will also work in an
ensemble of fixed $\mathfrak{j}_{1,2}$ charges

$\displaystyle
Z(\beta,\mathfrak{j}_{1},\mathfrak{j}_{2},\mathfrak{q}_{1},\mathfrak{q}_{2},\alpha)$
$\displaystyle=\mathrm{Tr}_{\mathfrak{j}_{1},\mathfrak{j}_{2},\mathfrak{q}_{1},\mathfrak{q}_{2}}[e^{-\beta\\{\mathcal{Q},\mathcal{Q}^{\dagger}\\}+2\pi
i\alpha R_{3}}]$ (3.35)
$\displaystyle=\int_{0}^{1}d\omega_{1}\int_{0}^{1}d\omega_{2}\int_{0}^{1}d\Delta_{1}\int_{0}^{1}d\Delta_{2}$
$\displaystyle\,e^{-2\pi i\Delta_{1}\mathfrak{q}_{1}-2\pi
i\Delta_{2}\mathfrak{q}_{2}-2\pi i\omega_{1}\mathfrak{j}_{1}-2\pi
i\omega_{2}\mathfrak{j}_{2}}Z(\beta,\omega_{1},\omega_{2},\Delta_{1},\Delta_{2},\alpha).$
(3.36)

To ease the notation we will use the same letter to denote the generators
$\mathfrak{j}_{1,2}$, $\mathfrak{q}_{1,2}$ and their fixed numerical values in
the ensemble. To work with equal charges $R_{1}=R_{2}=R_{3}$, we set
$\mathfrak{q}_{1}=\mathfrak{q}_{2}=0$, which leaves us with one free $U(1)$
charge which we have previously called $R$ in the gravity solution. As
mentioned above, solutions with more general charges are increasingly
difficult to construct [79, 73, 64, 7, 77] because one must include scalar
moduli corresponding to deformations of the $S^{5}$ metric, and are outside
the scope of the present work. The mixed ensemble at equal $U(1)^{3}$ charges
is then:

$\displaystyle Z(\beta,\mathfrak{j}_{1},\mathfrak{j}_{2},\alpha)$
$\displaystyle=\mathrm{Tr}_{\mathfrak{j}_{1,2},\mathfrak{q}_{1,2}=0}[e^{-\beta\\{\mathcal{Q},\mathcal{Q}^{\dagger}\\}+2\pi
i\alpha R_{3}}]$ (3.37) $\displaystyle=\int_{0}^{1}$ $\displaystyle
d\omega_{1}\int_{0}^{1}d\omega_{2}\int_{0}^{1}d\Delta_{1}\int_{0}^{1}d\Delta_{2}\,e^{-2\pi
i\omega_{1}\mathfrak{j}_{1}-2\pi
i\omega_{2}\mathfrak{j}_{2}}Z(\beta,\omega_{1},\omega_{2},\Delta_{1},\Delta_{2},\alpha).$
(3.38)

Now we can put everything together and write an expression for the
semiclassical limit of the partition function in the mixed ensemble we are
interested in, with

$\displaystyle\text{fixed }\alpha,\,\,\mathfrak{j}_{1,2}\text{ and
}R_{1}-R_{2}=R_{1}-R_{3}=0\,.$ (3.39)

The answer is a sum over a single integer-valued family of saddles

$\displaystyle Z(\beta,\mathfrak{j}_{1},\mathfrak{j}_{2},\alpha)$
$\displaystyle=\sum_{n\in\mathbb{Z}}Z_{\text{one-loop}}e^{-I_{\rm
GCE}(\mathfrak{q}_{1}=0,\mathfrak{q}_{2}=0,\mathfrak{j}_{1},\mathfrak{j}_{2},\alpha+n)-2\pi
i\omega_{1}\mathfrak{j}_{1}-2\pi i\omega_{2}\mathfrak{j}_{2}}$ (3.40)
$\displaystyle\equiv\sum_{n\in\mathbb{Z}}Z_{\text{one-loop}}e^{-I_{\rm
ME}(\beta,\mathfrak{j}_{1},\mathfrak{j}_{2},\alpha+n)}.$ (3.41)

In the first line we write the action in this mixed ensemble as the grand-
canonical action $I_{\rm
GCE}(\mathfrak{q}_{1}=0,\mathfrak{q}_{2}=0,\mathfrak{j}_{1},\mathfrak{j}_{2},\alpha+n)$
plus the evaluation of the boundary terms needed to make the variational
problem well-defined (adapting [72] to our problem) resulting in the extra
terms $2\pi i\omega_{1}\mathfrak{j}_{1}+2\pi i\omega_{2}\mathfrak{j}_{2}$. In
these last two terms $\omega_{1}$ and $\omega_{2}$ should be written in terms
of the quantities that are fixed in this ensemble. In the second line we
defined the total value of the action as $I_{\rm
ME}(\beta,\mathfrak{j}_{1},\mathfrak{j}_{2},\alpha)$.

We now have only the sum over the integer shifts of the holonomy corresponding
to the BPS potential in (3.40). The sum over saddles has the schematic form of
(2.14) in which the integer parameter $r=1$. Different values of $m$ can be
interpreted as different winding sectors of the Schwarzian $U(1)$ boson (seen
in section 2.3), or in the bulk, as sectors of an abelian gauge field in JT
gravity. Here however we see they are the holonomies of a particular linear
combination of 10D Kaluza-Klein gauge fields.

Either from the Schwarzian theory perspective or from the AdS5/$\mathcal{N}=4$
Yang Mills perspective, a black hole where we set $\alpha=1/2$ is
supersymmetric and this is equivalent to inserting $(-1)^{F}=e^{i\pi R_{3}}$
in the path integral. At these values of chemical potentials we will see the
gravity answer matches with the superconformal index. We would like to
emphasize that the index (evaluated in the ensemble we are working with) is
always in a deconfined phase in the large $N$ limit. The standard
superconformal index is studied in the fully grand-canonical ensemble, defined
as

$\widetilde{\mathcal{I}}(\beta,\omega_{1},\omega_{2},\Delta_{1},\Delta_{2})\equiv\mathrm{Tr}[e^{i\pi
R_{3}}e^{-\beta\\{\mathcal{Q},\mathcal{Q}^{\dagger}\\}+2\pi
i\omega_{1}\mathfrak{j}_{1}+2\pi i\omega_{2}\mathfrak{j}_{2}+2\pi
i\Delta_{1}\mathfrak{q}_{1}+2\pi i\Delta_{2}\mathfrak{q}_{2}}],$ (3.42)

and computed for $\mathcal{N}=4$ Yang Mills in [24, 25]. In our ensemble, the
index obtained from evaluating
$\mathcal{I}(\beta,\mathfrak{j}_{1},\mathfrak{j}_{2})\equiv
Z(\beta,\mathfrak{j}_{1},\mathfrak{j}_{2},\alpha=1/2)$ is related to the
superconformal index of [24, 25] by the following exact relation161616We can
generalize this calculation to an ensemble with fixed $\mathfrak{q}_{1,2}\neq
0$ by a similar formula. This should also display always a deconfined phase,
although the bulk analysis is more complicated.

$\displaystyle\mathcal{I}(\beta,\mathfrak{j}_{1},\mathfrak{j}_{2})$
$\displaystyle=\mathrm{Tr}_{\mathfrak{j}_{1,2},\mathfrak{q}_{1,2}=0}[(-1)^{F}e^{-\beta\\{\mathcal{Q},\mathcal{Q}^{\dagger}\\}}]$
(3.43) $\displaystyle=\int_{0}^{1}$ $\displaystyle
d\omega_{1}\int_{0}^{1}d\omega_{2}\int_{0}^{1}d\Delta_{1}\int_{0}^{1}d\Delta_{2}\,e^{-2\pi
i\omega_{1}\mathfrak{j}_{1}-2\pi
i\omega_{2}\mathfrak{j}_{2}}\widetilde{\mathcal{I}}(\beta,\omega_{1},\omega_{2},\Delta_{1},\Delta_{2}).$
(3.44)

While the superconformal index
$\widetilde{\mathcal{I}}(\beta,\omega_{1},\omega_{2},\Delta_{1},\Delta_{2})$
can be in a confined phase for some ranges of parameters, hiding the black
hole behavior [21, 22, 19], the index
$\mathcal{I}(\beta,\mathfrak{j}_{1},\mathfrak{j}_{2})$ defined above (with
some charges fixed) is always dominated by the black hole saddle. We leave for
future work verifying this from a boundary perspective.

In section 3.5, we will see that not only does (3.40) share the same sum over
$U(1)$ saddles of the $\mathcal{N}=2$ super-Schwarzian partition function, but
it also has the same classical action as a function of $\beta$ and $\alpha$ in
the limit of low temperatures and fixed $\alpha$. From this we will determine
the one-loop determinant around each saddle in section 3.6.

###  3.4 Near horizon geometry and supersymmetries

Having discussed the general black hole solution, the mixed ensemble partition
function, and the saddles that contribute to it, we turn our attention to the
near-extremal (and specifically near-BPS) limits. For the AdS5 black hole
(3.3), the exact supersymmetric and extremal limits have a near-horizon
geometry described by AdS2, and upon uplifting to 10D, the solution is a
specific fibration of AdS${}_{2}\times S^{3}\times S^{5}$ which we describe in
detail.

In the strict BPS near-horizon limit, we show the solution develops emergent
superconformal isometries. On the bosonic side, there is both an emergent
$SL(2,\mathbb{R})$ for the AdS2 as well as a $U(1)$ which rotates the solution
along a particular combination of angles on $S^{3}\times S^{5}$. For the
fermions, we find as expected a doubling of the number of supersymmetries
corresponding to the appearance of a conformal Killing spinor.

Altogether, the BPS gravity theory has a local $SU(1,1|1)$ superconformal
algebra of symmetries in the near horizon region, but finite temperature
quantum corrections explicitly break this local symmetry to a global
$SU(1,1|1)$ which acts as isometries of the near-AdS2 saddles. The
$\mathcal{N}=2$ Schwarzian theory captures the breaking of these symmetries.
In order to demonstrate that we have identified the correct pattern of
symmetry breaking (and thus the correct Schwarzian), we consider the BPS limit
of the leading saddle and compute explicitly the global super-isometries of
the 10D solution explained above.

In contrast to other parts of section 3, in this subsection we will set the
rotation parameters $a=b$ for simplicity. This will not change the general
structure of the symmetry algebra, but it will impact the specific form of the
Killing vectors and Killing spinors. Much of this section is based on the
analysis of [33], but see also [80]. Details of the supergravity analysis are
given in Appendix B.

In the original coordinates $(t,r,\theta,\phi,\psi)$ the metric is
asymptotically static and the Killing horizon is generated by

$\displaystyle V=\frac{\partial}{\partial
t}+\Omega_{1}\frac{\partial}{\partial\phi}+\Omega_{2}\frac{\partial}{\partial\psi}.$
(3.45)

For the purpose of determining the near horizon geometry, it is convenient to
switch to corotating coordinates, in which $V=\frac{\partial}{\partial t}$
becomes null at the horizon. This amounts to changing angles $(\phi,\psi)$ to
$(\tilde{\phi},\tilde{\psi})$, such that $\phi=\tilde{\phi}+\Omega_{1}\,t$ and
$\psi=\tilde{\psi}+\Omega_{2}\,t$. In these coordinates the horizon becomes
static and the asymptotic metric is now rotating. In the BPS limit, going to
corotating coordinates amounts to

$\displaystyle\phi=\tilde{\phi}+\Omega_{1}^{*}\,t=\tilde{\phi}+t,\qquad\psi=\tilde{\psi}+\Omega_{2}^{*}\,t=\tilde{\psi}+t.$
(3.46)

The near-horizon limit is now taken by setting

$\displaystyle r\rightarrow\sqrt{a(a+2)}+r,$ (3.47)

and expanding for $r\ll r^{*}$. In what follows, we introduce parameters

$\displaystyle\omega=\sqrt{\frac{2a}{1-a}}\,,\qquad\lambda=\sqrt{1+3\omega^{2}}\,.$
(3.48)

We also make a further general coordinate transformation in which we scale

$\displaystyle
r=\frac{\omega}{4\lambda}(1+a)\sqrt{\frac{a}{2+a}}\,\tilde{r}\,,$ (3.49)

then drop the tilde for notational convenience. This leads to a particular
fibration of AdS${}_{2}\times S^{3}$:

$\displaystyle ds^{2}_{5}$
$\displaystyle=\left(\frac{\omega}{2\lambda}\right)^{2}\left(-r^{2}dt^{2}+\frac{dr^{2}}{r^{2}}\right)+3\left(\frac{\omega^{2}}{4}\sigma_{3}^{L}+\frac{\omega}{4\lambda}rdt\right)^{2}+\omega^{2}d\Omega_{3}^{2},$
(3.50) $\displaystyle A$
$\displaystyle=-\frac{3}{2}\left(\frac{\omega^{2}}{4}\sigma_{3}^{L}+\frac{\omega}{4\lambda}rdt\right),$
(3.51)

where $d\Omega_{3}^{2}=d\theta^{2}+\cos^{2}\theta
d\tilde{\psi}^{2}+\sin^{2}\theta d\tilde{\phi}^{2}$ is the metric on $S^{3}$,
and $\sigma_{3}^{L}=2(\cos^{2}\theta d\tilde{\psi}+\sin^{2}\theta
d\tilde{\phi})$.

To determine the Killing spinors and Killing vectors in this geometry, we
follow [33] and work with a 10D lift of the above metric. While it is not
strictly necessary to work in 10D, this presentation serves to set our
conventions and gives a geometrical origin for all the symmetries of the
solution. Working with the leading order Type IIB supergravity theory, the
massless fields are the NS-NS sector fields $(G_{MN},B^{(2)}_{MN},\Phi)$, the
RR-form fields $(C^{(0)},C^{(2)},C^{(4)})$, as well as complex Weyl spinors
and gravitinos $(\lambda,\Psi_{M})$ of the same chirality. The field strength
$F^{(5)}=dC^{(4)}$ is self-dual, $F^{(5)}=\star_{(10)}F^{(5)}$. In the empty
AdS5 $\times$ S5 as well as the $1/16$-BPS AdS backgrounds of IIB
supergravity, we may set the axio-dilaton $(\Phi,C^{(0)})$ to a constant, all
fermions $(\lambda,\Psi_{M})$ to zero, and only turn on a supersymmetric
background for the metric and the 5-form flux $(G_{MN},F_{MNPQR})$. In
Einstein frame, there is no explicit dependence on the scalars, no Chern-
Simons terms, and the action takes the simple form:

$S_{IIB}=\frac{1}{16\pi G_{10}}\int
d^{10}x\sqrt{-G}\left(\mathcal{R}_{(10)}-\frac{1}{4\cdot 5!}F_{M_{1}\dots
M_{5}}F^{M_{1}\dots M_{5}}\right)\,.$ (3.52)

As is standard for theories with self-dual gauge fields, above we wrote the
kinetic term for $F^{(5)}$, but one should always impose the $F=\ast F$
equation of motion by hand171717In practice, one can work with the covariant
equations of motion[81], perform a dimensional reduction on $S^{5}$ and then
use the 5D action (3.1). The generation of the Chern-Simons term comes from
reducing the $F^{(5)}$ equation of motion[82, 83]..

The near-horizon metric and gauge field may now be uplifted to a 10D solution.
This is now given by [66, 84]:

$\displaystyle ds^{2}_{10}$
$\displaystyle=ds^{2}_{5}+\sum_{i=1}^{3}\left[(d\mu_{i})^{2}+\mu_{i}^{2}\left(d\xi_{i}^{2}-\frac{2}{3}A\right)^{2}\right],$
(3.53) $\displaystyle F^{(5)}$
$\displaystyle=(1+\ast_{(10)})\left[-4\,\text{vol}_{(5)}-\frac{1}{3}\sum_{i=1}^{3}d(\mu_{i}^{2})\wedge
d\xi_{i}\wedge\ast_{(5)}F^{(2)}\right],$ (3.54)

where we introduced $\mu_{1}=\sin\tilde{\alpha},$
$\mu_{2}=\cos\tilde{\alpha}\sin\tilde{\beta},$
$\mu_{3}=\cos\tilde{\alpha}\cos\tilde{\beta}$. The angles
$\tilde{\alpha}\in[0,\frac{\pi}{2}]$, $\tilde{\beta}\in[0,\frac{\pi}{2}]$,
$\xi_{i}\in[0,2\pi]$, parametrize the sphere $S^{5}$.

In the 10D background, the supersymmetry transformation of the gravitino is

$\delta_{\epsilon}\Psi_{M}=(\partial_{M}+\frac{1}{4}\omega^{AB}_{M}\Gamma_{AB})\epsilon+\frac{i}{16\cdot
5!}F_{N_{1}\dots N_{5}}\Gamma^{N_{1}\dots N_{5}}\Gamma_{M}\epsilon\,.$ (3.55)

Using this, the independent Killing spinors are found to be

$\displaystyle\epsilon_{1}$
$\displaystyle=\sqrt{\frac{8\lambda}{\omega^{2}r}}e^{-\frac{i}{2}(\xi_{1}+\xi_{2}+\xi_{3})}\left[\epsilon_{0}^{+}-\left(\frac{irt}{2}+\frac{3\omega}{2\lambda}\right)\Gamma^{4}\Gamma^{9}\epsilon_{0}^{+}\right],$
(3.56) $\displaystyle\epsilon_{2}$
$\displaystyle=\sqrt{\frac{\omega^{2}r}{8\lambda}}e^{-\frac{i}{2}(\xi_{1}+\xi_{2}+\xi_{3})}\epsilon_{0}^{-},$
(3.57)

where $\epsilon_{0}^{+}$ and $\epsilon_{0}^{-}$ are constant Majorana-Weyl
spinors determined from integrability condition for Killing spinors (B.22).
They are parametrized by two arbitrary real parameters each, giving four real
parameters in total. This means that the near-horizon region preserves four
supersymmetries, in contrast to full geometry of the $\frac{1}{16}$ BPS black
hole we started with [85, 86].
Knowing the Killing spinors of the near-horizon metric, we can immediately
find its Killing vectors by computing linearly independent Killing spinor
bilinears $\overline{\epsilon}_{I}\Gamma^{a}\epsilon_{J}$. After appropriately
normalizing the constant spinors $\epsilon_{0}^{\pm}$ we find Killing vectors

$\displaystyle D$ $\displaystyle=-t\partial_{t}+r\partial_{r},$ (3.58)
$\displaystyle\mathcal{Z}$
$\displaystyle=\frac{i\omega^{2}}{\lambda^{2}}(\partial_{\xi_{1}}+\partial_{\xi_{2}}+\partial_{\xi_{3}})-\frac{i}{2\lambda^{2}}(\partial_{\tilde{\phi}}+\partial_{\tilde{\psi}}),$
(3.59) $\displaystyle E_{+}$
$\displaystyle=-\frac{i\omega^{2}}{4\lambda}\partial_{t},$ (3.60)
$\displaystyle E_{-}$
$\displaystyle=\frac{8i\lambda}{\omega^{2}}rt\partial_{r}+\left(-\frac{4i\lambda}{\omega^{2}}t^{2}+\frac{4i(9\omega^{2}-4\lambda^{2})}{\omega^{2}\lambda}\frac{1}{r^{2}}\right)\partial_{t}$
(3.61)
$\displaystyle+\frac{8i}{\lambda^{2}\omega}\frac{1}{r}(\partial_{\xi_{1}}+\partial_{\xi_{2}}+\partial_{\xi_{3}})+\frac{12i}{\lambda^{2}\omega}\frac{1}{r}(\partial_{\tilde{\phi}}+\partial_{\tilde{\psi}}).$
(3.62)

One can verify that with the above normalization they satisfy the following
commutation relations

$\displaystyle[D,E_{\pm}]$ $\displaystyle=\pm E_{\pm},$
$\displaystyle[D,\mathcal{Z}]$ $\displaystyle=0,$ (3.63)
$\displaystyle[\mathcal{Z},E_{\pm}]$ $\displaystyle=0,$
$\displaystyle[E_{+},E_{-}]$ $\displaystyle=2D\,.$ (3.64)

We can thus identify $D,\,E_{\pm}$ as the generators of $SL(2,\mathbb{R})$ and
$\mathcal{Z}$ as a $U(1)$ R-symmetry generator. Using a standard procedure of
determining spacetime isometry superalgebras [87, 88, 89, 90] we can interpret
the above commutation relations as giving the bosonic part of the isometry
superalgebra through

$\displaystyle[\mathcal{Q}_{B}(k_{i}),\mathcal{Q}_{B}(k_{j})]$
$\displaystyle=\mathcal{Q}_{B}([k_{i},k_{j}]),$ (3.65)

with $\mathcal{Q}_{B}(k_{i})$ denoting the bosonic generator associated to
Killing vector $k_{i}$. To determine the rest of the superalgebra we follow
the prescription

$\displaystyle[\mathcal{Q}_{B}(k),\mathcal{Q}_{F}(\epsilon)]$
$\displaystyle=\mathcal{Q}_{F}(\mathbb{L}_{k}\epsilon),$ (3.66)
$\displaystyle\\{\mathcal{Q}_{F}(\epsilon_{I}),\mathcal{Q}_{F}(\epsilon_{J})\\}$
$\displaystyle=\mathcal{Q}_{B}(\epsilon_{I}\Gamma\epsilon_{J}),$ (3.67)

where $\mathcal{Q}_{F}(\epsilon_{I})$ denotes now the fermionic generators
associated to Killing spinor $\epsilon_{I}$, and the Killing vectors act on
the Killing spinors through the spinorial Lie derivative [90], shown
explicitly in (B.52). With that, we obtain

$\displaystyle\\{\overline{\mathcal{Q}}_{2},\mathcal{Q}_{1}\\}$
$\displaystyle=\mathcal{Z}+D,$
$\displaystyle\\{\overline{\mathcal{Q}}_{1},\mathcal{Q}_{2}\\}$
$\displaystyle=\mathcal{Z}-D,$ (3.68)
$\displaystyle\\{\overline{\mathcal{Q}}_{2},\mathcal{Q}_{2}\\}$
$\displaystyle=E_{+},$
$\displaystyle\\{\overline{\mathcal{Q}}_{1},\mathcal{Q}_{1}\\}$
$\displaystyle=E_{-},$ (3.69)

together with

$\displaystyle[\mathcal{Q}_{1},\mathcal{Z}]$
$\displaystyle=-\frac{1}{2}\mathcal{Q}_{1},$
$\displaystyle[\overline{\mathcal{Q}}_{1},\mathcal{Z}]$
$\displaystyle=\frac{1}{2}\overline{\mathcal{Q}}_{1},$
$\displaystyle[\mathcal{Q}_{2},\mathcal{Z}]$
$\displaystyle=-\frac{1}{2}\mathcal{Q}_{2},$
$\displaystyle[\overline{\mathcal{Q}}_{2},\mathcal{Z}]$
$\displaystyle=\frac{1}{2}\overline{\mathcal{Q}}_{2},$
$\displaystyle[\mathcal{Q}_{1},D]$ $\displaystyle=\frac{1}{2}\mathcal{Q}_{1},$
$\displaystyle[\overline{\mathcal{Q}}_{1},D]$
$\displaystyle=\frac{1}{2}\overline{\mathcal{Q}}_{1},$
$\displaystyle[\mathcal{Q}_{2},D]$
$\displaystyle=-\frac{1}{2}\mathcal{Q}_{2},$
$\displaystyle[\overline{\mathcal{Q}}_{2},D]$
$\displaystyle=-\frac{1}{2}\overline{\mathcal{Q}}_{2},$
$\displaystyle[\mathcal{Q}_{1},E_{+}]$ $\displaystyle=\mathcal{Q}_{2},$
$\displaystyle[\overline{\mathcal{Q}}_{1},E_{+}]$
$\displaystyle=-\overline{\mathcal{Q}}_{2},$
$\displaystyle[\mathcal{Q}_{2},E_{+}]$ $\displaystyle=0,$
$\displaystyle[\overline{\mathcal{Q}}_{2},E_{+}]$ $\displaystyle=0,$
$\displaystyle[\mathcal{Q}_{1},E_{-}]$ $\displaystyle=0,$
$\displaystyle[\overline{\mathcal{Q}}_{1},E_{-}]$ $\displaystyle=0,$
$\displaystyle[\mathcal{Q}_{2},E_{-}]$ $\displaystyle=\mathcal{Q}_{1},$
$\displaystyle[\overline{\mathcal{Q}}_{2},E_{-}]$
$\displaystyle=-\overline{\mathcal{Q}}_{1}\,,$ (3.70)

which can be identified as the $SU(1,1|1)$ superalgebra that the extremal
black holes exhibit in their near-horizon region.

It is worth mentioning, as explained in [33] (which was specialized to the
case $J_{1}=J_{2}$, $R_{i}=R$), that the solution has additional bosonic
isometries $SU(2)\times U(3)$ which do not mix with $SU(1,1|1)$. In principle
we could introduce more general charges or chemical potentials corresponding
to these symmetries, but as we have already explained in the discussion
surrounding (3.39), we are working in a mixed ensemble in which some
combination of the rotations and R-charges is held fixed, so these extra
symmetries play no role in the thermodynamics.

In the standard example of AdS2 at low but non-zero temperature, the boundary
modes of the metric become strongly coupled, and the gravitational path
integral reduces to that of JT gravity at leading order in $e^{S_{0}}$ and
$\beta$. JT gravity may itself by written as an $SL(2,\mathbb{R})$ $BF$ theory
in the first order formulation of gravity, subject to the correct boundary
conditions from the second order formalism [91]. In the present example,
superconformal symmetry of the 10D solution implies that both the boundary
modes of the metric and its superpartners become strongly coupled, and the
effective two-dimensional theory in the AdS2 throat should instead be based on
a $SU(1,1|1)$ $BF$ theory. In the present work, we have not attempted to
derive the full two-dimensional dilaton supergravity, however, this point of
view is not essential for our argument. A reader interested in dimensional
reductions to JT supergravity as well as the relation to super-$BF$ theory may
consult [29, 37, 92, 93, 94, 95, 96].

###  3.5 The low-temperature expansion of the action

The purpose of this section is to show the emergence of the $\mathcal{N}=2$
Jackiw-Teitelboim mode in the near $1/16$-BPS limit. This was anticipated in
the previous section based on the pattern of symmetry breaking of nearly
$1/16$-BPS states. We will work in an ensemble of fixed $\mathfrak{j}_{1}$,
$\mathfrak{j}_{2}$ and $\mathfrak{q}_{1}=\mathfrak{q}_{2}=0$. For simplicity
we focus first on saddles with integer $n=0$ since the general case can be
obtained by a simple shift of the chemical potential $\alpha\to\alpha+n$, with
$n\in\mathbb{Z}$. Instead of implementing a dimensional reduction of type IIB
supergravity near the horizon to obtain JT gravity, we will take a shortcut
and match the classical action. We leave a full treatement of the reduction
for future work.

We now turn to expansion of the on-shell action around the $1/16$-BPS values
of parameters $T=0$ and $\varphi=0$. We turn on non-zero values of $T$ and
chemical potential $\varphi$ and assume $T,\varphi\ll 1$, keeping
$\alpha=\frac{\beta}{4\pi i}\varphi$ fixed. As explained above, in the fixed
$(\beta,\mathfrak{j}_{1,2},\mathfrak{q}_{1,2}=0,\alpha)$ ensemble the
classical action involves new boundary terms compared with the grand canonical
ensemble

$I_{\rm ME}(\beta,\mathfrak{j}_{1},\mathfrak{j}_{2},\alpha)=I_{\rm
GCE}(\mathfrak{q}_{1}=0,\mathfrak{q}_{2}=0,\mathfrak{j}_{1},\mathfrak{j}_{2},\alpha)+2\pi
i\omega_{1}\mathfrak{j}_{1}+2\pi i\omega_{2}\mathfrak{j}_{2},$ (3.71)

where the GCE action is evaluated at necessary chemical potentials and angular
momenta such that it is dominates by the quantum numbers
$\mathfrak{q}_{1}=0,\,\mathfrak{q}_{2}=0,\,\mathfrak{j}_{1},$ and
$\mathfrak{j}_{2}$. To set up the calculation, we begin picking
$a^{\star},b^{\star}$ to correspond to the zero temperature $1/16$-BPS black
hole. The parameters $a^{\star},b^{\star}$ determines the fixed values of
$\mathfrak{j}_{1,2}$ in the mixed ensemble through

$\mathfrak{j}_{1}=N^{2}\frac{\left(a^{*}+b^{*}\right)\left(a^{*}+1\right)\left(b^{*}+1\right)}{2\left(a^{*}-1\right)^{2}\left(1-b^{*}\right)},~{}~{}~{}~{}\mathfrak{j}_{2}=N^{2}\frac{\left(a^{*}+b^{*}\right)\left(a^{*}+1\right)\left(b^{*}+1\right)}{2\left(1-a^{*}\right)\left(b^{*}-1\right)^{2}},$
(3.72)

where we restored the $N$ dependence by replacing $G_{5}=\frac{\pi}{2N^{2}}$
to make the comparison with $\mathcal{N}=4$ Yang Mills more transparent. Next
we go away from the $1/16$-BPS black hole by expanding around these values
$(r^{\star},q^{\star},a^{\star},b^{\star})\to(r^{\star}+\delta
r,q^{\star}+\delta q,a^{\star}+\delta a,b^{\star}+\delta b)$ where
$(r^{\star},q^{\star},a^{\star},b^{\star})$ are determined in terms of
$\mathfrak{j_{1}}$ and $\mathfrak{j_{2}}$ due to the nonlinear relation
imposed for BPS black holes. Imposing then that we work with fixed
$(T,\varphi,\mathfrak{j}_{1},\mathfrak{j}_{2})$, fixes the expansion
parameters $\delta r$, $\delta q$, $\delta a$ , $\delta b$ in terms of $T$ and
$\varphi$, using the expressions given in section 3.1. We carry this out in
detail in Appendix A. It is convenient to write all formulas in terms of
$a^{\star}$ and $b^{\star}$, understanding they should be thought of as
functions of $\mathfrak{j}_{1}$ and $\mathfrak{j}_{2}$.

The final answer for the action in the low temperature expansion in the mixed
ensemble is given by

$\displaystyle I_{\rm ME}(\beta,\mathfrak{j}_{1},\mathfrak{j}_{2},\alpha)$
$\displaystyle=$ $\displaystyle\beta E_{0}-S^{*}-\frac{1}{2}\beta\varphi
R^{*}-\frac{2\pi^{2}}{\beta
M_{SU(1,1|1)}}\left(1+\left(\frac{\beta\varphi}{2\pi}\right)^{2}\right),$
(3.73) $\displaystyle=$ $\displaystyle\beta E_{0}-S^{*}-2\pi i\alpha
R^{*}-\frac{2\pi^{2}}{\beta M_{SU(1,1|1)}}(1-4\alpha^{2}),$

where we defined the BPS Bekenstein-Hawking entropy and R-charge by the
expressions

$\displaystyle S^{*}$ $\displaystyle\equiv
N^{2}\frac{\pi(a^{*}+b^{*})\sqrt{a^{*}b^{*}+a^{*}+b^{*}}}{(1-a^{*})(1-b^{*})},$
(3.74) $\displaystyle R^{*}$ $\displaystyle\equiv
N^{2}\frac{(a^{*}+b^{*})}{(1-a^{*})(1-b^{*})},$ (3.75)

and as in (2.1), we defined the parameter $M_{SU(1,1|1)}$ explicitly as

$\displaystyle\frac{1}{M_{SU(1,1|1)}}$
$\displaystyle=N^{2}\frac{\left(a^{*}+b^{*}\right)^{2}\left(3-a^{*}b^{*}+a^{*}+b^{*}\right)}{8\left(1-a^{*}\right)\left(1-b^{*}\right)\left(3a^{*}b^{*}+3a^{*}+a^{*2}+b^{*2}+3b^{*}+1\right)}.$
(3.76)

Other evidence of the dynamics in the near-extremal limit being controlled by
the breaking of $SU(1,1|1)$ was given in [38], which also identified (3.76) as
the relevant energy scale.

Now we can put everything together and write a formula giving an approximation
of the mixed ensemble partition function in the nearly $1/16$-BPS limit after
including the sum over saddles

$Z(\beta,\mathfrak{j}_{1},\mathfrak{j}_{2},\alpha)=e^{2\pi i\alpha
R^{\star}}e^{-\beta E_{0}}\sum_{n\in\mathbb{Z}}Z_{\text{one-
loop}}~{}e^{S^{\star}+\frac{2\pi^{2}}{\beta
M_{SU(1,1|1)}}(1-4(\alpha+n)^{2})},$ (3.77)

where for now we ignored the one-loop corrections. This is precisely the
classical partition function as computed by the $\mathcal{N}=2$ Schwarzian
action, as given in (2.14). In this identification $S^{\star}$ corresponds to
the topological term $S_{0}$ in the Schwarzian theory. The only difference is
the first term $e^{2\pi i\alpha R^{\star}}$. Since for the theory under
consideration the $R$ charge is an integer, that phase is not affected by the
sum over saddles and can be pulled in front of the sum in (3.77). This term
comes from the fact that there is a mismatch between $R$, (the R-charge in the
UV) and the Schwarzian $U(1)_{R}$ charge $Z$ given by $Z=R-R^{\star}$. This is
analogous to how in section 2.1, the grand-canonical ensemble was dominated by
some charge $Q_{*}$ around which excitations were captured by a $U(1)$ whose
contribution once again consisted of a sum over saddles (2.1). Additionally,
we observe that the R-charge of the supercharge is the same as the fundamental
charge, so we have $r=1$ compared to (2.14). Finally, there are no Schwarzian
$\theta$ terms arising from our UV theory which would have appeared through an
imaginary contribution to the action, $i\theta n$ for each saddle in (3.77).
Such a topological term might have been present due to the Chern-Simons term
in the 5d action (3.1), but turns out to be absent here.

In the general case, the equations above depend non-trivially on the angular
momenta $\mathfrak{j}_{1,2}$ through the complicated dependence on $a^{\star}$
and $b^{\star}$. This dependence becomes simpler in the limit where for
example $a^{\star}\to 1$ and $b^{\star}\to 1$. In this limit
$\mathcal{J}_{1,2}\equiv\mathfrak{j}_{1,2}/N^{2}\to\infty$, and we get

$S^{\star}\sim
N^{2}\sqrt{3}\Big{(}\frac{\mathcal{J}_{1}\mathcal{J}_{2}}{2}\Big{)}^{1/3},~{}~{}~{}R^{\star}\sim
N^{2}\Big{(}\frac{\mathcal{J}_{1}\mathcal{J}_{2}}{2}\Big{)}^{1/3},~{}~{}~{}\frac{1}{M_{SU(1,1|1)}}\sim
N^{2}\frac{2}{3}\Big{(}\frac{\mathcal{J}_{1}\mathcal{J}_{2}}{2}\Big{)}^{1/3}\sim\frac{2R^{\star}}{3}.$
(3.78)

Since in this limit $R^{\star}\sim N^{2}\mathcal{O}(\mathcal{J}^{2/3})$, we
see that in defining $\mathfrak{j}_{1,2}$, the contribution from $R^{\star}$
is subleading compared with $J_{1,2}^{\star}$. Therefore in this limit we can
also identify $\mathcal{J}_{1,2}$ with the angular momentum along
$S^{3}\subset AdS_{5}$ directly.

###  3.6 The black hole spectrum including quantum corrections

The classical analysis above determined that the relevant effective theory
which describes the near-horizon region (as a function of the temperature and
chemical potential) is the $\mathcal{N}=2$ super-Schwarzian theory. Following
the conventions in section 2.3, the low-temperature expansion of the action
yields the coupling $M_{SU(1,1|1)}^{-1}$ in (3.76), as well as the $r$ unit-
charge and $\theta$-angle:

$\displaystyle\ r=1\,,\qquad\theta=0\,.$ (3.79)

Combining the classical analysis above with the quantum corrections from the
Schwarzian mode gives the following prediction for the mixed ensemble
partition function from gravity:

$Z(\beta,\mathfrak{j}_{1},\mathfrak{j}_{2},\alpha)=e^{2\pi i\alpha
R^{\star}}e^{-\beta
E_{0}}\sum_{n\in\mathbb{Z}}\frac{2\cos\left(\pi(\alpha+n)\right)}{\pi\left(1-4(\alpha+n)^{2}\right)}~{}e^{S^{\star}+\frac{2\pi^{2}}{\beta
M_{SU(1,1|1)}}(1-4(\alpha+n)^{2})}.$ (3.80)

As explained in section 2.1, if we work in an ensemble where we fix all
charges except the one that couples to the Schwarzian mode, the one-loop
determinant coming from all additional fields is a constant proportional to
$N^{\\#}$. These corrections were recently studied in the context of AdS5
black holes in, for example, [97]. These appear in the exponential as $\log N$
corrections. These will not be the focus of our discussion and therefore in
the equation above we absorbed them into $S^{\star}$ itself. Following 2.3,
the quantum corrected partition function can also be written

$\displaystyle Z(\beta,\mathfrak{j}_{1},\mathfrak{j}_{2},\alpha)=e^{-\beta
E_{0}}e^{2\pi i\alpha R^{\star}}\sum_{Z_{\rm
Sch}\in\mathbb{Z}}\Big{(}\delta_{Z_{\rm Sch},0}~{}e^{S^{\star}}$ (3.81)
$\displaystyle+\left(e^{2\pi i\alpha Z_{\rm Sch}}+e^{2\pi i\alpha(Z_{\rm
Sch}-1)}\right)\int_{E_{\text{gap}}(Z_{\rm Sch})}^{\infty}dE_{\rm
Sch}e^{-\beta E_{\rm
Sch}}\frac{e^{S^{\star}}\sinh{\big{(}2\pi\sqrt{\frac{2}{M_{SU(1,1|1)}}(E_{\rm
Sch}-E_{\text{gap}}(Z_{\rm Sch}))}\big{)}}}{2\pi E_{\rm Sch}}\Big{)},$

where we defined $E_{\rm gap}(Z_{\rm Sch})=\frac{M_{SU(1,1|1)}}{8}(Z_{\rm
Sch}-\frac{1}{2})^{2}$. We introduce in the equation above the variables
$E_{\rm Sch}$ and $Z_{\rm Sch}$ when performing the Laplace transform. These
are to be interpreted as the energy and charge respectively of the effective
$\mathcal{N}=2$ Schwarzian mode arising in the IR.

We can relate the Schwarzian charge and energy defined above to
$\mathcal{N}=4$ Yang Mills data in a simple way. First by taking into account
in (3.81) the $e^{2\pi i\alpha R^{\star}}$ prefactor, and remembering from the
discussion in section 3.3 that $\alpha$ couples to $R$, we see the
$\mathcal{N}=4$ Yang Mills R-charge is shifted by $R^{\star}$ with respect to
the Schwarzian charge. Second, we observe that temperature in (3.81) couples
to $E_{0}+E_{\rm Sch}$. We can use $E=E_{0}+\Delta$ to relate the Schwarzian
energy to the $\mathcal{N}=4$ Yang Mills scaling dimension. It is useful to
first define

$\Delta_{\rm
BPS}\equiv\mathfrak{j}_{1}+\mathfrak{j}_{2}+\frac{1}{2}R^{\star},$ (3.82)

which is the scaling dimension of the $1/16$-BPS extremal states. It is also
useful to define the scaling dimension obtained from imposing the
supersymmetric constraints as

$\Delta_{\rm SUSY}(R)\equiv\mathfrak{j}_{1}+\mathfrak{j}_{2}+\frac{1}{2}R,$
(3.83)

with $\Delta_{\rm SUSY}(R^{\star})=\Delta_{\rm BPS}$. Combining these
definitions with the fact that temperature in
$Z(\beta,\mathfrak{j}_{1},\mathfrak{j}_{2},\alpha)$ couples to
$E-J_{1}-J_{2}-\frac{1}{2}(R_{1}+R_{2}+R_{3})$ allows us to find $\Delta$ in
terms of $E_{\rm Sch}$. To summarize, the $\mathcal{N}=4$ Yang Mills data is
given by

$\displaystyle\Delta$ $\displaystyle=$ $\displaystyle\Delta_{BPS}+E_{\rm
Sch}+\frac{1}{2}Z_{\rm Sch},$ (3.84) $\displaystyle R$ $\displaystyle=$
$\displaystyle R^{\star}+Z_{\rm Sch}.$ (3.85)

Using this identification, we can interpret both lines of equation (3.81) in
terms of the spectrum of nearly $1/16$-BPS states in $\mathcal{N}=4$ Yang
Mills. The first line represent the $1/16$-BPS states. They have $Z_{\rm
Sch}=0$ and therefore $R=R^{\star}$ and $\Delta=\Delta_{\rm BPS}$, and their
degeneracy is equal to $\exp{\left(S^{\star}\right)}$ in the large $N$ limit.
The superconformal index corresponds to $\alpha=1/2$, and this only gets
contributions from the BPS states since otherwise the two states in the
supermultiplet with charge $Z_{\rm Sch}$ and $Z_{\rm Sch}-1$ cancel in the
second line of (3.81). The answer then for the index is, as usually assumed,
given by

$\displaystyle\mathcal{I}(\beta,\mathfrak{j}_{1},\mathfrak{j}_{2})$
$\displaystyle\equiv$ $\displaystyle e^{\beta
E_{0}}Z(\beta,\mathfrak{j}_{1},\mathfrak{j}_{2},\alpha=1/2),$ (3.86)
$\displaystyle=$
$\displaystyle(-1)^{R^{\star}}e^{S^{\star}(\mathfrak{j}_{1},\mathfrak{j}_{2})}.$
(3.87)

The first term is the overall sign depending on whether $R^{\star}$ is even or
odd. Consequently, if $R^{\star}$ is even we find that the BPS states are
bosonic, while when $R^{\star}$ is odd we find that the BPS states are
fermionic. In either case, there is no cancellation (at least to this order in
$N$) in the superconformal index and thus the matching between the index and
the entropy of BPS black holes is accurate. The change in sign of the index as
$R^{\star}$ goes from even to odd was also observed explicitly in the boundary
theory, in the calculations of [15].

$R-R^{\star}$$\Delta-\Delta_{\rm BPS}$(b) $R=R^{\star}$(a) Regions with black
holes(b) $R=R^{\star}$(c) $R\neq R^{\star}$ Figure 1: (a) The figure shows a
sketch of the $(R,\Delta)$ plane for fixed $\mathfrak{j}_{1}$ and
$\mathfrak{j}_{2}$. The red region indicates the forbidden region with
$\Delta<\Delta_{\rm SUSY}(R)$. The vertical gray lines denote the continuum
sector of the black hole spectrum starting at the quantum corrected extremal
value $\Delta_{\rm extremal}(R)$ for $R\neq R^{\star}$. The blue line denotes
the classical extremality bound. When $R=R^{\star}$ the lightest black hole
has dimension $\Delta_{\rm BPS}$ (black dot) and then a continuum above a gap
$\Delta_{\rm gap}$. (b) Density of states with charge $R=R^{\star}$
corresponding to the vertical line at the origin of figure a. (c) Density of
state for other charges $R\neq R^{\star}$.

Going beyond the BPS states, the second line of (3.81) gives the spectrum of
non supersymmetric black holes. These can be extremal or not. For example when
$R\neq R^{\star}$ the spectrum presents extremal black holes when

$\displaystyle\Delta_{\rm extremal}(R\neq R^{\star})={\rm
min}_{\pm}\left[\Delta_{\rm
BPS}+\frac{1}{2}\Big{(}R-R^{\star}+\frac{1}{2}\pm\frac{1}{2}\Big{)}+\frac{M_{SU(1,1|1)}}{8}\Big{(}R-R^{\star}\pm\frac{1}{2}\Big{)}^{2}\right],$
(3.88)

that have zero-temperature but are not supersymmetric and only when
$R=R^{\star}$ the two notions match $\Delta_{\rm
extremal}(R=R^{\star})=\Delta_{\rm BPS}$. The minimum is taken between the two
supermultiplets with charge $R$ states. The degeneracy for these black holes
predicted by (3.81) goes to zero instead of being exponential in
$N^{2}$.181818This is similar to the behavior for non-supersymmetric black
holes studied in [28]. The result for the energy of the extremal states should
be compared to the naive classical answer, which in an
$(R-R^{\star})/R^{\star}$ expansion is given by,

$\displaystyle\Delta_{\text{extremal}}^{\text{classical}}=\Delta_{\rm
BPS}+\frac{1}{2}(R-R^{\star})+\frac{M_{SU(1,1|1)}}{8}(R-R^{\star})^{2}.$
(3.89)

Interestingly, for instance when $R\geq R^{\star}$, the quantum corrected
extremal energy in (3.88) is lower than the naive classical value in
(3.89).191919This also occurs for sufficiently negative values of
$R-R^{\star}$.  Having a quantum corrected energy ground state below the
extremality bound has a possible interpretation in the context of the weak
gravity conjecture [98]. While usually such corrections are seen to come from
higher-derivative terms in the action202020See [99] and references therein for
a review. (which we will in fact discuss in section 3.7), in this case, the
correction comes from the temperature dependence of the one-loop determinant
in the gravitational path integral. We summarize all these results about the
spectrum of extremal black holes states in figure 1(a).

With the expression above we can derive the result quoted in the introduction
regarding the spectrum of black hole states in $\mathcal{N}=4$ Yang Mills for
states with charge $R=R^{\star}$. These come from the $1/16$-BPS states
counted by the first line of (3.81), but also from supermultiplets in the
second line with maximal R-charge $R^{\star}+1$. These states appear at a
scaling dimension in the range

$\Delta>\Delta_{\rm BPS}+\Delta_{\text{gap}}\,,\qquad\text{ with
}\qquad\Delta_{\rm gap}\equiv\frac{M_{SU(1,1|1)}}{32}\,.$ (3.90)

Figure 2: The energy above the supersymmetric bound (left column) and the
entropy (right column) as a function of temperature in the canonical ensemble
with $R=R^{\star}$ (top line) and an example of $R\neq R^{\star}$ (bottom
line). The black curve represents the result which includes quantum
corrections, while the blue curve is obtained by computing the naive semi-
classical answer.

From this expression we can determine the gap between the $1/16$-BPS black
hole and the lowest black hole state with charge $R^{\star}$, which we denote
by $\Delta_{\rm gap}$. Using (3.76), the quantum corrections at low energies
in the gravitational path integral thus gives

$\Delta_{\rm
gap}=\frac{\tilde{\Delta}(\mathcal{J}_{1},\,\mathcal{J}_{2})}{N^{2}},~{}~{}~{}~{}\tilde{\Delta}(\mathcal{J}_{1},\,\mathcal{J}_{2})=\frac{\left(1-a^{*}\right)\left(1-b^{*}\right)\left(3a^{*}b^{*}+3a^{*}+a^{*2}+b^{*2}+3b^{*}+1\right)}{4\left(a^{*}+b^{*}\right)^{2}\left(3-a^{*}b^{*}+a^{*}+b^{*}\right)}\,,$
(3.91)

where $a^{\star},b^{\star}$ are functions of
$\mathcal{J}_{1,2}\equiv\mathfrak{j}_{1,2}/N^{2}$ given above. Moreover, the
density of states at charge $R^{\star}$ above this gap is given by

$\rho(\Delta,\mathfrak{j}_{1},\mathfrak{j}_{2},R^{\star})=e^{S^{\star}}\delta(\Delta-\Delta_{\rm
BPS})+\frac{e^{S^{\star}}\sinh{\left(\pi\sqrt{\frac{\Delta-\Delta_{\rm
BPS}-\Delta_{\rm gap}}{4\Delta_{\rm gap}}}\right)}}{\pi(\Delta-\Delta_{\rm
BPS})}\Theta(\Delta-\Delta_{\rm BPS}-\Delta_{\rm gap}),$ (3.92)

where $S^{\star}$, $\Delta_{\rm gap}$ and $\Delta_{\rm BPS}$ are the functions
of $\mathfrak{j}_{1}$ and $\mathfrak{j}_{2}$ given above. A similar formula
can be rewritten for states with $R\neq R^{\star}$, although the delta
function is not present for these states

$\displaystyle\rho(\Delta,\mathfrak{j}_{1},\mathfrak{j}_{2},R)$
$\displaystyle=$
$\displaystyle\frac{e^{S^{\star}}\sinh{\left(\pi\sqrt{\frac{\Delta-\Delta_{\rm
extremal}(R)}{4\Delta_{\rm gap}}}\right)}}{2\pi(\Delta-\Delta_{\rm
SUSY}(R))}\Theta(\Delta-\Delta_{\rm extremal}(R))$ (3.93)
$\displaystyle~{}~{}+\frac{e^{S^{\star}}\sinh{\left(\pi\sqrt{\frac{\Delta-\Delta_{\rm
extremal}(R+1)}{4\Delta_{\rm gap}}}\right)}}{2\pi(\Delta-\Delta_{\rm
SUSY}(R+1))}\Theta(\Delta-\Delta_{\rm extremal}(R+1)),$

where we have suppressed the dependence of all parameters on
$\mathfrak{j}_{1}$ and $\mathfrak{j}_{2}$. For both cases, we plot the results
in figures 1(b) and (c). As in section 2.1, the continuum part of the density
of states (3.92) and (3.93) receives perturbative and non-perturbative
corrections in $N$. We expect that in the continuum region these corrections
could lead to non-perturbative gaps in the spectrum, which we expect to be
exponentially small in $N$.

We can also compare the quantum corrected entropies and energies at fixed $R$
with corresponding semiclassical results. We show this in figure 2.

As a final comment, it is reasonable to ask whether there are other BPS states
with general charge $R$ not necessarily equal to $R^{\star}$. There is
evidence that there are BPS black hole solutions with generic charges [100,
101]. These are found by incorporating scalar hair, and can be interpreted as
a black hole with a condensate of BPS particles outside the horizon. The
entropy of these solutions is expected to be subleading since they do not
dominate the superconformal index, but this deserves further study.

###  3.7 Type IIB string corrections to the Schwarzian Result

A basic essential feature of the original AdS/CFT correspondence between Type
IIB string theory on AdS${}_{5}\times S^{5}$ and $\mathcal{N}=4$ Super Yang-
Mills is that there are two expansion parameters on both sides of the duality.
In the bulk, this is because there are two different length scales
corresponding to the string scale and the Planck scale:

$\displaystyle\frac{L_{AdS_{5}}^{4}}{\ell_{\textrm{planck}}^{4}}\sim
N\,,\qquad\quad\frac{L_{AdS_{5}}^{4}}{\ell_{\textrm{string}}^{4}}\sim
g_{YM}^{2}N\equiv\lambda\,.$ (3.94)

With $L_{AdS_{5}}=1$, $\ell_{\textrm{string}}=\sqrt{\alpha^{\prime}}$, we have

$G_{5}=\frac{\pi}{2N^{2}}\qquad\Rightarrow\qquad
G_{10}=8\pi^{6}g_{s}^{2}(\alpha^{\prime})^{4}\,,\qquad
g_{s}=\frac{g_{YM}^{2}}{2\pi}\,,\qquad\alpha^{\prime}=\frac{1}{\sqrt{2g_{YM}^{2}N}}=\frac{1}{\sqrt{2\lambda}}\,.$
(3.95)

In the preceding sections, we have analyzed the euclidean gravity partition
function of black holes only in the limit of $N\rightarrow\infty$,
$\lambda\rightarrow\infty$. In the bulk, this is the limit of classical
supergravity with all higher derivative terms due to string corrections and
loop corrections suppressed. To our knowledge, this is the only limit in which
the rotating black hole solutions such as those in section 3.1 are known. The
spectrum of these black holes in this limit is dual to the spectrum of
$\mathcal{N}=4$ SYM at large $N$ and strong coupling. However, to first order
in the ‘t Hooft expansion, we can actually evaluate the first correction to
the Schwarzian result using only the leading gravity solution and knowledge of
the corrected 10-dimensional action.

In the 10D uplift of the black hole solution (3.53), the only IIB fields
turned on are the metric and the five form; therefore we may use the
$\alpha^{\prime}$ corrected action:

$S_{IIB}=\frac{1}{16\pi G_{10}}\int
d^{10}x\sqrt{G}\left(\mathcal{R}_{(10)}-\frac{1}{4\cdot
5!}(F_{5})^{2}+\gamma\mathcal{W}\right)\,,$ (3.96)

where in a particular scheme the 8-derivative term can be written in terms of
the Weyl tensor $C$ and:

$\displaystyle\gamma=\frac{\pi^{3}}{128}\zeta(3)(\alpha^{\prime})^{3}\,,\qquad\,\,\mathcal{W}\sim
C^{4}+\textrm{(supersymmetric completion)}\,.$ (3.97)

In [102], the action of this form was used to compute corrections to the free
energy of the near-extremal D3-brane, but the full supersymmetric completion
of $\mathcal{W}$ was not known. Following earlier work of [103, 104], [105]
found $\mathcal{W}$ to be a sum of 20 monomials built from the Weyl tensor
$C_{MNPQ}$ and a certain polynomial of derivatives of $F_{5}$ denoted
$\mathcal{T}_{MNPQRS}$. In the application to AdS5 black holes, [106]
carefully computed the effective action and its valuation on the AdS5 black
hole. In perturbation theory, the shift in the on-shell action due to the new
terms comes entirely from evaluating the leading solution on the perturbation
[107], and [106] found the final result in the case of equal rotation to be
relatively simple212121The final expression given in equation 16 of [106] v2
is off by a factor of $N^{-4}$. We thank J. Melo and J. Santos for confirming
it.:

$\frac{1}{16\pi
G_{10}}\gamma\int\mathcal{W}\Big{|}_{BH}=-\frac{\alpha^{\prime}{}^{3}}{G_{5}}\frac{3\pi^{4}\zeta(3)\beta(m+q)^{2}(m-q-2aq)(m-q+2aq)}{2(1-a^{2})^{\frac{1}{2}}(r_{+}^{2}+a^{2})^{15/2}}.$
(3.98)

Evaluating this correction in the near-BPS limit in terms of $\mathcal{N}=4$
Yang Mills data we find

$\displaystyle\frac{1}{16\pi G_{10}}\gamma\int\mathcal{W}\Big{|}_{BH}$
$\displaystyle=$ $\displaystyle\frac{2\pi^{2}\delta M_{SU(1,1|1)}}{\beta
M_{SU(1,1|1)}^{2}}(1-4\alpha^{2})+\ldots,$ (3.99) $\displaystyle\frac{\delta
M_{SU(1,1|1)}}{M_{SU(1,1|1)}}$ $\displaystyle=$
$\displaystyle-\frac{1}{\lambda^{3/2}}\frac{3\pi^{3}\zeta(3)(1+a^{*})^{3}(1-a^{*})^{3/2}}{2a^{*}{}^{7/2}(3-a^{*})},$
(3.100)

where $M_{SU(1,1|1)}\sim N^{-2}$ is the answer found above in (3.76) evaluated
at $b^{*}=a^{*}$. From this expression it is easy to find the correction to
the gap $\delta\Delta_{\rm gap}=\Delta_{\rm gap}\delta
M_{SU(1,1|1)}/M_{SU(1,1|1)}\sim N^{-2}\lambda^{-3/2}$. To write a more
explicit expression, we can take the limit $1-a^{\star}$ small, such that
$\mathcal{J}\equiv\mathfrak{j}_{1}/N^{2}=\mathfrak{j}_{2}/N^{2}\gg 1$. In this
limit

$\Delta_{\rm gap}(\lambda)=\Delta_{\rm
gap}(\lambda=\infty)\left(1-\frac{1}{\lambda^{3/2}}\frac{12\pi^{3}\zeta(3)}{\sqrt{\mathcal{J}}}+\ldots\right).$
(3.101)

In particular we find that the gap above the supersymmetric bound in figure
1(a) becomes smaller due to $\alpha^{\prime}$ corrections and the extremal
states deviate even further from their naive classical value. As mentioned in
the previous section, stringy corrections to the extremality bound, in
particular their sign (which in this case proved to be negative) are typically
important in the context of the weak gravity conjecture program (once again,
see [99] for a review).

To get a sense about the importance of stringy corrections it is useful to
compare their magnitude to that of the quantum corrections discussed in
section 3.6, when it comes to the energy of the extremal black hole states.
Writing

$\displaystyle\Delta_{\text{extremal}}=\Delta_{\rm
BPS}+\frac{1}{2}(R-R^{\star})+4\bigg{(}\Delta_{\text{gap}}+\underbrace{\delta\Delta_{\text{gap}}}_{\begin{subarray}{c}\text{Stringy}\\\
\text{corrections}\end{subarray}}\bigg{)}\bigg{(}R-R^{\star}\underbrace{-\,\,\frac{1}{2}\,}_{\begin{subarray}{c}\text{Schwarzian}\\\
\text{quantum}\\\ \text{corrections}\end{subarray}}\bigg{)}^{2},$ (3.102)

for $R>R^{\star}$.222222There is a similar expression for $R<R^{\star}$ which
will lead to similar conclusions.  In such a case, both the quantum
corrections and the stringy corrections push the extremal black hole states
below the classical extremality bound. Nevertheless, when it comes to the
energy at extremality, the stringy corrections can become more important then
the Schwarzian quantum corrections, when

$\displaystyle\frac{|\delta\Delta_{\text{gap}}|}{\Delta_{\text{gap}}}>\frac{1}{R-R^{\star}}\,.$
(3.103)

This can occur when we can scale $R^{\star}\to\infty$ while making
$R-R^{\star}\gg 1$ with $(R-R^{\star})/R^{\star}\to 0$. Thus, the contribution
of the Schwarzian quantum corrections and the stringy corrections to the
extremal energy could exchange dominance when $R-R^{\star}\sim\lambda^{3/2}$.

##  4 Further examples: black holes in $\mathcal{N}=(2,2)$ supergravity in
AdS3

In this section we will consider a simpler theory with similar properties as
those of the one studied above. We consider $(2,2)$ supergravity in an
asymptotically AdS3 spacetime. Regardless of the matter content, it has
rotating and charged black holes solutions which at low temperatures, in a
certain charge sector, also present an emergent $SU(1,1|1)$ symmetry in the
IR. We will show this is true using the Virasoro symmetry following the
approach of [108, 29].

We will focus on the contribution from supergravity only. The reason is that
the pure gravity answer is universal and an approximation of the full spectrum
for holographic theories, either for large central charge [109] or for large
angular momentum sector [110]. We will focus mostly on the role that integer
spectral flow has on the spectrum, using the technical results derived in
[111].

###  4.1 States and representations of 2D $\mathcal{N}=(2,2)$ CFT

We consider a theory with a gravity sector in asymptotically AdS3 given by
Chern-Simons theory with group $SU(1,1|1)_{L}\times SU(1,1|1)_{R}$, and an
asymptotic symmetry algebra given by $\mathcal{N}=(2,2)$ Virasoro symmetry.
The left- and right-mover generators contain a bosonic Virasoro algebra
generated by stress tensor $T_{L}$ and $T_{R}$, the complex supercurrent
$G^{\pm}_{L}$ and $G^{\pm}_{R}$ and a $U(1)$ current $J_{L}$ and $J_{R}$. We
denote the charges associated to the $U(1)$ currents by $Q_{L}$ and $Q_{R}$
respectively. The Virasoro central charge $c_{L}=c_{R}=c$ is given by
$\widehat{c}\equiv c/3=1+2b^{-2}$, which defines the parameters $\widehat{c}$
and $b$.

We will consider this theory when the boundary is a complexified torus with
moduli $\tau$ and $\overline{\tau}$ and define $q=e^{2\pi i\tau}$ and
$\overline{q}=e^{-2\pi i\overline{\tau}}$. We also introduce a chemical
potential $z$ and $\overline{z}$ conjugated to the $U(1)$ charges $Q_{L}$ and
$Q_{R}$. A state with charge $(Q_{L},Q_{R})$ contributes then with a weight
$y^{Q_{L}}\overline{y}^{Q_{R}}$, where we define $y=e^{2\pi iz}$.

Before analyzing the black hole spectrum in this theory, we will first quickly
review the different types of representations of the $(2,2)$ Virasoro algebra
that can appear. We will describe the left-moving case in the NS sector for
concreteness. A general representation is labeled by a scaling dimension
$\Delta$ and charge $Q$. Unitarity implies $\Delta>|Q|/2$ and
$0\leq|Q|<\widehat{c}-1$. The character of this representation, after summing
over descendants, is given by

${\rm
ch}_{NS}(\Delta,Q;\tau,z)=q^{\Delta-\frac{\widehat{c}-1}{8}}y^{Q}\frac{\theta_{3}(\tau,z)}{\eta(\tau)^{3}}.$
(4.1)

When the bound on dimension is saturated $\Delta=|Q|/2$ and
$0<|Q|<\widehat{c}$, the representations are BPS and preserve one supercharge.
The character of these representations is given by

${\rm
ch}_{NS}(Q;\tau,z)=q^{\frac{|Q|}{2}-\frac{\widehat{c}-1}{8}}y^{Q}\frac{1}{1+y^{{\rm
sgn}(Q)}q^{1/2}}\frac{\theta_{3}(\tau,z)}{\eta(\tau)^{3}}.$ (4.2)

The denominator comes from the preserved supercharge. Finally, we have the
vacuum representation with $\Delta=Q=0$, preserving two supercharges. The
character is given by

${\rm ch}_{NS}(\tau,z)\equiv
q^{-\frac{\widehat{c}-1}{8}}\frac{1-q}{(1+yq^{1/2})(1+y^{-1}q^{1/2})}\frac{\theta_{3}(q,y)}{\eta(q)^{3}}.$
(4.3)

The denominator comes form the preserved supersymetry, while the numerator
comes from the preserved ${\rm SL}(2,\mathbb{R})$ symmetry.

This can be extended to other sectors by simple transformations. First of all,
we can insert a $(-1)^{F}$ by simply shifting $y\to-y$. We can also go to the
Ramond sector by shifting $z\to z+\tau/2$. In the next section, we will be
interested in computing the partition function with Ramond boundary conditions
such that fermions are periodic along the spatial circle. Since we will study
black holes, this circle is not contractible.

###  4.2 Naive Spectrum

We now move to computing the contribution to the black hole partition function
from the supergravity sector. We will study it in the RR sector, as explained
above. This can be related to the partition function in vacuum AdS3 by a
modular transformation $\tau\to-1/\tau$ and $z\to z/\tau$. After this
transformation, the spatial direction becomes time-like and therefore it
involves a $(-1)^{F}$ insertion. On the other hand, time becomes space and the
boundary conditions are anti-periodic. Therefore the black hole partition
function is given by the vacuum characters in the NS sector, with a $(-1)^{F}$
insertion $Z_{\rm BTZ,RR}=|e^{-i\pi\frac{\widehat{c}z^{2}}{\tau}}{\rm
ch}_{NS}(-1/\tau,z/\tau+1/2)|^{2}$. The origin of the prefactor is explained
in [112]. After replacing the explicit form of this character we get

$Z_{\rm BTZ,RR}=e^{\frac{\pi i}{\tau}\frac{\widehat{c}}{4}(1-4z^{2})-\frac{\pi
i}{\overline{\tau}}\frac{\widehat{c}}{4}(1-4\overline{z}^{2})}~{}\Big{|}\frac{1-q^{\prime}}{\eta(q^{\prime})}\frac{1}{\eta(q^{\prime})}\frac{\theta_{3}(q^{\prime},-y^{\prime})}{\prod_{\pm}(1-y^{\prime}{}^{\pm
1}q^{\prime}{}^{1/2})}\Big{|}^{2},$ (4.4)

where $q^{\prime}=e^{2\pi i(-1/\tau)}$ and $y^{\prime}=e^{2\pi i(z/\tau)}$.
The first term is the exponential of the classical action on the black hole