# Black holes and modular forms in string theory

Sameer Murthy

Department of Mathematics, King’s College London,

Strand, London UK WC2R 2LS.

Summary

The study of black holes in string theory has led to the discovery of deep and
surprising connections between black holes and modular forms—which are two
classical, a priori unrelated, subjects. This article explains the main
physical and mathematical ideas behind these connections.

It is known from the pioneering work of J. Bekenstein and S. Hawking in the
1970s that black holes have thermodynamic entropy, and should therefore be
made up of a collection of microscopic quantum states. Superstring theory
provides a framework wherein one can associate a number of microscopic states
that make up the quantum-statistical system underlying a black hole, thus
explaining their thermodynamic behavior from a more fundamental point of view.
The basic connection to modular forms arises from the observation that, in the
simplest superstring-theoretic construction, the generating function of the
number of microscopic states is a modular form. In one direction, modular
symmetry acts as a powerful guide to the calculation of quantum-gravitational
effects on the black hole entropy. In the other direction, the connection has
led to the discovery of surprising relations between Ramanujan’s _mock_
modular forms and a class of string-theoretic black holes, thus providing an
infinite number of new examples of mock modular forms.

Keywords

Black holes, black hole thermodynamics, string theory, modular forms, mock
modular forms

##  1 Introduction

_Black holes_ (BH) are objects that exist in the physical universe: they are
regions of spacetime, typically formed by the collapse of heavy stars. At the
theoretical level they are described as solutions to the theory of General
Relativity (GR). This theory is one of the pillars underlying the paradigm of
modern physics—it determines all large-scale structure in our universe and
underlies the dynamics of spacetime itself. A black hole is characterized by
the existence of a one-way surface surrounding it, called the _event horizon_.
The one-way nature of the horizon means that things can fall into the black
hole but nothing—not even light—can come out, thus leading to its name. The
theoretical description of black holes has been spectacularly confirmed by
many recent observations in astronomy. See the textbook Schutz (2022) for an
introduction to GR and BHs.

_Modular forms_ are objects that exist in the mathematical universe: they are
a class of functions that naturally arise in the field of number theory. They
are characterized by their symmetry properties under the action of the
_modular group_ $SL(2,\mathbb{Z})$ (the group of $2\times 2$ integer matrices
with unit determinant). The related term _automorphic form_ is used for
functions with symmetry properties under more general arithmetic groups.
Starting from the study of theta functions in the 19th century and its classic
manifestations in number theory, the field of modular and automorphic forms
has grown to include ramifications in topology, algebraic geometry, algebraic
and analytic number theory, representation theory, as well as physics, and has
been the subject of intense study over the last 100 years. See Bruinier .
(2008) for an introduction to modular forms.

The work of Bekenstein (1973) and Hawking (1975) showed that, when one takes
into account quantum mechanical effects, black holes have thermodynamic
entropy. This leads to the conclusion that a black hole must be made up of
microscopic states, just like the entropy of air in a room is explained by the
fact it consists of a large number of molecules. The natural consequent
question is:  _What are the microstates (the “molecules”) of a black hole?_

Since no classical-mechanical process can probe the interior of a black hole,
the answer to the above question needs a framework which consistently takes
into account quantum effects in the presence of a black hole. Precisely such a
framework is provided by superstring theory, wherein one can construct models
of quantum black holes in which the number of microscopic states can be
counted. The basic connection to modular forms arises from the observation
that, in the simplest string-theoretic constructions, the number of these
microscopic states is the Fourier coefficient of a certain modular form.

The underlying modular symmetry allows us to make a simple asymptotic estimate
for the logarithm of these Fourier coefficients, which agrees precisely with
the Bekenstein-Hawking entropy of the corresponding BHs, as was first
discovered in the breakthrough work of Strominger Vafa (1996). In fact, the
power of modular symmetry allows us to go much further. The well-known Hardy-
Ramanujan-Rademacher formula of analytic number theory, which expresses the
Fourier coefficient of the modular form as an infinite sum of $I$-Bessel
functions, turns out to be intimately related to quantum-gravitational
corrections of the black hole entropy. The connection to black holes also
informs the field of modular forms in mathematics. For example, it has led to
new theorems and constructions of mock modular forms, (which were first
introduced by Ramanujan in his last letter to Hardy in 1920).

##  2 Black holes and their thermodynamics

The structure and evolution of spacetime at large scales in our universe is
described by the theory of General Relativity. Spacetime—the set of all points
in space and time—is considered to be a pseudo-Riemannian 4-manifold of
signature $(-,+,+,+)$, the $-$ sign indicating the single time coordinate
$x^{0}$ and the three $+$ signs related to the three spatial coordinates
$x^{1},x^{2},x^{3}$. The spacetime manifold comes equipped with a metric (a
rank-2 tensor field with components $g_{\mu\nu}(x)$ $\mu,\nu=0,1,2,3$) which
can be thought of, roughly, as encoding the local shape and size of the
manifold. At large scales, physical matter in the universe is effectively
described by a rank-2 tensor field on the spacetime manifold, called the
stress-energy tensor with components $T_{\mu\nu}(x)$.

The basic equations of general relativity, also referred to as the Einstein
equations, are written in terms of differential geometric quantities, i.e. the
Ricci tensor field $R_{\mu\nu}(g_{\mu\nu}(x))$ and its trace called the Ricci
scalar field $R(g_{\mu\nu}(x))$, which are measures of the curvature of the
spacetime metric. The equations relate the spacetime curvature to the stress-
energy tensor (for zero cosmological constant) as follows,

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R\;=\;8\pi G\,T_{\mu\nu}\,,$ (1)

where $G$ is Newton’s gravitational constant.

All physical structures at large scales in our universe, i.e. galaxies, stars,
etc, are solutions to the Einstein equations (1). Black holes are a very
special class of solutions to these equations with the property that the
resulting spacetime metric has a region from within which no classical signal
can propagate out to an external observer. The boundary of this region is
called the event horizon, the area of which is a measure of the size of the
black hole. Black holes can have very complicated dynamics such as binary
mergers111See e.g. https://www.ligo.org/, but the solutions of Einstein’s
equations that describe stationary black holes are very simple.222Stationary
means that they do not change as a function of time. One can think of these as
the end point of the complicated dynamical processes, when the black holes
“settle”. This is the limit when they can be studied as essentially
independent objects.  These solutions are characterized by their conserved
charges, namely their mass, spin, and electromagnetic charge.333Indeed, the
dynamics of stationary charge-neutral black holes that are observed in the sky
(see e.g. https://www.cfa.harvard.edu/research/topic/black-holes) are well-
described by the Kerr BH metric solution which is completely characterized by
mass and spin.

Theoretical investigations on black holes in the 1970s, primarily due to
Bekenstein (1973) and Hawking (1975) established the following remarkable
fact. Although nothing can come out of a black hole in the classical theory of
general relativity, a black hole in the quantum theory actually behaves like a
thermodynamic object with associated temperature and entropy.

The main arguments to demonstrate the thermodynamic behavior of black holes
are simple but profound, and proceed as follows. Consider a system (“the
universe”) consisting of a black hole of mass $M$ and a bucket of water (which
carries energy and entropy) outside the black hole horizon. Now imagine a
process where one throws the water into the black hole, say in a perfectly
radial direction so as not to generate angular momentum. After the water has
entered the horizon, the external region loses some energy $E_{\text{water}}$
and some entropy $S_{\text{water}}$. Once the perturbations of the system have
died down, the system settles into a new state with a new black hole solution
and nothing outside. The mass of the black hole can be measured from the
outside and it increases to ($\widetilde{M}=M+E_{\text{water}}$), such that
the total energy of the system remains the same (as it should by the principle
of conservation of energy!). On the other hand, it would seem that the total
entropy of the system decreases by $S_{\text{water}}$—thus violating the
second law of thermodynamics—unless one assigns an intrinsic entropy to the
black hole itself.

Now, if a black hole has entropy, and given that it has energy (= its mass
$M$), it must have a temperature, according to the first law of
thermodynamics. Building on the fact that the black hole area always increases
in physical processes Bardeen . (1973), Bekenstein further argued that the
thermodynamic entropy of black holes must be proportional to the area of its
event horizon. In a tour de force, Hawking then set up a thought experiment
involving scattering of particles (or, equivalently, waves in the quantum
theory) off a black hole using the formalism of quantum field theory. Within
this setting he calculated the ratio of the amplitude of the reflected wave
and the incident wave, which directly leads to the rate of radiation—from
which he could deduce the temperature using the Planck distribution formula,
and therefore a precise formula for the entropy.

These considerations result in the following succinct formula for the
thermodynamic entropy of a BH, that applies universally to any BH solution in
general relativity444In fact this entropy formula has a remarkably wide range
of applicability, and holds in theories of general relativity coupled to
matter fields in various dimensions (with the appropriate generalization of
the horizon area).:

$S^{\\!\text{class}}_{\\!\text{BH}}\;=\;k_{B}\,\frac{1}{4}\frac{A_{H}}{\ell_{\text{Pl}}^{2}}\,,\qquad\ell_{\text{Pl}}^{2}\;=\;\frac{G\hbar}{c^{3}}\,,$
(2)

where $A_{H}$ is the area of the BH horizon.555The superscript refers to the
fact that this is a semi-classical formula, which will later be promoted to a
quantum formula. The Bekenstein-Hawking black hole entropy formula (2) is one
of the most profound equations in theoretical physics, involving the three
fundamental constants: the speed of light $c$, Planck’s constant $\hbar$, and
Newton’s gravitational constant $G$, which determine the fundamental scales of
special relativity, quantum mechanics, and gravitation, respectively, as well
as the Boltzmann constant $k_{B}$ which determines the scale of thermodynamic
entropy. These constants will often be suppressed in the following formulas
for ease of presentation.

From the theory of quantum-statistical mechanics, one knows that the entropy
of a physical system is really a statistical property arising from an
underlying collection of quantum-mechanical microscopic states (or
_microstates_). This property is quantified by the Boltzmann equation which
expresses the thermodynamic entropy as the logarithm of the number of
microscopic states in which the system can exist for a given macroscopic
state. Now, upon combining the black hole entropy formula with the Boltzmann
equation one obtains the Boltzmann equation for black holes

$k_{B}\log d_{\text{micro}}\;=\;S^{\\!\text{class}}_{\\!\text{BH}}+\dots\,.$
(3)

The dots here indicate that the Boltzmann equation is an approximate equation,
which is valid in the so-called _thermodynamic limit_ of very large black hole
size.

One thus reaches the profound conclusion that a black hole should be made up
of a collection of a large number $d_{\text{micro}}$ of microscopic states.

Finding a theory of quantum gravity—a theory which brings together Quantum
Mechanics and General Relativity into one consistent framework—has remained an
important open question of fundamental physics for the last 60 years. One of
the reasons it is such a hard problem is the lack of an experimental guide due
to constraints on current technology. In this situation it is very useful to
have a quantitative criterion which can be used to test or falsify a given
theory. The Bekenstein-Hawking entropy formula plays this role: a consistent
theory of quantum gravity should be able to produce a microscopic quantum-
statistical ensemble of black hole microstates which satisfies the Boltzmann
equation (3) for black holes.

##  3 Two pictures of black holes in string theory

A series of developments in the 1990s pioneered by Sen (1995) and Strominger
Vafa (1996), building on previous work by ’t Hooft (1990); Susskind (1993);
Susskind Uglum (1994), led to a quantum-statistical explanation of the
thermodynamic entropy of black holes in string theory. The basic idea is that
there are actually two pictures of a black hole in string theory (see Figure
1), which allows us to separately calculate and compare its thermodynamic and
statistical entropy.

Let us first consider the basic physical question: what determines whether an
object of mass $M$ is a black hole or not? The criterion, roughly speaking, is
whether the intrinsic size of the object is smaller or larger than the size of
a black hole of the same mass. Consider, for the sake of simplicity,
spherically symmetric objects. Then if the radius of the object is larger than
the _Schwarzschild radius_ $R_{\text{Sch}}=2GM$, then it is _not_ a black
hole, while if its radius is smaller than the Schwarzschild radius then there
is pure vacuum outside the horizon and one cannot distinguish the object from
a black hole. (Recall that one cannot observe anything behind the horizon.)
For composite objects like stars, the intrinsic size is determined by the
radiation pressure of electrons, and the above logic leads to the so-called
Chandrashekar limit.666See https://en.wikipedia.org/wiki/Chandrasekhar_limit.

Now we apply this criterion to string theory. The consistency of
supersymmetric string theory imposes that space-time is ten dimensional. In
order to obtain an effective theory in $\mathbb{R}^{1,3}$, six of the
dimensions should describe a compact manifold. The resulting geometry is
called a string compactification, and it determines the spectrum of particles
and their interactions at low energies. (See e.g. the classic textbook Green .
(1988) for more details.) The spectrum is characterized by the conserved
charges of the theory, also called  _quantum numbers_ , which are quantized as
integer multiples of fundamental units of charge in the quantum theory. In the
simplest case the spectrum is labelled by a single quantum number
$N\in\mathbb{N}$.

The gravitational coupling (the effective Newton’s constant as appears in all
equations) in string theory is determined by the value of the string coupling
$g_{\text{s}}$ as $G=g_{\text{s}}^{2}\ell_{\text{s}}^{2}$ where
$\ell_{\text{s}}$ is the fundamental scale of a string. In this context, the
criterion of whether the collection of states labelled by $N$ is a black hole
or not takes the following form. When $g_{\text{s}}N\gg 1$ the effective
description is that of weakly-coupled general relativity interacting with a
specified set of matter fields, and one has a black hole solution of this
effective theory with quantum number $N$. This is the _macroscopic_ picture.
When $g_{\text{s}}N\ll 1$, there is a completely different dual microscopic
picture: one has a set of fundamental excitations of string theory with the
same quantum number $N$, consisting of bound states of fluctuations of
strings, branes, and other fundamental objects of string theory.

The microscopic picture gives us the possibility of calculating the
statistical entropy. However, for a given value of $g_{s}$, only one of the
pictures should hold, and one still cannot make a comparison between the two
types of entropies.777Note, however, that it leads to the idea of a transition
between black holes and microstates Horowitz Polchinski (1997). The
breakthrough of Sen (1995); Strominger Vafa (1996) was to consider the black
hole entropy problem in the context of _supersymmetric compactifications_ of
string theory. In a class of such compactifications, the string coupling
constant $g_{s}$ turns out to be not fixed by the equations of motion, and
therefore is a tunable parameter.

Now one can test whether black hole entropy is a statistical entropy as
follows. Start with a particular supersymmetric compactification of string
theory which admits black hole solutions with some set of conserved charges.
Tune the coupling so that the effect of gravity is arbitrarily small and,
therefore, the fluctuations of strings and branes can be described using usual
quantum field theoretic methods. Count the number of microstates
$d_{\text{micro}}$ in the theory as a function of the quantum numbers. Then
crank up the coupling and therefore $G$, so that the horizon radius is much
larger than the fundamental scale of the microscopic constituents. Now the
collection of microstates gravitate and form a black hole. Measure its horizon
area and obtain the thermodynamic entropy
$S^{\\!\text{class}}_{\\!\text{BH}}$. Finally, compare
$S^{\\!\text{class}}_{\\!\text{BH}}$ and $\log d_{\text{micro}}$ as a function
of the quantum numbers.

However, one is faced with yet another problem—the number of states of the
theory can change as one changes the coupling, so it is not clear that one
should be comparing the entropy of states calculated at weak coupling with the
black hole entropy calculated at strong coupling. The resolution of Sen
(1995); Strominger Vafa (1996) is to consider a special set of states which
exist in supersymmetric theories called BPS (after Bogomolny (1976); Prasad
Sommerfield (1975)) states which are “protected”, which is to say that the
number of BPS states—counted with a certain weighting, called the
supersymmetric index—does not change under change of small parameters of the
theory and their entropy can be compared with the entropy of supersymmetric
BHs. These ideas are explained in the following section.

Figure 1: Microscopic and macroscopic pictures of a black hole in string
theory

##  4 Supersymmetric states and supersymmetric black holes

In order to explain the supersymmetric index it is useful to first briefly
review the basic ideas underlying supersymmetry. All particles in nature comes
in two types—bosons and fermions. The bosonic or fermionic nature of particles
determines their fundamental properties such as their spin—which can be
integer multiples of $\hbar$ for bosons and integer+$\frac{1}{2}$ multiples
for fermions, and statistical properties: bosons tend to cluster in the same
state, i.e. the probability for a boson to be in a certain quantum state
increases with the number of bosons already present in that state, while
fermions repel each other, i.e. the probability of two fermions to be in the
same quantum state is zero. Supersymmetry is a symmetry, first proposed by
particle physicists in the 1970s, that relates fermions and bosons.888Although
it has not been observed in nature, it is very useful to construct model
systems where one can perform controllable analytic calculations (which are
usually out of reach) in order to probe physical properties of quantum
theories e.g. Seiberg Witten (1994). Here we use it to test the Boltzmann
equation for black holes.

The main ideas of supersymmetric state counting can be illustrated in the
context of a simple type of system called supersymmetric quantum mechanics
Witten (1982). Recall that a quantum mechanical system has a Hilbert space
$\mathcal{H}$ whose elements are called (quantum) states, and a self-adjoint
operator $H$ on the Hilbert space, called the Hamiltonian. The eigenvalues of
the Hamiltonian are interpreted as possible energies of the states. Now,
consider a quantum mechanical system whose Hilbert space $\mathcal{H}$
consists of states which can be bosonic or fermionic. The _fermion number
operator_ denoted $(-1)^{F}$ is a multiplicative operator defined to have
value $+1$ for bosonic states and $-1$ for fermionic states. The
supersymmetric quantum mechanical system comes equipped with a pair of complex
conjugate fermionic operators $Q,\overline{Q}$, called the supercharges, which
are nilpotent (i.e. $Q^{2}=\overline{Q}^{2}=0$), and obey the algebra

$\\{Q,\overline{Q}\\}\;=\;H\,,\qquad[H,Q]\;=\;0\,,\qquad[H,\overline{Q}]\;=\;0\,.$
(4)

The spectrum of states of the supersymmetric system falls into representations
of this algebra, which come in two types. The first type is a two-dimensional
_long representation_ $(|b\rangle,|f\rangle)$ with $Q|b\rangle=|f\rangle$,
with $Q|f\rangle=0$ by the nilpotency of $Q$. Since $Q$ has fermion number
$-1$, the two states have opposite fermion numbers, which are denoted by $b$
(bosonic) and $f$ (fermionic). If $|b\rangle$ is an eigenstate of $H$ with
eigenvalue $E$, the algebra (4) implies that $E\geq 0$ and that $|f\rangle$ is
also an eigenstate with eigenvalue $E$. The second type is a one-dimensional
_short representation_ $|\cdot\rangle$, which can be either bosonic or
fermionic, which obeys $Q|\cdot\rangle=\overline{Q}|\cdot\rangle=0$. The the
algebra (4) implies that $|\cdot\rangle$ is an eigenstate of $H$ with
eigenvalue $E=0$. Thus the complete spectrum of the theory consists of
positive energy states which come in boson-fermion pairs, and zero energy
states which need not be paired.

The basic supersymmetric index, called the _Witten index_ Witten (1982), is
defined as

$Z\;=\;\mbox{Tr}_{\mathcal{H}}\,(-1)^{F}\,e^{-\beta H}\,.$ (5)

This quantity is similar to the thermal partition function that one encounters
in statistical physics, with $\beta$ being the inverse temperature. In
particular, states with large energies are suppressed by the exponential
factor, which usually leads to a convergent trace. There is, however, a big
difference compared to the thermal partition function. Since states with $E>0$
come in pairs with opposite values of $(-1)^{F}$, they do not contribute to
the spectrum. The only contribution to the index comes from zero-energy
states, i.e.

$Z\;=\;n_{0}^{b}-n_{0}^{f}\,,$ (6)

the difference between the number of bosons and fermions with zero energy,
which, in particular, is independent of $\beta$!

Now consider the effect of changing some parameter of the system in a
supersymmetric manner, that is to say, the supercharges and the Hamiltonian
are functions of the parameter, but obey the algebra (4) for all values of the
parameter. The energy levels (the eigenvalues of the Hamiltonian)
correspondingly change, but bosons and fermions with $E>0$ are still paired.
Any change in the $E=0$ spectrum consists of states moving down from $E>0$ to
$E=0$, or leaving it in the opposite direction, but this can only happen in
boson-fermion pairs. Thus, although the total number of (bosonic + fermionic)
states at $E=0$ could change, their difference is invariant.

In this simple example, one sees that the supersymmetric index only receives
contributions from the one-dimensional zero-energy representations, which are
annihilated by the supercharge operators. In the more general setting of
supersymmetric quantum field theories, one has multiple supercharges. The
types of representations of the supersymmetry algebra in this case are
labelled by the fraction of the total number of supercharge operators that are
annihilated in that representation. This fraction can range from 0, the so-
called non-BPS or long representations, to 1, which are the zero energy vacuum
states as in the simple example above. However, now one also has BPS states
with non-zero energy, which are annihilated by only some of the supercharges.
Correspondingly, one has more general types of supersymmetric indices, which
receive contributions only from states that are annihilated by at least the
same fraction of the supercharges. The precise details of these supersymmetric
indices depends on the full symmetry algebra of the theory. In the context of
asymptotically flat space (as discussed in this article) the symmetry algebra
is super-Poincaré, in which case the relevant indices are called helicity
supertraces Kiritsis (1998).

The micro/macro picture revisited

Now consider superstring theory and a particular supersymmetric index that
counts a certain type of BPS state. Since the index does not change under
changes of coupling, one can revisit the idea of Figure 1 and ask what happens
to these states in the macroscopic regime $g_{\text{s}}N\gg 1$? The complete
answer depends on the details of the system but, happily, there are situations
where one has a black hole solution of string theory with the same charges as
those of the original microscopic quantum states. In such a situation, one can
try to compare the microscopic and macroscopic pictures more carefully.

Upon reviewing the above logic carefully, one notices that the original
problem seems to have been changed at two levels. Firstly, the states that are
being considered are annihilated by a certain fraction of supercharges, and
this should be reflected in the gravitational theory. Indeed, the effective
macroscopic theory—called supergravity—also has a notion of supersymmetry that
applies to the gravitational variables, and there are black hole solutions to
supergravity—called BPS black holes—that are annihilated by some fraction of
the set of supercharges. BPS black holes are a very special type of black hole
solutions. One of their peculiarities is that they are necessarily have zero
temperature. In this respect they are different from Schwarzschild black holes
(as initially mentioned in the introduction). Nevertheless, BPS black holes
have an event horizon and do carry entropy!999The apparent violation of the
lore/notion of the third law of thermodynamics is avoided because the
supersymmetry of such solutions allows for a large degeneracy.

The second level at which the initial problem seems to have changed is that
one is no longer discussing the total number of states (and the consequent
Boltzmann entropy), but a modified counting of the index of the type (6). In
the early days of black hole microstate counting in string theory this aspect
was not quite understood fully and it was even questioned whether the
agreement obtained by Strominger Vafa (1996) was only some kind of accidental
or approximate agreement in the limit of large charges. More recent
developments have made it clear that this is not an accident and there is a
more detailed and beautiful mechanism underlying the system.

The basic physics picture is as follows. Typically, boson-fermion pairs do
pair up and leave the spectrum upon reaching strong coupling. Assuming for the
moment that the initial microscopic system has more bosons than fermions, one
is left with a final ensemble in which $n^{f}=0$ and the index only counts
bosonic states, i.e. a genuine count in the sense of Boltzmann. On the
macroscopic side, it is a non-trivial fact Sen (20091), Dabholkar, Gomes,
Murthy Sen (2011), Iliesiu . (2022) that the BPS black holes are indeed made
up of only bosonic states (this is sometimes referred to as a vanishing
theorem)!101010 The bosonic nature of the states has been verified in
Dabholkar, Gomes, Murthy Sen (2011) for various examples of BPS black holes in
string theory. The theoretical argument assumes the decoupling of the quantum
states of the BH from the fluctuations of the fields outside the horizon. This
was recently shown to hold for supersymmetric BHs in string theory Iliesiu .
(2022). Thus, one has a logic of the following sort:

microscopic index = gravitational index = BH entropy ,

(supersymmetry) (vanishing theorem)

so that the supersymmetric index should indeed capture the Boltzmann entropy
of supersymmetric black holes.

With the formalism in place, one proceeds to calculate the supersymmetric
index at large charges and check whether it agrees with the Bekenstein-Hawking
area formula for the entropy.

##  5 The counting of black hole microstates in string theory

As mentioned in the introduction, black hole microstates in string theory are
quantum-mechanical bound states of fluctuations of fundamental objects of
string theory like strings and branes. We first consider a toy model,
consisting of a string wrapping a circle of radius $R$, which nicely
illustrates a number of features of the microscopic counting of black hole
microstates in string theory.

The string is a $1$-space-dimensional object evolving in time, whose points
are labelled by the coordinates $x^{0}$ (time) and $x^{1}$ (space). The
position on the circle is parameterized by the map
$X(x^{0},x^{1})\in\mathbb{R}/2\pi\mathrm{i}R\mathbb{Z}$. The classical
equation of motion on the free string is the free Laplacian equation (with
$\partial_{i}\equiv\partial/\partial x^{i}$),

$\bigl{(}\partial_{0}^{2}-\partial_{1}^{2}\bigr{)}X(x^{0},x^{1})\;=\;0\,.$ (7)

The general solution of this equation is

$X(x^{0},x^{1})\;=\;\frac{n}{R}x^{0}+wRx^{1}+\,\sum_{k\in\mathbb{Z}\atop k\neq
0}\,\frac{\mathrm{i}}{\sqrt{k}}\Bigl{(}\alpha_{k}\,e^{-\mathrm{i}k(x^{0}-x^{1})}\,+\,\widetilde{\alpha}_{k}\,e^{-\mathrm{i}k(x^{0}+x^{1})}\,\Bigr{)}\,.$
(8)

Here $n,w\in\mathbb{Z}$ are interpreted as the momentum of the center of mass
of the string around the circle and the winding of the string around the
circle, respectively. The terms proportional to
$e^{-\mathrm{i}k(x^{0}-x^{1})}$ and $e^{-\mathrm{i}k(x^{0}+x^{1})}$ are called
left- and right-moving excitations, respectively. We focus on the left-moving
excitations, and the right-movers are treated in an analogous manner. In the
classical-mechanical theory, the coefficients $\alpha_{k}$ are complex numbers
obeying $\overline{\alpha_{k}}=\alpha_{-k}$. In the quantum theory, the field
$X$ and therefore the coefficients $\alpha_{k}$, $\widetilde{\alpha}_{k}$ are
promoted to operators on a Hilbert space obeying
$\alpha_{k}^{\dagger}=\alpha_{-k}$. The quantization process leads to the
following commutation relations

$[\alpha_{k},\alpha_{-k}]\;=\;1\,,\quad k\;=\;1,2,3,\dots\,,$ (9)

with all other commutators vanishing.

The algebra obeyed by the pair $(\alpha_{k},\alpha_{-k})$, defines the quantum
bosonic oscillator, which is one of the simplest known quantum-mechanical
systems. The Hilbert space $\mathcal{H}_{k}$ associated to this system is
described as follows. There is a special _vacuum_ state $|0\rangle_{k}$ of
this system obeying $\alpha_{k}|0\rangle_{k}=0$, $k>0$, and the Hilbert space
is given by the span over $\mathbb{C}$ of the following tower of states,

$|m\rangle_{k}\;=\;\alpha_{-k}^{m}|0\rangle_{k}\,,\qquad m=0,1,2,\dots\,.$
(10)

The Hamiltonian of this system is given by

$H_{k}\;=\;k\bigl{(}\alpha_{k}\,\alpha_{-k}+\tfrac{1}{2}\bigr{)}\,,$ (11)

and the energy eigenvalues of the states (10) can be easily calculated using
the commutation relation in (9) to be

$H_{k}|m\rangle_{k}\;=\;k(m+\tfrac{1}{2})|m\rangle_{k}\,.$ (12)

The thermal partition function of this harmonic oscillator is given by

$\mbox{Tr}_{\mathcal{H}_{k}}\,e^{-\beta
H_{k}}\;=\;\sum_{m=0}^{\infty}e^{-\beta k(m+\frac{1}{2})}\;=\;\frac{e^{-\beta
k/2}}{1-e^{-\beta k}}\,.$ (13)

The full left-moving Hilbert space $\mathcal{H}$ of the theory is the Fock
space built out of these individual Hilbert spaces $k\in\mathbb{Z}^{+}$, and
the total left-moving Hamiltonian is given by $H=\sum_{k=1}^{\infty}H_{k}$
where $H_{k}$ is now interpreted to act as the identity operator on all the
Hilbert subspaces $k^{\prime}\neq k$. The thermal partition function on this
Hilbert space is the product over all $k>0$ of the thermal partition functions
(13). The exponents of the numerators of (13) add up to give the energy
eigenvalue of the vacuum state, which, according to the above discussion, is
naively given by $\sum_{k=1}^{\infty}\frac{k}{2}$. This sum is interpreted
using the zeta-function regulator111111Recall that
$\zeta(s)=\sum_{k=1}^{\infty}k^{-s}$ is defined for $\mbox{Re}(s)>1$, and can
be analytically continued to a meromorphic function on the whole complex
plane. to be $\zeta(-1)=-1/24$. Setting $\beta=-2\pi\mathrm{i}\tau$,
$q=e^{2\pi\mathrm{i}\tau}$, one obtains,

$\begin{split}\mbox{Tr}_{\mathcal{H}}\,q^{H}&\;=\;q^{-\frac{1}{24}}\prod_{k=1}^{\infty}\,\frac{1}{1-q^{k}}\;=\;\sum_{n=0}^{\infty}\,p(n)\,q^{n-\frac{1}{24}}\\\
&\;=\;q^{-\frac{1}{24}}\bigl{(}1+q+2q^{2}+3q^{3}+5q^{4}+\dots\bigr{)}\,.\end{split}$
(14)

The coefficient $p(n)$ in the above expansion is the number of partitions of
the integer $n$, and can be expressed as the integral

$p(n)\;=\;\frac{1}{2\pi\mathrm{i}}\oint_{C}\frac{dq}{q^{n+1}}\,\prod_{k=1}^{\infty}\,\frac{1}{1-q^{k}}$
(15)

over a contour lying inside the unit disk and surrounding the origin. The
asymptotic behavior of $p(n)$ as $n\to\infty$ is well-known, and can be
calculated using the well-known relation of the above infinite product to the
Dedekind $\eta$-function

$\frac{1}{\eta(\tau)}\;=\;q^{-\frac{1}{24}}\prod_{k=1}^{\infty}\,\frac{1}{1-q^{k}}\,,\qquad
q=e^{2\pi\mathrm{i}\tau}\,.$ (16)

As $n\to\infty$, the integral (15) is dominated by the singular behavior of
the integrand near the unit circle, which is governed by the amazing symmetry
property obeyed by the $\eta$-function, which is expressed as the functional
equation

$\eta(-1/\tau)\;=\;\sqrt{-\mathrm{i}\tau}\,\eta(\tau)\,.$ (17)

Using this, one obtains

$\log p(n)\sim 2\pi\sqrt{n/6}+\dots$ (18)

The actual counting of supersymmetric microstates in string theory is more
complicated than the above toy example, but retains many of its basic
features. As mentioned above, we consider compactifications of ten-dimensional
string theory on a six-manifold $\mathcal{M}_{6}$ to obtain a geometry of the
form $\mathbb{R}^{1,3}\times\mathcal{M}_{6}$. There are, in fact, multiple
versions of string theory, and we focus on one theory called Type IIB string
theory. It is known that Type IIB string theory compactified on a Calabi-Yau
3-fold leads to a supersymmetric macroscopic theory in four dimensions
described by General Relativity coupled to multiple gauge fields, scalar
fields, and fermions.

This theory admits supersymmetric black hole solutions, labelled by their
charges under all the gauge fields of the theory. The corresponding
microscopic states involves excitations of string theory called branes, which
are higher-dimensional generalizations of strings, wrapping cycles of the
Calabi-Yau manifold. Counting the number of such microscopic states for a
given vector of charges is a complicated problem, and in general one only has
estimates for this number given in Maldacena . (1997). However, it can be
solved exactly when the Calabi-Yau is sufficiently simple. Typically, in these
cases, the generating functions of microstates consist of combinations of
Dedekind $\eta$-functions and classical $\theta$-functions.

The simplest situation is when the Calabi-Yau manifold is $T^{6}$. The
resulting macroscopic theory is called $\mathcal{N}=8$ supergravity, which
admits supersymmetric black hole solutions labelled by one positive integer
$N$ (cf. Fig. 1), whose Bekenstein-Hawking entropy is given by

$S^{\text{class}}_{\text{BH}}(N)\;=\;\pi\sqrt{N}\,,$ (19)

The corresponding microstates are described by left-moving oscillations of an
effective string wrapping some cycles in $T^{6}$. The single circle in the toy
example is now replaced by multiple free bosons and also fermions, and the
generating function of the microscopic number of states121212As explained
above, one really calculates the microscopic index. In this case, it is known
that the microscopic index is related to the number of black hole microstates
as $(-1)^{N+1}d_{\text{micro}}(N)$, due to fermionic zero-energy modes that
one has to factor out Sen (20091), Dabholkar, Gomes, Murthy Sen (2011). The
resulting generating functions $(-1)^{a}\theta_{a}(\tau)/\eta(\tau)^{6}$,
$a=1,2$ transform as a vector under $SL(2,\mathbb{Z})$ modular
transformations, and they can be assembled into a single Jacobi form Eichler
Zagier (2013). is given by

$\sum_{N=-1\atop
N\;\equiv\;(a+1)\;\text{mod}2}^{\infty}\,d_{\text{micro}}(N)\,q^{N/4}\;=\;\frac{\theta_{a}(\tau)}{\eta(\tau)^{6}}\,,$
(20)

where $\theta_{a}(\tau)=\sum_{n\in\mathbb{Z}+a/2}q^{n^{2}}$, $a=0,1$. The
asymptotic behaviour of states can be calculated in a similar manner as for
integer partitions in (15), using the symmetry property of the $\theta$ and
$\eta$ functions relating the behaviors at $\tau$ and $-1/\tau$, and the
result is, as $N\to\infty$,

$\log d_{\text{micro}}(N)\;\sim\;\pi\sqrt{N}+\dots\,.$ (21)

The agreement of the statistical (Boltzmann) entropy of microstates (21) and
the thermodynamic entropy of black holes (19), as pictured in Figure 1, is a
major success of string theory. The generating function of the microscopic
degeneracy of states in this example is built out of very special functions
like the Dedekind $\eta$-function and the $\theta$-functions, whose symmetry
under $\tau\to-1/\tau$ is very useful to estimate the growth of the degeneracy
of states. These functions are examples of a very special type of functions
called modular forms which are symmetric under an infinite group of symmetries
called the modular group.

##  6 Modular forms and the Hardy-Ramanujan-Rademacher expansion

In this section we switch gears and discuss some mathematical aspects of the
theory of modular forms. In the following section we discuss some consequences
of this theory which have implications on the quantum aspects of black hole
entropy.

The theory of modular and automorphic forms is a vast field of mathematics,
and we only discuss a very small part of it which is needed for the sequel. In
particular, we discuss holomorphic functions of one variable
$\tau\in\mathbb{H}$, the upper half-plane. Recall that the modular group
$SL(2,\mathbb{Z})$, the group of $2\times 2$ matrices with integer entries and
determinant one, acts on $\mathbb{H}$ by fractional linear transformations.
The central defining property of a modular form $f(\tau)$ is its
transformation under this action,

$f\biggl{(}\frac{a\tau+b}{c\tau+d}\biggr{)}\;=\;(c\tau+d)^{k}\,f(\tau)\,,\qquad\biggl{(}\,\begin{matrix}a&b\\\
c&d\end{matrix}\,\biggr{)}\in SL(2,\mathbb{Z})\,.$ (22)

Here $k$ is called the  _weight_ of the modular form. In writing (22), the
weight is assumed to be integer. The theory of modular forms exists for half-
integer and even more general values of $k$, and also for congruence subgroups
of $SL(2,\mathbb{Z})$. The details in each case vary: for example, there could
be roots of unity appearing in the transformation law.

Taking the $SL(2,\mathbb{Z})$ matrix in (22) to be
$\left(\begin{smallmatrix}1&1\\\ 0&1\end{smallmatrix}\right)$, one obtains the
periodicity property $f(\tau+1)=f(\tau)$ which, together with the holomorphy,
leads to the Fourier expansion

$f(\tau)\;=\;\sum_{n\in\mathbb{Z}}d(n)\,q^{n}\,,\qquad
q\;=\;e^{2\pi\mathrm{i}\tau}\,.$ (23)

This expansion, called the $q$-expansion in this context, is at the heart of
many of the relations of modular forms with other problems in mathematics and
physics. In particular, the coefficients $d(n)$ are often integers, and can
sometimes be interpreted as (virtual) dimensions of certain Hilbert spaces.
Indeed, in the context of our main interest here, the coefficients are
supersymmetric indices corresponding to black hole microstates in string
theory, as explained in the previous section.

When the sum in (23) is restricted to $n\geq 0$, the corresponding functions
are holomorphic on the upper half-plane, and are called holomorphic modular
forms. Those with the stronger restriction $n>0$ are called cusp forms.
However, it can be proved that the growth of coefficients $d(n)$ as
$n\to\infty$ of holomorphic modular forms is bounded by a polynomial function.
The exponential growth of $d(n)$ that is characteristic of black hole
microstates appears in modular forms with negative powers of $n$ in its
$q$-expansion (such functions are called _weakly holomorphic_) in which case
the function itself diverges as $\tau\to\mathrm{i}\infty$. In order to obtain
a controllable analytic theory of such functions, one restricts the sum in
(23) in the negative direction to $n\geq n_{0}$ for some fixed $n_{0}<0$. This
is indeed the case for the examples in the previous section, e.g.
$n_{0}=-1/24$ in (16)131313In fact, it is $1/\eta(\tau)^{24}$ that is a
modular form on $SL(2,\mathbb{Z})$, and the transformation (22) of
$1/\eta(\tau)$ involves a $24^{\text{th}}$ root of unity..

Asymptotic expansion of coefficients of modular forms

The leading asymptotic behavior of the coefficients of weakly holomorphic
modular forms can be estimated quite easily, as seen in the examples of the
previous section. In fact, the modular symmetry is much more powerful, and
leads to an exact analytic formula for the coefficients of functions of the
type (16) and (20), called the _Hardy-Ramanujan-Rademacher formula_ after
Hardy Ramanujan (1918), Rademacher (1937). The formula for the coefficients of
(20) is:

$d_{\text{micro}}(N)\;=\;2{\pi}\,\Bigl{(}\frac{\pi}{2}\Bigr{)}^{7/2}\,\sum_{c=1}^{\infty}c^{-9/2}\,K_{c}(N)\;\widetilde{I}_{7/2}\Bigl{(}\frac{\pi\sqrt{N}}{c}\Bigr{)}\,.$
(24)

Here $\widetilde{I}_{\rho}$, a modification of the standard $I$-Bessel
function, is given by the following integral formula,

$\widetilde{I}_{\rho}(z)\;=\;\Bigl{(}\frac{z}{2}\Bigr{)}^{-\rho}I_{\rho}(z)\;=\;\frac{1}{2\pi\mathrm{i}}\int_{\epsilon-\mathrm{i}\infty}^{\epsilon+\mathrm{i}\infty}\,\frac{d\sigma}{\sigma^{\rho+1}}\exp\Bigl{(}\sigma+\frac{z^{2}}{4\sigma}\Bigr{)}\,.$
(25)

The Kloosterman sum $K_{c}$ in (24) is a sum over phases, which is a class of
functions appearing in many number-theoretic contexts. The reader is referred
to Iliesiu . (2022) for the details of this example.

##  7 Quantum corrections to black hole entropy

The discussion of Sections 5 and 6 showed that the macroscopic, thermodynamic
black hole entropy (2) is indeed reproduced by the statistical Boltzmann
entropy of the corresponding black hole microstates in the thermodynamic limit
when the black hole size becomes very large, i.e.,

$\log d_{\text{micro}}(N)\;=\;S^{\text{class}}_{\text{BH}}(N)+\dots\,,$ (26)

where the integer $N$ is the quantized charge of the black hole. This match is
one of the big successes of string theory.

The quantities appearing on the two sides of (26) have very different
mathematical natures as well as physical origins: the left-hand side is
defined by the arithmetic function $d_{\text{micro}}:\mathbb{N}\to\mathbb{N}$
that counts the dimension of the microscopic Hilbert space, and has its origin
in quantum-statistical mechanics, while the right-hand side is defined by the
smooth real function $A_{H}:\mathbb{R}^{+}\to\mathbb{R}^{+}$ that measures the
area of the BH horizon in semi-classical General Relativity, and is
intrinsically geometric in origin. It is therefore very surprising that the
two quantities are equal to each other! Is there a deeper explanation of this
equality?

One can try to sharpen this question as follows. The dots on the right-hand
side of (26) refer to the fact that the area formula is really an
approximation, as $N\to\infty$, to the logarithm of the exact integer given by
the left-hand side.141414The difference in the nature of the domain of the two
functions is well-understood: it is explained by the fact that the charge of
the black hole in classical general relativity is continuous, while in the
quantum theory it should be quantized in units of the fundamental charge.  One
is thus led to ask: is it an approximation to an exact formula? Note that the
left-hand side is perfectly well-defined for any integer $N$, so the question
is whether there is an exact quantity defined in the macroscopic,
gravitational theory, which agrees with the integer $d_{\text{micro}}(N)$
appearing on the left-hand side. The pursuit of the answer to this question
has led to a very beautiful story spanning a few decades of research in string
theory, and has resulted in non-trivial relations with number theory referred
to in the introduction. The rest of this section presents a sketch of the
answer using the prototype system mentioned in the previous section, along
with references to the literature for more detailed explanations.

The Hardy-Ramanujan-Rademacher expansion (24) already suggests a possible
answer to the above questions at a mathematical level: the integer
$d_{\text{micro}}(N)$ is expressed as the sum of a series of analytic
functions evaluated at discrete values of the argument controlled by $N$. This
series can be interpreted precisely as arising from quantum gravitational
corrections to the Bekenstein-Hawking area formula for the thermodynamic
entropy of the black hole, as anticipated in Dijkgraaf . (2000); Ooguri .
(2004); Dabholkar . (2005) and as explained below.

Recalling that $I_{\rho}(z)\sim e^{z}$, $z\to+\infty$, it is clear from the
form of the expansion (24) that the $c>1$ terms are all exponentially
suppressed compared to the $c=1$ term. One already saw that the leading
asymptotic growth of $d_{\text{micro}}(N)$, given in Equation (21), agrees
precisely with the Bekenstein-Hawking entropy formula (19) for the BH entropy.

The starting point for the quantum corrections to this formula is the
observation that the spacetime geometry has a special form near the horizon of
supersymmetric black holes. In particular, the manifold is a product of AdS2
(two-dimensional constant negative curvature spacetime) and a compact manifold
(S2 for the BH under consideration here). Recall that the equations of general
relativity are a set of coupled non-linear second-order PDEs. Black hole
solutions usually depend on the values of the various fields at the asymptotic
boundary of spacetime as well as the conserved charges carried by the black
hole (which play the role of integration constants). The near-horizon geometry
of supersymmetric black holes, by contrast, is governed by the fixed points of
a set of first-order differential equations, and is completely fixed by the
charges of the black hole. This is the _black hole attractor mechanism_ of
Ferrara . (1995).

The leading-order quantum correction comes from summing the virtual loops of
all the quantum excitations (graviton, photons,…) in the AdS${}_{2}\times$ S2
background, using a non-trivial adaptation of standard quantum field theory
methods in Banerjee . (2011). The result agrees precisely with the first
correction to the leading order result in the asymptotic expansion of the
Bessel function in (24):

$\log\,d_{\text{micro}}(N)\;=\;\pi\sqrt{N}-2\log
N+\text{O}\bigl{(}1/\sqrt{N}\bigr{)}\,.$ (27)

The next corrections to $d_{\text{micro}}$ coming from (24) are in the form of
negative powers of $N$. The physical origin of these terms are corrections to
the local effective action of two-derivative supergravity Lopes Cardoso .
(1999) arising from integrating out the massive modes of the string theory in
the background of the black hole. They affect the BH entropy in two ways:
firstly, in the presence of higher-derivative operators in the gravitational
effective action the Bekenstein-Hawking formula needs to be replaced by the
Wald entropy formula Wald (1993); Iyer Wald (1994) and, secondly, the black
hole solution itself gets corrected by the higher-order effects. The reader is
referred to Sen (20081) for a nice review of these ideas and calculations. The
summary is that all such quantum-gravitational calculations in string theory
agree with the corresponding microscopic results!

Continuing on to a better approximation to the integer degeneracy, one
considers the first ($c=1$) term in the Rademacher expansion (24), i.e.,

$d_{\text{micro}}(N)\;=\;\frac{\pi^{9/2}}{8}\,\widetilde{I}_{7/2}\bigl{(}\pi\sqrt{N}\bigr{)}+\text{O}\bigl{(}e^{\sqrt{N}/2}\bigr{)}\,.$
(28)

This result is interpreted in gravity as the result of summing up the entire
perturbation series for the quantum entropy of the $\frac{1}{8}$-BPS BH. Such
an interpretation is made possible in the framework of the quantum black hole
entropy expressed as a functional integral over the fluctuating fields of
supergravity around the near-horizon AdS${}_{2}\times$ S2 background Sen
(20082, 20092). The calculation of this functional integral to all orders in
perturbation theory is performed by using the technique of supersymmetric
localization applied to the gravitational path integral Dabholkar, Gomes
Murthy (2011); Dabholkar . (2013); de Wit . (2018); Jeon Murthy (2019). This
reduces the integral over the infinite-dimensional field space to (in this
case) a one-dimensional integral which, with a reasonable assumption about the
measure, agrees with the representation (28) of the $I$-Bessel function.

Now that the asymptotic series coming from perturbation theory has been summed
up into the Bessel function, one is now in a position to rigorously discuss
the exponentially suppressed corrections coming from the terms $c=2,3,\dots$
in (24). It was shown in Dabholkar . (2015) that the $c^{\text{th}}$ term has
the same form as the contribution of a $\mathbb{Z}/c\mathbb{Z}$ orbifold in
string theory which contribute to the functional integral around asymptotic
AdS${}_{2}\times$ S2. The Kloosterman sum $K_{c}(N)$ is precisely reproduced
by the functional integral of a certain Chern-Simons theory which emerges in
this background.

Finally, Iliesiu . (2022) revisited and clarified various aspects of the
calculation of the supergravity path integral using supersymmetric
localization, and applied it to the fluctuations of the gravitational fields
in the background of the $\mathbb{Z}_{c}$ orbifold. It is important in this
calculation to take into account the subtle effects of a certain mode of the
gravitational called the Schwarzian mode151515See Mertens Turiaci (2023) for a
review of recent developments including references, and Heydeman . (2022) for
the details of the supersymmetric version of the Schwarzian relevant for the
discussion in this article.. This gives a contribution $1/c$ to the functional
integral (and, therefore, does not affect the $c=1$ answer). Upon putting
everything together, one obtains precisely the Bessel function
$\widetilde{I}_{7/2}(\pi\sqrt{N/c})$. In this manner, the physical quantum-
gravitational calculation reproduces, term-by-term, the Hardy-Ramanujan-
Rademacher expansion.

##  8 Mock modular forms

The results described in the previous section show that the modular symmetry
of the microscopic partition function acts as a powerful guiding principle for
the quantum-gravitational corrections to the macroscopic black hole. This
power is based on the agreement of the microscopic and macroscopic pictures,
which in turn relies on the invariance of the supersymmetric index under a
change of parameters of the theory as explained earlier. Thus one has a
perfect one-to-one correspondence between the microstates of the strings and
branes, and those of the black hole.

However, there is a subtlety in the statement of invariance of the index.
Recall that the main argument underlying its invariance is that a change of
parameters shifts the energy levels of the supersymmetric theory, but states
enter or exit the BPS spectrum in boson-fermion pairs, thus not affecting the
index $n^{B}-n^{F}$. It turns out that there are special codimension-one
surfaces of the parameter space of the theory, called _walls_ , upon crossing
which new BPS solutions enter the spectrum of the theory, which could be
purely bosonic or fermionic. Typically, the corresponding field configurations
are normalizable states in the Hilbert space on one side of the wall and
become non-normalizable on the other side. The index therefore jumps at these
walls and this phenomenon is called _wall-crossing_.

The wall-crossing phenomenon manifests itself in supergravity as multi-black
hole bound-state configurations carrying the same total charges as the
original black hole (Fig. 2), that appear on moving across certain co-
dimension-one surfaces (walls) in the gravitational parameter space Denef
Moore (2011); Sen (20081). Our desired single-black-hole states are therefore
only part of the full partition function, and therefore are not, a priori,
expected to preserve the symmetries of the full theory. For the highly
symmetric situation of black holes in $T^{6}$ compactifications of string
theory discussed earlier, it was shown in Dabholkar . (2010) that BH bound
state solutions do not contribute to the relevant supersymmetric index, but in
general the wall-crossing phenomenon breaks the modular symmetry of the
theory. This seems like a potential disaster for the guiding principle!

Figure 2: Wall-crossing in supergravity. The vertical line denotes a co-
dimension one wall in the parameter space of solutions. On the left side of
the wall the only solution is the single black hole. On the right side there
is an additional solution which is a bound state of two black holes, which is
a stationary solution of supergravity with angular momentum carried by
electromagnetic fields.

The natural question is: can one calculate the supersymmetric index
corresponding to the single BH? Does that have any interesting modular-like
property? In general, this is a very difficult question to answer, but in the
context of the next simplest theory called $\mathcal{N}=4$ supergravity
(coming from string compactifications on $K3\times T^{2}$), one has a complete
solution to this problem (see the review Sen (20081)). In this case one can
show that there is only a specific type of 2-center BH bound state that
contributes to the index Dabholkar . (2010). In fact, at any point in the
parameter space of string theory, the total index can be written as

$d_{\text{micro}}(n)\;=\;d_{\text{1-BH}}(n)+d_{\text{2-BH}}(n)$ (29)

where one can explicitly calculate the three components of this equation
separately.

The remarkable fact found in Dabholkar . (2012) is that $d_{\text{1-BH}}(n)$
is the Fourier coefficient of a _mock modular form_. What this means is that
one can add a specific correction term (called the _shadow_) to recover
modular symmetry, analogous to quantum anomalies in physics. Examples of such
functions (without a definition!) were first discovered by Ramanujan in his
famous last letter to Hardy, and S. Zwegers in his PhD thesis Zwegers (2008)
gave a definition and explained the structure hidden in Ramanujan’s examples
(see Zagier (2009); Bringmann . (2017)).

The relations of black hole partition functions and mock modular forms were
formalized and generalized in Dabholkar . (2012) into a set of theorems that
apply to a very large class of functions called meromorphic Jacobi forms (one
sub-family of which is given by the black hole partition functions of
$\mathcal{N}=4$ string theory). These theorems have been used to provide exact
analytic formulas for the black hole degeneracies extending the Hardy-
Ramanujan-Rademacher expansion Bringmann Mahlburg (2011); Murthy Reys (2016);
Ferrari Reys (2017); Chowdhury . (2020); Lopes Cardoso . (2021); Cardoso .
(2021), and to prove and analyze conjectures about the positivity of black
hole microstate degeneracies Sen (2011); Bringmann Murthy (2013);
Chattopadhyaya David (2019, 2021); Govindarajan . (2023).

These theorems also led to the construction of an infinite number of mock
modular forms in Dabholkar . (2012), generalizing the known special examples
developed in mathematics starting from the $q$-series of Ramanujan. The above
developments also played a role in the remarkable discoveries in Cheng .
(2014) of _Umbral moonshine_ relations between mock modular forms and discrete
groups.

##  9 Conclusion, and broader relations of modular forms and string theory

We have described one small corner of the relations of modular forms and
modular symmetry to the physics of black holes in string theory, and discussed
how they (i) provide a guiding principle for the quantum-gravitational effects
in black holes, (ii) are a powerful tool for calculating black hole entropy,
and (iii) have fed back and led to new developments in mathematics.

There are many other very interesting relations of modular forms and string
theory (and, more generally, physics) that could not be covered in this brief
article. In particular, modular forms have made an appearance in string theory
from the very early days through the world-sheet torus amplitudes of free
strings, and to estimate the Hagedorn growth of states of strings (see Green .
(1988)). Even before that, the simplest manifestation of modular invariance
had appeared in the analysis of 2d CFT in Cardy (1986) and led to the famous
Cardy formula.

In the 35 years since then, modular, and more generally automorphic forms,
have made multiple appearances in string theory (see D’Hoker Kaidi (2022) for
a recent summary). They have been used as a powerful tool e.g. in the
description of string worldsheet amplitudes (see Fleig . (2018)). The
corresponding symmetry has also played the role of a fundamental principle as
dualities e.g. the $SL(2,\mathbb{Z})$ duality of supersymmetric Yang-Mills
theory (generalizing Montonen-Olive duality) in Vafa Witten (1994), and of
type-IIB string theory in Sen (1994), which has given rise to strong
constraints on the structure of the effective action of string and M-theory
(see Green Vanhove (2006), Green, Miller . (2010), Green, Russo Vanhove
(2010), Pioline (2010), Bossard . (2020), Liu . (2022), and references
therein).

As discussed above, black hole microstate degeneracies are captured precisely
by a modular form in $T^{6}$ compactifications of string theory
($\mathcal{N}=8$ supergravity) and by mock modular forms in $K3\times T^{2}$
compactifications ($\mathcal{N}=4$ supergravity). Modular and Jacobi
structures in more general Calabi-Yau compactifications ($\mathcal{N}=2$
supergravity) have been discussed in Gaiotto . (2007), Klemm . (2012), Huang .
(2015), Haghighat . (2016), Bouchard . (2017). Relations between black hole
partition functions, functions called indefinite theta series (which are
closely related to mock modular forms), and mock modular forms of “higher
depth” have been found in Manschot (2010), Manschot . (2011), Alexandrov .
(2013), Alexandrov Pioline (2019).

Mock modular forms have also made other appearances in conformal field theory
and string theory, seemingly unrelated to black holes. This includes the early
work on $\mathcal{N}=4$ superconformal characters Eguchi Taormina (1988),
Mathieu moonshine Eguchi . (2011) and Umbral moonshine and Niemeier lattices
Cheng . (2014, 2013), and relations to K3 surfaces Gaberdiel . (2012);
Taormina Wendland (2015). Another important physical situation where mock
modular forms have made an appearance is in the context of the completion of
the partition function of Vafa-Witten theory Dabholkar . (2020) , and more
generally four-dimensional $\mathcal{N}=2$ gauge theories Korpas . (2019);
Manschot Moore (2021); Aspman . (2022); Korpas (2022).

Physically, the appearance of mock modular behavior is related to the non-
compactness of field space, and consequent holomorphic anomalies. This has
been studied in the context of supersymmetric quantum mechanics with non-
compact target Murthy Pioline (2018); Dabholkar . (2019), non-compact sigma
models and CFT Eguchi Sugawara (2011); Troost (2010); Ashok Troost (2011),
Murthy (2014); Ashok . (2014); Harvey, Lee Murthy (2015); Gupta Murthy (2017);
Nazaroglu (2018); Kumar Gupta . (2019), and string theory Harvey Murthy
(2014); Cheng Harrison (2015); Harvey, Murthy Nazaroglu (2015), and AdS/CFT
Manschot Moore (2010).

More broadly, there are other extremely interesting topics at the modular-
forms-physics interface, which are slowly being uncovered. These include the
discussion of black holes and class groups Benjamin . (2018), the relation of
quantum modular forms and 3-manifold invariants Gukov . (2020); Cheng .
(2019); Garoufalidis Zagier (2021), modularity of periods at special points of
Calabi-Yau moduli spaces Moore (1998); Candelas . (2020), and probably many
other treasures that are waiting to be discovered!

## References

  * Alexandrov . (2013) Alexandrov:2012auAlexandrov, S., Manschot, J. Pioline, B.  2013\.  D3-instantons, Mock Theta Series and Twistors D3-instantons, Mock Theta Series and Twistors.  JHEP04002.  10.1007/JHEP04(2013)002
  * Alexandrov Pioline (2019) Alexandrov:2018lgpAlexandrov, S. Pioline, B.  2019\.  Black holes and higher depth mock modular forms Black holes and higher depth mock modular forms.  Commun. Math. Phys.3742549–625.  10.1007/s00220-019-03609-y
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