July 2022

Black Hole Entropy from Quantum Mechanics

Nick Dorey, Rishi Mouland and Boan Zhao
DAMTP, Centre for Mathematical Sciences
University of Cambridge, Wilberforce Road
Cambridge CB3 0WA, UK
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

We provide evidence for a holographic duality between superconformal quantum
mechanics on the moduli space of Yang-Mills instantons and M-theory in certain
asymptotically $AdS_{7}\times S^{4}$ backgrounds with a plane-wave boundary
metric. We show that the gravitational background admits a supersymmetric
black hole solution whose entropy is precisely reproduced by the
superconformal index of the dual quantum mechanics.

## Introduction

A key task for any candidate theory of quantum gravity is to provide a
microscopic explanation for the thermodynamic properties of black holes.
String theory provides a concrete setting in which black hole microstates can
be counted, reproducing the classical Bekenstein-Hawking entropy formula, at
least for supersymmetric black holes [1]. The AdS/CFT correspondence yields
further insight by providing a dual description of black hole microstates in
quantum field theory. In particular, recent work has lead to rapid progress in
accounting for the entropy of supersymmetric black holes in Anti-de-Sitter
space (see [2] and references therein). However, all currently understood
examples where this analysis is possible involve dual field theories in
spacetime dimension $D\geq 2$. Holographic dualities involving one-dimensional
theories or, in other words, finite-dimensional quantum mechanics, are still
poorly understood [3] and there are few working examples. A notable exception
is the BFSS matrix model [4] which is holographically dual to the near horizon
geometry of D0 branes in Type IIA string theory. Such finite dimensional
examples are particularly interesting conceptually as they have a fully non-
perturbative definition. This is also important in practical terms as as they
can be simulated on a computer111For a survey of recent progress and prospects
in simulations of holographic matrix quantum mechanics on classical and
quantum computers see [5]. avoiding many of the difficulties associated with
lattice field theory in higher dimensions.

####

So far there has been little intersection between the two strands of research
described above. In this paper we propose a precise duality between a
superconformal quantum mechanical model and a gravitational theory which
admits a family of supersymmetric black hole solutions. We test the duality by
computing the index degeneracies of BPS states from first principles in
quantum mechanics and find an exact match to the Bekenstein-Hawking entropy of
the black hole. As far as we know, this is the first example where black hole
entropy can be reproduced analytically in a finite-dimensional model. The
model therefore potentially provides a novel setting where analytical and
numerical approaches may coexist.

####

The correspondence described here between quantum mechanics and gravity is
self-contained and can be stated without reference to field theory or
string/M-theory. However it is also related to the more familiar holographic
duality between the six dimensional $(2,0)$ CFT and M-theory on $AdS_{7}\times
S^{4}$. In particular, building on the ideas of [6], it corresponds to a
special limit in which the boundary theory reduces to quantum mechanics. This
will be explored further in two follow up papers [7, 8]. In the rest of the
paper we provide an overview of our results. Additional details are provided
in a series of appendices.

## The quantum mechanical model

We consider superconformal quantum mechanics [9, 10] on the ADHM moduli space,
$\mathcal{M}_{K,N}$, of $K$ Yang-Mills instantons in a $U(N)$ gauge theory
defined on $\mathbb{R}^{4}$. The moduli space is a hyperKähler cone of complex
dimension $d_{\mathbb{C}}=2KN$. A $(0+1)$-dimensional supersymmetric non-
linear $\sigma$-model with target space $\mathcal{M}_{K,N}$ therefore admits
an $\mathfrak{osp}(4^{*}|4)$ superconformal symmetry [11]. The theory we seek
has a discrete222As usual in superconformal quantum mechanics we are working
in a Hilbert space appropriate for diagonalising the dilatation operator $D$
rather than the Laplacian denoted $H$. By a standard argument this is
equivalent to the spectrum of the operator $\Delta=H+C$ where $C$ is certain a
harmonic potential on $\mathcal{M}_{K,N}$, yielding a discrete spectrum. See
Appendix A.1 for further details. spectrum consisting of unitary, irreducible,
lowest-weight representations of this algebra analogous to the spectrum of
local operators described by radial quantization of a higher dimensional CFT.
As $\mathcal{M}_{K,N}$ is a singular space, a proper definition of the theory
requires further elaboration. The instanton moduli space has a smooth
resolution $\tilde{\mathcal{M}}_{K,N}$ characterised by a real parameter
$\xi_{\mathbb{R}}$ with the dimensions of length. Although the resolution
breaks the superconformal symmetry described above, it preserves ordinary
supersymmetry as well as (most of) the global and $R$-symmetries of the model.
As discussed in [12, 13, 14, 15], supersymmetric quantum mechanics on the
smooth manifold $\tilde{\mathcal{M}}_{K,N}$ equipped with a certain potential
provides a regulated version of the model. We expect to recover an
$\mathfrak{osp}(4^{*}|4)$ invariant spectrum in the “continuum” limit
$\xi_{\mathbb{R}}\rightarrow 0$. We also note that the both the superconformal
model and its regulated counterpart can be embedded in a manifestly finite
matrix quantum mechanical gauge theory. The latter model can be put in a form
suitable for numerical simulation on a computer333Indeed computer simulations
of closely related models have already appeared [16]..

####

States in the model are labelled by a set of Cartan generators for
$\mathfrak{osp}(4^{*}|4)$ denoted $\Delta$, $J_{1}+J_{2}$, $Q_{1}$, $Q_{2}$.
Here the quantum number444We will use the same notation throughout to denote
both a Cartan generator and its eigenvalue. $\Delta$ is a scaling dimension
which obeys a BPS bound of the form,

$\displaystyle\Delta$ $\displaystyle\geq$ $\displaystyle
J_{1}+J_{2}+2Q_{1}+2Q_{2}$ (1)

Although the full spectrum of the model is out of reach, analytic methods can
provide some control over the spectrum of BPS states which saturate the bound.
As in more familiar higher-dimensional examples [17, 18], these states belong
to special short multiplets of the superconformal algebra and they can be
counted by defining a superconformal index [12] (see also [19]). It is useful
to define the following linear combinations of the charges,

$\displaystyle L_{t}$ $\displaystyle:=$ $\displaystyle
J_{1}+J_{2}+Q_{1}+Q_{2}$ $\displaystyle L_{y}$ $\displaystyle:=$
$\displaystyle Q_{1}-Q_{2}$

which commute with the supercharges used to define the index. The remaining
linear combination $F:=-2Q_{2}$ instead plays the role of fermion number. In
addition to superconformal invariance, the theory also has an
$\mathfrak{su}(2)\oplus\mathfrak{su}(N)$ global symmetry with Cartan
generators denoted $L_{x}:=J_{1}-J_{2}$ and $n_{a}$, $a=1,2\ldots,N-1$. The
resulting index corresponds to the following trace over the space of BPS
states,

$\displaystyle\mathcal{I}_{\rm SC}[\mathcal{M}_{K,N}]$ $\displaystyle:=$
$\displaystyle{\rm Tr}_{\rm
BPS}\left[(-1)^{F}t^{L_{t}}x^{L_{x}}y^{L_{y}}\prod_{a=1}^{N}w_{a}^{n_{a}}\right]$
(2)

####

Importantly the same index can also be defined in the regulated version of the
model, where it corresponds to the Witten index of supersymmetric quantum
mechanics on the resolved space, with a potential corresponding to the norm of
a holomorphic Killing vector [20]. The resulting index is independent of the
resolution parameter $\xi_{\mathbb{R}}$ and serves as a regulated definition
of the superconformal index. It also has a nice geometric interpretation as an
Euler characteristic for equivariant sheaf cohomology on the resolved space
[14].

$\displaystyle\mathcal{I}_{\rm SC}\left[\mathcal{M}_{K,N}\right]$
$\displaystyle=$
$\displaystyle\sum_{p=0}^{d_{\mathbb{C}}}\left(-\frac{y}{t}\right)^{p-d_{\mathbb{C}}/2}\,\chi\left(\tilde{\mathcal{M}}_{K,N},\mathcal{A}^{p}\right)$
(3)

Here $\chi(Y,\mathcal{A}^{p})$ denotes the equivariant Euler character of the
sheaf of $p$-forms on a smooth variety $Y$. The latter can be computed by
standard localisation theorems in equivariant K-theory giving a closed formula
for the index as a sum over fixed points of the maximal torus of a group
action on $\tilde{\mathcal{M}}_{K,N}$. These are enumerated by $N$-coloured
partitions [21] $\vec{\lambda}\in\mathcal{P}^{N}$ of total weight
$||\vec{\lambda}||=K$. The resulting expression for the superconformal index
is,

$\displaystyle\mathcal{I}_{\rm SC}\left[\mathcal{M}_{K,N}\right]$
$\displaystyle=$
$\displaystyle\sum_{\\{\lambda\in\mathcal{P}^{N}:\,\,||\vec{\lambda}||=K\\}}\,\,\prod_{i,j=1}^{N}\,\,\prod_{s\in
Y(\lambda^{(i)})}\,\,{\rm
Pexp}\left(\frac{z_{i}}{z_{j}}\,t^{g_{ij}(s)}\,x^{f_{ij}(s)}\left[ty\right]\left[t/y\right]\right)$
(4)

where $w_{a}=z_{i+1}/z_{i}$ for $a=i=1,\ldots,N-1$. See Appendix A.2 for
definitions and notation. The consistency of this expression with the original
definition (2) as trace over an $\mathfrak{osp}(4^{*}|4)$-invariant spectrum
is non trivial. It requires, for example, that the resulting rational function
of the fugacities appearing in (4) has an expansion in positive powers of $t$
with coeficients which are Laurent polynomials in the remaining
variables555This property can be checked to high order by explicit calculation
and a proof will be given in [22].

####

The index computed above coincides exactly with the $K$-instanton contribution
to the Nekrasov partition function of an auxiliary five-dimensional $U(N)$
gauge theory, a fact we will use extensively below. In this context the
$\mathfrak{su}(N)$ symmetry of the model is inherited from the global part of
the $U(N)$ gauge group in five dimensions. As usual, for comparison with a
gravitational dual, we will focus on the $\mathfrak{su}(N)$ invariant sector
of the model. We project onto the $\mathfrak{su}(N)$ singlets by integrating
over the maximal torus of $SU(N)$ with the Haar measure666In fact, the
agreement we find between the model and a gravitational dual does not seem to
depend on this projection. A related issue in the context of holography has
recently been discussed in [23].. It is also convenient sum over the instanton
number $K$, with fugacity $q_{\tau}=\exp(2\pi i\tau)=\exp(-\beta)$ although
ultimately we will project back onto sectors of fixed $K$. The final form of
our index is,

$\displaystyle\mathcal{I}_{\rm QM}\left[t,x,y,q_{\tau}\right]$
$\displaystyle=$
$\displaystyle\sum_{K=0}^{\infty}\,q_{\tau}^{K}\,\int\,d\mu_{\rm
Haar}[{Z}]\,\,\mathcal{I}_{\rm SC}\left[\mathcal{M}_{K,N}\right]$ (5)

####

As mentioned above, the index at each instanton number has an expansion in
positive powers of the complex fugacity $t$. The coefficients in this
expansion encode information about the degeneracies of BPS states in the
model. Concretely, if we define,

$\displaystyle\mathcal{C}(L_{t},L_{x},L_{y},K)$ $\displaystyle\in$
$\displaystyle\mathbb{Z}$

as the coefficient of,

$t^{L_{t}}x^{L_{x}}y^{L_{y}}q_{\tau}^{K}$ (6)

in this expansion, then the definition of the index as a trace over BPS states
given above implies,

$\displaystyle\mathcal{C}(L_{t},L_{x},L_{y},K)$ $\displaystyle=$
$\displaystyle\sum_{F\in\mathbb{Z}}\,\left(-1\right)^{F}\,d(L_{t},L_{x},L_{y},K,F)$
(7)

where $d(\ldots)\in\mathbb{Z}_{\geq 0}$ denote the corresponding degeneracies.

####

Brute force evaluation of the coefficients $\mathcal{C}(L_{t},L_{x},L_{y},K)$
on a computer777Some sample data for the most computationally tractable case
$N=1$ is given is Appendix B. Preliminary data for $N>1$ reveals a
qualitatively similar behaviour. reveals that they grow rapidly in magnitude
as we increase $L_{t}$, $L_{x}$ $L_{y}$ and $K$ and oscillate with a sign that
is not correlated in any obvious way with the values of these charges.
Obtaining an analytic understanding of this growth is one the main goals of
this paper. The numbers $\mathcal{C}$ can also be related to alternating sums
of the dimensions of certain subspaces of fixed grade in the equivariant sheaf
cohomology of the resolution $\tilde{\mathcal{M}}_{K,N}$ [14, 22].

## Building the duality

In order to identify a gravitational dual of the superconformal quantum
mechanics defined above, we will first exploit its relation to the $(2,0)$
superconformal field theory in six dimensions. The most familiar form of this
relation arises from compactification of the the $(2,0)$ theory of type
$A_{N-1}$ on a spacelike circle of radius $R$. The resulting effective theory
in five dimensions is maximally supersymmetric $U(N)$ Yang-Mills theory with
coupling $g_{5}^{2}\sim R$. Yang-Mills instantons give rise to classical BPS
solitons in this gauge theory which correspond to Kaluza-Klein modes on
$S^{1}$ in the original six dimensional description. The quantum mechanical
$\sigma$-model with target space $\mathcal{M}_{K,N}$ considered above then
provides a low-energy effective description of the sector of the $(2,0)$
theory with $K$ units of momentum on $S^{1}$.

####

A more profound relation between the two theories emerges when we perform a
large boost along the compactified direction (together with a suitable
rescaling). In the Seiberg limit [24] where the parameter of the boost becomes
infinite, the low-energy description becomes exact. The result is a precise
duality between the $(2,0)$ theory compactified on a null circle with $K$
units of momentum, and superconformal quantum mechanics on
$\mathcal{M}_{K,N}$. A key point underlying this duality is the fact that (the
simple part of) the stabilizer of the six-dimensional $(2,0)$ superconformal
algebra under such a null compactification coincides with the quantum
mechanical superconformal algebra $\mathfrak{osp}(4^{*}|4)$. In this context,
the Cartan generators $J_{1}$ and $J_{2}$ correspond to angular momenta in two
orthogonal planes transverse to the lightcone, while $Q_{1}$ and $Q_{2}$
correspond to R-charges. The eigenvalues of the Hamiltonian $\Delta=H+C$
coincide with scaling dimensions under $D$, which is the the so-called
lightcone or Lifschitz dilatation operator of the $(2,0)$ theory. Such null
compactifications are examples of Discrete Light Cone Quantization (DLCQ), and
the fact that they lead to finite dimensional non-relativistic quantum
mechanics for fixed values of the null momentum $K$ is familiar in this
context. Indeed the model considered here was first introduced [25, 26] as a
DLCQ formulation of the $(2,0)$ superconformal field theory.

####

The operator $\Delta=H+C$, which is the natural Hamiltonian for superconformal
quantum mechanics, does not act in a simple way in Minkowski space. For the
present purposes, the duality can be put in a more useful form by a conformal
transformation [27] which maps $\mathbb{R}^{5,1}$ to a pp-wave geometry with
metric,

$\displaystyle ds^{2}=-2dx^{+}dx^{-}-x^{i}x^{i}(dx^{-})^{2}+dx^{i}dx^{i}$ (8)

with $i=1,\dots,4$, subject to null compactification via the identification
$x^{+}\sim x^{+}+2\pi$. In these coordinates, $\Delta$ simply generates
translations of $x^{-}$, while $J_{1},J_{2}$ are a pair of commuting rotations
in the $x^{i}$ directions. Due to the gravitational potential well in the
transverse directions of the metric (8), the resulting spectrum is discrete
and, thanks to the residual superconformal invariance, consists of multiplets
of $\mathfrak{osp}(4^{*}|4)$. The map thus provides a lightcone analogue of
the usual radial quantization map [28, 29]. Following [26, 6], we then have,

####

Duality I: Superconformal quantum mechanics on $\mathcal{M}_{K,N}$ is
equivalent to the six-dimensional $(2,0)$ theory of type $A_{N-1}$ in the
null-compactified pp-wave geometry (8) with $K$ units of null momentum in the
compact direction.

####

We are able to provide an immediate check of this duality at the level of BPS
states. As is well known, the six-dimensional pp-wave geometry (without null
compactification) can be obtained as a Penrose limit of the Lorentzian
cylinder $S^{5}\times\mathbb{R}_{t}$. If we replace the $S^{5}$ factor with a
suitable orbifold $S^{5}/\mathbb{Z}_{n}$ and coordinate the usual Penrose
limit with the limit $n\rightarrow\infty$, we instead obtain the null-
compactified version of the pp-wave (8) as a limiting geometry. Fortunately,
an appropriate supersymmetric index for the $(2,0)$-theory on the orbifold
$S^{5}/\mathbb{Z}_{n}$ has already been computed via reduction to 5d gauge
theory on a spatial $\mathbb{CP}^{2}$ and subsequent localisation [30]. The
resulting index is expressed as a convolution of contributions from instantons
and perturbation theory on three distinct coordinate patches of
$\mathbb{CP}^{2}$ as well as a sum over flux sectors. By applying the combined
Penrose and $n\rightarrow\infty$ limits we can obtain an index for the $(2,0)$
theory on a null compactified pp-wave background. We find that the orbifold
index undergoes dramatic simplification in this limit and coincides exactly
with the index $\mathcal{I}_{\rm QM}$ for superconformal quantum mechanics
defined above [8].

####

The main motivation for relating our quantum mechanical model to the $(2,0)$
theory is that the latter has a known gravitational dual. The theory of type
$A_{N-1}$ is dual to M-theory on $AdS_{7}\times S^{4}$ with $N$ units of four-
form flux through $S^{4}$. When $N>>1$, M-theory reduces to eleven-dimensional
supergravity on the same background. As noted above, the six dimensional pp-
wave spacetime is related to flat Minkowski space by a conformal
transformation. Correspondingly we can find coordinates on $AdS_{7}$ that
manifestly realise pp-wave asymptotics; explicitly, we have

$\displaystyle
ds^{2}_{\text{AdS}}=\frac{dr^{2}}{g^{2}r^{2}}+r^{2}\Big{(}-2dx^{+}dx^{-}-x^{i}x^{i}(dx^{-})^{2}+dx^{i}dx^{i}\Big{)}-\frac{1}{g^{2}}(dx^{-})^{2}$
(9)

where $g=R^{-1}_{\text{AdS}}$ is the inverse AdS radius. We define $X$ as this
spacetime subject to the identification $x^{+}\sim x^{+}+2\pi$. It is clear
then that $X$ realises null-compactified pp-wave asymptotics. Then, applying
the standard holographic logic and using Duality I, we have our second
proposal [6],

####

Duality II: Superconformal quantum mechanics on $\mathcal{M}_{K,N}$ is
equivalent to M-theory on spacetimes asymptotic to $X\times S^{4}$ . More
precisely we restrict to the sector of this theory with with $K$ units of
momentum around the null boundary circle, as well as $N$ units of four-form
flux through $S^{4}$.

####

As usual the basic kinematics underlying the duality ensure that the
superconformal algebra $\mathfrak{osp}(4^{*}|4)$ of the boundary theory is
linearly realised as the algebra of (super)-isometries in the bulk888In fact,
the full group of super-isometries of the bulk is the maximal subgroup of
$\mathfrak{osp}(8^{*}|4)$ preserved by the null compactification. This is the
supersymmetrised Schrödinger group in four spatial dimensions, of which
$\mathfrak{osp}(4^{*}|4)$ is a subgroup. This full symmetry group is indeed
realised in the superconformal quantum mechanical model we construct, albeit
non-linearly, as first noted in [31].. The scaling dimension $\Delta$
corresponds to translations in the bulk time coordinate corresponding to
boundary lightcone time $x^{-}$. The Cartan generators $J_{1}$ and $J_{2}$
correspond to transverse angular momenta in the AdS pp-wave part of the bulk
space time while $Q_{1}$ and $Q_{2}$ are associated with rotations on the
internal $S^{4}$. Furthermore, the parameters on either side of the duality
are fixed by the relation

$\displaystyle\frac{R_{\text{AdS}}}{l_{p}}=\frac{2R_{S^{4}}}{l_{p}}=2(\pi
N)^{1/3}$ (10)

where $l_{p}$ denotes the 11-dimensional Planck length.

## A black hole and its entropy

One can then seek M-theory solutions with the desired asymptotics. As first
proposed in [6], beginning with some asymptotically AdS${}_{7}\times S^{4}$
solution, one can take the AdS dual of the Penrose limit discussed above to
arrive at solutions with $X\times S^{4}$ asymptotics. We will call this the
boundary Penrose limit to distinguish it from the conventional Penrose limit
of the bulk theory (ie the BMN limit [32]). In more detail, the $(2,0)$ theory
on the Lorentzian cylinder $S^{5}\times\mathbb{R}_{t}$ is dual to M-theory on
$AdS_{7}\times S^{4}$ in global coordinates. After taking the boundary Penrose
limit, the asymptotic geometry takes the form (9); a subsequent identification
of a null coordinate results in precisely the spacetime geometry $X$ defined
above. In particular, this procedure was applied in [6] to (non-
supersymmetric) black hole solutions in $AdS_{7}\times S^{4}$ to obtain new
black hole solutions with $X\times S^{4}$ asymptotics. For brevity, we will
refer to any such black hole supergravity solutions with $X\times S^{4}$
asymptotics as an $X$ black hole.

####

Crucial to the duality is the compact null direction in the asymptotic
geometry $X\times S^{4}$. As a null circle is equivalent to a spacelike circle
of vanishing size this seems to preclude a description in terms of classical
gravity. In particular, the bulk configuration of pure $X\times S^{4}$ should
be dual to the vacuum state in the boundary theory. Because the boundary
circle remains null throughout the bulk, there is no reliable approximation of
this state in terms of classical gravity. However, the situation is somewhat
better for $X$ black holes. Such solutions should be dual to thermal ensembles
of (generically non-BPS) states of large charge in the boundary quantum
mechanics. Then, as first noted in [6], the null circle on the boundary
becomes spacelike in the bulk, and for sufficiently large momentum along it,
this circle is large enough in some regions of the bulk spacetime to allow a
classical gravity description.

####

As the index of the quantum mechanical model exhibits a degeneracy of BPS
states which grows rapidly with the charges, it’s natural to seek an
explanation in terms of the entropy of a supersymmetric black hole in the dual
theory; that is, an $X$ black hole whose charges satisfy the BPS bound for
$\mathfrak{osp}(4^{*}|4)$. Fortunately, although such solutions have not been
considered before, they can be constructed from known supersymmetric
asymptotically AdS7 Kerr-Newman black hole solutions following the method
outlined above.

####

The general non-extremal black hole solution in AdS7 is expected to depend on
six parameters, which can be exchanged in favour of conserved charges
corresponding to the six Cartan generators of $SO(2,6)\times SO(5)$. These are
made up of a mass $M$ and three angular momenta $J_{1},J_{2},J_{3}$
corresponding respectively to time translations and $S^{5}$ rotations on the
$\mathbb{R}_{t}\times S^{5}$ conformal boundary, and two electric charges
$Q_{1},Q_{2}$ corresponding to the $SO(5)$ gauge fields arising after Kaluza-
Klein reduction on $S^{4}$. However, the most general solutions currently
known are constrained to have either equal angular momenta $J_{1}=J_{2}=J_{3}$
[33], or equal charges $Q_{1}=Q_{2}$ [34]. We will focus exclusively on the
latter solutions here. Such solutions then admit a BPS limit, in which two
degrees of freedom are lost. Firstly, a certain relation amongst parameters
ensures that the BPS bound

$\displaystyle M\geq J_{1}+J_{2}+J_{3}+4Q$ (11)

is saturated, while an additional non-linear relation amongst
$\\{J_{1},J_{2},J_{3},Q\\}$ is also required to avoid naked closed time-like
curves. The entropy $\mathcal{S}_{\text{BH}}>0$ of such BPS black holes is
found to be

$\displaystyle\bigg{(}\frac{\mathcal{S}_{\text{BH}}}{2\pi}\bigg{)}^{2}$
$\displaystyle=\frac{Q^{3}-(N^{3}/6)\left(J_{1}J_{2}+J_{2}J_{3}+J_{1}J_{3}\right)}{Q-N^{3}/6}$
(12)

and the non-linear constraint mentioned above can be recast as the statement
that $\mathcal{S}_{\text{BH}}$ must also satisfy

$\displaystyle\left(\frac{\mathcal{S}_{\text{BH}}}{2\pi}\right)^{4}-2A\left(\frac{\mathcal{S}_{\text{BH}}}{2\pi}\right)^{2}+B=0$
(13)

with

$\displaystyle
A\,=\,\frac{N^{3}}{3}\left(J_{1}+J_{2}+J_{3}\right)\,\,+\,\,3Q^{2},$
$\displaystyle B\,=\,\frac{2}{3}N^{3}J_{1}J_{2}J_{3}\,\,+\,\,Q^{4}$

Then, following the method of [6] as outlined above, we obtain from Chow’s
solutions a new class of black holes precisely realising $X\times S^{4}$
asymptotics. Our solutions retain the five degrees of freedom of the initial
solution of [34], which can be exchanged for a set of charges
$\\{\Delta,K,J_{1},J_{2},Q\\}$ corresponding to the isometries of $X\times
S^{4}$ as detailed above. The fact that Chow’s black hole configurations are
indeed solutions of the bulk supergravity ensures that our solutions are too;
nonetheless, we have independently checked using Mathematica that this is
indeed the case. Here, we present the key relevant details of our solutions,
while their full details can be found in Appendix C. The derivation of our
solutions in the boundary Penrose limit of those of Chow will appear in a
forthcoming paper [7].

####

In order to match with the index of the quantum mechanics, we then specialise
to solutions which saturate the BPS bound (1) for $\mathfrak{osp}(4^{*}|4)$.
We also find an additional non-linear constraint on the charges which arises
as the smooth boundary Penrose limit of the corresponding constraint on Chow’s
solution described above. We are then left with a three parameter family of
supersymmetric $X$ black holes. To describe the resulting thermodynamics, it’s
useful work in terms of $K$ and the two linear combinations
$L_{t}=J_{1}+J_{2}+2Q$ and $L_{x}=J_{1}-J_{2}$ introduced above. Then, the
constraint allows us to express the remaining independent charge as
$Q=Q(L_{t},L_{x},K)$, while we can also write the Bekenstein-Hawking entropy
as $\mathcal{S}_{\text{BH}}=\mathcal{S}_{\text{BH}}(L_{t},L_{x},K)$. These
take the forms

$\displaystyle\mathcal{S}_{\text{BH}}(L_{t},L_{x},K)=2\pi\sqrt{\delta}\,\,\mathcal{F}_{\text{BH}}\\!\left[L_{t}/\sqrt{\delta},L_{x}/\sqrt{\delta}\right],\qquad
Q(L_{t},L_{x},K)=\sqrt{\delta}\,\,q\\!\left[L_{t}/\sqrt{\delta},L_{x}/\sqrt{\delta}\right]$
(14)

where we define $\delta=KN^{3}/3$. The functions
$\mathcal{F}_{\text{BH}}[u,v]$ and $q[u,v]$ are determined as follows. One can
show that for our solutions, $u,v$ are such that the quartic equation

$\displaystyle s^{4}=2(s+i(u+v)/2)(s+i(u-v)/2)$ (15)

for complex variable $s$ has precisely one root strictly lying within each
quadrant of the complex plane. Let $s_{\star}$ denote the unique root with
positive real part and negative imaginary part. Then,

$\displaystyle s_{\star}=\mathcal{F}_{\text{BH}}[u,v]-iq[u,v].$ (16)

In the simplifying case of equal angular momenta where $J_{1}=J_{2}=J$, the
black hole entropy can be written explicitly as,

$\displaystyle\mathcal{S}_{\rm BH}(L,0,K)$ $\displaystyle=$ $\displaystyle
2\pi\sqrt{K}\sqrt{\frac{N^{3}}{12}}\,\left[\sqrt{1+\sqrt{1+\frac{6L^{2}}{KN^{3}}}}-\sqrt{2}\right]$
(17)

with $L=L_{t}=2(J+Q)$. Although a more precise derivation will be given in [7]
(see also Appendix C.4), we note that these results can be obtained as a
simple scaling limit of the corresponding relations for Chow’s solutions
(12,13) in which $J_{3}\rightarrow\infty$ with the remaining charges scaling
like $\sqrt{J_{3}}$.

####

Before comparing with the boundary theory, we must determine the range of
parameters and charges for which we can reliably approximate M-theory by
supergravity. This is only the case if the curvature is small everywhere in
units of the Planck length. We must also ensure that the radius of the compact
$x^{+}$ direction is large enough to avoid the presence of additional light
states. As noted above, as we approach the conformal boundary, the invariant
length of the compact circle shrinks towards zero, and thus any classical
description breaks down. Nonetheless, we can still seek conditions under which
the classical description makes sense within the bulk, and in particular near
the black hole horizon. We find that these conditions are satisfied for
$K>>N^{7/3}>>1$, with the charges $J_{1}$, $J_{2}$ and $Q$ growing like
$\sqrt{KN^{3}}$. In particular, this means that we may hold fixed the
arguments $L_{t}/\sqrt{\delta}$ and $L_{x}/\sqrt{\delta}$ appearing in (14) as
we take $K,N$ large.

####

## Black hole entropy from quantum mechanics

The holographic duality proposed above suggest that the degeneracy of BPS
states in superconformal quantum mechanics with fixed values of the charges
$L_{t}$, $L_{x}$ (and $L_{y}=0$) should be related to the black hole entropy
provided we work inside the regime of validity of the gravitational
description. In the following we will test this by computing the asymptotics
of the index coefficients $\mathcal{C}[L_{t},L_{x},L_{y},K]$ in the limit
$K\rightarrow\infty$. More precisely we will calculate the asymptotic growth
of the index coefficients for fixed $N>>1$ in the $K\rightarrow\infty$ with
the remaining charges $J_{1}$, $J_{2}$, $Q_{1}$, $Q_{2}$ scaling like
$\sqrt{K}$ (or, equivalently, $L_{t}$, $L_{x}$, $L_{y}\sim\sqrt{K}$). As
expressed in (7), these integer coefficients correspond to alternating sums
over degeneracies of BPS states with different values of $F=-2Q_{2}$.

####

In determining the asymptotics of the index coefficients we will adapt a
standard strategy of analytic combinatorics which originates Hardy and
Ramanujan’s classic analysis [35] of the growth of partition numbers. The
first step is to represent the coefficients as an appropriate contour
integral,

$\displaystyle\mathcal{C}[L_{t},L_{x},L_{y},K]$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi
i)^{4}}\,\oint_{C_{\beta}}\,\frac{dq_{\tau}}{q_{\tau}^{K+1}}\,\oint_{C_{+}}\,\frac{dt}{t^{L_{t}+1}}\,\oint_{C_{-}}\,\frac{dx}{x^{L_{x}+1}}\,\oint_{C_{m}}\,\frac{dy}{y^{L_{y}+1}}\,\,\,\mathcal{I}_{\rm
QM}\left[t,x,y,q_{\tau}\right]$

with contours specified in Appendix D.5 below. Continuing with the standard
strategy, the next step is to determine the analytic properties of the
integrand. In particular, we will show that the integrand has an essential
singularity in $\beta=-\log q_{\tau}$ at $\beta=0$. Holding the remaining
parameters, $t=\exp(-\epsilon_{+})$, $x=\exp(-\epsilon_{-})$ and $y=\exp(-m)$,
at fixed complex values with ${\rm Im}[\epsilon_{+}]<0$ in the scaling limit,
the relevant asymptotic of the index derived in our analysis is999We also find
asymptotics for ${\rm Im}[\epsilon_{+}]>0$ which are related to (22) by
$\epsilon_{1}$, $\epsilon_{2}\rightarrow-\epsilon_{1}$, $-\epsilon_{2}$.

$\displaystyle\log\,\mathcal{I}_{QM}$
$\displaystyle\begin{array}[]{c}_{\beta\rightarrow 0}\\\ \sim\\\ \end{array}$
$\displaystyle-\frac{N^{3}}{24}\,\frac{\Delta_{1}^{2}\Delta_{2}^{2}}{\epsilon_{1}\epsilon_{2}\beta}\,\,\,+\,\,\,O\left(\beta^{0}\right)$
(22)

where $\epsilon_{\pm}=(\epsilon_{1}\pm\epsilon_{2})/2$,
$m=(\Delta_{1}-\Delta_{2})/2+i\pi$ and,

$\displaystyle\Delta_{1}+\Delta_{2}-\epsilon_{1}-\epsilon_{2}$
$\displaystyle=$ $\displaystyle 2\pi i$

More precisely, these asymptotics hold up to parametrically small corrections
if101010Our result in the latter case of small $\epsilon_{1}$, $\epsilon_{2}$
agrees with that of [36], although their result is derived for real values of
the parameters $\epsilon_{1}$ and $\epsilon_{2}$ with opposite sign. either
$N>>1$ or $|\epsilon_{1}|$, $|\epsilon_{2}|<<1$. Similar asymptotics appear in
the large-$N$ and Cardy limits for indices of field theories in higher
dimensions where they are related to the supersymmetric Casimir energy [37]
and also the the entropy of asymptotically AdS black holes111111Indeed such a
relation exists [38, 39] between the superconformal index of the $(2,0)$
theory itself and the Chow black hole solutions discussed above which is
clearly related to the match found here. [40].

####

Our study is aided by the coincidence noted above between the superconformal
index and the much studied Nekrasov partition function [41] of $U(N)$
supersymmetric Yang-Mills theory on $\mathbb{R}^{4}\times S^{1}$ with a
massive adjoint hypermultiplet. The index defined above can be written as,

$\displaystyle\mathcal{I}_{\rm QM}\left[t,x,y,q_{\tau}\right]:=\oint d\mu_{\rm
Haar}\left[{Z}\right]\,\mathcal{Z}^{U(N)}_{\rm
inst}\left[t,x,y,q_{\tau};{Z}\right]$ (23)

Here $\mathcal{Z}^{U(N)}_{\rm inst}$ is the non-perturbative part of the
Nekrasov partition $\mathcal{Z}_{\rm Nek}$ (see Appendix D.1 for a full
definition), which is naturally defined as an infinite series of instanton
contributions. However our main goal in this section is to understand the
behaviour of the index in the limit $\beta=-\log q_{\tau}\rightarrow 0$, when
the contributions from large instanton number become unsupressed. In order to
make progress, we need to find an effective way of resumming this series to
obtain analytic function of the fugacity $q_{\tau}=\exp(-\beta)$. For fixed
values of the other parameters, we expect the series to have a finite radius
of convergence in $q_{\tau}$. The sum then defines $\mathcal{Z}^{U(N)}_{\rm
Nek}$ as a holomorphic function within this radius which can then be
analytically continued to whole of the disk $|q_{\tau}|<1$. Indeed this has
been proven rigorously for some special values of the parameters [42]. An
important special case is provided by the $U(1)$ theory where the instanton
moduli space coincides with the Hilbert scheme of points on $\mathbb{C}^{2}$.
In this case only, one finds a completely explicit resummation of the
instanton series in Plethystic form [43, 44]. In particular, for $|q_{\tau}|$,
$|q_{1}|$, $|q_{2}|<1$, we have,

$\displaystyle\log\mathcal{Z}^{U(1)}_{\rm inst}\left[t,x,y,q_{\tau}\right]$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\,\frac{1}{n}\,\frac{q_{\tau}^{n}}{1-q_{\tau}^{n}}\,\frac{t^{n}}{y^{n}}\,\frac{\left(1-y^{n}q_{1}^{n}/t^{n}\right)\left(1-y^{n}q_{2}^{n}/t^{n}\right)}{\left(1-q_{1}^{n}\right)\left(1-q_{2}^{n}\right)}$
(24)

Provided $|q_{\tau}|<{\rm min}\\{|yt^{-1}|,|y^{-1}t^{-1}|\\}$, this sum is
convergent and defines a holomorphic function which can be extended by
analytic continuation to the whole of the domain,

$\displaystyle D^{U(1)}$ $\displaystyle:=$
$\displaystyle\left\\{q_{1},q_{2},q_{\tau},y\in\mathbb{C}^{\star}\,:|q_{1}|,|q_{2}|,|q_{\tau}|\,\,<1\right\\}$
(25)

The series (24) for $\log\mathcal{Z}^{U(1)}_{\rm inst}$ easily yields the
asymptotic [45],

$\displaystyle\log\mathcal{Z}^{U(1)}_{\rm inst}\left[t,x,y,q_{\tau}\right]$
$\displaystyle\sim$ $\displaystyle-\frac{1}{24}\frac{m^{2}(2\pi
i-m)^{2}}{\epsilon_{1}\epsilon_{2}\beta}$ (26)

for $|\epsilon_{1}|$, $|\epsilon_{2}|<<1$. These asymptotics allow for a
rigorous saddle-point analysis of the resulting integral (LABEL:int1intro)
which will be given in [22].

####

As in the original analysis of Hardy and Ramanujan, the key step in the
general case $N\geq 1$ is to exploit the modular properties of the integrand
to determine its behaviour near the singular point. Our analysis follows
closely a related discussion in [36]. The modular properties of the Nekrasov
partition function have a clear physical origin which can be understood by
embedding the 5d gauge theory in string/M-theory. As discussed at various
points in this paper, maximally supersymmetric five dimensional $U(N)$ gauge
theory arises from the the $S^{1}$ compactification of the $(2,0)$ theory of
type $A_{N-1}$. The latter is the worldvolume theory of $N$ M5 branes in
M-theory. The Nekrasov partition function is itself an index which counts BPS
states of this system or, equivalently, a twisted Euclidean partition function
for the $(2,0)$ CFT on $T^{2}\times\mathbb{R}^{4}$ where the $T^{2}$ is a
torus of complex structure $\tau$ with an associated $SL(2,\mathbb{Z})$ group
of modular transformations.

####

The usual definition of the Nekrasov partition function as an infinite series
of instanton corrections comes from a limit where the M-theory circle is small
and M5 branes reduce to D4 branes in the Type IIA theory. In this picture
instantons correspond to D0 branes and the $K$-instanton contribution counts
BPS boundstates of $K$ D0 branes and $N$ D4 branes. However, the index is
deformation invariant and can also be computed in an M-theory regime where the
relevant degrees of freedom are M2 branes stretched between M5 branes or
”M-strings” [46]. In this regime the same partition function can be evaluated
by calculating the elliptic genus of the M-string worldsheet theory in each
charge sector. The resulting expression is,

$\displaystyle\mathcal{Z}^{U(N)}_{\rm ell}$ $\displaystyle:=$
$\displaystyle\left[\mathcal{Z}^{U(1)}_{\rm Nek}\right]^{N}\,Z_{S}$

Where the M-string partition function
$Z_{S}=Z_{S}[\epsilon_{1},\epsilon_{2},m,\tau;\mathbf{v}]$ is written as a
function of the chemical potentials. To avoid clutter in our formulae, we will
often suppress dependence on the remaining parameters and write
$Z_{S}=Z_{S}[\mathbf{v}]$. Here $Z_{S}[\mathbf{v}]$ is defined as an expansion
in powers of the $SU(N)$ fugacities $W_{a}:=\exp(-v_{a})$, for
$a=1,2,\ldots,N-1$.

$\displaystyle Z_{S}[\mathbf{v}]$ $\displaystyle:=$
$\displaystyle\sum_{\mathbf{n}\in\mathbb{Z}_{\geq
0}^{N-1}}\,Z^{(\mathbf{n})}_{S}\,\exp\left(-\mathbf{n\cdot v}\right)$ (27)

####

The coefficents in this expansion come from a localisation computation of the
M-string worldsheet elliptic genus [47, 48, 49] and are given explicitly in
Appendix D.2. In this expression the instanton expansion has been resummed at
each order in $w_{a}=\exp(-v_{a})$, to give an analytic function of the
complex parameter $\tau=-2\pi i\beta$ defined for ${\rm Im}\tau>0$, which is
elliptic in the parameters $\epsilon_{1}$, $\epsilon_{2}$ and $m$. The
resulting coefficient functions are manifestly modular under
$SL(2,\mathbb{Z})$ transformations acting on $\tau$. In particular, the
$S$-generator of the modular group maps $\tau$ to $\tilde{\tau}:=-1/\tau$ and
acts on the other parameters of the partition function as,

$\displaystyle\epsilon_{1}\rightarrow\tilde{\epsilon}_{1}:=\epsilon_{1}/\tau\qquad{}$
$\displaystyle\qquad\epsilon_{2}\rightarrow\tilde{\epsilon}_{2}:=\epsilon_{2}/\tau\qquad{}$
$\displaystyle\qquad m\rightarrow\tilde{m}:=m/\tau$ (28)

In the following we will use a ”$\,\,\tilde{}\,\,$” to denote this replacement
in functions of the parameters $\\{\epsilon_{1},\epsilon_{2},m,\tau\\}$. The
modular transformation property for the coefficents $Z_{S}^{(\mathbf{n})}$ is
[45],

$\displaystyle Z_{S}^{(\mathbf{n})}$ $\displaystyle=$
$\displaystyle\mathcal{K}_{+}^{(\mathbf{n})}\,\tilde{Z}_{S}^{(\mathbf{n})}$
(29)

with

$\displaystyle\mathcal{K}^{(\mathbf{n})}_{+}$ $\displaystyle=$
$\displaystyle\exp\left[\frac{\epsilon_{1}\epsilon_{2}}{2\beta}\,\sum_{a,b=1}^{N-1}\,n_{a}\Omega_{a,b}n_{b}\,\,+\,\,\frac{1}{\beta}\left(m^{2}-\epsilon_{+}^{2}\right)\,\sum_{a=1}^{N-1}\,n_{a}\right]$
(30)

where $\Omega_{a,b}=2\delta_{a,b}-\delta_{|a-b|,1}$ is the $A_{N-1}$ Cartan
matrix. The exponent in this expression is proportional to the central charge
of the M-string worldsheet theory.

####

The physical discussion given above strongly suggests that that the two
different representations of the M5-brane partition function described above
correspond to different expansions of the same analytic function,

$\displaystyle\mathcal{Z}^{U(N)}$ $\displaystyle:=$
$\displaystyle\mathcal{Z}^{U(N)}_{\rm Nek}=\mathcal{Z}^{U(N)}_{\rm ell}$ (31)

The same assumption is implicit in the modular bootstrap approach [50, 51, 52]
to determining refined BPS invariants of elliptically-fibred Calabi-Yau
manifolds and its predictions have been tested in this context. The equality
of $\mathcal{Z}^{U(N)}_{\rm Nek}$ and $\mathcal{Z}^{U(N)}_{\rm ell}$ is also
closely related to the invariance of the refined topological vertex formalism
[53] under a change of preferred direction [46]. We have checked this equality
for $N=2$ by expanding both sides in $q_{\tau}$ and $\exp(-v_{a})$ at the
first few orders, but we do not know of a general proof.

####

Through the relation (31) the Nekrasov partition function inherits the modular
properties of $Z_{\rm ell}$. In the present context, the $S$-transformation
discussed above relates the desired asymptotics in the strong-coupling limit
$\beta\rightarrow 0$ to the corresponding behaviour in the weak-coupling limit
$\beta\rightarrow\infty$. In Appendix D.3/D.4, we exploit this modular
transformation property to derive the asymptotic form (22). Using these
results, we find that the dominant contribution to the integral
(LABEL:int1intro) in the scaling limit of large $K$ comes from the region near
$\beta=0$ allowing its evaluation by steepest descent. We rewrite the integral
in the form121212The parameters $\mu_{\beta}\ldots$ appearing in the limits of
integration are related to the choice of contour in (LABEL:int1intro).,

$\displaystyle\mathcal{C}(L_{t},L_{x},L_{y},K)$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi i)^{4}}\,\int_{\mu_{\beta}}^{\mu_{\beta}+2\pi
i}\,d\beta\,\int_{\mu_{+}}^{\mu_{+}+2\pi
i}\,d\epsilon_{+}\,\int_{\mu_{-}}^{\mu_{-}+2\pi
i}\,d\epsilon_{-}\,\int_{\mu_{m}}^{2\pi
i+\mu_{m}}\,dm\,\,\,\exp\left(\mathcal{S}\right)$

where the exponent is given as,

$\displaystyle\mathcal{S}+2\pi iQ_{2}$ $\displaystyle=$
$\displaystyle-\frac{N^{3}}{24}\,\frac{\Delta_{1}^{2}\Delta_{2}^{2}}{\epsilon_{1}\epsilon_{2}\beta}\,+\,\Delta_{1}Q_{1}\,+\,\Delta_{2}Q_{2}\,+\,\epsilon_{1}J_{1}\,+\,\epsilon_{2}J_{2}\,+\,\beta
K$

We find that the integral is dominated by the contribution of a complex
conjugate pair of saddle-points at $\mathcal{S}=2\pi\sqrt{\delta}s_{\star}$
and $2\pi\sqrt{\delta}\bar{s}_{\star}$, where $\delta=KN^{3}/3$ as above, and
$s_{\star}$ is the unique root with positive real part and negative imaginary
part of the quartic equation,

$\displaystyle s^{2}(s-iw)^{2}$ $\displaystyle=$ $\displaystyle
2\left(s+i(u+v)/2\right)\left(s+i(u-v)/2\right)$ (33)

Hence, we find the exponential growth,

$\displaystyle\log\,\left|\mathcal{C}[L_{t},L_{x},L_{y},K]\right|$
$\displaystyle\sim$ $\displaystyle 2\pi\sqrt{\delta}\mathcal{F}_{\rm
QM}\left[L_{t}/\sqrt{\delta},L_{x}/\sqrt{\delta},L_{y}/\sqrt{\delta}\right]$
(34)

and $\mathcal{F}_{\rm QM}[u,v,w]=\text{Re}\,s_{\star}$. In the case $w=0$
applicable to comparison with our gravity, we find an exact match with the
function which determines the black hole entropy.

$\displaystyle\mathcal{F}_{\rm QM}[u,v,0]$ $\displaystyle=$
$\displaystyle\mathcal{F}_{\rm BH}[u,v]$

####

It follows that the growth of the index coefficients in the case $L_{y}=0$ is
determined by the entropy of the corresponding supersymmetric $X$ black hole,

$\displaystyle\log\,\left|\mathcal{C}[L_{t},L_{x},0,K]\right|$
$\displaystyle\sim$ $\displaystyle\mathcal{S}_{\rm BH}(L_{t},L_{x},K)$

This agreement is clearly in line with the ”central dogma” [54] which suggests
that the black hole should contribute a factor $\exp\,\mathcal{S}_{\rm BH}$ to
the partition function of the gravitational theory. It is striking that, for
each value of the charges $L_{t}$, $L_{x}$ and $K$, the full exponential
growth of the index is exactly captured by the unique supersymmetric black
hole solution which obeys the non-linear constraint $Q=Q(L_{t},L_{x},K)$
described above131313Of course we cannot rule out the presence of additional
supersymmetric black holes which do not obey the non-linear contraint like
those found in the $AdS_{5}$ case in [55]. However there are no known
solutions of this type in $AdS_{7}$ which are relevant for the boundary
Penrose limit considered here. If additional solutions do exist in our case,
then their contribution to the growth of the index must be subleading.. More
speculatively, we can extend the comparison to include the phase of
$\mathcal{C}$. Keeping only the leading-order contribution at each saddle
point we find,

$\displaystyle\mathcal{C}[L_{t},L_{x},0,K]$ $\displaystyle\sim$ $\displaystyle
2\cos\left(2\pi Q(L_{t},L_{x},K)\right)\,\exp\,\mathcal{S}_{\rm
BH}(L_{t},L_{x},K)$

Imposing the quantization condition $Q\in\frac{1}{2}\mathbb{Z}$ we find the
sign $\cos(2\pi Q)=(-1)^{-2Q}=(-1)^{F}$ in agreement with the expected
contribution of the black hole microstates to the index.

####

Finally, note that a version of this analysis, valid for
$L_{t}^{2}>>N^{3}K/6$, holds for all values of $N$. This regime is analgous to
the Cardy limit of higher dimensional CFTs and resulting picture is very
similar to recent studies of the superconformal index of $\mathcal{N}=4$ SUSY
Yang-Mills in this context[56, 57]. The complex conjugate pair of saddle-point
actions can then be written as $\mathcal{S}(L)\pm i\mathcal{S}(L)$ with,

$\displaystyle\mathcal{S}(L)$ $\displaystyle=$ $\displaystyle
2\pi\,\left(\frac{N^{3}}{24}L^{2}K\right)^{\frac{1}{4}}$ (35)

Adding together the contribution of both saddle points, and including the
prefactor from small fluctuations around the stationary point, we obtain a
formula for the asymptotic growth of the index coefficients in the Cardy
regime,

$\displaystyle\mathcal{C}\left(L_{,}0,0,K\right)$ $\displaystyle\sim$
$\displaystyle\frac{3\sqrt{2}}{16\pi}\,\frac{\mathcal{S}(L)}{L^{2}K}\,\exp\left[\mathcal{S}(L)\right]\cos\left(\mathcal{S}(L)+\frac{\pi}{4}\right)$

with $\mathcal{S}(L)$ given by (35). In the Cardy regime, this formula holds
for all $N$. In the case $N=1$, we have tested this formula against a direct
expansion of the localisation formula on a computer up to $K\sim L\sim 60$,
finding good agreement141414In particular the formula agrees the data in
Appendix B up to errors of around $1$-$2\%$ in the exponent. The accuracy is
in agreement with error estimates for subleading terms..

####

We would like to thank Dario Martelli, Jorge Santos and Chiung Hwang for
useful discussions. We would also like to thank Sam Crew for collaboration in
the early stages of this project. This work has been partially supported by
STFC consolidated grant ST/T000694/1.

## Appendices

##  Appendix A Model and Index details

###  A.1 Defining the model

The simplest description of the model considered in this paper is provided by
embedding it in gauged supersymmetric matrix quantum mechanics, which can in
turn be obtained by dimensional reduction from supersymmetric gauge theory in
higher dimension. An efficent starting point is to consider $\mathcal{N}=4$
supersymmetric $U(K)$ gauge theory in three spacetime dimensions with a
particular choice of matter content. In addition to the $\mathcal{N}=4$ vector
multiplet, the theory contains an adjoint hypermultiplet whose lowest
components are two complex $K\times K$ matrix-valued scalar fields $X$ and
$\tilde{X}$. We also include $N$ additional hypermultiplets in the fundamental
representation of $U(K)$. The lowest components of these multiplets are scalar
fields $Q_{i}$, $\tilde{Q}_{i}$ which are $K$-component complex vectors and
also carry an aditional flavour index $i=1,2,\ldots,N$ together with a $U(K)$
gauge field. By writing the theory in terms of complex fields, we are picking
an $\mathcal{N}=2$ subalgebra of $\mathcal{N}=4$ supersymmetry in three
dimensions. With respect to the chosen $\mathcal{N}=2$ subalgebra, each
complex scalar field is the lowest component of a chiral multiplet. Apart from
the 3d gauge coupling, the parameters of the theory are a triplet of masses
for each hypermultiplet and a triplet of real FI parameters. In the
$\mathcal{N}=2$ language it is natural to split each of these parameters into
a single complex parameter and a real parameter.

####

In the case of vanishing masses, the three-dimensional theory has a Higgs
branch151515The 3d theory also has a Coulomb branch which is not relevant to
our discussion. which is defined by solving the F- and D-term equations. In
case of vanishing FI parameters, these take the form

$\displaystyle\left[X,\tilde{X}\right]\,+\,\sum_{i=1}^{N}Q_{i}\tilde{Q}_{i}$
$\displaystyle=$ $\displaystyle 0$
$\displaystyle\left[X,X^{\dagger}\right]+\left[\tilde{X},\tilde{X}^{\dagger}\right]\,+\,\sum_{i=1}^{N}\,|Q_{i}|^{2}-|\tilde{Q}_{i}|^{2}$
$\displaystyle=$ $\displaystyle 0$ (36)

To find the space of inequivalent vacua we must also mod out by the action of
the $U(K)$ gauge group. These steps coincide precisely with the definition of
the moduli space $\mathcal{M}_{K,N}$ appearing in the ADHM contruction [58] of
$K$ instantons of an $SU(N)$ gauge theory defined on $\mathbb{R}^{4}$. This is
a singular space of complex dimension $2KN$. Turning on the FI parameters
introduces non-zero constant terms on the RHS of the F- and D-term equations.
This replaces the singular space $\mathcal{M}_{K,N}$ by a resolved space
$\tilde{\mathcal{M}}_{K,N}$ which is a smooth hyper-Kähler manifold of the
same dimension. Both the singular space and its resolution also have
descriptions in algebraic geometry as Nakajima quiver varieties. In the
abelian case $N=1$, this coincides with the Hilbert scheme of $K$ points on
$\mathbb{C}^{2}$.

####

The next step is to dimensionally reduce the three-dimensional gauge theory
described above to obtain a theory in $(0+1)$ dimensions. The resulting
quantum mechanics has a gauge coupling $e^{2}$ of mass dimension $3$. At
energies high compared to the mass scale $\Lambda=e^{\frac{2}{3}}$ the theory
is weakly coupled. At energies much lower than $\Lambda$, gauge interactions
become strong and the theory flows to a non-linear $\sigma$-model with target
$\mathcal{M}_{K,N}$. As the target space is hyper-Kähler, the $\sigma$-model
has $\mathcal{N}=(4,4)$ supersymmetry161616Strictly speaking this refers to
the supersymmetry of the corresponding $\sigma$-model in $(1+1)$ dimensions
but we refer here to its dimensional reduction to QM.. The quantum mechanical
gauge theory theory also has FI parameters and masses inherited from the
corresponding parameters of the three dimensional theory. Introducing non-zero
values for these parameters in the UV results in supersymmetry preserving
deformations of the IR $\sigma$-model. As already mentioned turning on FI
parameters replaces the singular target space $\mathcal{M}_{K,N}$ by its
smooth resolution $\tilde{\mathcal{M}}_{K,N}$. Mass parameters in the UV
theory are associated to global symmetries which are realised isometries of
the target manifold in the IR. Turning on non-zero masses in the UV introduces
a potential in the IR $\sigma$-model proportional to the norm of the
corresponding Killing vector.

####

In the following, we will be mainly interested in the IR limit of the matrix
model described above. This corresponds to focussing on energies far below the
mass scale set by gauge coupling or simply taking the limit
$e^{2}\rightarrow\infty$. The resulting theory is a $(0+1)$-dimensional non-
linear $\sigma$-model with target space $\mathcal{M}_{K,N}$. The singular
space $\mathcal{M}_{K,N}$ is a hyper-Kähler cone. In other words, it admits a
triholomorphic homothety. This condition ensures that the Poincare
supersymmetry of the $\sigma$-model is enhanced to full superconformal
invariance. The corresponding superconformal algebra is the simple Lie super-
algebra $\mathfrak{g}:=\mathfrak{osp}(4^{*}|4)$ [11]. Thus matrix quantum
mechanics flows in the IR to a one dimensional superconformal theory with this
symmetry. In analogy with the more familiar case of higher dimensional SCFT,
we expect to find a spectrum of states transforming in irreducible, unitary
representations of $\mathfrak{g}:=\mathfrak{osp}(4^{*}|4)$.

####

The maximal bosonic subalgebra of $\mathfrak{g}:=\mathfrak{osp}(4^{*}|4)$ is,

$\displaystyle\mathfrak{g}_{B}$ $\displaystyle=$
$\displaystyle\mathfrak{so}(2,1)\oplus\mathfrak{su}(2)\oplus\mathfrak{so}(5)$
(37)

The first factor is the quantum mechanical superconformal algebra with
generators $\\{H,C,D\\}$ obeying,

$\displaystyle[D,H]\,=\,2iH$ $\displaystyle\qquad{}[D,C]\,=\,-2iC\qquad{}$
$\displaystyle[C,H]\,=\,-iD$ (38)

The remaining factors in the bosonic subalgebra $\mathfrak{g}_{B}$, correspond
to R-symmetries which act on the bosonic and fermionic factors respectively.
The generators of the Lie superalgebra are completed by eight ”Poincare”
supercharges $\mathbb{Q}_{A}$ of positive dimension and eight superconformal
charges $\mathbb{S}_{A}$ of negative dimension.

####

As usual for supersymmetric quantum mechanics, the wavefunctions correspond to
differential forms on the target manifold and the Hamiltonian $H$ is naturally
identified with the corresponding Laplacian operator. The action of the full
superconformal algebra on differential forms is constructed in [11]. In this
geometrical picture, the dilatation operator $D$ coincides with the Lie
derivative corresponding to the homothetic Killing vector [59] while the
special conformal generator $C$ corresponds to a harmonic potential on the
moduli space. The ADHM construction corresponds to a hyper-Kähler quotient
construction of the moduli-space and one can define an action of
$\mathfrak{so}(2,1)$ on the pre-quotient space spanned by the ADHM matrices
which descends through the quotient to $\mathcal{M}_{K,N}$. In this context,
the special conformal generator takes the form,

$\displaystyle C$ $\displaystyle=$
$\displaystyle|X|^{2}\,+\,|\tilde{X}|^{2}\,+\,\sum_{i=1}^{N}\,\left(|Q_{i}|^{2}\,+\,|\tilde{Q}_{i}|^{2}\right)$

As the instanton moduli space is non-compact, the spectrum of $H$ is
continuous when evaluated on plane wave normalisable states. However, by a
standard argument, the operator $-iD$ has a discrete spectrum which coincides
with that of the oscillator Hamiltonian $\Delta:=H+C$.

####

Any non-flat hyper-Kähler cone is necessarily singular and hence the
discussion given above is not precise in the general case. As mentioned above,
the instanton moduli space $\mathcal{M}_{K,N}$ has a smooth resolution
$\tilde{\mathcal{M}}_{K,N}$ induced by turning on FI parameters in the
underlying $U(K)$ gauge theory. Here, in our chosen complex structure, we will
only consider the real FI parameter $\xi_{\mathbb{R}}$ which deforms the
D-term equation in (36)171717In this complex structure, the resolved moduli-
space $\tilde{\mathcal{M}}_{K,N}$ is a holomorphic symplectic manifold known
as the symplectic resolution of the singular space $\mathcal{M}_{K,N}$.. This
deformation breaks conformal symmetry while preserving the usual Poincare
supersymmetry of the $\sigma$-model. In particular, as the resolved space
admits a smooth hyper-Kähler metric, $\mathcal{N}=(4,4)$ supersymmetry remains
unbroken. The $\mathfrak{su}(2)$ factor in the bosonic subalgebra
$\mathfrak{g}_{B}$ is broken to a $\mathfrak{u}(1)$ which corresponds to a
holomorphic isometry of the resolved space. The harmonic potential $C$,
corresponds to the norm of the corresponding Killing vector and can therefore
also be introduced in a way which preserves supersymmetry. With the FI
parameter, $\xi_{\mathbb{R}}$ turned on, the model is well defined and the
Hamiltonian as $\Delta=H+C$ has a discrete spectrum which is bounded below. In
the limit that the the resolution parameters go to zero, it is natural to
expect that the corresponding states will lie in multiplets of the full
superconformal algebra. Following [26], we will consider the resolution as UV
regulator for the theory analogous to the usual situation in higher
dimensional field theory.

####

In our discussion, the algebra $\mathfrak{osp}(4^{*}|4)$ arises in three
different ways which are related by dualities I and II described in the
introduction. The first is as the superconformal algebra of quantum mechanics
as described above. The second is as the light-cone reduction of the
superconformal algebra $\mathfrak{osp}(8^{*}|4)$ of the $(2,0)$ theory in six
dimensions. Finally, the same algebra characterises the (super-)isometries of
the dual gravitational background which corresponds to a certain null
compactification of $AdS_{7}\times S^{4}$. We will now introduce some global
conventions for the generators of the algebra and the corresponding chemical
potentials/fugacities.

We define Cartan subgroup generators $\Delta$, $J_{R}$ and $Q_{1}$, $Q_{2}$
for each factor in the maximal bosonic subgroup,

$SO(2,1)\times SU(2)_{R}\times SO(5)\subset OSp(4^{*}|4)$ (39)

of the superconformal group. Here the label $R$ distinguishes the $SU(2)$
factor in the superconformal group from the global symmetry $SU(2)_{L}$ of the
instanton moduli space which commutes with all superconformal generators. The
Cartan generator for the latter symmetry is denoted $J_{L}$. The
$SU(2)_{L}\times SU(2)_{R}\simeq SO(4)$ symmetry originates in the rotational
symmetry of the $\mathbb{R}^{4}$ on which the instantons are defined.
Sometimes it will be convenient to work instead with the generators $J_{1}$
and $J_{2}$ of rotations in two orthogonal places of $\mathbb{R}^{4}$ in terms
of which $J_{R}=J_{1}+J_{2}$, $J_{L}=J_{1}-J_{2}$. We also define similar
combinations for the Cartan generators of $SO(5)$: $Q_{R}:=Q_{1}+Q_{2}$,
$Q_{L}:=Q_{1}-Q_{2}$. For convenience, we will use the same notation for the
generators and for the corresponding eigenvalues. We work in a normalisation
such that $J_{1}$, $J_{2}$, $Q_{1}$ and $Q_{2}$ are, in general, half-integer
valued. However, for all states which contribute to the index it turns out
that the combinations $J_{L}$, $J_{R}$, $Q_{L}$, $Q_{R}$ defined above take
integer values.

####

In addition to the superconformal invariance described above the quantum
mechanical model also has global symmetries corresponding to the isometries of
the instanton moduli space. The resulting global symmetry group is,

$SU(2)_{L}\times SU(N)$ (40)

where $SU(2)_{L}$ corresponding to remaining factor in the rotation group on
$\mathbb{R}^{4}$ and the $SU(N)$ factor originates from the Yang-Mills gauge
symmetry. We introduce Cartan generators/eigenvalues $n_{a}$, for
$a=1,2,\ldots N-1$. In the following we will also introduce chemical
potentials $\epsilon_{-}=(\epsilon_{1}-\epsilon_{2})/2$ for the Cartan
generator of $SU(2)_{L}$ and chemical potentials $v_{a}$ for generators
$n_{a}$ of SU(N), with $a=1,2,\ldots,n$. We will typically use vector notation
for the $SU(N)$ charges and chemical potentials., with
$\mathbf{n}:=(n_{1},n_{2},\ldots,n_{N-1})\in\mathbb{Z}^{N-1}$ and
$\mathbf{v}:=(v_{1},v_{2},\ldots,v_{N-1})\in\mathbb{C}^{N-1}$.

###  A.2 Localisation Formulae

We establish the following conventions in relation to the localisation
formulae for the superconformal index, the M-string elliptic genus and the
Nekrasov partition function.

####

A partition $\lambda$ is a finite sequence of nonincreasing positive integers
$\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}...\geq\lambda_{\ell(\lambda)}>\lambda_{\ell(\lambda)+1}=0$.
Here $\ell(\lambda)$ is called the length of the partition and
$|\lambda|=\sum_{p}\lambda_{p}$ is its weight. The set of partitions is
denoted $\mathcal{P}$.

####

One usually visualizes a partition $\lambda$ by drawing the associated Young
diagram $Y(\lambda)$. A box $s$ in a young diagram $Y(\lambda)$ is labelled by
its cooordinates $(p,q)$ where $p=1,...,l(\lambda)$. Arm and leg lengths of a
box $s=(p,q)\in Y(\lambda)$ are defined as,

$\displaystyle a(s)=\lambda_{p}-q,$ $\displaystyle l(s)=\lambda^{\vee}_{q}-p$
(41)

where the dual $\lambda^{\vee}$ of a partition $\lambda$ is obtained by
interchanging rows and columns in the Young diagram. This definition can
extended to boxes $s=(p,q)$ which lie outside the Young diagram.

####

We also consider $N$-component vectors of partitions denoted
$\vec{\lambda}=(\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(N)})$ with total
weight,

$\displaystyle||\vec{\lambda}||$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\,|\lambda^{(i)}|$ (42)

For each box $s\in Y(\lambda^{(i)})$ we define,

$\displaystyle g_{ij}(s)$ $\displaystyle=$ $\displaystyle-a_{i}(s)+l_{j}(s)$
$\displaystyle f_{ij}(s)$ $\displaystyle=$ $\displaystyle-a_{i}(s)-l_{j}(s)-1$
(43)

Here $a_{i}(s)$ and $l_{j}(s)$ are the arm and leg lengths of box $s$,
relative to the Young Tableaux $Y(\lambda^{(i)})$ and $Y(\lambda^{(j)})$
respectively.

####

We define a Plethystic exponental for any function $f$ of $r$ formal variables
$\\{x_{1},x_{2},\ldots,x_{r}\\}$ as,

$\displaystyle{\rm Pexp}\left[f(x_{1},x_{2},\ldots,x_{r})\right)$
$\displaystyle=$ $\displaystyle\exp\left[\sum_{n\geq
1}\frac{1}{n}f\left(x^{n}_{1},x^{n}_{2},\ldots,x^{n}_{r}\right)\right]{}$

For any $X$ we also define,

$\displaystyle\left[X\right]$ $\displaystyle:=$
$\displaystyle\sqrt{X}-\frac{1}{\sqrt{X}}$

With these definitions the explicit formula for the superconformal index can
be written in the compact form (4).

##  Appendix B Computational Results

Evaluation of the index coefficients $\mathcal{C}[L,0,0,K]$ for various values
of $K$ and $L$ in the abelian case $N=1$. The second table focusses
specifically on the case $K=L$.

$(L,K)$ | $\mathcal{C}(L,0,0,K)$
---|---
(24, 25) | -4548136426
(26, 26) | 7935209206
(28, 27) | 88800402896
(30, 28) | -4887654890
(32, 29) | -1425581403152
(40, 45) | 7337290205677620
(42, 46) | 23800263998384620
(44, 47) | -51625798313702826
(46, 48) | -429211479407800616
(48, 49) | -288354194415296772
(50, 50) | 4773158006473089778
(52, 51) | 14870285533157146362
(54, 52) | -21630735101481854366

| $L$
---|---
|  | 40 | 42 | 44 | 46
| 45 | 7337290205677620 | 22476503716465140 | 9610110542310490 | -143942357454816860
| 46 | 10640057177590492 | 23800263998384620 | -14596846924592666 | -231844859695908664
K | 47 | 14121688544604138 | 21607000602591350 | -51625798313702826 | -330802093776910478
| 48 | 17342715906788200 | 14262589686082038 | -102521859977682242 | -429211479407800616
| 49 | 19682978250039222 | 60266617136788 | -166700677839463202 | -509237288538994006

| $L$
---|---
|  | 50 | 52 | 54
| 50 | 4773158006473089778 | 13722136087430823474 | 10728665632616173124
| 51 | 6943033937905529622 | 14870285533157146362 | -1493086283031736666
| 52 | 9190525712121239144 | 13822179413239343452 | -21630735101481854366
| 53 | 11174425419671147488 | 9447220680249748082 | -50744695186842694114
K | 54 | 12412734295210394496 | 543115812956557290 | -88938229446341455870
| 57 | 4868162317987965318 | -63009089217696929546 | -233897399529627516932
| 58 | -4069732338637176522 | -97265673400284395108 | -272448778843705818614
| 59 | -17508654288324952938 | -136216858133084019088 | -288990007308923696954
| 60 | -35960421557696977470 | -176744583922449466508 | -268821840220720476958

##  Appendix C Details of black hole solution

In this Appendix, we present an explicit supergravity black hole solution
realising the desired $X\times S^{4}$ asymptotics, where recall, $X$ is a
particular null compactification of AdS7. Following [33, 34], we will proceed
by seeking a solution to $\mathcal{N}=4$ $SO(5)$ gauged supergravity in seven
dimensions, obtained by reduction of eleven-dimensional supergravity on
$S^{4}$ [60, 61], which is further truncated to include only a pair of Abelian
gauge fields in the $U(1)^{2}\subset SO(5)$ Cartan subgroup. In addition to
the metric and this pair $A_{(1)}^{1,2}$ of 1-form gauge fields, the bosonic
field content also contains a pair of scalars $X_{1,2}$ and a 3-form
$A_{(3)}$. There is also a gauge coupling $g$. Finally, there is additionally
a self-duality condition imposed upon the fields, written in terms of an
auxiliary 2-form $A_{(2)}$. We use a convenient formulation of the Lagrangian
as first set out in [62].

####

Our solution has equal scalars $X=X_{1}=X_{2}$ and equal 1-forms
$A_{(1)}=A_{(1)}^{1}=A_{(1)}^{2}$. Writing then $F_{(2)}=dA_{(1)}$ and
$F_{(4)}=dA_{(3)}$, the bosonic field equations are found by variation of the
Lagrangian

$\displaystyle\mathcal{L}$ $\displaystyle=R\star 1-5X^{-2}dX\wedge\star
dX-X^{-2}F_{(2)}\wedge\star F_{(2)}-\frac{1}{2}X^{4}F_{(4)}\wedge\star
F_{(4)}$ $\displaystyle\qquad+2g^{2}(8X^{2}+8X^{-3}-X^{-8})\star
1+F_{(2)}\wedge F_{(2)}\wedge A_{(3)}+gF_{(4)}\wedge A_{(3)}$ (44)

while we must also satisfy the self-duality equation

$\displaystyle X^{4}\star F_{(4)}=2gA_{(3)}-F_{(3)}+F_{(2)}\wedge A_{(1)}$
(45)

where $F_{(3)}=dA_{(2)}$ is the field strength of some 2-form $A_{(2)}$.

###  C.1 Statement of the solution

Our solution is in terms of coordinates $(u,x^{+},x^{-},y,z,\phi_{\alpha})$,
$\alpha=1,2$, with $\phi_{\alpha}\sim\phi_{\alpha}+2\pi$. In addition to the
gauge coupling $g$, it depends on five parameters
$(m,\delta,a_{1},a_{2},\lambda)$, where the correct signature requires
$a_{1}g,a_{2}g\in(-1,1)$. Crucially, our solution realises $X$ asymptotics,
given by the metric (9), where we see that the gauge coupling $g$ is
identified as the inverse AdS radius $g=R_{\text{AdS}}^{-1}$ of the asymptotic
geometry, and thus is further determined in terms of the rank $N$ as in (10).

####

The metric is

$\displaystyle ds^{2}=H^{2/5}\Bigg{[}$
$\displaystyle\frac{(u^{2}+y^{2})(u^{2}+z^{2})}{U}du^{2}+\frac{(u^{2}+y^{2})(y^{2}-z^{2})}{Y}dy^{2}+\frac{(u^{2}+z^{2})(z^{2}-y^{2})}{Z}dz^{2}$
$\displaystyle\quad-\frac{U}{H^{2}(u^{2}+y^{2})(u^{2}+z^{2})}\mathcal{B}[y^{2},z^{2},0]^{2}$
$\displaystyle\quad+\frac{Y}{(u^{2}+y^{2})(y^{2}-z^{2})}\left(\mathcal{B}[-u^{2},z^{2},0]-\frac{q}{H(u^{2}+y^{2})(u^{2}+z^{2})}\mathcal{B}[y^{2},z^{2},0]\right)^{2}$
$\displaystyle\quad+\frac{Z}{(u^{2}+z^{2})(z^{2}-y^{2})}\left(\mathcal{B}[-u^{2},y^{2},0]-\frac{q}{H(u^{2}+y^{2})(u^{2}+z^{2})}\mathcal{B}[y^{2},z^{2},0]\right)^{2}$
$\displaystyle\quad+\frac{a_{1}^{2}a_{2}^{2}}{g^{2}u^{2}y^{2}z^{2}}\left(\mathcal{B}[-u^{2},y^{2},z^{2}]-\frac{q}{H(u^{2}+y^{2})(u^{2}+z^{2})}\left(1-\frac{g^{2}y^{2}z^{2}}{a_{1}a_{2}}\right)\mathcal{B}[y^{2},z^{2},0]\right)^{2}\,\,\Bigg{]}$
(46)

where

$\displaystyle U(u)$
$\displaystyle=\frac{(1+g^{2}u^{2})^{2}}{g^{2}u^{2}}(u^{2}+a_{1}^{2})(u^{2}+a_{2}^{2})+qg^{2}\left(2u^{2}+a_{1}^{2}+a_{2}^{2}+\frac{1}{g^{2}}\right)+\frac{2qa_{1}a_{2}}{u^{2}}+\frac{q^{2}g^{2}}{u^{2}}-2m$
$\displaystyle Y(y)$
$\displaystyle=\frac{\left(1-g^{2}y^{2}\right)^{2}}{g^{2}y^{2}}(a_{1}^{2}-y^{2})(a_{2}^{2}-y^{2}),\qquad
Z(z)=\frac{\left(1-g^{2}z^{2}\right)^{2}}{g^{2}z^{2}}(a_{1}^{2}-z^{2})(a_{2}^{2}-z^{2})$
$\displaystyle H(u,y,z)$
$\displaystyle=1+\frac{q}{(u^{2}+y^{2})(u^{2}+z^{2})},\quad
q=2m\sinh^{2}\delta$ (47)

and, given any three arguments $v_{1},v_{2},v_{3}$, we define the one-form

$\displaystyle\mathcal{B}[v_{1},v_{2},v_{3}]$
$\displaystyle=\frac{\lambda}{g}\frac{(1-g^{2}v_{1})(1-g^{2}v_{2})(1-g^{2}v_{3})}{(1-a_{1}^{2}g^{2})(1-a_{2}^{2}g^{2})}\,dx^{+}$
$\displaystyle\quad+\frac{1}{2g}\frac{1}{(1-a_{1}^{2}g^{2})^{2}(1-a_{2}^{2}g^{2})^{2}}\bigg{(}(1-3a_{1}^{2}g^{2}-3a_{2}^{2}g^{2}+5a_{1}^{2}a_{2}^{2}g^{4})$
$\displaystyle\hskip
170.71652pt+g^{2}(1+a_{1}^{2}g^{2}+a_{2}^{2}g^{2}-3a_{1}^{2}a_{2}^{2}g^{4})(v_{1}+v_{2}+v_{3})$
$\displaystyle\hskip
170.71652pt+g^{4}(-3+a_{1}^{2}g^{2}+a_{2}^{2}g^{2}+a_{1}^{2}a_{2}^{2}g^{4})(v_{1}v_{2}+v_{2}v_{3}+v_{3}v_{1})$
$\displaystyle\hskip
170.71652pt+g^{6}(5-3a_{1}^{2}g^{2}-3a_{2}^{2}g^{2}+a_{1}^{2}a_{2}^{2}g^{4})v_{1}v_{2}v_{3}\bigg{)}dx^{-}$
$\displaystyle\quad+\frac{g^{2}(a_{1}^{2}-v_{1})(a_{1}^{2}-v_{2})(a_{1}^{2}-v_{3})}{a_{1}(1-a_{1}^{2}g^{2})^{2}(a_{1}^{2}-a_{2}^{2})}d\phi_{1}+\frac{g^{2}(a_{2}^{2}-v_{1})(a_{2}^{2}-v_{2})(a_{2}^{2}-v_{3})}{a_{2}(1-a_{2}^{2}g^{2})^{2}(a_{2}^{2}-a_{1}^{2})}d\phi_{2}$
(48)

The solutions for the scalars and 1-forms are

$\displaystyle X_{1}=X_{2}=H^{-1/5},\quad
A_{(1)}^{1}=A_{(1)}^{2}=\frac{2m\sinh\delta\cosh\delta}{H(u^{2}+y^{2})(u^{2}+z^{2})}\mathcal{B}[y^{2},z^{2},0]$
(49)

while for the 3-form we have

$\displaystyle A_{(3)}$
$\displaystyle=-qg^{3}a_{1}a_{2}\Big{(}\mathcal{B}[y^{2},z^{2},0]-\mathcal{B}[y^{2},z^{2},g^{-2}]\Big{)}\wedge\left(\frac{1}{(u^{2}+y^{2})z}dz\wedge\Big{(}\mathcal{B}[y^{2},0,0]-\mathcal{B}[y^{2},0,g^{-2}]\Big{)}\right.$
$\displaystyle\hskip
227.62204pt+\left.\frac{1}{(u^{2}+z^{2})y}dy\wedge\Big{(}\mathcal{B}[z^{2},0,0]-\mathcal{B}[z^{2},0,g^{-2}]\Big{)}\right)$
$\displaystyle\quad+qg^{3}\mathcal{B}[y^{2},z^{2},0]\wedge\left(\frac{z}{(u^{2}+y^{2})}dz\wedge\Big{(}\mathcal{B}[y^{2},0,0]-\mathcal{B}[y^{2},0,g^{-2}]\Big{)}\right.$
$\displaystyle\hskip
128.0374pt+\left.\frac{y}{(u^{2}+z^{2})}dy\wedge\Big{(}\mathcal{B}[z^{2},0,0]-\mathcal{B}[z^{2},0,g^{-2}]\Big{)}\right)$
(50)

The 2-form $A_{(2)}$ appearing in the self-duality equation (45) can be taken
as

$\displaystyle A_{(2)}$
$\displaystyle=\frac{q}{H(u^{2}+y^{2})(u^{2}+z^{2})}\mathcal{B}[y^{2},z^{2},0]\wedge\Bigg{(}(1-a_{1}a_{2}g^{2})\mathcal{B}[y^{2},z^{2},g^{-2}]$
$\displaystyle\hskip
199.16928pt+\frac{(a_{1}^{2}-y^{2})(a_{1}^{2}-z^{2})}{a_{1}(a_{1}^{2}-a_{2}^{2})}d\phi_{1}+\frac{(a_{2}^{2}-y^{2})(a_{2}^{2}-z^{2})}{a_{2}(a_{2}^{2}-a_{1}^{2})}d\phi_{2}\Bigg{)}$
(51)

Our solution was derived in a particular parameter coordinate limit of the
solution of Chow [34], following the procedure put forward in [6]. The fact
that the Chow configuration is indeed a supergravity solution implies that our
solution is too; nonetheless, we have performed an independent check in
Mathematica that confirms that this is indeed the case.

####

For generic values of the parameters $(m,\delta,a_{1},a_{2},\lambda)$, the
black hole (46) is non-extremal and breaks all supersymmetry. However, we can
preserve $\frac{1}{8}$ supersymmetry, if the parameters are constrained to
satisfy

$\displaystyle e^{2\delta}=1+\frac{2}{(1-a_{1}g-a_{2}g)}$ (52)

The black holes of [34]—of which our solutions are a Penrose limit—have an
additional constraint amongst parameters, required in order to avoid the
presence of naked closed timelike curves. In particular, this constraint
admits a smooth Penrose limit, and thus we assume it must also be imposed upon
our solutions. It has the effect of fixing the parameter $m$, as

$\displaystyle
q=2m\sinh^{2}\delta=-\frac{2}{g^{3}}\frac{(a_{1}+a_{2})(1-a_{1}g)^{2}(1-a_{2}g)^{2}}{(2-a_{1}g-a_{2}g)^{2}}$
(53)

In line with much of the literature, we distinguish a black hole satisfying
both of these conditions as a “BPS” black hole. In particular, these two
conditions are sufficient to ensure the vanishing of the temperature of the
black hole; BPS black holes are extremal.

The event horizon of such a BPS black hole is a surface of constant $u=u_{0}$,
where $u_{0}$ is the largest root of $U(u)$. Indeed, the extremality of the
BPS solution corresponds to this root being a double root. We have explicitly,

$\displaystyle u_{0}^{2}=\frac{2ga_{1}a_{2}-a_{1}-a_{2}}{(2-a_{1}g-a_{2}g)g}$
(54)

A real solution of the supersymmetry condition (52) for the parameters
$(m,\delta,a_{1},a_{2},\lambda)$ requires $e^{2\delta}>0$, while if we want a
horizon we need $u_{0}^{2}>0$. These two constraints are satisfied precisely
if, in addition to the existing $a_{1}g,a_{2}g\in(-1,1)$, we have

$\displaystyle 2(a_{1}g)(a_{2}g)-a_{1}g-a_{2}g>0$ (55)

Note, this condition then implies $g(a_{1}+a_{2})<0$, which by (53) ensures
that $m>0$.

###  C.2 Asymptotic geometry

The metric (46) realises the AdS7 pp-wave slicing (9) asymptotically. More
precisely, there exists coordinates $(r,x^{+},x^{-},n_{\alpha},\phi_{\alpha})$
such that as we approach the conformal boundary $r\to\infty$ (corresponding to
$u\to\infty$), we have

$\displaystyle
ds^{2}\longrightarrow\frac{dr^{2}}{g^{2}r^{2}}+r^{2}\left(-2dx^{+}dx^{-}-\rho^{2}(dx^{-})^{2}+\sum_{\alpha}\left(dn_{\alpha}^{2}+n_{\alpha}^{2}d\phi_{\alpha}^{2}\right)\right)$
(56)

To get to this coordinate system, we exchange the coordinates $(u,y,z)$ with
$(r,n_{1},n_{2})$ using the relations

$\displaystyle(1-a_{1}^{2}g^{2})^{2}r^{2}n_{1}^{2}$
$\displaystyle=-\frac{(a_{1}^{2}-y^{2})(a_{1}^{2}-z^{2})}{(a_{1}^{2}-a_{2}^{2})}g^{2}(u^{2}+a_{1}^{2})$
$\displaystyle(1-a_{2}^{2}g^{2})^{2}r^{2}n_{2}^{2}$
$\displaystyle=-\frac{(a_{2}^{2}-y^{2})(a_{2}^{2}-z^{2})}{(a_{2}^{2}-a_{1}^{2})}g^{2}(u^{2}+a_{2}^{2})$
$\displaystyle r^{2}$
$\displaystyle=\frac{(1-g^{2}y^{2})(1-g^{2}z^{2})}{(1-a_{1}^{2}g^{2})(1-a_{2}^{2}g^{2})}\frac{\lambda}{g^{2}}(1+g^{2}u^{2})$
(57)

Indeed, it is this implicit transformation that fixes the range of $(y,z)$.
Assuming $a_{1}^{2}<a_{2}^{2}<g^{-2}$, we can take $y,z\geq 0$ with

$\displaystyle a_{1}^{2}\leq y^{2}\leq a_{2}^{2}\leq z^{2}\leq g^{-2}$ (58)

Clearly, the transformation (57) breaks down if $a_{1}\to a_{2}$ and/or
$a_{2}\to g^{-1}$; indeed, the entire black hole solution (46) becomes
singular in either limit. However, one can remedy this by first making a
further coordinate transformation, replacing $(y,z)$ with a pair of angles
$(\theta_{1},\theta_{2})$ with $0\leq\theta\leq\pi/2$ and

$\displaystyle y^{2}=a_{1}^{2}+(a_{2}^{2}-a_{1}^{2})\sin^{2}\theta_{1},\qquad
z^{2}=a_{2}^{2}+(g^{-2}-a_{2}^{2})\sin^{2}\theta_{2}$ (59)

Then the transformation (57) is perfectly regular when either (or both)
problematic limits are taken, and further the solution (46) is regular in
either or both limits when written in terms of
$(u,x^{+},x^{-},\theta_{1},\theta_{2},\phi_{1},\phi_{2})$.

####

Thus, in order to define a black hole solution with $X$ asymptotics, we simply
identify the coordinate $x^{+}\sim x^{+}+2\pi$ throughout the spacetime.

###  C.3 Length of the compact circle

The norm of the vector field $\partial_{+}$ approaches zero as we approach the
conformal boundary; indeed, the identification $x^{+}\sim x^{+}+2\pi$ is a
null compactification throughout all of pure AdS7, as is evident from the
slicing (9).

In contrast, for the black hole solution (46), $\partial_{+}$ is spacelike
everywhere, with a norm-squared going like $\mathcal{O}(u^{-4})$ as we
approach $u\to\infty$. Crucially, in the vicinity of the black hole horizon,
the norm-squared can be made large in units of the Planck length as first
noted in [6], and thus we are able to trust semiclassical supergravity at
large $N$.

Let us see in more detail why this is true. We compute everywhere in the black
hole solution,

$\displaystyle||V||^{2}=g_{++}$
$\displaystyle=\frac{\lambda^{2}}{g^{2}}\frac{(1-g^{2}y^{2})^{2}(1-g^{2}z^{2})^{2}}{(1-a_{1}^{2}g^{2})^{2}(1-a_{2}^{2}g^{2})^{2}}\frac{H^{-8/5}}{(u^{2}+y^{2})^{2}(u^{2}+z^{2})^{2}}$
$\displaystyle\qquad\times\Big{(}\left(2m+q-(a_{1}-a_{2})^{2}g^{2}q\right)(u^{2}+y^{2})(u^{2}+z^{2})-(a_{1}-a_{2})^{2}g^{2}q^{2}\Big{)}$
(60)

We do indeed see that $||V||^{2}$ vanishes like $u^{-4}$ as we approach the
conformal boundary. Let us further focus on BPS black holes, for which $m,q$
can be replaced by functions of $a_{1},a_{2}$, and let us consider the value
of $||V||^{2}$ at the black hole horizon $u=u_{0}$. For brevity, let us
further specialise to the case of equal angular momenta. Then, replacing
$(y,z)$ with $(\theta_{1},\theta_{2})$ as in (59) and setting

$\displaystyle
a_{1}=a_{2}=-\frac{1}{g}\left(\frac{1-\alpha^{2}}{1+\alpha^{2}}\right),\quad\alpha\in(0,1)$
(61)

we find simply

$\displaystyle\left.\frac{||V||^{2}}{l_{p}^{2}}\right|_{u=u_{0}}=16\lambda^{2}(\pi
N)^{2/3}\frac{\cos^{2}(\theta_{2})(1-\alpha^{2}\cos(2\theta_{2}))}{(1+\alpha^{2})(1+\alpha^{2}\sin(\theta_{2})^{2})^{2}}$
(62)

where we have utilised (10). Thus, for suitably large $N$ and $\lambda$ and
away from181818Note, there is a co-dimension $1$ hypersurface in the horizon,
and indeed throughout the spacetime, at $\theta_{2}=\pi/2$ along which $||V||$
vanishes and thus the circle is null. This was also noted in [6]. Like those
authors, we argue that this region of spacetime does not contribute
appreciably to the thermodynamics, and thus can be neglected.
$\theta_{2}=\pi/2$, the circle is large in Planck units near the horizon.
Below, we will use this to derive a classical gravity regime for $N$ and $K$.

###  C.4 Thermodynamics

Let us consider the black hole solution (46) in the BPS limit, in which
$\delta$ and and $m$ are fixed as functions of the remaining parameters
$(\lambda,a_{1},a_{2})$. In additional to its Bekenstein-Hawking entropy
$\mathcal{S}_{\text{BH}}=A/4$ for horizon area $A$, we have thermodynamics
quantities $\\{K,\Delta,J_{1},J_{2}\\}$ corresponding to a choice of commuting
conformal Killing vectors of the boundary pp-wave metric, as well as a pair of
equal R-charges $Q=Q_{1}=Q_{2}$.

The values of these quantities are

$\displaystyle\mathcal{S}_{\text{BH}}$
$\displaystyle=\frac{1}{G_{N}^{(7)}}\frac{\pi^{3}\lambda(a_{1}+a_{2})(1-a_{1}g)(1-a_{2}g)(a_{1}+a_{2}-2a_{1}a_{2}g)}{2g^{4}(1+a_{1}g)(1+a_{2}g)(2-a_{1}g-a_{2}g)^{2}u_{0}}$
$\displaystyle>0$ $\displaystyle K$
$\displaystyle=-\frac{1}{G_{N}^{(7)}}\frac{\pi^{2}\lambda^{2}(a_{1}+a_{2})(1-a_{1}g)^{2}(1-a_{2}g)^{2}}{g^{4}(1+a_{1}g)(1+a_{2}g)(2-a_{1}g-a_{2}g)^{2}}$
$\displaystyle>0$ $\displaystyle\Delta$
$\displaystyle=-\frac{1}{G_{N}^{(7)}}\frac{\pi^{2}\lambda(a_{1}+a_{2})(1-a_{1}g)(1-a_{2}g)(2-a_{1}^{2}g^{2}-a_{2}^{2}g^{2})}{g^{4}(1+a_{1}g)^{2}(1+a_{2}g)^{2}(2-a_{1}g-a_{2}g)^{2}}$
$\displaystyle>0$ $\displaystyle J_{1}$
$\displaystyle=\frac{1}{G_{N}^{(7)}}\frac{\pi^{2}\lambda(a_{1}+a_{2})(1-a_{1}g)(1-a_{2}g)(3a_{1}+a_{2}-a_{1}^{2}g-3a_{1}a_{2}g)}{4g^{3}(1+a_{1}g)^{2}(1+a_{2}g)(2-a_{1}g-a_{2}g)^{2}}$
$\displaystyle J_{2}$
$\displaystyle=\frac{1}{G_{N}^{(7)}}\frac{\pi^{2}\lambda(a_{1}+a_{2})(1-a_{1}g)(1-a_{2}g)(3a_{2}+a_{1}-a_{2}^{2}g-3a_{1}a_{2}g)}{4g^{3}(1+a_{2}g)^{2}(1+a_{1}g)(2-a_{1}g-a_{2}g)^{2}}$
$\displaystyle Q$
$\displaystyle=-\frac{1}{G_{N}^{(7)}}\frac{\pi^{2}\lambda(a_{1}+a_{2})(1-a_{1}g)(1-a_{2}g)}{4g^{4}(1+a_{1}g)(1+a_{2}g)(2-a_{1}g-a_{2}g)}$
$\displaystyle>0$ (63)

where the definite signs follow from (55). Additionally, while $J_{1},J_{2}$
do not individually have definite sign over the physical parameter space, we
do have $J_{1}+J_{2}>0$ for all physical solutions. $G_{N}^{(7)}$ denotes the
seven-dimensional Newton’s constant, related to the eleven-dimensional
Newton’s constant by
$G_{N}^{(7)}=\text{Vol}(S^{4})^{-1}G_{N}^{(11)}=6\pi^{5}l_{p}^{5}(\pi
N)^{-4/3}$, where $G_{N}^{(11)}=16\pi^{7}l_{p}^{9}$. The holographic relation
(10) then tells us that

$\displaystyle\frac{R_{\text{AdS}}^{5}}{G_{N}^{(7)}}=\frac{1}{g^{5}G_{N}^{(7)}}=\frac{16N^{3}}{3\pi^{2}}$
(64)

These thermodynamics can be deduced [7] from a Penrose limit of the known
thermodynamics of the AdS7 black hole solution of [34]. They satisfy the BPS
condition

$\displaystyle\Delta-J_{1}-J_{2}-4Q=0$ (65)

precisely matching (1) as expected. They additionally satisfy a non-linear
constraint, whose origin can be traced back to the condition (53) on
parameters; this constraint is compactly expressed by the determination of $Q$
in terms of $J_{1},J_{2}$ and $K$ as explained in the main text. Then,
$\mathcal{S}_{\text{BH}}$ can be expressed as a function of only $J_{1},J_{2}$
and $K$, as appears in the main text.

###  C.5 Range of classical validity

As discussed in the main text, there are a pair of conditions under which
M-theory is well described by supergravity. Let us check these now.

####

Firstly, we should have that the AdS radius and the radius of the internal
$S^{4}$ are both large in Planck units. We have

$\displaystyle\frac{R_{\text{AdS}}}{l_{p}}=2\frac{R_{S^{4}}}{l_{p}}=2(\pi
N)^{1/3}$ (66)

and thus we require $N>>1$.

####

Secondly, we must ensure that the compact circle is large in Planck units, at
least at the horizon. From our expression (62), we find that it is sufficient
that the parameter $\lambda$ satisfies

$\displaystyle\lambda N^{1/3}>>1$ (67)

Let us then convert this to statement about the charges
$\\{K,J_{1},J_{2},Q\\}$. We can write

$\displaystyle K$ $\displaystyle=\lambda^{2}N^{3}F_{K}(a_{1}g,a_{2}g),$
$\displaystyle J_{1}$ $\displaystyle=\lambda N^{3}F_{J_{1}}(a_{1}g,a_{2}g),$
$\displaystyle J_{2}$ $\displaystyle=\lambda N^{3}F_{J_{2}}(a_{1}g,a_{2}g),$
$\displaystyle Q$ $\displaystyle=\lambda N^{3}F_{Q}(a_{1}g,a_{2}g),$ (68)

for some functions $F_{K},F_{J_{1}},F_{J_{2}},F_{Q}$ of the $\mathcal{O}(1)$
parameters $a_{1}g,a_{2}g$. By then considering $\lambda$ satisfying (67)
while holding $a_{1},a_{2}$ fixed191919More generally, one could consider
regions of charge space corresponding to also taking some limit of
$a_{1},a_{2}$, towards either $0$ or $-1/g$. However, one can show that any
such limit corresponds either to even faster growth for
$\\{K,J_{1},J_{2},Q\\}$, or else a horizon area that becomes small in Planck
units., we find that the full set of sufficient conditions for the classical
approximation to hold near the horizon is recast as

$\displaystyle K>>N^{7/3}>>1$ (69)

with $J_{1},J_{2},Q$ growing like

$\displaystyle J_{1}\sim\sqrt{\delta},\qquad J_{2}\sim\sqrt{\delta},\qquad
Q\sim\sqrt{\delta}$ (70)

where recall $\delta=KN^{3}/3$.

##  Appendix D Entropy from quantum mechanics

###  D.1 Nekrasov partition function

####

The Nekrasov partition function for five-dimensional gauge theory defined on
$\mathbb{R}^{4}\times S^{1}$ will play a central role in our analysis.
Specifically we consider the five dimensional lift of $\mathcal{N}=2$
supersymmetric Yang-Mills theory with gauge group $U(N)$ with a single
hypermultiplet of mass $m$ in the adjoint representation. The resulting four-
dimensional theory has an effective coupling constant $\tau$. The theory is
defined in a an $\Omega$ background on $\mathbb{R}^{4}$ with rotation
parameters $\epsilon_{1}$ and $\epsilon_{2}$. The Coulomb branch of the $5d$
theory is parameterized by complex coordinates $z_{i}$, for $i=1,,\ldots N$,
on the maximal torus of $U(N)$. We will denote the $\\{z_{i}\\}$ collectively
as $Z$. As before, we exponentiate these parameters to define,

$\displaystyle t:=\exp(-\epsilon_{+})\qquad{}$
$\displaystyle\qquad{}x:=\exp(-\epsilon_{-})\qquad{}$
$\displaystyle\qquad{}q_{\tau}:=\exp(2\pi i\tau)=\exp(-\beta)$ $\displaystyle
y:=\exp(-m)\qquad{}$
$\displaystyle\qquad{}z_{i}:=\exp(-\sum_{a=1}^{i-1}v_{a})\qquad$ (71)

where $\mathbf{v}=(v_{1},v_{2},\ldots,v_{N-1})$ is the $SU(N)$ chemical
potential introduced in Section 2. As above we have
$\epsilon_{\pm}:=(\epsilon_{1}\pm\epsilon_{2})/2$. The various partition
function we discuss below will each be defined either as functions of the
fugacities $\\{t,x,y,q_{\tau};Z\\}$ or as functions of the chemical potentials
$\\{\epsilon_{1},\epsilon_{2},m,\tau;\mathbf{v}\\}$. The relations (71)
between the two sets of variables is assumed throughout in the following.

####

We define the Nekrasov partition function for this 5d gauge theory,

$\displaystyle\mathcal{Z}^{U(N)}_{\rm Nek}$ $\displaystyle:=$
$\displaystyle\mathcal{Z}^{U(N)}_{\rm Nek}\left[t,x,y,q_{\tau};Z\right]$ (72)

as the product of perturbative and instanton contributions,

$\displaystyle\mathcal{Z}^{U(N)}_{\rm Nek}$ $\displaystyle=$
$\displaystyle\mathcal{Z}^{U(N)}_{\rm pert}\,\mathcal{Z}^{U(N)}_{\rm inst}$
(73)

As we review below, only the instanton piece is directly related to the
superconformal index we study. In fact the perturbative piece will also be
relevant to our discussion once modular transformations are taken into
account. Here we will use the slightly non-standard definition for the
perturbative part introduced in [45]. This is given as,

$\displaystyle\mathcal{Z}^{U(N)}_{\rm pert}$ $\displaystyle=$
$\displaystyle\hat{\mathcal{Z}}_{\rm pert}\,\left[\mathcal{Z}^{U(1)}_{\rm
pert}\right]^{N}$

with,

$\displaystyle\hat{\mathcal{Z}}_{\rm pert}$ $\displaystyle:=$
$\displaystyle{\rm
Pexp}\left[\left(\sum_{i>j}\,\frac{Z_{i}}{Z_{j}}\right)\,\frac{\left[yt\right]\,\left[y/t\right]}{\left[tx\right]\,\left[t/x\right]}\right]$
$\displaystyle\mathcal{Z}^{U(1)}_{\rm pert}$ $\displaystyle:=$
$\displaystyle{\rm
Pexp}\left[\frac{1}{2}\frac{\left[yx\right]\,\left[y/x\right]}{\left[tx\right]\,\left[t/x\right]}\right]$
(74)

where, for any $X$ we define,

$\displaystyle\left[X\right]$ $\displaystyle:=$
$\displaystyle\sqrt{X}-\frac{1}{\sqrt{X}}$

Here the rational functions appearing in the argument of each Plethystic
exponential are expanded as an infinite Taylor series in the variables $t$,
$x$, $y$ and $z_{i}$. Different expansions are convergent in different
chambers of the parameter space. Using the rules introduced above, we can then
rewrite the Plethystic exponentials as infinite products. For example, for
$t\leq 1$, we expand the argument in powers of $q_{1}=tx=\exp(-\epsilon_{1})$
and $q_{2}=t/x=\exp(-\epsilon_{2})$ to get,

$\displaystyle\hat{\mathcal{Z}}_{\rm
pert}=\Pi^{(+)}\left[\epsilon_{1},\epsilon_{2},m;\mathbf{v}\right]$
$\displaystyle:=$
$\displaystyle\prod_{i>j=1}^{N}\,\prod_{l_{1},l_{2}=0}^{\infty}\,\,\frac{\left(1-\frac{z_{i}}{z_{j}}q_{1}^{l_{1}}q_{2}^{l_{2}}\right)\left(1-\frac{z_{i}}{z_{j}}t^{2}q_{1}^{l_{1}}q_{2}^{l_{2}}\right)}{\left(1-\frac{z_{i}}{z_{j}}ty^{-1}q_{1}^{l_{1}}q_{2}^{l_{2}}\right)\left(1-\frac{z_{i}}{z_{j}}tyq_{1}^{l_{1}}q_{2}^{l_{2}}\right)}$
(75)

On the other hand, for $|t|>1$, we can expand the argument of the Plethystic
exponential in inverse powers of $q_{1}$ and $q_{t}$ to get,

$\displaystyle\hat{\mathcal{Z}}_{\rm
pert}=\Pi^{(-)}\left[\epsilon_{1},\epsilon_{2},m;\mathbf{v}\right]$
$\displaystyle:=$
$\displaystyle\prod_{i>j=1}^{N}\,\prod_{l_{1},l_{2}=0}^{\infty}\,\,\frac{\left(1-\frac{z_{i}}{z_{j}}q_{1}^{-l_{1}}q_{2}^{-l_{2}}\right)\left(1-\frac{z_{i}}{z_{j}}t^{-2}q_{1}^{-l_{1}}q_{2}^{-l_{2}}\right)}{\left(1-\frac{z_{i}}{z_{j}}t^{-1}yq_{1}^{-l_{1}}q_{2}^{-l_{2}}\right)\left(1-\frac{z_{i}}{z_{j}}t^{-1}y^{-1}q_{1}^{-l_{1}}q_{2}^{-l_{2}}\right)}$

The instanton part of the Nekrasov partition function is defined as an
infinite sum over instanton number $K$,

$\displaystyle\mathcal{Z}^{U(N)}_{\text{inst}}$ $\displaystyle=$
$\displaystyle\sum_{K=0}^{\infty}q_{\tau}^{K}\mathcal{Z}_{\rm inst}^{(K)}$
(77)

The rescaled coupling $\beta:=-2\pi i\tau$ serves as a chemical potential for
instanton number $K$ and $q_{\tau}=\exp(-\beta)$ is the corresponding
fugacity. For each $K$ the contribution $\mathcal{Z}_{\rm inst}^{(K)}$ is
expressed as a sum over $N$-coloured Young Tableaux,

$\displaystyle\mathcal{Z}^{(K)}_{\rm inst}\left[t,x,y,;{Z}\right]$
$\displaystyle=$
$\displaystyle\sum_{\vec{\lambda}\in\mathcal{P}^{N}:||\vec{\lambda}||=K}\,\,\prod_{i,j=1}^{N}\,\,\prod_{s\in
Y(\lambda^{(i)})}\,\,{\rm
Pexp}\left(\frac{z_{i}}{z_{j}}\,t^{g_{ij}(s)}\,x^{f_{ij}(s)}\left[ty\right]\left[t/y\right]\right)$

with notation described in Appendix A.2. This yields a rational function of
the fugacities which coincides precisely with the superconformal index of the
instanton moduli space introduced in Section 2,

$\displaystyle\mathcal{I}_{\rm SC}\left(\mathcal{M}_{K,N}\right)$
$\displaystyle=$ $\displaystyle\mathcal{Z}_{\rm inst}^{(K)}$ (79)

We project onto the $\mathfrak{su}(n)$ singlet sector of the model by
integrating against the Haar measure,

$\displaystyle\int\,d\mu_{\rm Haar}[{Z}]$ $\displaystyle=$
$\displaystyle\left(\prod_{i=1}^{N}\,\frac{1}{2\pi
i}\,\oint_{C}\,\frac{dz_{i}}{z_{i}}\right)\,\,\prod_{i\neq
j}\,\left(1-\frac{z_{i}}{z_{j}}\right)$ (80)

Combining the above relations, the quantum mechanical index defined in the
text can be expressed as,

$\displaystyle\mathcal{I}_{\rm QM}\left[t,x,y,q_{\tau}\right]:=\oint d\mu_{\rm
Haar}\left[{Z}\right]\,\mathcal{Z}^{U(N)}_{\rm
inst}\left[t,x,y,q_{\tau};{Z}\right]$ (81)

To ensure convergence of the $t$ expansion, at each instanton number we should
work with $|t|<1$, keeping the other fugacities $x$ and $y$ on or near the
unit circle. The Haar measure is also define above as an integration around
the unit circle for each $z_{i}$.

###  D.2 Elliptic resummation formula

####

The explicit expression for $Z_{S}^{(\mathbf{n})}$ involves a sum over the set
$S(\mathbf{n})$ of N-coloured partitions
$\vec{\lambda}=(\lambda^{(1)},\ldots,\lambda^{(N)})$ with
$|\lambda^{(i)}|=n_{a}$ for $i=a=1,2,\ldots,N-1$ and $|\lambda^{(N)}|=0$,

$\displaystyle Z_{S}^{(\mathbf{n})}$ $\displaystyle=$
$\displaystyle\sum_{\vec{\lambda}\in
S(\mathbf{n})}\,\prod_{i=1}^{N}\,\prod_{s\in
Y(\lambda^{(i)})}\,\,\frac{t}{y}\,\,\frac{\vartheta\left(\tau|U_{1}^{(i)}(s)\right)\vartheta\left(\tau|U_{2}^{(i)}(s)\right)}{\vartheta\left(\tau|V_{1}^{(i)}(s)\right)\vartheta\left(\tau|V_{2}^{(i)}(s)\right)}$
(82)

For each box $s\in Y(\lambda^{(i)})$, we define,

$\displaystyle
U_{1}^{(i)}(s):=\left(m-\epsilon_{+}+(l_{i+1}(s)+1)\epsilon_{2}-a_{i}(s)\epsilon_{1}\right)/2\pi
i$ $\displaystyle
V_{1}^{(i)}(s):=\left(-l_{i}(s)\epsilon_{2}+(a_{i}(s)+1)\epsilon_{1}\right)/2\pi
i$ $\displaystyle
U_{2}^{(i)}(s):=\left(m-\epsilon_{+}-l_{i-1}(s)\epsilon_{2}+(a_{i}(s)+1)\epsilon_{1}\right)/2\pi
i$ $\displaystyle
V_{2}^{(i)}(s):=\left((l_{i}(s)+1)\epsilon_{2}-a_{i}(s)\epsilon_{1}\right)/2\pi
i$

and we extend the range of the index $i$ by the identification $\lambda^{(i\pm
N)}:=\lambda^{(i)}$.

####

The theta function $\vartheta(\tau|z)$ appearing in (82) is defined for all
$\tau$, $z\in\mathbb{C}$ with ${\rm Im}\tau>0$ as,

$\vartheta(\tau|z)=\sum_{n\in\mathbf{Z}}\exp((n^{2}+n)i\pi\tau)(-1)^{n}\exp(2\pi
inz)$ (83)

It has a modularity property under $\tau\to-1/\tau,z\to z/\tau$:

$\vartheta(\tau|z)=\vartheta(-\frac{1}{\tau}|\frac{z}{\tau})\frac{1}{\sqrt{-i\tau}}\exp\left(\frac{(z+\tau/2-1/2)^{2}\pi}{i\tau}\right)$
(84)

The modular transformation property (29,30) quoted in the text can be derived
by applying this formula to each occurence of $\vartheta$ in (82).

###  D.3 Asymptotics of $Z_{S}$

Our next aim is to find the asymptotic behaviour of $Z_{S}[\mathbf{v}]$ in the
limit $\beta=-2\pi i\tau\rightarrow 0$. In particular the saddle point
analysis will require us to take this limit with fixed complex values of the
remaining chemical potentials. We will work indside the domain of analyticity
of the index with ${\rm Re}[\beta]$, ${\rm Re}[\epsilon_{1}]$, ${\rm
Re}[\epsilon_{2}]>0$. Note that the convegence of the sum (27) defining
$Z_{S}$ typically requires additional conditions. However as explained in the
text we assume that we can extend the definition of $Z_{S}$ to the whole
domain by analytic continuation in $\beta$ so that the relation,

$\displaystyle Z_{S}\left[\epsilon_{1},\epsilon_{2},m,\tau;\mathbf{v}\right]$
$\displaystyle:=$ $\displaystyle\frac{\mathcal{Z}^{U(N)}_{\rm
Nek}}{\left[\mathcal{Z}^{U(1)}_{\rm Nek}\right]^{N}}$ (85) $\displaystyle=$
$\displaystyle\hat{\mathcal{Z}}_{\rm pert}\,\frac{\mathcal{Z}^{U(N)}_{\rm
inst}}{\left[\mathcal{Z}^{U(1)}_{\rm inst}\right]^{N}}$

is an equality between analytic functions. With this in mind, it is useful to
extract the coeficients in the expansion (27) of the partition function and
its dual by integrating the sum against the following measure,

$\displaystyle\oint d\hat{\mu}\left[\mathbf{v}\right]$ $\displaystyle:=$
$\displaystyle\prod_{a=1}^{N-1}\,\left(\frac{1}{2\pi
i}\,\oint_{C_{a}}\,\frac{dw_{a}}{w_{a}}\right){}$

where, $w_{a}=\exp(-v_{a})$ for $a=1,2,\ldots,N-1$, and the contour $C_{a}$ is
a small circle in the complex $w_{a}$-plane enclosing the origin. We then
have,

$\displaystyle Z^{(\mathbf{n})}_{S}$ $\displaystyle=$ $\displaystyle\oint
d\hat{\mu}\left[\mathbf{v}\right]\exp\left(+\mathbf{n\cdot
v}\right)Z_{S}\left[\mathbf{v}\right]$ (86)

and

$\displaystyle\tilde{Z}^{(\mathbf{n})}_{S}$ $\displaystyle=$
$\displaystyle\oint d\hat{\mu}\left[\mathbf{v}\right]\exp\left(+\mathbf{n\cdot
v}\right)\tilde{Z}_{S}\left[\mathbf{v}\right]$ (87)

with the S-duality relation
$Z_{S}^{(\mathbf{n})}=\mathcal{K}_{+}^{(\mathbf{n})}\,\tilde{Z}_{S}^{(\mathbf{n})}$.

####

The first step is to find the asymptotics of the dual coefficients
$\tilde{Z}_{S}^{(\mathbf{n)}}:=Z^{(\mathbf{n})}_{S}[\tilde{\epsilon}_{1},\tilde{\epsilon}_{2},\tilde{m},\tilde{\tau}]$.
A direct approach to this problem starting from the explicit expression
(82,D.2) is not straightforward for general values of the parameters so we
will take a different approach. We start from the equality (85) evaluated on
the S-dualized parameters,

$\displaystyle\tilde{Z}_{S}[\mathbf{v}]$ $\displaystyle=$
$\displaystyle\tilde{\hat{\mathcal{Z}}}_{\rm
pert}^{U(N)}\,\,\frac{\tilde{\mathcal{Z}}^{U(N)}_{\rm
inst}}{\left[\tilde{\mathcal{Z}}^{U(1)}_{\rm inst}\right]^{N}}$

where, as above, $\tilde{}$ denotes the replacement (28). Thus, for example,

$\displaystyle\tilde{\mathcal{Z}}^{U(N)}_{\rm inst}$ $\displaystyle=$
$\displaystyle\mathcal{Z}^{U(N)}_{\rm
inst}\left[\tilde{t},\tilde{x},\tilde{y},\tilde{q}_{\tau};{Z}\right]$

with

$\displaystyle\tilde{q}_{\tau}\,$ $\displaystyle:=$
$\displaystyle\,\exp\left(-\tilde{\beta}\right)\,=\,\exp\left(-\frac{4\pi^{2}}{\beta}\right)$
$\displaystyle\tilde{y}\,$ $\displaystyle:=$
$\displaystyle\,\exp\left(-\tilde{m}\right)\,=\,\exp\left(-2\pi
i\frac{m}{\beta}\right)$ $\displaystyle\tilde{t}\,$ $\displaystyle:=$
$\displaystyle\,\exp\left(-\tilde{\epsilon}_{+}\right),=\,\exp\left(-2\pi
i\frac{\epsilon_{+}}{\beta}\right)$ $\displaystyle\tilde{x}\,$
$\displaystyle:=$
$\displaystyle\exp\left(-\tilde{\epsilon}_{-}\right),=\,\exp\left(-2\pi
i\frac{\epsilon_{-}}{\beta}\right)$ (88)

####

For small $\beta$, $|\tilde{q}_{\tau}|<<1$, and we have,

$\displaystyle\tilde{Z}_{S}$ $\displaystyle=$
$\displaystyle\tilde{\hat{\mathcal{Z}}}^{U(N)}_{\rm
pert}\,\,\left[\,1\,\,+\,\,O\left(\tilde{q}_{\tau}\right)\,\right]{}$

It is tempting to conclude that $\tilde{Z}_{S}$ can be replaced by
$\tilde{\hat{\mathcal{Z}}}_{\rm pert}$ which is in line with our expectation
that the theory becomes weakly-coupled in the S-dual frame. In fact this
conclusion depends sensitively on the values of the other parameters appearing
in $\tilde{Z}_{S}$. We will adopt this as a hypothesis and then describe some
checks that this indeed the case for the saddle point values of the
parameters. Making this assumption in (86) we have,

$\displaystyle\tilde{Z}_{S}^{(\mathbf{n})}$ $\displaystyle\simeq$
$\displaystyle\int\,d\hat{\mu}\left[\mathbf{v}\right]\,\exp\left(+\,\mathbf{n}\cdot\mathbf{v}\right)\,\Pi^{(\pm)}\left[\frac{\epsilon_{1}}{\tau},\frac{\epsilon_{2}}{\tau},\frac{m}{\tau};\mathbf{v}\right]$
(89)

where $\Pi^{(\pm)}$ are the infinite products defined in (75,LABEL:infprod2)
above, choosing the ”$+$” and ”$-$” signs when ${\rm Re}[\epsilon_{+}/\tau]$
is positive and negative respectively.

####

The integral (89) can be evaluated formally as an infinite sum over the
residues of the product $\Pi^{(\pm)}$. For each $\mathbf{n}\in\mathbb{Z}_{\geq
0}^{N-1}$, the leading term in the limit $\beta\rightarrow 0$, comes from the
pole which maximises the value of

$\displaystyle{\rm Re}\left[\mathbf{n}\cdot\mathbf{v}\right]$ (90)

####

To investigate this further we must make some assumptions about the values of
the other chemical potentials in the problem. Ultimately these values will be
fixed by the saddle point equations studied in the next section. Although a
more general analysis is possible, we will mainly focus on the special case of
equal charges $Q_{1}=Q_{2}$ and equal angular momenta $J_{1}=J_{2}$ where the
saddle point equations will impose $\epsilon_{1}=\epsilon_{2}=\epsilon_{+}$
and $m\simeq i\pi$. In the limit $\beta\rightarrow 0$ the dual parameters
scale like,

$\displaystyle\tilde{q}_{\tau}\,:=\,\exp\left(-\tilde{\beta}\right)\,=\,\exp\left(-\frac{4\pi^{2}}{\beta}\right)$
$\displaystyle\rightarrow$ $\displaystyle 0$
$\displaystyle\tilde{y}\,:=\,\exp\left(-\tilde{m}\right)\,\simeq\,\exp\left(-\frac{2\pi^{2}}{\beta}\right)\,=\,\tilde{q}_{\tau}^{\frac{1}{2}}$
$\displaystyle\rightarrow$ $\displaystyle 0$
$\displaystyle\tilde{t}\,:=\,\exp\left(-\tilde{\epsilon}_{+}\right)$
$\displaystyle\rightarrow$ $\displaystyle 0\qquad{}\qquad{}\,\,{\rm
Re}\left[\tilde{\epsilon}_{+}\right]>0$ (91) $\displaystyle\rightarrow$
$\displaystyle\infty\qquad{}\qquad{}{\rm
Re}\left[\tilde{\epsilon}_{+}\right]<0$

In this limit we find that the dominant pole lies at
$v_{a}=\tilde{m}-\tilde{\epsilon}_{+}$
($v_{a}=\tilde{m}+\tilde{\epsilon}_{+}$), with $a=1,2,\ldots,N-1$, for ${\rm
Re}\left[\tilde{\epsilon}_{+}\right]>0$ (${\rm
Re}\left[\tilde{\epsilon}_{+}\right]<0$). Evaluating the residue of the
integrand in (89) at these points immediately gives

$\displaystyle\log\tilde{Z}_{S}^{(\mathbf{n})}$
$\displaystyle\begin{array}[]{c}_{\beta\rightarrow 0}\\\ \sim\\\ \end{array}$
$\displaystyle\frac{m\mp\epsilon_{+}}{\tau}\sum_{a=1}^{n}\,n_{a}$ (95)

with the $\mp$ sign depending on the sign of ${\rm Re}[\epsilon_{+}/\tau]$.

####

We have performed several checks that the asymptotics derived above are not
altered by instanton corrections in powers of $\tilde{q}_{\tau}$. In
particular one can check that instanton corrections to the residue of the
poles considered above are indeed suppressed at the first few orders. It is
also possible to check that no competing poles arise at any finite order in
the instanton expansion. It would be interesting to investigate whether this
remains true in the full non-perturbative partition function using its
Gopakumar-Vafa reformulation as in [50, 51, 52].

####

The S-duality relation (29) then allows us to determine the asymptotics of
coefficients $Z_{S}^{(\mathbf{n})}$ as,

$\displaystyle\log Z_{S}^{(\mathbf{n})}$
$\displaystyle\begin{array}[]{c}_{\beta\rightarrow 0}\\\ \sim\\\ \end{array}$
$\displaystyle\frac{\epsilon_{1}\epsilon_{2}}{2\beta}\sum_{a,b=1}^{N-1}n_{a}\Omega_{ab}n_{b}\,\,-\,\,\frac{1}{\beta}\Delta^{(\pm)}_{1}\Delta^{(\pm)}_{2}\sum_{a=1}^{N-1}\,n_{a}$
(99)

with,

$\displaystyle\Delta^{(\pm)}_{1}:=m\pm\epsilon_{+}$
$\displaystyle\qquad{}\Delta^{(\pm)}_{2}:=2\pi i-m\pm\epsilon_{+}{}$

Inserting this result in (27) we find tha the modular partition function
$Z_{S}$ asymptotes term by term to the series,

$\displaystyle Z_{S}\left[\mathbf{v}\right]$ $\displaystyle\sim$
$\displaystyle\sum_{\mathbf{n}\in\mathbb{Z}_{\geq
0}^{N-1}}\,\,\exp\left[\frac{\epsilon_{1}\epsilon_{2}}{2\beta}\sum_{a,b=1}^{N-1}n_{a}\Omega_{ab}n_{b}\,\,-\,\,\frac{1}{\beta}\sum_{a=1}^{N-1}\left(\Delta^{(\pm)}_{1}\Delta^{(\pm)}_{2}+\beta
v_{a}\right)n_{a}\right]$ (101)

As the Cartan matrix $\Omega_{ab}$ has positive eigenvalues, the series is
convergent for ${\rm Re}[\epsilon_{1}\epsilon_{2}/\beta]<0$ and gives an
asymptotic formula for $Z_{S}[\mathbf{v}]$ as $\beta\rightarrow 0$ in this
case.

####

To obtain a more general result we can return to equation (86) which can be
analytically continued to define $Z_{S}[\mathbf{v}]$ beyond the domain of
convergence of the original series (27). Substituting for
$Z_{S}^{(\mathbf{n})}$ with its asymptotic form (LABEL:as), this equation then
takes an asymptotic form,

$\displaystyle\oint\,d\hat{\mu}[\mathbf{v}]\,\,\exp\left(+\mathbf{n}\cdot\mathbf{v}\right)\,Z_{S}[\mathbf{v}]$
$\displaystyle\begin{array}[]{c}_{\beta\rightarrow 0}\\\ \sim\\\ \end{array}$
$\displaystyle\exp\left[\frac{\epsilon_{1}\epsilon_{2}}{2\beta}\sum_{a,b=1}^{N-1}n_{a}\Omega_{ab}n_{b}\,\,-\,\,\frac{\Delta^{(\pm)}_{1}\Delta^{(\pm)}_{2}}{\beta}\sum_{a=1}^{N-1}n_{a}\right]$
(105)

Its easy to check that this constraint is satisfied for all $\mathbf{n}$ by
the following asymptotic form for $Z_{S}[\mathbf{v}]$,

$\displaystyle\log\,Z_{S}\left[\mathbf{v}\right]$
$\displaystyle\begin{array}[]{c}_{\beta\rightarrow 0}\\\ \sim\\\ \end{array}$
$\displaystyle\,-\frac{1}{2\epsilon_{1}\epsilon_{2}\beta}\,\,\sum_{a,b}^{N-1}\,\left(\Delta^{(\pm)}_{1}\Delta_{2}^{(\pm)}+\beta
v_{a}\right)\Omega^{-1}_{ab}\left(\Delta^{(\pm)}_{1}\Delta_{2}^{(\pm)}+\beta
v_{b}\right)$ (110)

In particular, using this asymptotic formula on the LHS of (LABEL:intzs), we
can evaluate the resulting contour integral by steepest descent to reprodice
the expression in the RHS. More precisely, the saddle-point approximation is
valid provided the action is large at the stationary point. We find that there
are then two overlapping regimes in which he asymptotic is valid. The first is
$N>>1$, we will call this the large-$N$ regime. The second case of interest is
the Cardy regime where $|\epsilon_{1}|$, $|\epsilon_{2}|<<1$ for arbitrary
$N$. We will consider the asymptotics of the index in both these cases in the
following.

####

We also note that the same asymptotic can be obtained [36] in the case ${\rm
Re}[\epsilon_{1}\epsilon_{2}/\beta]<0$ directly from the convergent series
(101). In either the Cardy or large-$N$ regime, the discrete summation over
$\mathbf{n}\in\mathbb{Z}^{N-1}_{\geq 0}$ can be traded for an integration over
a continuous variable which can also be evaluated in saddle-point
approximation giving. The above analysis suggests that the same asymptotic
form (LABEL:zsfinal) is also valid for ${\rm
Re}[\epsilon_{1}\epsilon_{2}/\beta]>0$.

####

One additional subtle point relates to the condition determining the two cases
$\pm$ in (LABEL:zsfinal). In our derivation, which of the two cases arise
depends on the sign of the dual parameter $\mathcal{A}:={\rm
Re}[\tilde{\epsilon}_{+}]=2\pi{\rm Im}[\epsilon_{+}/\beta]$. Thus the relevant
condition depends on both $\beta$ and $\epsilon_{+}$. When ${\rm
Re}[\epsilon_{1}\epsilon_{2}/\beta]<0$ and the series converges, we can check
that the condition $\mathcal{A}>0$ (or $\mathcal{A}<0$) can always be replaced
by a simpler one involving only $\epsilon_{+}$. In particular, it is easy to
show that, for ${\rm Re}[\epsilon_{1}\epsilon_{2}/\beta]<0$, we have, ${\rm
Sign}(\mathcal{A})={\rm Sign}({\rm Im}[\epsilon_{+}])$. In extending our
analysis as described above, it will suffice to discuss analytic continuation
in $\beta$ with the other parameters held fixed. Then the $\pm$ cases of the
asymptotic (LABEL:zsfinal) are determined by the condition ${\rm
Im}[\epsilon_{+}]<0$ (+ case) and ${\rm Im}[\epsilon_{+}]>0$ (- case)
throughout the domain of analyticity in $\beta$.

###  D.4 Asymptotics of the Index

####

Finally we can consider the implications of he above analysis for the
asymptotics of the supersymmetric index $\mathcal{I}_{QM}$ as
$\beta\rightarrow 0$. The above analysis provides an effective resummation of
the instanton series for our quantum mechanical index. Inverting the relation
(85), we can express the index in terms of the modular partition function
$Z_{S}$. Suppressing the arguments of the various partition functions we have,

$\displaystyle\mathcal{I}_{\rm QM}$ $\displaystyle=$
$\displaystyle\int\,d\mu_{\rm Haar}\,\mathcal{Z}^{U(N)}_{\rm inst}$
$\displaystyle=$ $\displaystyle\left[Z^{U(1)}_{\rm
inst}\right]^{N}\int\,d\mu_{\rm
Haar}\,\,\frac{1}{\hat{\mathcal{Z}}^{U(N)}_{\rm pert}}\,Z_{S}$

In either the Cardy or large-$N$ regime, we can use asymptotic form
(LABEL:zsfinal) for $Z_{S}$ to get,

$\displaystyle\log\mathcal{I}_{\rm QM}$
$\displaystyle\begin{array}[]{c}_{\beta\rightarrow 0}\\\ \sim\\\ \end{array}$
$\displaystyle-\frac{N(N^{2}-1)}{24}\frac{\left(\Delta^{(\pm)}\Delta^{(\pm)}\right)^{2}}{\epsilon_{1}\epsilon_{2}\beta}\,\,+\,\,N\log\mathcal{Z}^{U(1)}_{\rm
inst}\,\,+\,\,\log\mathcal{R}\,\,\ $ (115)

where $\mathcal{R}$ is a residual integral,

$\displaystyle\mathcal{R}$ $\displaystyle:=$ $\displaystyle\int d\mu_{\rm
Haar}[\mathbf{v}]\frac{1}{\hat{\mathcal{Z}}_{\rm
pert}}\,\exp\left[\frac{\Delta^{(\pm)}\Delta^{(\pm)}}{\epsilon_{1}\epsilon_{2}}\mathbf{\rho}\cdot\mathbf{v}\right]\times\exp\left[-\frac{\beta}{2\epsilon_{1}\epsilon_{2}}\,\,\sum_{a,b}^{N-1}\,v_{a}\Omega^{-1}_{ab}v_{b}\right]$

where $\mathbf{\rho}=(\rho_{1},\ldots,\rho_{n})$ is the $SU(N)$ Weyl vector
with components $\rho_{a}=a(N-a)/2$, for $a=1,\ldots,N-1$ and we have used the
identity $\sum_{a,b=1}^{N}\Omega^{-1}_{ab}=N^{2}(N-1)/12$. Further, in the
limit $\beta\rightarrow 0$, the quadratic term of $O(\beta)$ can be dropped,
and $\mathcal{R}$ reduces to,

$\displaystyle\mathcal{R}_{0}[\epsilon_{1}\epsilon_{2},m]$ $\displaystyle=$
$\displaystyle\int d\mu_{\rm
Haar}[\mathbf{v}]\left(\Pi^{(+)}[\epsilon_{1},\epsilon_{2},m;\mathbf{v}]\right)^{-1}\,\,\exp\left[\frac{\Delta^{(\pm)}\Delta^{(\pm)}}{\epsilon_{1}\epsilon_{2}}\mathbf{\rho}\cdot\mathbf{v}\right]$

where $\Pi^{(+)}$ is the convergent infinite product representing
$\hat{\mathcal{Z}}_{\rm pert}$ in the region ${\rm Re}[\epsilon_{+}]>0$. This
integral is finite and independent of the coupling $\beta=-2\pi i\tau$. The
quantity $\mathcal{R}_{0}$ can be cast in the form of a perturbative index
with scaling $\log\mathcal{R}_{0}\sim N^{2}$ at large $N$.

####

Finally we can determine the leading asymptotics of our index in the two
regimes of interest. For large $N$, the first term in (115) scales like
$N^{3}$, the second term involving the $U(1)$ instanton partition function
provides a correction of order $N$. The residual integral has a finite limit
as $\beta\rightarrow 0$ and does not contribute to the leading order growth of
the exponent which goes like $\beta^{-1}$. Thus for $N>>1$ we have,

$\displaystyle\log\mathcal{I}_{\rm QM}$
$\displaystyle\begin{array}[]{c}_{\beta\rightarrow 0}\\\ \sim\\\ \end{array}$
$\displaystyle-\frac{N^{3}}{24}\frac{\left(\Delta^{(\pm)}\Delta^{(\pm)}\right)^{2}}{\epsilon_{1}\epsilon_{2}\beta}$
(119)

####

In the Cardy regime, $|\epsilon_{1}|$, $|\epsilon_{2}|<<1$, for fixed $N$, we
have $\Delta^{(\pm)}_{1}\simeq m$ and $\Delta^{(\pm)}_{2}\simeq 2\pi i-m$. We
can also use the asymptotic form (26) for $\mathcal{Z}^{U(1)}_{\rm inst}$.
Combining this with the first term in (115) we find,

$\displaystyle\log\mathcal{I}_{\rm QM}$
$\displaystyle\begin{array}[]{c}_{\beta\rightarrow 0}\\\ \sim\\\ \end{array}$
$\displaystyle-\frac{N^{3}}{24}\frac{m^{2}(2\pi
i-m)^{2}}{\epsilon_{1}\epsilon_{2}\beta}$ (123)

The two regimes overlap and we can summarise our results by concluding that
the asymptotic form (119) is valid in both the Cardy and large-$N$ regimes.

###  D.5 Saddle-point analysis

We start from the representation of the index coefficients as a contour
integral,

$\displaystyle\mathcal{C}(L_{t},L_{x},L_{y},K)$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi
i)^{4}}\,\oint_{C_{\beta}}\,\frac{dq_{\tau}}{q_{\tau}^{K+1}}\,\oint_{C_{+}}\,\frac{dt}{t^{L_{t}+1}}\,\oint_{C_{-}}\,\frac{dx}{x^{L_{x}+1}}\,\oint_{C_{m}}\,\frac{dy}{y^{L_{y}+1}}\,\,\,\mathcal{I}_{\rm
QM}\left[t,x,y,q_{\tau}\right]$

where the contours $C_{\beta}$, $C_{+}$, $C_{-}$, $C_{m}$ are circles,
centered at the origin of the complex $q_{\tau}$, $t$, $x$ $y$ planes of radii
$R_{\beta}$, $R_{+}$, $R_{-}$ and $R_{m}$ respectively. The specific expansion
which yields the superconformal index corresponds to choosing,

$\displaystyle 0<R_{\beta}<R_{+}<<1$ $\displaystyle R_{-}=R_{y}=1$

This corresponds to expanding the rational function arising at each instanton
number in powers of $t$ with $|\mathcal{M}|<1$ for arbitrary monomials of the
form $\mathcal{M}=tx^{a}y^{b}\prod z_{i}^{c_{i}}$.

####

In this section we will show that integral (LABEL:int1) can be evaluate by
saddle point in the scaling limit. Replacing the integrand with its asymptotic
form near $\beta=0$ (for the case ${\rm Im}[\epsilon_{+}]<0$),

$\displaystyle\mathcal{C}(L_{t},L_{x},L_{y},K)$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi i)^{4}}\,\int_{\mu_{\beta}}^{\mu_{\beta}+2\pi
i}\,d\beta\,\int_{\mu_{+}}^{\mu_{+}+2\pi
i}\,d\epsilon_{+}\,\int_{\mu_{-}}^{\mu_{-}+2\pi
i}\,d\epsilon_{-}\,\int_{\mu_{m}}^{2\pi
i+\mu_{m}}\,dm\,\,\,\exp\left(\mathcal{S}\right)$ (125)

with $\mu_{+}=\log R_{+}$ with similar definitions for the other integration
variables. Here the exponent is given as,

$\displaystyle\mathcal{S}+2\pi iQ_{2}$ $\displaystyle=$
$\displaystyle-\frac{N^{3}}{24}\,\frac{\Delta_{1}^{2}\Delta_{2}^{2}}{\epsilon_{1}\epsilon_{2}\beta}\,+\,\Delta_{1}Q_{1}\,+\,\Delta_{2}Q_{2}\,+\,\epsilon_{1}J_{1}\,+\,\epsilon_{2}J_{2}\,+\,\beta
K$

where, as above,

$\displaystyle L_{t}$ $\displaystyle=$ $\displaystyle Q_{1}+Q_{2}+J_{1}+J_{2}$
$\displaystyle L_{x}$ $\displaystyle=$ $\displaystyle J_{1}-J_{2}$
$\displaystyle L_{y}$ $\displaystyle=$ $\displaystyle Q_{1}-Q_{2}$

and

$\displaystyle\Delta_{1}=\Delta_{1}^{(+)}=m+\epsilon_{+}$
$\displaystyle\Delta_{2}=\Delta_{2}^{(+)}=2\pi i-m+\epsilon_{+}$

####

To identify the dominant contribtion to the integral, investigate the
stationary points of $\mathcal{S}$, satisfying,

$\displaystyle\frac{\partial\mathcal{S}}{\partial\epsilon_{1}}\,=\,\frac{\partial\mathcal{S}}{\partial\epsilon_{2}}\,=\,\frac{\partial\mathcal{S}}{\partial
m}\,=\,\frac{\partial\mathcal{S}}{\partial\beta}$ $\displaystyle=$
$\displaystyle 0$

####

The chemical potentials can be eliminated using the stationary conditions to
obtain a quartic equation for the saddle-point action.

$\displaystyle\mathcal{S}^{2}\left(\mathcal{S}+2\pi i(Q_{2}-Q_{1})\right)^{2}$
$\displaystyle=$ $\displaystyle\frac{8\pi^{2}}{3}N^{3}K\left(\mathcal{S}+2\pi
i(J_{1}+Q_{2})\right)\left(\mathcal{S}+2\pi i(J_{2}+Q_{2})\right)$ (126)

####

For the rest of this section we will mainly focus on the simplest case of
equal charges $Q_{1}=Q_{2}=Q$ and angular momenta $J_{1}=J_{2}=J$ and
determine degeneracies as a function of $L:=2(J+Q)$. The following analysis
can easily be generalised to the case of generic charges. In this special case
the saddle point action solves

$\displaystyle\mathcal{S}^{4}$ $\displaystyle=$
$\displaystyle\frac{8\pi^{2}}{3}N^{3}K\,\left(\mathcal{S}+\pi iL\right)^{2}$
(127)

To determine the dominant saddle-point, we define rescaled variables $s$ and
$\lambda$ by,

$\displaystyle\mathcal{S}=2\pi\,\sqrt{K}\,\sqrt{\frac{N^{3}}{3}}\,s$
$\displaystyle L=\sqrt{K}\,\sqrt{\frac{N^{3}}{3}}\,\lambda$

and split the rescaled action into its real and imaginary part as $s=:x+iy$,
$x$, $y\in\mathbb{R}$. The complex equation (127), then yields a pair of real
equations for $x$ and $y$,

$\displaystyle x^{2}$ $\displaystyle=$ $\displaystyle
y^{2}+\frac{y+\lambda/2}{y}$ (128) $\displaystyle
2y^{4}+y^{2}-\frac{\lambda^{2}}{4}$ $\displaystyle=$ $\displaystyle 0$ (129)

Solutions with $x>0$ correspond to saddle-points which contribute
exponentially large terms to the integral. We find two real roots of (129)
$y=y^{(\pm)}=\pm y_{\star}$ with,

$\displaystyle y_{\star}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(-1+\sqrt{1+2\lambda^{2}}\right)^{\frac{1}{2}}\,\,>\,\,0{}$

which give,

$\displaystyle x=x^{(\pm)}$ $\displaystyle:=$
$\displaystyle\left(y_{\star}^{2}+\frac{y_{\star}\pm\lambda/2}{y_{\star}}\right)^{\frac{1}{2}}\,\,>\,\,0$

####

For either root, the corresponding saddle point is is located at $m=i\pi$ and,

$\displaystyle\epsilon_{1}=\epsilon_{2}=\epsilon_{+}$ $\displaystyle=$
$\displaystyle i\pi\frac{x+iy}{x+i(y+\lambda)}$

thus we find,

$\displaystyle{\rm Re}\left[\epsilon_{+}\right]$ $\displaystyle=$
$\displaystyle\pi\frac{\lambda x}{x^{2}+(y+\lambda)^{2}}$ $\displaystyle{\rm
Im}\left[\epsilon_{+}\right]$ $\displaystyle=$
$\displaystyle\pi\frac{x^{2}+y(y+\lambda/2)}{x^{2}+(y+\lambda)^{2}}$

Thus ${\rm Re}[\epsilon_{+}]$ is positive for both saddle-points, and it is
easy to check using (128,129) that ${\rm Im}[\epsilon_{+}]$ is positive
(negative) for $(x,y)=(x^{(+)},y^{(+)})$, ($(x,y)=(x^{(-)},y^{(-)})$). It
follows that only the ”$-$” saddle-point lies within the domain of validity of
the ”$+$” case of the asymptotic form (LABEL:zsfinal). From now on we restrict
our discussion to this saddle.

####

At the saddle point, the coupling $\beta$ takes the value

$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\pi\sqrt{\frac{N^{3}}{3K}}\,\frac{(x+iy)\left(x+i\left(y+i\lambda/2\right)\right)}{x+i(y+\lambda)}$

and therefore has real part,

$\displaystyle{\rm Re}[\beta]$ $\displaystyle=$
$\displaystyle\pi\sqrt{\frac{N^{3}}{3K}}\frac{x(y+\lambda/2)}{x^{2}+(y+\lambda)^{2}}\left(\frac{1}{y}+\lambda+2y\right)$
(131)

which remains positive provided that
$\lambda>\lambda_{cr}:=2\sqrt{1+\sqrt{5}/2}\simeq 2.91$. Here the critical
value, $\lambda_{cr}$, is determined by the largest zero of last factor in
(131).

####

Combining the above results for $\beta$ and $\epsilon$, we therefore have
$|t|<1$ and $|q_{\tau}|<1$ at the saddle point provided that202020Our analysis
cannot easily be extended to lower values of $\lambda$ without additional
assumptions. $\lambda>\lambda_{cr}$. Hence, in this range, the saddle-point
lies inside the assumed domain of analyticity for the index. Note that one
also has,

$\displaystyle{\rm Re}\left[\frac{\epsilon^{2}}{\beta}\right]$
$\displaystyle=$
$\displaystyle-\pi\,\sqrt{\frac{3K}{N^{3}}}\,\frac{x(x^{2}+y^{2}-\lambda^{2}/2)}{(x^{2}+(y+\lambda)^{2})(x^{2}+(y+\lambda/2)^{2})}$

and one may check that this is always positive at the remaining saddle point.
Thus the stationary point lies outside the region of convergence of the series
(101). However, as discussed above the ”$+$” case of the asymptotic
(LABEL:zsfinal) still holds in this regime provided ${\rm Im}[\epsilon]<0$.

####

Finally the analysis can be repeated with $\Delta^{(+)}_{i}$ replaced by
$\Delta^{(-)}_{i}$, for $i=1,2$, and yields a second consistent saddle-point,
with ${\rm Im}[\epsilon_{+}]>0$, related to the one discussed above by complex
conjugation: $x\rightarrow x$, $y\rightarrow-y$ with $\lambda$ fixed. The two
consistent saddle points each contribute an exponentially growing term to the
integral. The resulting contribution to the asymptotics of the index
coefficients can be written as,

$\displaystyle\log\,\left|\mathcal{C}[L_{,}0,0,K]\right|$ $\displaystyle\sim$
$\displaystyle
2\pi\sqrt{K}\,\sqrt{\frac{N^{3}}{12}}\,\left[\sqrt{1+\sqrt{1+6L^{2}/KN^{3}}}-\sqrt{2}\right]$
(132)

Where we take the $K\rightarrow\infty$ limit with $L\sim\sqrt{K}$. The above
analysis is easily generalised to the case where $J_{1}\neq J_{2}$, $Q_{1}\neq
Q_{2}$. The result has the scaling form,

$\displaystyle\log\,\left|\mathcal{C}[L_{t},L_{x},0,K]\right|$
$\displaystyle\sim$ $\displaystyle 2\pi\sqrt{\delta}\,\mathcal{F}_{\rm
QM}\left[L_{t}/\sqrt{\delta},L_{x}/\sqrt{\delta},L_{y}/\sqrt{\delta}\right]$
(133)

with $\delta=KN^{3}/3$. The function $\mathcal{F}_{\rm QM}[u,v,w]$ is
determined by the smaller positive root $s=s_{\star}$ of the quartic equation,

$\displaystyle s^{2}(s-iw)^{2}$ $\displaystyle=$ $\displaystyle
2\left(s+i(u+v)/2\right)\left(s+i(u-v)/2\right)$ (134)

as $\mathcal{F}_{\rm QM}={\rm Re}\,s_{\star}$.

Returning to the special case of equal charges and angular momenta, a more
precise analysis is possible (for any $N$) in the Cardy regime corresponding
to $L^{2}>>\delta/2=N^{3}K/6$. The complex conjugate pair of saddle-point
actions can then be written as $\mathcal{S}(L)\pm i\mathcal{S}(L)$ with,

$\displaystyle\mathcal{S}(L)$ $\displaystyle=$ $\displaystyle
2\pi\,\left(\frac{N^{3}}{24}L^{2}K\right)^{\frac{1}{4}}$ (135)

Adding together the contribution of both saddle points, and including the
prefactor from small fluctuations around the stationary point, we obtain a
formula for the asymptotic growth of the index coefficients in the Cardy
regime,

$\displaystyle\mathcal{C}\left(L_{,}0,0,K\right)$ $\displaystyle\sim$
$\displaystyle\frac{3\sqrt{2}}{16\pi}\,\frac{\mathcal{S}(L)}{L^{2}K}\,\exp\left[\mathcal{S}(L)\right]\cos\left(\mathcal{S}(L)+\frac{\pi}{4}\right)$

with $\mathcal{S}(L)$ given by (35). In the Cardy regime, this formula holds
for all $N$. In the case $N=1$, we have tested this formula against a direct
expansion of the localisation formula on a computer up to $K\sim L\sim 60$,
finding good agreement.

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