background. The second term is the one-loop determinant around this solution,
where we wrote first the graviton contribution, then the $U(1)_{R}$ Chern-
Simons one, and finally that from the gravitini. This is exact for pure
gravity, but also reproduces a universal feature of the spectrum in the large
central charge for holographic theories [109] or large angular momentum for
theories with a twist gap [110].

From this expression for the partition function, we can extract the black hole
spectrum expanding it in a sum over the characters introduced above.
Explicitly, we define the density of states from

$\displaystyle Z_{{\rm BTZ,RR}}$ $\displaystyle=$
$\displaystyle\sum_{E_{L},E_{R}}\sum_{Q_{L},Q_{R}}~{}\rho_{Q_{L},Q_{R}}(E_{L},E_{R})~{}{\rm
ch}_{R}(E_{L},Q_{L};\tau,z){\rm
ch}_{R}(E_{R},Q_{R};\overline{\tau},\overline{z}),$ (4.5) $\displaystyle+$
$\displaystyle\sum_{E_{R}}\sum_{Q_{L},Q_{R}}~{}\rho_{Q_{L},Q_{R}}(E_{R})~{}{\rm
ch}_{R}(Q_{L};\tau,z){\rm
ch}_{R}(E_{R},Q_{R};\overline{\tau},\overline{z})+({\rm L}\leftrightarrow{\rm
R}),$ $\displaystyle+$
$\displaystyle\sum_{Q_{L},Q_{R}}~{}\rho_{Q_{L},Q_{R}}~{}{\rm
ch}_{R}(Q_{L};\tau,z){\rm ch}_{R}(Q_{R};\overline{\tau},\overline{z}).$

We defined the energy of the states in terms of dimensions as
$E\equiv\Delta-\widehat{c}/8$. There are three types of terms from each line,
the first involves left- and right-moving generic representations, the second
either left or right BPS representations, and the final one involves both left
and right BPS representations. To ease notation, the density of states in each
sector is differentiated only by the arguments in $\rho$.

The density of states as a function of charge and energy can be extracted from
the following modular transformation for the vacuum character

$\displaystyle e^{-i\pi\frac{\widehat{c}z^{2}}{\tau}}{\rm
ch}_{NS}(-1/\tau,z/\tau+1/2)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dP\int_{-\infty}^{\infty}dQS(P,Q){\rm
ch}_{R}\Big{(}\frac{\widehat{c}}{8}+\frac{b^{2}Q^{2}}{4}+P^{2},Q;\tau,z\Big{)}$
(4.6) $\displaystyle+2\int_{0}^{1}dQ\sin\left[\pi
Q\right]\sum_{n\in\mathbb{Z}}q^{\frac{k}{2}n^{2}}y^{kn}{\rm
ch}_{R}(Q;\tau,z+n\tau)\,,$

where we defined following [113] the modular S-matrix

$S(P,Q)\equiv\frac{\sinh(2\pi bP)\sinh\left(\frac{2\pi
P}{b}\right)}{b^{-1}\sinh\left(\pi bP+\frac{i\pi
b^{2}\omega}{2}\right)\sinh\left(\pi bP-i\frac{\pi b^{2}Q}{2}\right)}.$ (4.7)

Taking the square of this expression we can compute
$\rho_{Q_{L},Q_{R}}(E_{L},E_{R})$ comparing with (4.5). The answer for the
fully degenerate states is given by a product of a left- and right-moving
modular S-matrices, together with the Jacobian appearing when replacing an
integral over $P$ to an integral over energies. The full answer is

$\rho_{Q_{L},Q_{R}}(E_{L},E_{R})=\frac{\sinh\Big{(}2\pi\sqrt{\frac{2}{\widehat{c}-1}(E_{L}-\frac{Q_{L}^{2}}{2(\widehat{c}-1)})}\Big{)}\sinh\Big{(}2\pi\sqrt{\frac{\widehat{c}-1}{2}(E_{L}-\frac{Q_{L}^{2}}{2(\widehat{c}-1)}})\Big{)}}{2\sqrt{\frac{\widehat{c}-1}{2}(E_{L}-\frac{Q_{L}^{2}}{2(\widehat{c}-1)}})\Big{|}\sinh\Big{(}\pi\sqrt{\frac{2}{\widehat{c}-1}(E_{L}-\frac{Q_{L}^{2}}{2(\widehat{c}-1)})}+\frac{i\pi
Q_{L}}{\widehat{c}-1}\Big{)}\Big{|}^{2}}\times({\rm L}\rightarrow{\rm R}),$
(4.8)

whenever $E_{L/R}>Q_{L/R}^{2}/(2(\widehat{c}-1))$ and zero otherwise. The
energy spectrum is continuous but which is fine since we believe that non-
perturbative corrections could fix that.

However, the first issue is that the spectrum involves an integral over
$Q_{L/R}\in(-\infty,+\infty)$. This corresponds to an R-symmetry group being
$\mathbb{R}$ instead of $U(1)$. This will be resolved in the next section by
including additional saddles we so far ignored. This is the problem that [111]
address. A second issue is the contribution from the BPS states. We see that
it involves a sum over integer spectral flowed characters and its not clear
how to interpret that sector of the spectrum. The sum over saddles in the next
section will correct both problems.

Before explaining the resolution of these issues we will take the near
extremal limit of the continuum part and show it matches the $\mathcal{N}=2$
Schwarzian answer. For theories with large $\widehat{c}$ and at large $E_{R}$
and fixed $E_{L}\sim\mathcal{O}(1/\widehat{c})$ the density of states is

$\rho_{Q_{L},Q_{R}}(E_{L},E_{R})\approx
e^{S_{0}}~{}\frac{\sinh\Big{(}2\pi\sqrt{2\Phi_{r}(E_{L}-\frac{Q_{L}^{2}}{8\Phi_{r}}})\Big{)}}{2\pi
E_{L}},$ (4.9)

where defined
$S_{0}=2\pi\sqrt{\frac{\widehat{c}-1}{2}(E_{R}-\frac{Q_{R}^{2}}{2(\widehat{c}-1)})}$
and $M_{SU(1,1|1)}=4/(\widehat{c}-1)$ to make a direct comparison with the
$\mathcal{N}=2$ JT gravity answer. Notice that given the standard 2D CFT
conventions, the $\mathcal{N}=2$ supermultiplet has charge assignment
$(Q+1/2)\oplus(Q-1/2)$, and therefore there is a shift of the charge $Z_{\rm
Sch}=Q_{CFT}+1/2$ needed in order to match with the Schwarzian answer.

###  4.3 Corrected Spectrum

As emphasized in [111] this cannot be the whole story since the spectrum
derived from this partition function has a continuum value of charges. From
the bulk perspective, we need to sum over saddles that recover the
discreteness of charge. From the boundary perspective, we need to include the
integer spectral flow generator as part of the algebra.

We will begin by describing the boundary perspective. The spectral flow
generators $U_{\eta}$ are defined more generally, in terms of a real parameter
$\eta$, by the following transformation of the currents

$\displaystyle U_{\eta}^{-1}L_{n}U_{\eta}$ $\displaystyle=$ $\displaystyle
L_{n}+\eta J_{n}+\frac{\widehat{c}}{2}\eta^{2}\delta_{0,n},$ (4.10)
$\displaystyle U_{\eta}^{-1}J_{n}U_{\eta}$ $\displaystyle=$ $\displaystyle
J_{n}+\widehat{c}\eta\delta_{0,n},$ (4.11) $\displaystyle
U_{\eta}^{-1}G_{s}^{\pm}U_{\eta}$ $\displaystyle=$ $\displaystyle
G_{s\pm\eta}^{\pm}.$ (4.12)

The extended algebra we will consider includes, besides the stress tensor,
$U(1)$ current and supergenerators, the integer spectral flow generator
$U_{\pm 1/r}$ for integer $1/r\in\mathbb{Z}$. The reason to parametrize this
integer by $r$ is to make a connection later with the spectrum described in
section 2.3. It was shown in [111] that the modular properties of
representations of this extended algebra are only consistent when the central
charge has the form

$\widehat{c}=1+2rk,~{}~{}~{}k\in\mathbb{Z},$ (4.13)

where $k$ is a positive integer that can be chosen independently of the
integer $1/r$. We will see below that when this generator is included in the
algebra the spectrum of charges is fractional $Q\in r\cdot\mathbb{Z}$. Before
showing that we will present the new set of extended characters, written in
the NS sector for concreteness, given by

$\displaystyle\chi_{NS}(\Delta,Q,s;\tau,z)$ $\displaystyle=$
$\displaystyle\sum_{n\in
s+r^{-1}\cdot\mathbb{Z}}q^{\frac{\widehat{c}n^{2}}{2}}y^{\widehat{c}n}{\rm
ch}_{NS}(\Delta,Q;\tau,z+n\tau),$ (4.14) $\displaystyle\chi_{NS}(Q,s;\tau,z)$
$\displaystyle=$ $\displaystyle\sum_{n\in
s+r^{-1}\cdot\mathbb{Z}}q^{\frac{\widehat{c}n^{2}}{2}}y^{\widehat{c}n}{\rm
ch}_{NS}(Q;\tau,z+n\tau),$ (4.15) $\displaystyle\chi_{NS}(s;\tau,z)$
$\displaystyle=$ $\displaystyle\sum_{n\in
s+r^{-1}\cdot\mathbb{Z}}q^{\frac{\widehat{c}n^{2}}{2}}y^{\widehat{c}n}{\rm
ch}_{NS}(\tau,z+n\tau),$ (4.16)

for generic BPS and vacuum representations respectively. The states are
further parametrized by an integer $s\in\mathbb{Z}_{1/r}$. Similar formulas
can be written for the R sector, and including a $(-1)^{F}$, after performing
half-integer spectral flow and shifting $y\to-y$ respectevely.

So far we discussed the boundary theory. In the bulk theory, the choice of $r$
corresponds to the choice of the size of $U(1)_{R}$. Picking the correct gauge
group is not trivial since it determines the allowed gauge transformations and
therefore which set of saddles are to be considered gauge equivalent or not.
When we fix $r$, new set of saddles appear under the shift of the chemical
potential $z\to z+n$, for $n\in r^{-1}\cdot\mathbb{Z}$. These saddles have the
same boundary conditions as $n=0$ up to a global gauge transformation. The sum
over these saddles is equivalent to implementing the sum over integer spectral
flow. For BTZ in the RR sector we have $Z_{\rm
BTZ,RR}=|e^{-i\pi\frac{\widehat{c}z^{2}}{\tau}}\chi_{NS}(0;-1/\tau,z/\tau+1/2)|^{2}$.
This can again be written explicitly in a way that makes it bulk
interpretations clear. The partition function is give by

$Z_{\rm BTZ,RR}=\sum_{n,\overline{n}\in r^{-1}\cdot\mathbb{Z}}e^{\frac{\pi
i}{\tau}\frac{\widehat{c}}{4}(1-4(z+n)^{2})-\frac{\pi
i}{\overline{\tau}}\frac{\widehat{c}}{4}(1-4(\overline{z}+\overline{n})^{2})}~{}\Big{|}\frac{1-q^{\prime}}{\eta(q^{\prime})}\frac{1}{\eta(q^{\prime})}\frac{\theta_{3}(q^{\prime},-y^{\prime}q^{\prime}{}^{n})}{\prod_{\pm}(1-y^{\prime}{}^{\pm
1}q^{\prime}{}^{\pm n}q^{\prime}{}^{1/2})}\Big{|}^{2}.$ (4.17)

Again the first term is the exponential of the classical action, while the
second term is the one-loop determinant in the new saddles.

As mentioned above, it was shown in [111] that the extended characters have
consistent modular transformations when the central charge is fractional. For
the vacuum representation case that we need their result is

$\displaystyle
e^{-i\pi\frac{\widehat{c}z^{2}}{\tau}}\chi_{NS}(0;-1/\tau,z/\tau+1/2)$
$\displaystyle=$ $\displaystyle\int_{0}^{\infty}dP\sum_{Q\in
r\cdot\mathbb{Z}_{\widehat{c}-1}}r~{}S(P,Q)~{}\chi^{R}\Big{(}\frac{\widehat{c}}{8}+\frac{b^{2}Q^{2}}{4}+P^{2},Q,0;\tau,z\Big{)}$
(4.18) $\displaystyle+\sum_{Q\in
r\cdot\mathbb{Z}_{\widehat{c}-1},0<Q<1}2r\sin\left[\pi
Q\right]\sum_{r\in\mathbb{Z}_{N}}\chi^{R}(Q,r;\tau,z),$

where $S(P,Q)$ is the same function defined above. Now we see that both
problems are resolved. The charge spectrum is discrete with a fractional unit
charge $r$ and the modular transformation involves the same representations
that are allowed by the extended algebra. The overall factor of $r$ in the
right-hand side is required to reproduce (4.6) in the $r\to 0$ limit.

Having the modular properties of the vacuum character of the extended algebra,
we can extract the improved spectrum for a finite fractional charge $r$. First
of all it is easy to see that other than the fact that charge is discrete the
continuum density of states $\rho_{Q_{L},Q_{R}}(E_{L},E_{R})$ is exactly the
same as in the previous case (4.8), up to an extra factor of $r$ from (4.18)
and the final answer is

$\rho_{Q_{L},Q_{R}}(E_{L},E_{R})=r\frac{\sinh\Big{(}2\pi\sqrt{\frac{2}{\widehat{c}-1}(E_{L}-\frac{Q_{L}^{2}}{2(\widehat{c}-1)})}\Big{)}\sinh\Big{(}2\pi\sqrt{\frac{\widehat{c}-1}{2}(E_{L}-\frac{Q_{L}^{2}}{2(\widehat{c}-1)}})\Big{)}}{2\sqrt{\frac{\widehat{c}-1}{2}(E_{L}-\frac{Q_{L}^{2}}{2(\widehat{c}-1)}})\Big{|}\sinh\Big{(}\pi\sqrt{\frac{2}{\widehat{c}-1}(E_{L}-\frac{Q_{L}^{2}}{2(\widehat{c}-1)})}+\frac{i\pi
Q_{L}}{\widehat{c}-1}\Big{)}\Big{|}^{2}}\times({\rm L}\rightarrow{\rm R}),$
(4.19)

whenever $E_{L/R}>Q_{L/R}^{2}/(2(\widehat{c}-1))$ and $Q_{L/R}\in
r\cdot\mathbb{Z}_{\widehat{c}-1}$ and zero otherwise. In the near extremal
limit, at large $\widehat{c}$, this is exactly the same as the $\mathcal{N}=2$
Schwarzian answer with coupling $M_{SU(1,1|1)}=2/(rk)$, for large $k$.
Moreover, the answer is still universal either for holographic theories at
large $\widehat{c}$ [109] or any theory with a twist gap at large angular
momentum [110]. We can also extract the density of states of BPS states, when
either $E_{R}=0$ or $E_{L}=0$, from the black hole spectrum. Then answer when
$E_{R}=0$ is

$\rho_{Q_{L},Q_{R}}(E_{R})=2r\sin(\pi
Q_{L})~{}2r\frac{\sinh\Big{(}2\pi\sqrt{\frac{2}{\widehat{c}-1}(E_{R}-\frac{Q_{R}^{2}}{2(\widehat{c}-1)})}\Big{)}\sinh\Big{(}2\pi\sqrt{\frac{\widehat{c}-1}{2}(E_{R}-\frac{Q_{R}^{2}}{2(\widehat{c}-1)}})\Big{)}}{2\sqrt{\frac{\widehat{c}-1}{2}(E_{R}-\frac{Q_{R}^{2}}{2(\widehat{c}-1)}})\Big{|}\sinh\Big{(}\pi\sqrt{\frac{2}{\widehat{c}-1}(E_{R}-\frac{Q_{R}^{2}}{2(\widehat{c}-1)})}+\frac{i\pi
Q_{R}}{\widehat{c}-1}\Big{)}\Big{|}^{2}}$ (4.20)

The answer when $E_{L}=0$ is completely analogous. In the near extremal limit
with large $E_{R}$, we reproduce the Schwarzian theory answer
$\rho_{Q_{L},Q_{R}}(E_{R})\approx e^{S_{0}}2r\sin(\pi Q_{L})$ where $S_{0}$ is
the same function of $E_{R}$ and $Q_{R}$ as defined in the previous section.
Finally we can look at full BPS states with $E_{L}=E_{R}=0$. The answer for
pure $\mathcal{N}=(2,2)$ gravity is

$\rho_{Q_{L},Q_{R}}=2r\sin(\pi Q_{L})~{}2r\sin(\pi Q_{R}).$ (4.21)

This example is beyond the near extremal regime and therefore is not universal
for $\mathcal{N}=(2,2)$ CFTs either from the regime considered by [109] nor
[110].

The elliptic genus of these theories vanishes. This can be resolved by
defining a refined elliptic genus analogous to [34], assuming an exact
$\mathbb{Z}_{1/r}$ symmetry. The situation simplifies when $r=1$, since in
that case BPS states come only with zero charge and the usual elliptic genus
does not vanish anymore.

Finally, even though this discussion has been done for the case of
$\mathcal{N}=(2,2)$ supersymmetry, this can be generalized to cases with
$\mathcal{N}=(0,2)$ using the results of [108] or $\mathcal{N}=(4,2)$ using
[29].

##  5 Discussion and future work

In this paper, we have argued that the spectrum of $1/16$-BPS and
near-$1/16$-BPS black holes can be obtained by studying the appropriate
version of the $\mathcal{N}=2$ super-Schwarzian theory. From this, we have
shown that the gravitational path integral correctly reproduces the
degeneracies and charges of BPS states in $\mathcal{N}=4$ super Yang-Mills,
while at the same time predicting various gaps between these BPS states and a
“dense set” of near-BPS black hole states. Above this gap we find a continuum
of states (to leading order in $S_{0}$) and predict what the density of states
in this region is. There are however numerous open questions, some of which we
hope to address in future work.

### Extension to theories with less supersymmetry

We expect similar conclusions to hold about the spectrum of BPS and near-BPS
black holes in bulk theories with less supersymmetry. For example, one can
consider type IIB string theory in a spacetime that is asymptotically
AdS${}_{5}\times M_{5}$, where $M_{5}$ is a Sasaki–Einstein space. For each
such $M_{5}$, the bulk has a dual description in terms of a four-dimensional
$\mathcal{N}=1$ superconformal field theory. The BPS and near-BPS black hole
solutions reviewed in section 3.1 and 3.2 are also valid in a space that is
asymptotically AdS${}_{5}\times M_{5}$ [114, 115]; however, in contrast to the
case discussed in 3.1 the bulk does not have an $SO(6)$ R-symmetry gauge field
with the three Cartans whose eigenvalues we labeled by $R_{1,2,3}$, but
instead generically has a single $U(1)$ R-symmetry gauge field. Since in
section 3.3 – 3.5 we have solely focused on the case when $R_{1,2,3}=R$, the
analysis of the isometry of the near-horizon region as well as that of the
low-temperature expansion of the on-shell action should similarly hold for
theories in AdS${}_{5}\times M_{5}$. Consequently, we expect that the spectrum
of near-BPS black holes is also controlled by the $\mathcal{N}=2$ super-
Schwarzian theory. Nevertheless, a detailed check, regarding the correct value
of $r$ and $\theta$-angle, needs to be performed in order to be certain that
the spectrum is described by precisely the same super-Schwarzian theory for
all Sasaki–Einstein spaces.

### Non-perturbative corrections

In this paper we have focused on the leading quantum corrections around the
leading near-BPS black hole geometry in $AdS_{5}\times S^{5}$ with a nearly
AdS2 throat. When computing quantities not protected by supersymmetry, such as
the partition function with all chemical potentials turned on, there can be a
a variety of non-perturbative geometries contributing, including for example
spacetime wormholes.

The situation from the perspective of the gravitational path integral is
different when computing protected quantities such as the index [76] or even
some quantities that are not protected by supersymmetry, such as the zero
temperature partition function which yields the degeneracy of $1/16$-BPS
states. In that case, we expect a reduced number of geometries to contribute
when they preserve supersymmetry. These have been considered in [76] in the
case of throats with emergent $PSU(1,1|2)$ symmetry (in contrast to the
$SU(1,1|1)$ near-horizon geometry considered in this paper), where it was
shown quantum corrections are described by a deformation of the
$\mathcal{N}=4$ Super-Schwarzian theory. This result from the $\mathcal{N}=4$
Super-Schwarzian theory is consistent with examples of black holes in type IIA
where the exact index was shown to be reproduced by a sum over these orbifolds
[116].

Nevertheless, similar results should hold for the black holes discussed in
this paper due to similarities between the effective theory of $\mathcal{N}=2$
JT gravity found in this paper and $\mathcal{N}=4$ JT gravity analyzed in
[76]. Both theories have vacuum BPS states, all with the same spin-statistics,
that are separated by a gap for the lightest near-BPS states. In both cases,
even though the boundary conditions are supersymmetric for $\alpha=1/2$, the
only geometry which preserves supersymmetry in the bulk is the one with $n=0$
(for the case discussed in this paper, see (3.80)). A similar calculation as
that in [76] shows that, from the perspective of the near-horizon geometry,
the only other geometries preserving supersymmetry in $\mathcal{N}=2$ JT
gravity are particular orbifolds of AdS2 (these were considered in bosonic
gravity previously in [117, 118, 119]). In particular, the contribution of
higher genus geometries or, more broadly, any near-horizon geometry that
involves spacetime wormhole can be seen to vanish in $\mathcal{N}=2$ JT
gravity. Consequently, in the full gravitational path integral only orbifolds
of $AdS_{2}$ that can be uplifted to smooth $AdS_{2}\times S^{3}\times S^{5}$
geometries contribute.

Have these geometries already been observed from the boundary side? When
expanding the $\mathcal{N}=4$ Yang Mills superconformal index in large $N$,
while the leading contribution is given by the black hole considered in
section 3.1, there are subleading contributions which corresponds to
supersymmetric orbifolds of the black hole geometry [23]. We leave a detailed
comparison of these contributions, including the effect of the $\mathcal{N}=2$
super-JT one-loop determinant, for future work. We should however stress that
as opposed to the case in [116], which have a $PSU(1,1|2)$ near-horizon
isometry, for the superconformal index in $\mathcal{N}=4$ Yang Mills there are
contributions at large $N$ which cannot be interpreted as orbifolds of the
black hole [23], such as contributions from wrapped D-branes or black hole
solutions which are yet unknown. It would be interesting to investigate if
these non-perturbative corrections can be analyzed also in the context of the
near-BPS limit.

$S^{3}$$S^{5}$$\neq\,\,0$$=\,\,0\,.$ Figure 3: Possible non-perturbative
contributions to the index of black holes in $AdS_{5}\times S^{5}$. Using
$\mathcal{N}=2$ super-JT gravity one can show that all geometries involving
“spacetime wormholes”, such as the one on the right side, vanish. The only
surviving contributions are orbifolds of the original $AdS_{2}\times
S^{3}\times S^{5}$ near-horizon geometry which we show on the left side. In
such a case the angle of the defect, the rotation on $S^{3}$ and that on
$S^{5}$ are all related in order for supersymmetry to be preserved and for the
spacetime to be smooth. These geometries were also explicitly seen in [23] as
subleading corrections to the superconformal index.

Beyond computing the index, the effect of geometries of higher topology or
with a larger number of defects in the near-horizon region (from higher-
dimensional generalizations of Seifert geometries [118]) on the
$\mathcal{N}=2$ super-JT path integral, shows that a gap is present in the
spectrum associated to each specific geometry.232323The gap was also found to
persist when accounting for other topologies in the $\mathcal{N}=4$ super-JT
path integral [76]. While such geometries are off-shell (for example, the
equation of motion for the dilaton cannot be satisfied on such geometries
[91]) they can nevertheless be systematically accounted for using the sum over
topologies in the $\mathcal{N}=2$ super-JT path integral. Thus, when summing
over all geometries we expect that they might yield non-perturbative
corrections (in $N^{2}$) to the value of the gap, but will not affect its
existence by inserting additional states between the extremal BPS state and
the lightest near-BPS black hole state coming from the original black hole
saddle.

One can additionally discuss non-perturbative effects coming from different
(possibly supersymmetric) black hole solutions which have not yet been
understood analytically. For instance, as previously mentioned, [100, 101]
found numerical evidence for black hole solutions that support scalar hair and
can be supersymmetric even away from $R\neq R^{\star}$. Such black holes would
yield corrections to the spectrum found in figure 1 that would, once again, be
non-perturbatively suppressed in $N^{2}$. Nevertheless, one might hope to
analyze such solutions at low-temperatures in Euclidean signature where we
expect there to still be an $AdS_{2}\times S^{3}\times S^{5}$ near-horizon
region. If at extremality the near-horizon super-isometry is still
$SU(1,1|1)$, we expect that the temperature dependence of quantum corrections
around these new saddles still be captured by the $\mathcal{N}=2$ super-
Schwarzian. Thus, we expect that even if sectors with $R\neq R^{\star}$ are
populated by new BPS solutions, energy gaps should still be present within
each charge sector; moreover, even if the degeneracy of the $1/16$-BPS black
states that we have found in the $R=R^{\star}$ sector is affected by such non-
perturbative corrections coming from hairy black holes, the gap should be
unaffected within that charged sector.

Thus, this (albeit incomplete) accounting for possible gravitational non-
perturbative effects, all of which preserve the existence of the mass gap,
along with the fact that the leading order stringy corrections also did not
affect its present, prompts us to conjecture that the gap persists within each
large charge sector of $\mathcal{N}=4$ Yang-Mills.

### An effective field theory for near-BPS states from the boundary-side

We derived the spectrum of nearly $1/16$-BPS black hole states from a bulk
$AdS_{5}\times S^{5}$ calculation. Due to AdS/CFT, this makes a prediction for
the spectrum of $\mathcal{N}=4$ Yang Mills. This raises the question of how to
derive the same spectrum from an independent boundary CFT argument. This would
be an extremely non-trivial check of holography that would help understand
quantum aspects of gravity in these black hole backgrounds better. At weak ’t
Hooft coupling some of the states with the black hole quantum numbers were
constructed in [74, 75] in the limit of large spin $J/N^{2}\gg 1$, and some
interaction effects were considered. We leave for future work to identify the
most relevant interactions in this limit, and to derive the emergence of a
softly broken $SU(1,1|1)$ symmetry. This would give the first example of a
quantum theory describing a local nearly AdS2 background, as opposed to SYK
which describes a highly non-local bulk.

### Acknowledgements

We thank W. Zhao for initial discussion. We also thank O. Aharony, F. Benini,
M. Berkooz, P. Caputa, A. Castro, J. Karlsson, M. Kolanowski, Z. Komargodski,
S. Komatsu, J. H. Lee, R. Mahajan, J. Maldacena, J. Melo, S. Murthy, B. Post,
J. Santos, J.H Schwarz, D. Stanford, L. Tizzano, M. Tomasevic and E. Witten
for valuable discussions. MTH is supported by Princeton University and the
Institute for Advanced Study under Grant No. DE-SC0009988 and the Corning
Glass Foundation. GJT is supported by the Institute for Advanced Study and the
National Science Foundation under Grant No. PHY-1911298, and by the Dipal and
Rupal Patel funds. LVI was supported by the Simons Collaboration on Ultra-
Quantum Matter, a Simons Foundation Grant with No. 651440. JB is supported by
the NCN Sonata Bis 9 2019/34/E/ST2/00123 grant. This work was partly done at
the Aspen Center for Physics, which is supported by National Science
Foundation grant PHY-1607611. JB wants to thank the organizers and
participants of Brussels – Paris – Amsterdam Doctoral School on Quantum Field
Theory, Strings and Gravity, for a wonderful workshop, during which a part of
this work was done.

##  Appendix A Details on the low temperature expansion

In this section we explain in more details how to expand the action

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

around its BPS limit [38]. We start with introducing four expansion parameters
$(\epsilon_{r},\epsilon_{q},\epsilon_{a},\epsilon_{b})$ such that

$r_{+}^{2}=r^{*2}+\epsilon_{r},~{}~{}~{}q=q^{*}+\epsilon_{q},~{}~{}~{}a=a^{*}+\epsilon_{a},~{}~{}~{}b=b^{*}+\epsilon_{b},$
(A.2)

where parameters $(a^{*},b^{*})$ are defined through the BPS configuration of
fixed charges

$\mathfrak{j}_{1}(a^{*},b^{*})=\frac{\pi\left(a^{*}+b^{*}\right)\left(a^{*}+1\right)\left(b^{*}+1\right)}{4G_{5}\left(a^{*}-1\right)^{2}\left(1-b^{*}\right)},~{}~{}~{}\mathfrak{j}_{2}(a^{*},b^{*})=\frac{\pi\left(a^{*}+b^{*}\right)\left(a^{*}+1\right)\left(b^{*}+1\right)}{4G_{5}\left(1-a^{*}\right)\left(b^{*}-1\right)^{2}}.$
(A.3)

We want to perform the expansion in such a way that the variables
$(T,\varphi,\mathfrak{j}_{1},\mathfrak{j}_{2})$ are fixed. This imposes the
following four relations, since the fixed parameters in the PCE ensemble are:

$\varphi=\varphi(r_{+},q,a,b),~{}~{}~{}T=T(r_{+},q,a,b),~{}~{}~{}\mathfrak{j}_{1}(a^{*},b^{*})=\mathfrak{j}_{1}(r_{+},q,a,b),~{}~{}~{}\mathfrak{j}_{2}(a^{*},b^{*})=\mathfrak{j}_{2}(r_{+},q,a,b).$
(A.4)

Using these relations, we can now express the expansion parameters
$(\epsilon_{r},\epsilon_{q},\epsilon_{a},\epsilon_{b})$ in terms of
$(T,\varphi)$. For our purposes, it will be enough to work to second order in
$\epsilon$’s. Inverting the above relations order by order, we find solutions
in the form

$\displaystyle\epsilon_{X}$
$\displaystyle=\epsilon_{X,\varphi}\varphi+\epsilon_{X,T}T+\epsilon_{X,\varphi^{2}}\varphi^{2}+\epsilon_{X,T^{2}}T^{2}+\epsilon_{X,\varphi
T}\,(\varphi T)\,,$ (A.5)

where $X=\,r,\,q,\,a$ or $b$. The first order coefficients are given through

$\displaystyle\epsilon_{r,\varphi}\varphi+\epsilon_{r,T}T$
$\displaystyle=\frac{\left(a^{*}+b^{*}\right)\left(a^{*}+1\right)\left(b^{*}+1\right)\left(2\pi
T\sqrt{a^{*}b^{*}+a^{*}+b^{*}}+\varphi\left(a^{*}b^{*}+a^{*}+b^{*}\right)\right)}{2\left(3a^{*}\left(b^{*}+1\right)+\left(a^{*}\right)^{2}+\left(b^{*}\right)^{2}+3b^{*}+1\right)},$
(A.6) $\displaystyle\epsilon_{q,\varphi}\varphi+\epsilon_{q,T}T$
$\displaystyle=\frac{3\varphi\left(a^{*}+b^{*}\right)^{2}\left(a^{*}+1\right)^{2}\left(b^{*}+1\right)^{2}}{4\left(3a^{*}\left(b^{*}+1\right)+\left(a^{*}\right)^{2}+\left(b^{*}\right)^{2}+3b^{*}+1\right)},$
(A.7) $\displaystyle\epsilon_{a,\varphi}\varphi+\epsilon_{a,T}T$
$\displaystyle=\frac{\varphi\left(a^{*}+b^{*}\right)\left(a^{*}-1\right)\left(a^{*}+1\right)\left(b^{*}+1\right)}{4\left(3a^{*}\left(b^{*}+1\right)+\left(a^{*}\right)^{2}+\left(b^{*}\right)^{2}+3b^{*}+1\right)},$
(A.8) $\displaystyle\epsilon_{b,\varphi}\varphi+\epsilon_{b,T}T$
$\displaystyle=\frac{\varphi\left(a^{*}+b^{*}\right)\left(a^{*}+1\right)\left(b^{*}-1\right)\left(b^{*}+1\right)}{4\left(3a^{*}\left(b^{*}+1\right)+\left(a^{*}\right)^{2}+\left(b^{*}\right)^{2}+3b^{*}+1\right)}.$
(A.9)

Following the same logic we derive second order coefficients. Their form is
more complicated for general $(a^{*},b^{*})$. For reader’s convenience we
provide simplified formulas for the case $a^{*}=b^{*}$ as a potential check

$\displaystyle\epsilon_{r,\varphi^{2}}$
$\displaystyle=\frac{a^{*2}\left(15a^{*5}+35a^{*4}+24a^{*3}-28a^{*2}+a^{*}+1\right)}{4\left(5a^{*}+1\right){}^{3}},$
(A.10) $\displaystyle\epsilon_{r,\varphi T}$ $\displaystyle=\frac{3\pi
a^{*2}\left(a^{*}+1\right){}^{3}\sqrt{a^{*}\left(a^{*}+2\right)}}{\left(5a^{*}+1\right){}^{3}},$
(A.11) $\displaystyle\epsilon_{r,T^{2}}$
$\displaystyle=\frac{4\pi^{2}a^{*2}\left(a^{*4}+7a^{*3}+5a^{*2}+9a^{*}+2\right)}{\left(5a^{*}+1\right){}^{3}},$
(A.12) $\displaystyle\epsilon_{q,\varphi^{2}}$
$\displaystyle=\frac{a^{*3}\left(30a^{*5}+84a^{*4}+105a^{*3}+7a^{*2}-11a^{*}+1\right)}{2\left(5a^{*}+1\right){}^{3}},$
(A.13) $\displaystyle\epsilon_{q,\text{$\varphi$T}}$
$\displaystyle=-\frac{3\pi
a^{*3}\left(a^{*}+1\right){}^{3}\left(7a^{*2}-10a^{*}-3\right)}{\sqrt{a^{*}\left(a^{*}+2\right)}\left(5a^{*}+1\right){}^{3}},$
(A.14) $\displaystyle\epsilon_{q,T^{2}}$
$\displaystyle=\frac{2\pi^{2}a^{*3}\left(6a^{*4}+51a^{*3}+13a^{*2}+a^{*}+1\right)}{\left(5a^{*}+1\right){}^{3}},$
(A.15) $\displaystyle\epsilon_{a,\varphi^{2}}$
$\displaystyle=\frac{\left(a^{*}-1\right){}^{2}a^{*3}\left(5a^{*2}+7a^{*}+6\right)}{4\left(a^{*}+1\right)\left(5a^{*}+1\right){}^{3}},$
(A.16) $\displaystyle\epsilon_{a,\varphi T}$
$\displaystyle=-\frac{\pi\left(a^{*}-1\right)a^{*2}\left(a^{*}+1\right)\left(7a^{*2}-10a^{*}-3\right)}{2\sqrt{a^{*}\left(a^{*}+2\right)}\left(5a^{*}+1\right){}^{3}},$
(A.17) $\displaystyle\epsilon_{a,T^{2}}$
$\displaystyle=\frac{\pi^{2}\left(a^{*}-1\right)a^{*2}\left(2a^{*3}+15a^{*2}+6a^{*}+1\right)}{\left(a^{*}+1\right)\left(5a^{*}+1\right){}^{3}},$
(A.18) $\displaystyle\epsilon_{b,\varphi^{2}}$
$\displaystyle=\frac{\left(a^{*}-1\right){}^{2}a^{*3}\left(5a^{*2}+7a^{*}+6\right)}{4\left(a^{*}+1\right)\left(5a^{*}+1\right){}^{3}},$
(A.19) $\displaystyle\epsilon_{b,\varphi T}$
$\displaystyle=-\frac{\pi\left(a^{*}-1\right)a^{*2}\left(a^{*}+1\right)\left(7a^{*2}-10a^{*}-3\right)}{2\sqrt{a^{*}\left(a^{*}+2\right)}\left(5a^{*}+1\right){}^{3}},$
(A.20) $\displaystyle\epsilon_{b,T^{2}}$
$\displaystyle=\frac{\pi^{2}\left(a^{*}-1\right)a^{*2}\left(2a^{*3}+15a^{*2}+6a^{*}+1\right)}{\left(a^{*}+1\right)\left(5a^{*}+1\right){}^{3}}.$
(A.21)

With this we can now expand the quantity of interest to second order in
$(\epsilon_{r},\epsilon_{q},\epsilon_{a},\epsilon_{b})$ and insert the
relations
$(\epsilon_{r}(T,\varphi),\epsilon_{q}(T,\varphi),\epsilon_{a}(T,\varphi),\epsilon_{b}(T,\varphi))$
derived above. As a last step, in the resulting expansion we keep only terms
up to second order in $(\varphi,T)$ (to get to higher orders we would also
need to solve for $\epsilon$’s to higher order). Applying this to procedure to
our partial canonical action leads us to the result (3.73).

##  Appendix B Details on Killing spinors

In this section we verify explicitly the Killing spinors (3.57) and Killing
vectors (3.58)–(3.62) of the near-horizon geometry lifted to ten-dimensions
(3.53). This has been first derived in [33]. Here we review the construction
in order to account for differences in conventions and provide reader with
necessary formulas.

Lets recall that we work with a ten dimensional lift [84, 66]

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

where the Hodge star is defined as
$\ast_{(n)}\omega_{\mu_{1}\dots\mu_{n-p}}=\frac{1}{p!}\epsilon_{\mu_{1}\dots\mu_{n-p}}^{\phantom{\mu_{1}\dots\mu_{n-p}}\nu_{1}\dots\nu_{p}}\omega_{\nu_{1}\dots\nu_{p}}$,
and we work with $\epsilon_{0123456789}=\epsilon_{01234}=1$. We introduced

$\displaystyle\mu_{1}=\sin\tilde{\alpha},\qquad\mu_{2}=\cos\tilde{\alpha}\sin\tilde{\beta},\qquad\mu_{3}=\cos\tilde{\alpha}\cos\tilde{\beta},$
(B.3)

and the angles which parametrize the $S^{5}$ are
$\tilde{\alpha}\in[0,\frac{\pi}{2}]$, $\tilde{\beta}\in[0,\frac{\pi}{2}]$,
$\xi_{i}\in[0,2\pi]$. The near-horizon 5d metric and the gauge field are given
by242424The relation between the near-horizon gauge field used here and the
one used in [33] is $A_{\text{here}}=-\sqrt{3}A_{\text{there}}$.

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

In the above $d\Omega_{3}^{2}=d\theta^{2}+\cos^{2}\theta
d\tilde{\psi}^{2}+\sin^{2}\theta d\tilde{\phi}^{2}$ is the metric on $S^{3}$,
$\sigma_{3}^{L}=2(\cos^{2}\theta d\tilde{\psi}+\sin^{2}\theta d\tilde{\phi})$,
$\tilde{\nu}=a\cos^{2}\theta d\tilde{\psi}+a\sin^{2}\theta d\tilde{\phi}$, and
the parameters are given through $a$ as

$\displaystyle\lambda=\sqrt{1+3\omega^{2}},\qquad\omega=\frac{\sqrt{2a}}{\sqrt{1-a}}\,.$
(B.6)

We work with corotating angles $(\tilde{\psi},\tilde{\phi})$, related to
original coordinates via $\psi=\tilde{\psi}+t$, $\phi=\tilde{\phi}+t$. The 5d
tetrads of the near-horizon metric are given by

$\displaystyle e^{0}$ $\displaystyle=\frac{\omega
rdt}{4\lambda}-\frac{3}{2}\omega^{2}\left(\cos^{2}\theta
d\tilde{\psi}+\sin^{2}\theta d\tilde{\phi}\right),$ (B.7) $\displaystyle
e^{1}$ $\displaystyle=\frac{\omega dr}{2\lambda r},$ (B.8) $\displaystyle
e^{2}$
$\displaystyle=\frac{1}{2}\omega\sin(2\theta)(d\tilde{\psi}-d\tilde{\phi})\sin(\tilde{\psi}+\tilde{\phi})+\omega
d\theta\cos(\tilde{\psi}+\tilde{\phi}),$ (B.9) $\displaystyle e^{3}$
$\displaystyle=\frac{1}{2}\omega\sin(2\theta)(d\tilde{\psi}-d\tilde{\phi})\cos(\tilde{\psi}+\tilde{\phi})-\omega
d\theta\sin(\tilde{\psi}+\tilde{\phi}),$ (B.10) $\displaystyle e^{4}$
$\displaystyle=\lambda\omega\left(\cos^{2}\theta d\tilde{\psi}+\sin^{2}\theta
d\tilde{\phi}\right).$ (B.11)

For tetrads in the lifted 10d metric, in addition to the above tetrads, we
also have

$\displaystyle e^{5}$ $\displaystyle=d\tilde{\alpha},$ (B.12) $\displaystyle
e^{6}$ $\displaystyle=\cos\tilde{\alpha}d\tilde{\beta},$ (B.13) $\displaystyle
e^{7}$
$\displaystyle=\sin\tilde{\alpha}\cos\tilde{\alpha}\left(d\xi_{1}-\sin^{2}\tilde{\beta}d\xi_{2}-\cos^{2}\tilde{\beta}d\xi_{3}\right),$
(B.14) $\displaystyle e^{8}$
$\displaystyle=\cos\tilde{\alpha}\sin\tilde{\beta}\cos\tilde{\beta}(d\xi_{2}-d\xi_{3}),$
(B.15) $\displaystyle e^{9}$
$\displaystyle=-\cos^{2}\tilde{\alpha}\left(\sin^{2}\tilde{\beta}d\xi_{2}+\cos^{2}\tilde{\beta}d\xi_{3}\right)-\sin^{2}\tilde{\alpha}d\xi_{1}+\frac{2A}{3}.$
(B.16)

In this basis, the electromagnetic fields are explicitly given by

$\displaystyle F^{(2)}$ $\displaystyle=-\frac{3}{2}\left(3e^{1}\wedge
e^{4}-\frac{2\lambda}{\omega}e^{0}\wedge e^{1}-e^{2}\wedge e^{3}\right),$
(B.17) $\displaystyle F^{(5)}$ $\displaystyle=-4(e^{0}\wedge e^{1}\wedge
e^{2}\wedge e^{3}\wedge e^{4}+e^{5}\wedge e^{6}\wedge e^{7}\wedge e^{8}\wedge
e^{9})$ (B.18) $\displaystyle\phantom{=}-\frac{2}{3}(e^{5}\wedge
e^{7}+e^{6}\wedge e^{8})\wedge(\ast_{(5)}F^{(2)}-e^{9}\wedge F^{(2)}).$ (B.19)

Our goal is to verify the solutions to the Killing spinor equation

$\displaystyle\widehat{\nabla}_{M}\epsilon_{K}\equiv(\partial_{M}+\frac{1}{4}\omega_{MAB}\Gamma^{A}\Gamma^{B})\epsilon_{K}+\frac{i}{1920}F^{(5)}_{A_{1}A_{2}A_{3}A_{4}A_{5}}\Gamma^{A_{1}A_{2}A_{3}A_{4}A_{5}}\Gamma_{M}\epsilon_{K}=0.$
(B.20)

Here $M,N$ denotes spacetime indices and $A_{i},B_{i}$ denotes frame indices;
$\epsilon_{K}=\epsilon_{R}+i\epsilon_{I}$ where $\epsilon_{R},\epsilon_{I}$
are both Majorana-Weyl spinors satisfying the chirality condition

$\displaystyle\Gamma^{11}\epsilon_{K}\equiv\Gamma^{0}\Gamma^{1}\dots\Gamma^{9}\epsilon_{K}=\epsilon_{K}.$
(B.21)

A common strategy for solving the Killing spinor equation, is to first use the
algebraic integrability condition
$[\widehat{\nabla}_{M_{1}},\widehat{\nabla}_{N_{1}}]\epsilon_{K}=0$ to
restrict the space of possible solutions, and then solve a simplified Killing
spinor equation on the restricted subspace. In our case, the integrability
condition takes the form [85]

$\displaystyle\bigg{[}R_{M_{1}N_{1}S_{1}S_{2}}$
$\displaystyle-\frac{1}{48}F^{(5)}_{\phantom{(5)}M_{1}S_{1}R_{1}R_{2}R_{3}}F_{\phantom{(5)}N_{1}S_{2}}^{(5)\phantom{N_{1}S_{2}}R_{1}R_{2}R_{3}}\bigg{]}\Gamma^{S_{1}S_{2}}\epsilon_{K}$
$\displaystyle+\left[\frac{i}{24}\nabla_{[M_{1}}F^{(5)}_{\phantom{(5)}N_{1}]S_{1}S_{2}S_{3}S_{4}}+\frac{1}{96}F^{(5)}_{\phantom{(5)}M_{1}N_{1}R_{1}R_{2}S_{1}}F^{(5)R_{1}R_{2}}_{\phantom{(5)R_{1}R_{2}}S_{2}S_{3}S_{4}}\right]\Gamma^{S_{1}S_{2}S_{3}S_{4}}\epsilon_{K}=0.$
(B.22)

This implies the following projection conditions

$\displaystyle\Gamma^{23}\epsilon_{K}=-i\epsilon_{K},\qquad\Gamma^{57}\epsilon_{K}=-i\epsilon_{K},\qquad\Gamma^{0149}\epsilon_{K}=i\epsilon_{K}.$
(B.23)

Let us now choose a general constant spinor
$\epsilon_{0}=\epsilon_{0,R}+i\epsilon_{0,I}$, subject to all four of the
above projection conditions. Such a spinor is now labeled by four independent
real parameters. We can split it into two chiralities under projector
$P_{\pm}=\frac{1}{2}(1\pm\Gamma^{09})$ such that

$\displaystyle\epsilon_{0}=\epsilon_{0}^{+}+\epsilon_{0}^{-},\qquad\Gamma^{09}\epsilon_{0}^{\pm}=\mp\epsilon_{0}^{\pm}.$
(B.24)

With these definitions, one can now verify that two independent spinors

$\displaystyle\epsilon_{1}$
$\displaystyle=\sqrt{\frac{8\lambda}{\omega^{2}r}}e^{-\frac{i}{2}(\xi_{1}+\xi_{2}+\xi_{3})}\left[\epsilon_{0}^{+}-\left(\frac{irt}{2}+\frac{3\omega}{2\lambda}\right)\Gamma^{4}\Gamma^{9}\epsilon_{0}^{+}\right],$
(B.25) $\displaystyle\epsilon_{2}$
$\displaystyle=\sqrt{\frac{\omega^{2}r}{8\lambda}}e^{-\frac{i}{2}(\xi_{1}+\xi_{2}+\xi_{3})}\epsilon_{0}^{-},$
(B.26)

satisfy the Killing spinor equation. Because the above spinors are labeled by
four independent real parameters, we conclude that the 10d lift of the near-
horizon geometry preserves four supersymmetries. Note that this is in contrast
with the full black hole geometry, which preserves only two supersymmetries
[85]. This means that there is a supersymmetry enhancement in the near-horizon
region.

Knowing the Killing spinors of the near-horizon geometry, we can now find its
Killing vectors by computing independent Killing spinor bilinears

$\displaystyle(\overline{\epsilon}_{I}\Gamma^{A}\epsilon_{J})\tilde{e}_{A},$
(B.27)

where $\tilde{e}_{a}$ denotes the dual tetrad basis, explicitly given by

$\displaystyle\tilde{e}_{0}$
$\displaystyle=\frac{4\lambda\partial_{t}-r\omega(\partial_{\xi_{1}}+\partial_{\xi_{2}}+\partial_{\xi_{3}})}{r\omega},$
(B.28) $\displaystyle\tilde{e}_{1}$ $\displaystyle=\frac{2\lambda
r\partial_{r}}{\omega},$ (B.29) $\displaystyle\tilde{e}_{2}$
$\displaystyle=\frac{\cos(\tilde{\psi}+\tilde{\phi})\partial_{\theta}+\tan\theta\,\sin(\tilde{\psi}+\tilde{\phi})\left(\partial_{\tilde{\psi}}-\cot^{2}\theta\,\partial_{\tilde{\phi}}\right)}{\omega},$
(B.30) $\displaystyle\tilde{e}_{3}$
$\displaystyle=\frac{-\sin(\tilde{\psi}+\tilde{\phi})\partial_{\theta}+\tan\theta\,\cos(\tilde{\psi}+\tilde{\phi})\partial_{\tilde{\psi}}-\cot\theta\,\cos(\tilde{\psi}+\tilde{\phi})\partial_{\tilde{\phi}}}{\omega},$
(B.31) $\displaystyle\tilde{e}_{4}$
$\displaystyle=\frac{6\partial_{t}}{r}+\frac{-2\omega^{2}(\partial_{\xi_{1}}+\partial_{\xi_{2}}+\partial_{\xi_{3}})+\partial_{\tilde{\psi}}+\partial_{\tilde{\phi}}}{\lambda\omega},$
(B.32)

together with

$\displaystyle\tilde{e}_{5}$ $\displaystyle=\partial_{\tilde{\alpha}},$ (B.33)
$\displaystyle\tilde{e}_{6}$
$\displaystyle=\text{sec}\tilde{\alpha}\,\partial_{\tilde{\beta}},$ (B.34)
$\displaystyle\tilde{e}_{7}$
$\displaystyle=\frac{1}{4}\sin^{2}(2\tilde{\alpha})\left(\csc^{3}\tilde{\alpha}\,\text{sec}\tilde{\alpha}\,\partial_{\xi_{1}}-\csc\tilde{\alpha}\,\text{sec}^{3}\tilde{\alpha}\,(\partial_{\xi_{2}}+\partial_{\xi_{3}})\right),$
(B.35) $\displaystyle\tilde{e}_{8}$
$\displaystyle=\text{sec}\tilde{\alpha}\cot\tilde{\beta}\partial_{\xi_{2}}-\text{sec}\tilde{\alpha}\tan\tilde{\beta}\partial_{\xi_{3}},$
(B.36) $\displaystyle\tilde{e}_{9}$
$\displaystyle=-\partial_{\xi_{1}}-\partial_{\xi_{2}}-\partial_{\xi_{3}}.$
(B.37)

The bilinears are found to be

$\displaystyle(\overline{\epsilon}_{2}\Gamma^{A}\epsilon_{1})\tilde{e}_{A}$
$\displaystyle=\overline{\epsilon}^{-}_{0}\Gamma^{4}\epsilon_{0}^{+}\frac{2i\lambda}{\omega}(\mathcal{Z}+D)=\mathcal{Z}+D,$
(B.38)
$\displaystyle(\overline{\epsilon}_{1}\Gamma^{A}\epsilon_{2})\tilde{e}_{A}$
$\displaystyle=\overline{\epsilon}^{+}_{0}\Gamma^{4}\epsilon_{0}^{-}\frac{2i\lambda}{\omega}(\mathcal{Z}-D)=\mathcal{Z}-D,$
(B.39)
$\displaystyle(\overline{\epsilon}_{2}\Gamma^{A}\epsilon_{2})\tilde{e}_{A}$
$\displaystyle=\overline{\epsilon}^{-}_{0}\Gamma^{0}\epsilon_{0}^{-}\frac{2i\lambda}{\omega}E_{+}=E_{+},$
(B.40)
$\displaystyle(\overline{\epsilon}_{1}\Gamma^{A}\epsilon_{1})\tilde{e}_{A}$
$\displaystyle=\overline{\epsilon}^{+}_{0}\Gamma^{0}\epsilon_{0}^{+}\frac{2i\lambda}{\omega}E_{-}=E_{-},$
(B.41)

where we have chosen a specific normalization for constant spinors
$\epsilon_{0}^{\pm}$ and introduced four independent Killing vectors
$(\mathcal{Z},D,E_{+},E_{-})$. They are defined as

$\displaystyle D$ $\displaystyle=-t\partial_{t}+r\partial_{r},$ (B.42)
$\displaystyle\mathcal{Z}$
$\displaystyle=\frac{i\omega^{2}}{\lambda^{2}}(\partial_{\xi_{1}}+\partial_{\xi_{2}}+\partial_{\xi_{3}})-\frac{i}{2\lambda^{2}}(\partial_{\tilde{\phi}}+\partial_{\tilde{\psi}}),$
(B.43) $\displaystyle E_{+}$
$\displaystyle=-\frac{i\omega^{2}}{4\lambda}\partial_{t},$ (B.44)
$\displaystyle E_{-}$
$\displaystyle=\frac{8i\lambda}{\omega^{2}}rt\partial_{r}+\left(-\frac{4i\lambda}{\omega^{2}}t^{2}+\frac{4i(9\omega^{2}-4\lambda^{2})}{\omega^{2}\lambda}\frac{1}{r^{2}}\right)\partial_{t}$
(B.45)
$\displaystyle+\frac{8i}{\lambda^{2}\omega}\frac{1}{r}(\partial_{\xi_{1}}+\partial_{\xi_{2}}+\partial_{\xi_{3}})+\frac{12i}{\lambda^{2}\omega}\frac{1}{r}(\partial_{\tilde{\phi}}+\partial_{\tilde{\psi}}).$
(B.46)

With these definitions one can verify the algebra

$\displaystyle[D,E_{\pm}]$ $\displaystyle=\pm
E_{\pm},\qquad[D,\mathcal{Z}]=0,$ (B.47) $\displaystyle[\mathcal{Z},E_{\pm}]$
$\displaystyle=0,\qquad[E_{+},E_{-}]=2D,$ (B.48)

which allows us to identify $(D,E_{+},E_{-})$ as $SL(2,\mathbb{R})$ generators
and $\mathcal{Z}$ as the $U(1)$ generator of R-symmetry.
We are now in a position to fully determine the isometry superalgebra of the
near-horizon geometry [87, 88, 89, 90]. A standard prescription is to
associate bosonic generators $\mathcal{Q}_{B}(k_{i})$ and fermionic generators
$\mathcal{Q}_{F}(\epsilon_{I})$ to Killing vectors and Killing spinors
respectively. The superalgebra is then determined from the relations

$\displaystyle[\mathcal{Q}_{B}(k_{i}),\mathcal{Q}_{B}(k_{j})]$
$\displaystyle=\mathcal{Q}_{B}([k_{i},k_{j}]),$ (B.49)
$\displaystyle[\mathcal{Q}_{B}(k),\mathcal{Q}_{F}(\epsilon)]$
$\displaystyle=\mathcal{Q}_{F}(\mathbb{L}_{k}\epsilon),$ (B.50)
$\displaystyle\\{\mathcal{Q}_{F}(\epsilon_{I}),\mathcal{Q}_{F}(\epsilon_{J})\\}$
$\displaystyle=\mathcal{Q}_{B}(\epsilon_{I}\gamma\epsilon_{J}),$ (B.51)

where we introduced the spinorial Lie derivative [90] through which Killing
vectors act on Killing spinors

$\displaystyle\mathbb{L}_{k}\epsilon=k^{M}(\partial_{M}\epsilon+\frac{1}{4}\omega_{MAB}\Gamma^{A}\Gamma^{B}\epsilon)+\frac{1}{4}D_{[A}k_{B]}\Gamma^{A}\Gamma^{A}\epsilon.$
(B.52)

In the above $\omega_{MAB}$ denotes the spin connection, and

$\displaystyle
D_{A}k_{B}=\tilde{e}^{M}_{A}(\partial_{M}k_{B}-\omega_{M\phantom{C}B}^{\phantom{M}C}k_{C}).$
(B.53)

From (B.49) we immediately read off the bosonic part of isometry algebra as
(B.47),(B.48). Similarly, (B.51) together with (B.38)-(B.41) imply non-
vanishing anticommutators

$\displaystyle\\{\overline{\mathcal{Q}}_{2},\mathcal{Q}_{1}\\}$
$\displaystyle=\mathcal{Z}+D,$
$\displaystyle\\{\overline{\mathcal{Q}}_{1},\mathcal{Q}_{2}\\}$
$\displaystyle=\mathcal{Z}-D,$ (B.54)
$\displaystyle\\{\overline{\mathcal{Q}}_{2},\mathcal{Q}_{2}\\}$
$\displaystyle=E_{+},$
$\displaystyle\\{\overline{\mathcal{Q}}_{1},\mathcal{Q}_{1}\\}$
$\displaystyle=E_{-}.$ (B.55)

Lastly, to determine the commutators of bosonic and fermionic generators we
can either compute the spinorial Lie derivatives or try to solve the super
Jacobi identity. Here we take the former approach. The Lie derivatives of the
Killing spinors in the directions of Killing vectors are found to be

$\displaystyle\mathbb{L}_{Z}\epsilon_{1}$
$\displaystyle=\frac{1}{2}\epsilon_{1},$
$\displaystyle\mathbb{L}_{Z}\overline{\epsilon}_{1}$
$\displaystyle=-\frac{1}{2}\overline{\epsilon}_{1},$
$\displaystyle\mathbb{L}_{Z}\epsilon_{2}$
$\displaystyle=\frac{1}{2}\epsilon_{2},$
$\displaystyle\mathbb{L}_{Z}\overline{\epsilon}_{2}$
$\displaystyle=-\frac{1}{2}\overline{\epsilon}_{2},$ (B.56)
$\displaystyle\mathbb{L}_{J}\epsilon_{1}$
$\displaystyle=-\frac{1}{2}\epsilon_{1},$
$\displaystyle\mathbb{L}_{J}\overline{\epsilon}_{1}$
$\displaystyle=-\frac{1}{2}\overline{\epsilon}_{1},$
$\displaystyle\mathbb{L}_{J}\epsilon_{2}$
$\displaystyle=\frac{1}{2}\epsilon_{2},$
$\displaystyle\mathbb{L}_{J}\overline{\epsilon}_{2}$
$\displaystyle=\frac{1}{2}\overline{\epsilon}_{2},$ (B.57)
$\displaystyle\mathbb{L}_{E_{+}}\epsilon_{1}$ $\displaystyle=-\epsilon_{2},$
$\displaystyle\mathbb{L}_{E_{+}}\overline{\epsilon}_{1}$
$\displaystyle=\overline{\epsilon}_{2},$
$\displaystyle\mathbb{L}_{E_{+}}\epsilon_{2}$ $\displaystyle=0,$
$\displaystyle\mathbb{L}_{E_{+}}\overline{\epsilon}_{2}$ $\displaystyle=0,$
(B.58) $\displaystyle\mathbb{L}_{E_{-}}\epsilon_{1}$ $\displaystyle=0,$
$\displaystyle\mathbb{L}_{E_{-}}\overline{\epsilon}_{1}$ $\displaystyle=0,$
$\displaystyle\mathbb{L}_{E_{-}}\epsilon_{2}$ $\displaystyle=-\epsilon_{1},$
$\displaystyle\mathbb{L}_{E_{-}}\overline{\epsilon}_{2}$
$\displaystyle=\overline{\epsilon}_{1},$ (B.59)

from which we identify commutation relations

$\displaystyle[\mathcal{Q}_{1},Z]$
$\displaystyle=-\frac{1}{2}\mathcal{Q}_{1},$
$\displaystyle[\overline{\mathcal{Q}}_{1},Z]$
$\displaystyle=\frac{1}{2}\overline{\mathcal{Q}}_{1},$
$\displaystyle[\mathcal{Q}_{2},Z]$
$\displaystyle=-\frac{1}{2}\mathcal{Q}_{2},$
$\displaystyle[\overline{\mathcal{Q}}_{2},Z]$
$\displaystyle=\frac{1}{2}\overline{\mathcal{Q}}_{2},$ (B.60)
$\displaystyle[\mathcal{Q}_{1},J]$ $\displaystyle=\frac{1}{2}\mathcal{Q}_{1},$
$\displaystyle[\overline{\mathcal{Q}}_{1},J]$
$\displaystyle=\frac{1}{2}\overline{\mathcal{Q}}_{1},$
$\displaystyle[\mathcal{Q}_{2},J]$
$\displaystyle=-\frac{1}{2}\mathcal{Q}_{2},$
$\displaystyle[\overline{\mathcal{Q}}_{2},J]$
$\displaystyle=-\frac{1}{2}\overline{\mathcal{Q}}_{2},$ (B.61)
$\displaystyle[\mathcal{Q}_{1},E_{+}]$ $\displaystyle=\mathcal{Q}_{2},$
$\displaystyle[\overline{\mathcal{Q}}_{1},E_{+}]$
$\displaystyle=-\overline{\mathcal{Q}}_{2},$
$\displaystyle[\mathcal{Q}_{2},E_{+}]$ $\displaystyle=0,$
$\displaystyle[\overline{\mathcal{Q}}_{2},E_{+}]$ $\displaystyle=0,$ (B.62)
$\displaystyle[\mathcal{Q}_{1},E_{-}]$ $\displaystyle=0,$
$\displaystyle[\overline{\mathcal{Q}}_{1},E_{-}]$ $\displaystyle=0,$
$\displaystyle[\mathcal{Q}_{2},E_{-}]$ $\displaystyle=\mathcal{Q}_{1},$
$\displaystyle[\overline{\mathcal{Q}}_{2},E_{-}]$
$\displaystyle=-\overline{\mathcal{Q}}_{1},$ (B.63)

As a consistency check, we can verify that the above superalgebra indeed
satisfies the super Jacobi identity

$\displaystyle(-1)^{|x||z|}[x,[y,z]]+(-1)^{|y||x|}[y,[z,x]]+(-1)^{|z||y|}[z,[x,y]]=0,$
(B.64)

where one has to choose $|x|=0$ for bosonic and $|x|=1$ for fermionic
generators, and the (anti)commutators are defined through

$\displaystyle[x,y]=-(-1)^{|x||y|}[y,x].$ (B.65)

## References

  * [1] J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Int. J. Theor. Phys. 38 (1999) 1113–1133, arXiv:hep-th/9711200 [hep-th]. [Adv. Theor. Math. Phys.2,231(1998)].
  * [2] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, “Gauge theory correlators from noncritical string theory,” Phys. Lett. B428 (1998) 105–114, arXiv:hep-th/9802109 [hep-th].
  * [3] E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. 2 (1998) 253–291, arXiv:hep-th/9802150.
  * [4] J. B. Gutowski and H. S. Reall, “Supersymmetric AdS(5) black holes,” JHEP 02 (2004) 006, arXiv:hep-th/0401042.
  * [5] J. B. Gutowski and H. S. Reall, “General supersymmetric AdS(5) black holes,” JHEP 04 (2004) 048, arXiv:hep-th/0401129.
  * [6] Z. W. Chong, M. Cvetic, H. Lu, and C. N. Pope, “General non-extremal rotating black holes in minimal five-dimensional gauged supergravity,” Phys. Rev. Lett. 95 (2005) 161301, arXiv:hep-th/0506029.
  * [7] H. K. Kunduri, J. Lucietti, and H. S. Reall, “Supersymmetric multi-charge AdS(5) black holes,” JHEP 04 (2006) 036, arXiv:hep-th/0601156.
  * [8] A. Cabo-Bizet, D. Cassani, D. Martelli, and S. Murthy, “Microscopic origin of the Bekenstein-Hawking entropy of supersymmetric AdS5 black holes,” JHEP 10 (2019) 062, arXiv:1810.11442 [hep-th].
  * [9] S. Choi, J. Kim, S. Kim, and J. Nahmgoong, “Large AdS black holes from QFT,” arXiv:1810.12067 [hep-th].
  * [10] F. Benini and P. Milan, “Black Holes in 4D $\mathcal{N}$=4 Super-Yang-Mills Field Theory,” Phys. Rev. X 10 no. 2, (2020) 021037, arXiv:1812.09613 [hep-th].
  * [11] M. Honda, “Quantum Black Hole Entropy from 4d Supersymmetric Cardy formula,” Phys. Rev. D 100 no. 2, (2019) 026008, arXiv:1901.08091 [hep-th].
  * [12] A. Arabi Ardehali, “Cardy-like asymptotics of the 4d $\mathcal{N}=4$ index and AdS5 blackholes,” JHEP 06 (2019) 134, arXiv:1902.06619 [hep-th].
  * [13] A. Cabo-Bizet and S. Murthy, “Supersymmetric phases of 4d $\mathcal{N}$ = 4 SYM at large $N$,” JHEP 09 (2020) 184, arXiv:1909.09597 [hep-th].
  * [14] A. Arabi Ardehali, J. Hong, and J. T. Liu, “Asymptotic growth of the 4d $\mathcal{N}$ = 4 index and partially deconfined phases,” JHEP 07 (2020) 073, arXiv:1912.04169 [hep-th].
  * [15] S. Murthy, “The growth of the $\frac{1}{16}$-BPS index in 4d $\mathcal{N}=4$ SYM,” arXiv:2005.10843 [hep-th].
  * [16] P. Agarwal, S. Choi, J. Kim, S. Kim, and J. Nahmgoong, “AdS black holes and finite N indices,” Phys. Rev. D 103 no. 12, (2021) 126006, arXiv:2005.11240 [hep-th].
  * [17] F. Benini, E. Colombo, S. Soltani, A. Zaffaroni, and Z. Zhang, “Superconformal indices at large $N$ and the entropy of AdS5 $\times$ SE5 black holes,” Class. Quant. Grav. 37 no. 21, (2020) 215021, arXiv:2005.12308 [hep-th].
  * [18] A. González Lezcano, J. Hong, J. T. Liu, and L. A. Pando Zayas, “Sub-leading Structures in Superconformal Indices: Subdominant Saddles and Logarithmic Contributions,” JHEP 01 (2021) 001, arXiv:2007.12604 [hep-th].
  * [19] C. Copetti, A. Grassi, Z. Komargodski, and L. Tizzano, “Delayed Deconfinement and the Hawking-Page Transition,” arXiv:2008.04950 [hep-th].
  * [20] A. Cabo-Bizet, “From multi-gravitons to Black holes: The role of complex saddles,” arXiv:2012.04815 [hep-th].
  * [21] A. Arabi Ardehali and S. Murthy, “The 4d superconformal index near roots of unity and 3d Chern-Simons theory,” JHEP 10 (2021) 207, arXiv:2104.02051 [hep-th].
  * [22] D. Cassani and Z. Komargodski, “EFT and the SUSY Index on the 2nd Sheet,” SciPost Phys. 11 (2021) 004, arXiv:2104.01464 [hep-th].
  * [23] O. Aharony, F. Benini, O. Mamroud, and P. Milan, “A gravity interpretation for the Bethe Ansatz expansion of the $\mathcal{N}=4$ SYM index,” arXiv:2104.13932 [hep-th].
  * [24] C. Romelsberger, “Counting chiral primaries in N = 1, d=4 superconformal field theories,” Nucl. Phys. B 747 (2006) 329–353, arXiv:hep-th/0510060.
  * [25] J. Kinney, J. M. Maldacena, S. Minwalla, and S. Raju, “An Index for 4 dimensional super conformal theories,” Commun. Math. Phys. 275 (2007) 209–254, arXiv:hep-th/0510251.
  * [26] J. Preskill, P. Schwarz, A. D. Shapere, S. Trivedi, and F. Wilczek, “Limitations on the statistical description of black holes,” Mod. Phys. Lett. A6 (1991) 2353–2362.
  * [27] J. Michelson and M. Spradlin, “Supergravity spectrum on AdS(2)$\times S^{2}$,” JHEP 09 (1999) 029, arXiv:hep-th/9906056 [hep-th].
  * [28] L. V. Iliesiu and G. J. Turiaci, “The statistical mechanics of near-extremal black holes,” JHEP 05 (2021) 145, arXiv:2003.02860 [hep-th].
  * [29] M. Heydeman, L. V. Iliesiu, G. J. Turiaci, and W. Zhao, “The statistical mechanics of near-BPS black holes,” J. Phys. A 55 no. 1, (2022) 014004, arXiv:2011.01953 [hep-th].
  * [30] A. Strominger and C. Vafa, “Microscopic origin of the Bekenstein-Hawking entropy,” Phys. Lett. B 379 (1996) 99–104, arXiv:hep-th/9601029.
  * [31] J. M. Maldacena and L. Susskind, “D-branes and fat black holes,” Nucl. Phys. B475 (1996) 679–690, arXiv:hep-th/9604042 [hep-th].
  * [32] J. M. Maldacena and A. Strominger, “Universal low-energy dynamics for rotating black holes,” Phys. Rev. D56 (1997) 4975–4983, arXiv:hep-th/9702015 [hep-th].
  * [33] A. Sinha, J. Sonner, and N. V. Suryanarayana, “At the horizon of a supersymmetric AdS(5) black hole: Isometries and half-BPS giants,” JHEP 01 (2007) 087, arXiv:hep-th/0610002.
  * [34] W. Fu, D. Gaiotto, J. Maldacena, and S. Sachdev, “Supersymmetric Sachdev-Ye-Kitaev models,” Phys. Rev. D 95 no. 2, (2017) 026009, arXiv:1610.08917 [hep-th]. [Addendum: Phys.Rev.D 95, 069904 (2017)].
  * [35] D. Stanford and E. Witten, “Fermionic Localization of the Schwarzian Theory,” JHEP 10 (2017) 008, arXiv:1703.04612 [hep-th].
  * [36] T. G. Mertens, G. J. Turiaci, and H. L. Verlinde, “Solving the Schwarzian via the Conformal Bootstrap,” JHEP 08 (2017) 136, arXiv:1705.08408 [hep-th].
  * [37] D. Stanford and E. Witten, “JT Gravity and the Ensembles of Random Matrix Theory,” arXiv:1907.03363 [hep-th].
  * [38] F. Larsen, J. Nian, and Y. Zeng, “AdS5 black hole entropy near the BPS limit,” JHEP 06 (2020) 001, arXiv:1907.02505 [hep-th].
  * [39] S. Choi and F. Larsen, Effective Field Theory of Quantum Black Holes.  8, 2021.  arXiv:2108.04028 [hep-th].
  * [40] F. Benini, K. Hristov, and A. Zaffaroni, “Black hole microstates in AdS4 from supersymmetric localization,” JHEP 05 (2016) 054, arXiv:1511.04085 [hep-th].
  * [41] D. Gaiotto, A. Kapustin, Z. Komargodski, and N. Seiberg, “Theta, Time Reversal, and Temperature,” JHEP 05 (2017) 091, arXiv:1703.00501 [hep-th].
  * [42] H. T. Lam, T. G. Mertens, G. J. Turiaci, and H. Verlinde, “Shockwave S-matrix from Schwarzian Quantum Mechanics,” JHEP 11 (2018) 182, arXiv:1804.09834 [hep-th].
  * [43] S. Sachdev, “Bekenstein-Hawking Entropy and Strange Metals,” Phys. Rev. X5 no. 4, (2015) 041025, arXiv:1506.05111 [hep-th].
  * [44] A. Almheiri and B. Kang, “Conformal Symmetry Breaking and Thermodynamics of Near-Extremal Black Holes,” JHEP 10 (2016) 052, arXiv:1606.04108 [hep-th].
  * [45] D. Anninos, T. Anous, and R. T. D’Agnolo, “Marginal deformations & rotating horizons,” JHEP 12 (2017) 095, arXiv:1707.03380 [hep-th].
  * [46] G. Turiaci and H. Verlinde, “Towards a 2d QFT Analog of the SYK Model,” JHEP 10 (2017) 167, arXiv:1701.00528 [hep-th].
  * [47] P. Nayak, A. Shukla, R. M. Soni, S. P. Trivedi, and V. Vishal, “On the Dynamics of Near-Extremal Black Holes,” JHEP 09 (2018) 048, arXiv:1802.09547 [hep-th].
  * [48] U. Moitra, S. P. Trivedi, and V. Vishal, “Extremal and near-extremal black holes and near-CFT1,” JHEP 07 (2019) 055, arXiv:1808.08239 [hep-th].
  * [49] S. Hadar, “Near-extremal black holes at late times, backreacted,” JHEP 01 (2019) 214, arXiv:1811.01022 [hep-th].
  * [50] A. Castro, F. Larsen, and I. Papadimitriou, “5D rotating black holes and the nAdS2/nCFT1 correspondence,” JHEP 10 (2018) 042, arXiv:1807.06988 [hep-th].
  * [51] F. Larsen and Y. Zeng, “Black hole spectroscopy and AdS2 holography,” JHEP 04 (2019) 164, arXiv:1811.01288 [hep-th].
  * [52] U. Moitra, S. K. Sake, S. P. Trivedi, and V. Vishal, “Jackiw-Teitelboim Gravity and Rotating Black Holes,” arXiv:1905.10378 [hep-th].
  * [53] S. Sachdev, “Universal low temperature theory of charged black holes with AdS2 horizons,” J. Math. Phys. no. 5, (2019) 052303, arXiv:1902.04078 [hep-th].
  * [54] J. Hong, F. Larsen, and J. T. Liu, “The scales of black holes with nAdS2 geometry,” JHEP 10 (2019) 260, arXiv:1907.08862 [hep-th].
  * [55] A. Castro and V. Godet, “Breaking away from the near horizon of extreme Kerr,” arXiv:1906.09083 [hep-th].
  * [56] A. M. Charles and F. Larsen, “A One-Loop Test of the near-AdS2/near-CFT1 Correspondence,” arXiv:1908.03575 [hep-th].
  * [57] F. Larsen and S. Paranjape, “Thermodynamics of Near BPS Black Holes in AdS4 and AdS7,” arXiv:2010.04359 [hep-th].
  * [58] L. V. Iliesiu, “On 2D gauge theories in Jackiw-Teitelboim gravity,” arXiv:1909.05253 [hep-th].
  * [59] S. Banerjee, R. K. Gupta, and A. Sen, “Logarithmic Corrections to Extremal Black Hole Entropy from Quantum Entropy Function,” JHEP 03 (2011) 147, arXiv:1005.3044 [hep-th].
  * [60] S. Banerjee, R. K. Gupta, I. Mandal, and A. Sen, “Logarithmic Corrections to N=4 and N=8 Black Hole Entropy: A One Loop Test of Quantum Gravity,” JHEP 11 (2011) 143, arXiv:1106.0080 [hep-th].
  * [61] A. Sen, “Logarithmic Corrections to N=2 Black Hole Entropy: An Infrared Window into the Microstates,” Gen. Rel. Grav. 44 no. 5, (2012) 1207–1266, arXiv:1108.3842 [hep-th].
  * [62] A. Sen, “Logarithmic Corrections to Rotating Extremal Black Hole Entropy in Four and Five Dimensions,” Gen. Rel. Grav. 44 (2012) 1947–1991, arXiv:1109.3706 [hep-th].
  * [63] S. W. Hawking, C. J. Hunter, and M. Taylor, “Rotation and the AdS / CFT correspondence,” Phys. Rev. D 59 (1999) 064005, arXiv:hep-th/9811056.
  * [64] Z. W. Chong, M. Cvetic, H. Lu, and C. N. Pope, “Five-dimensional gauged supergravity black holes with independent rotation parameters,” Phys. Rev. D 72 (2005) 041901, arXiv:hep-th/0505112.
  * [65] S. M. Hosseini, K. Hristov, and A. Zaffaroni, “An extremization principle for the entropy of rotating BPS black holes in AdS5,” JHEP 07 (2017) 106, arXiv:1705.05383 [hep-th].
  * [66] A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myers, “Charged AdS black holes and catastrophic holography,” Phys. Rev. D60 (1999) 064018, arXiv:hep-th/9902170 [hep-th].
  * [67] J. P. Gauntlett and J. B. Gutowski, “All supersymmetric solutions of minimal gauged supergravity in five-dimensions,” Phys. Rev. D 68 (2003) 105009, arXiv:hep-th/0304064. [Erratum: Phys.Rev.D 70, 089901 (2004)].
  * [68] G. W. Gibbons and S. W. Hawking, “Action Integrals and Partition Functions in Quantum Gravity,” Phys. Rev. D 15 (1977) 2752–2756.
  * [69] V. Balasubramanian and P. Kraus, “A Stress tensor for Anti-de Sitter gravity,” Commun. Math. Phys. 208 (1999) 413–428, arXiv:hep-th/9902121 [hep-th].
  * [70] W. Chen, H. Lu, and C. N. Pope, “Mass of rotating black holes in gauged supergravities,” Phys. Rev. D 73 (2006) 104036, arXiv:hep-th/0510081.
  * [71] D. Cassani and L. Papini, “The BPS limit of rotating AdS black hole thermodynamics,” JHEP 09 (2019) 079, arXiv:1906.10148 [hep-th].
  * [72] S. W. Hawking and S. F. Ross, “Duality between electric and magnetic black holes,” Phys. Rev. D52 (1995) 5865–5876, arXiv:hep-th/9504019 [hep-th].
  * [73] M. Cvetic, G. W. Gibbons, H. Lu, and C. N. Pope, “Rotating black holes in gauged supergravities: Thermodynamics, supersymmetric limits, topological solitons and time machines,” arXiv:hep-th/0504080.
  * [74] M. Berkooz, D. Reichmann, and J. Simon, “A Fermi Surface Model for Large Supersymmetric AdS(5) Black Holes,” JHEP 01 (2007) 048, arXiv:hep-th/0604023.
  * [75] M. Berkooz and D. Reichmann, “Weakly Renormalized Near 1/16 SUSY Fermi Liquid Operators in N=4 SYM,” JHEP 10 (2008) 084, arXiv:0807.0559 [hep-th].
  * [76] L. V. Iliesiu, M. Kologlu, and G. J. Turiaci, “Supersymmetric indices factorize,” arXiv:2107.09062 [hep-th].
  * [77] S.-Q. Wu, “General Nonextremal Rotating Charged AdS Black Holes in Five-dimensional $U(1)^{3}$ Gauged Supergravity: A Simple Construction Method,” Phys. Lett. B 707 (2012) 286–291, arXiv:1108.4159 [hep-th].
  * [78] D. Harlow, L. Iliesiu, H. Ooguri, and G. Turiaci work in progress .
  * [79] M. Cvetic, H. Lu, and C. N. Pope, “Charged rotating black holes in five dimensional U(1)**3 gauged N=2 supergravity,” Phys. Rev. D 70 (2004) 081502, arXiv:hep-th/0407058.
  * [80] M. David, J. Nian, and L. A. Pando Zayas, “Gravitational Cardy Limit and AdS Black Hole Entropy,” JHEP 11 (2020) 041, arXiv:2005.10251 [hep-th].
  * [81] J. H. Schwarz, “Covariant Field Equations of Chiral N=2 D=10 Supergravity,” Nucl. Phys. B 226 (1983) 269.
  * [82] M. Pernici, K. Pilch, and P. van Nieuwenhuizen, “Gauged N=8 D=5 Supergravity,” Nucl. Phys. B 259 (1985) 460.
  * [83] M. Gunaydin, L. J. Romans, and N. P. Warner, “Compact and Noncompact Gauged Supergravity Theories in Five-Dimensions,” Nucl. Phys. B 272 (1986) 598–646.
  * [84] M. Cvetic, M. J. Duff, P. Hoxha, J. T. Liu, H. Lu, J. X. Lu, R. Martinez-Acosta, C. N. Pope, H. Sati, and T. A. Tran, “Embedding AdS black holes in ten-dimensions and eleven-dimensions,” Nucl. Phys. B 558 (1999) 96–126, arXiv:hep-th/9903214.
  * [85] J. P. Gauntlett, J. B. Gutowski, and N. V. Suryanarayana, “A Deformation of AdS(5) x S**5,” Class. Quant. Grav. 21 (2004) 5021–5034, arXiv:hep-th/0406188.
  * [86] A. H. Chamseddine, S. Ferrara, G. W. Gibbons, and R. Kallosh, “Enhancement of supersymmetry near 5-d black hole horizon,” Phys. Rev. D 55 (1997) 3647–3653, arXiv:hep-th/9610155.
  * [87] S. Beck, U. Gran, J. Gutowski, and G. Papadopoulos, “All Killing Superalgebras for Warped AdS Backgrounds,” JHEP 12 (2018) 047, arXiv:1710.03713 [hep-th].
  * [88] J. P. Gauntlett, R. C. Myers, and P. Townsend, “Supersymmetry of rotating branes,” Phys. Rev. D 59 (1998) 025001, arXiv:hep-th/9809065.
  * [89] J. P. Gauntlett, R. C. Myers, and P. K. Townsend, “Black holes of D = 5 supergravity,” Class. Quant. Grav. 16 (1999) 1–21, arXiv:hep-th/9810204.
  * [90] J. M. Figueroa-O’Farrill, “On the supersymmetries of Anti-de Sitter vacua,” Class. Quant. Grav. 16 (1999) 2043–2055, arXiv:hep-th/9902066.
  * [91] P. Saad, S. H. Shenker, and D. Stanford, “JT gravity as a matrix integral,” arXiv:1903.11115 [hep-th].
  * [92] S. Forste, A. Gerhardus, and J. Kames-King, “Supersymmetric Black Holes and the SJT/nSCFT${}_{1}\\!$ Correspondence,” arXiv:2007.12393 [hep-th].
  * [93] M. Cárdenas, O. Fuentealba, H. A. González, D. Grumiller, C. Valcárcel, and D. Vassilevich, “Boundary theories for dilaton supergravity in 2D,” JHEP 11 (2018) 077, arXiv:1809.07208 [hep-th].
  * [94] L. V. Iliesiu, S. S. Pufu, H. Verlinde, and Y. Wang, “An exact quantization of Jackiw-Teitelboim gravity,” JHEP 11 (2019) 091, arXiv:1905.02726 [hep-th].
  * [95] A. Blommaert, T. G. Mertens, and H. Verschelde, “Fine Structure of Jackiw-Teitelboim Quantum Gravity,” arXiv:1812.00918 [hep-th].
  * [96] Y. Fan and T. G. Mertens, “Supergroup Structure of Jackiw-Teitelboim Supergravity,” arXiv:2106.09353 [hep-th].
  * [97] M. David, A. G. Lezcano, J. Nian, and L. A. P. Zayas, “Logarithmic Corrections to the Entropy of Rotating Black Holes and Black Strings in AdS5,” arXiv:2106.09730 [hep-th].
  * [98] N. Arkani-Hamed, L. Motl, A. Nicolis, and C. Vafa, “The String landscape, black holes and gravity as the weakest force,” JHEP 06 (2007) 060, arXiv:hep-th/0601001.
  * [99] D. Harlow, B. Heidenreich, M. Reece, and T. Rudelius, “The Weak Gravity Conjecture: A Review,” arXiv:2201.08380 [hep-th].
  * [100] J. Markeviciute and J. E. Santos, “Evidence for the existence of a novel class of supersymmetric black holes with AdS${}_{5}\times$S5 asymptotics,” Class. Quant. Grav. 36 no. 2, (2019) 02LT01, arXiv:1806.01849 [hep-th].
  * [101] J. Markeviciute, “Rotating Hairy Black Holes in AdS${}_{5}\times$S5,” JHEP 03 (2019) 110, arXiv:1809.04084 [hep-th].
  * [102] S. S. Gubser, I. R. Klebanov, and A. A. Tseytlin, “Coupling constant dependence in the thermodynamics of N=4 supersymmetric Yang-Mills theory,” Nucl. Phys. B 534 (1998) 202–222, arXiv:hep-th/9805156.
  * [103] M. B. Green and C. Stahn, “D3-branes on the Coulomb branch and instantons,” JHEP 09 (2003) 052, arXiv:hep-th/0308061.
  * [104] M. B. Green and S. Sethi, “Supersymmetry constraints on type IIB supergravity,” Phys. Rev. D 59 (1999) 046006, arXiv:hep-th/9808061.
  * [105] M. F. Paulos, “Higher derivative terms including the Ramond-Ramond five-form,” JHEP 10 (2008) 047, arXiv:0804.0763 [hep-th].
  * [106] J. a. F. Melo and J. E. Santos, “Stringy corrections to the entropy of electrically charged supersymmetric black holes with $\mathrm{AdS}_{5}\times S^{5}$ asymptotics,” Phys. Rev. D 103 no. 6, (2021) 066008, arXiv:2007.06582 [hep-th].
  * [107] H. S. Reall and J. E. Santos, “Higher derivative corrections to Kerr black hole thermodynamics,” JHEP 04 (2019) 021, arXiv:1901.11535 [hep-th].
  * [108] A. Ghosh, H. Maxfield, and G. J. Turiaci, “A universal Schwarzian sector in two-dimensional conformal field theories,” arXiv:1912.07654 [hep-th].
  * [109] T. Hartman, C. A. Keller, and B. Stoica, “Universal Spectrum of 2d Conformal Field Theory in the Large c Limit,” JHEP 09 (2014) 118, arXiv:1405.5137 [hep-th].
  * [110] N. Benjamin, H. Ooguri, S.-H. Shao, and Y. Wang, “Light-cone modular bootstrap and pure gravity,” Phys. Rev. D 100 no. 6, (2019) 066029, arXiv:1906.04184 [hep-th].
  * [111] T. Eguchi and Y. Sugawara, “Modular bootstrap for boundary N = 2 Liouville theory,” JHEP 01 (2004) 025, arXiv:hep-th/0311141.
  * [112] P. Kraus, “Lectures on black holes and the AdS(3) / CFT(2) correspondence,” Lect. Notes Phys. 755 (2008) 193–247, arXiv:hep-th/0609074.
  * [113] T. Eguchi and A. Taormina, “Unitary Representations of $N=4$ Superconformal Algebra,” Phys. Lett. B196 (1987) 75.
  * [114] A. Buchel and J. T. Liu, “Gauged supergravity from type IIB string theory on Y**p,q manifolds,” Nucl. Phys. B 771 (2007) 93–112, arXiv:hep-th/0608002.
  * [115] J. P. Gauntlett and O. Varela, “Consistent Kaluza-Klein reductions for general supersymmetric AdS solutions,” Phys. Rev. D 76 (2007) 126007, arXiv:0707.2315 [hep-th].
  * [116] A. Dabholkar, J. Gomes, and S. Murthy, “Nonperturbative black hole entropy and Kloosterman sums,” JHEP 03 (2015) 074, arXiv:1404.0033 [hep-th].
  * [117] T. G. Mertens and G. J. Turiaci, “Defects in Jackiw-Teitelboim Quantum Gravity,” JHEP 08 (2019) 127, arXiv:1904.05228 [hep-th].
  * [118] H. Maxfield and G. J. Turiaci, “The path integral of 3D gravity near extremality; or, JT gravity with defects as a matrix integral,” arXiv:2006.11317 [hep-th].
  * [119] E. Witten, “Matrix Models and Deformations of JT Gravity,” arXiv:2006.13414 [hep-th].