BPS Spectra, Barcodes and Walls

Michele Cirafici

Center for Mathematical Analysis, Geometry and Dynamical Systems,
Instituto Superior Técnico, Universidade de Lisboa,
Av. Rovisco Pais, 1049-001 Lisboa, Portugal
and
Institut des Hautes Études Scientifiques,
Le Bois-Marie, 35 route de Chartres, F-91440 Bures-sur-Yvette, France
Email<EMAIL_ADDRESS>
BPS spectra give important insights into the non-perturbative regimes of
supersymmetric theories. Often from the study of BPS states one can infer
properties of the geometrical or algebraic structures underlying such
theories. In this paper we approach this problem from the perspective of
persistent homology. Persistent homology is at the base of topological data
analysis, which aims at extracting topological features out of a set of
points. We use these techniques to investigate the topological properties
which characterize the spectra of several supersymmetric models in field and
string theory. We discuss how such features change upon crossing walls of
marginal stability in a few examples. Then we look at the topological
properties of the distributions of BPS invariants in string compactifications
on compact threefolds, used to engineer black hole microstates. Finally we
discuss the interplay between persistent homology and modularity by
considering certain number theoretical functions used to count dyons in string
compactifications and by studying equivariant elliptic genera in the context
of the Mathieu moonshine.

###### Contents

  1. 1 Introduction
  2. 2 BPS states and wall-crossing
  3. 3 Persistent homology
    1. 3.1 Homology of simplicial complexes
    2. 3.2 Persistent homology and barcodes
    3. 3.3 Barcodes and topological features
    4. 3.4 An example
  4. 4 BPS spectra in theories of class ${\cal S}[A_{K}]$
    1. 4.1 Weak coupling chamber
    2. 4.2 Wild chamber
  5. 5 Witness and lazy witness
  6. 6 BPS invariants on the conifold
    1. 6.1 Chambers $\mathscr{C}[\mathsf{MS}_{n+1}^{1},\mathsf{MS}_{n}^{1}]$
    2. 6.2 Large radius
    3. 6.3 Noncommutative crepant resolution
    4. 6.4 Beyond the NCCR chamber
    5. 6.5 Topological analysis
  7. 7 Quantum geometry of compact threefolds
    1. 7.1 Topological strings on one parameter CY
    2. 7.2 BPS invariants
    3. 7.3 Topological analysis
  8. 8 ${\cal N}=4$ dyons and quantum black holes
  9. 9 Mathieu moonshine
  10. 10 Discussion
  11. A Moonshine barcodes

##  1 Introduction

In supersymmetric theories one often can gain deep insights by studying the
properties of protected states. States which preserve a fraction or all of the
supersymmetries can be used to get exact results about quantities of physical
interest. Such states are usually directly related to geometrical quantities,
such as enumerative invariants, or to the mathematical structures underlying
the physical models.

For example the BPS spectral problem in quantum field theories is deeply
related to the structure of the quantum vacuum and plays an important role in
understanding various dualities [1]. In black hole physics, the exact
enumeration of microstates is a problem of prime importance as it provides a
quantum statistical derivation of gravitational thermodynamics [2, 3].
Furthermore in many cases the duality properties of a theory directly imply
modular properties of the partition functions. In all these cases the counting
problem has roots in various areas of mathematics and has important physical
consequences.

In general one is lead to investigate the structures underlying such counting
problems. For example if the counting can be organized according to the
representation theory of some group or algebra, then one has identified a
fundamental principle in the physical theory. The typical situation is however
less direct and often a consequence of several structures simultaneously.

Consider for example the case of ${\cal N}=2$ SU(N) super Yang-Mills theories
on ${}^{3}\times S^{1}$. The moduli space of vacua is the Hitchin moduli space
and is hyperKähler [4]. The latter condition is guaranteed by the fact that
the BPS spectrum of the theory obeys a wall-crossing formula [5]. The latter
originates from the theory of generalized Donaldson-Thomas invariants, which
encodes the behavior of stable objects in a (derived) category under a change
of stability conditions [6]. As a consequence the moduli space of vacua is
locally a (generalized) cluster variety, the overlap transformations between
charts being dictated by a cluster algebra [7]. The BPS spectrum can also be
seen as the set of stable representations of the quiver underlying this
cluster algebra [8] and has a deep connection with integrable systems [9].

In many physical cases the situation is similar and the study of the structure
of the set of supersymmetric states leads to several layers of increasing
complexity. In this paper we take a step back and ask the following question:
is there any structure at the topological level? In particular we can consider
a collection of supersymmetric states simply as a set and study its properties
using topological methods. The purpose of this note is to investigate the
presence (or absence) of any noticeable topological feature in certain samples
of supersymmetric states. In particular we will be interested in how these
features change as the parameters of the theory are changed, or if one
considers a similar problem in different settings.

To be more precise, with topological features we mean the properties of the
spectra as seen from the perspective of persistent homology [10, 11].
Persistence is a relatively new approach to homological features of a space or
a set, and is at the core of what is by now known as topological data analysis
[12, 13, 14]. This field proposes to handle multidimensional and large sets of
data using methods based on topology. This approach has been quite useful in
disparate fields, such as biology, neuroscience or complex systems [15, 16,
17, 18, 19]. In this note we will apply such methods to supersymmetric
spectra, computed directly or extracted from certain number theoretical
functions.

In essence topological data analysis is a multi-scale approach to extracting
homological features out of a set of data, focused on identifying those
features which persist over a long range of scales. The idea consists in
defining a family of simplicial complexes which depend on a continuous
proximity parameter $\epsilon$. For each value of $\epsilon$ one can pass to
the homology of the complex and study how it varies as a function of
$\epsilon$. At each length scale the homology is characterized by its homology
classes; as the length scale changes new homology classes can form or already
existing classes can disappear, depending on the evolution of the underlying
simplicial complex. The set of data is characterized by the lifespans, or
persistence, of said homology classes. These lifespans can be more easily
visualized as a collection of intervals on the $\epsilon$ line, which begin at
the value of $\epsilon$ at which the homology class appears and end when it
disappear. Such collections of intervals are called barcodes.

We will consider a supersymmetric spectrum as a dataset, where each point is
labelled by the charges or the relevant quantum numbers of a state, and by its
degeneracy, or BPS enumerative invariant. We will then proceed to apply the
methods of topological data analysis to compute the homology of this set as a
function of a proximity parameter $\epsilon$ and compute its barcodes for each
non-trivial homology group. The set of barcodes gives a complete
characterization of the persistent homological features of the supersymmetric
spectra. We will then discuss how these topological characteristics vary
between different datasets. Roughly speaking we will do so in two ways: or by
comparing spectra obtained within the same physical theory but as the
parameters are varied; or by studying different models which can be however
associated with a very similar physical problem.

We will discuss at length how the expected physical features show up in the
topological analysis. In many cases this will be apparent from the barcodes,
in others a bit more care will be required. Overall we will learn how to apply
the methods of topological data analysis to BPS counting problems and argue
which kind of information we can hope to extract. We will do so with many
examples. In this paper we will focus more in detail on supersymmetric
spectra, but the formalism is more general and can be applied to other
counting problems. Applications of persistent homology to the study of string
vacua will appear in [20].

To keep this note readable we have included very brief reviews of the physical
problems at hand. This material is known, but we have chosen to present it in
such a way as to highlight the role played by BPS invariants. All the
computations in persistent homology have been done with the program matlab
using the library javaplex, made available in [22]. The accompanying software
and datasets are available in [21]. For the extraction of the BPS invariants
and the manipulations of the relevant series we have used mathematica.

This paper is organized as follows. Section 2 will give some background about
BPS states, their wall-crossing behavior and their enumerative interpretation.
In Section 3 we will give an elementary introduction to the ideas of
persistence and discuss the methods of topological data analysis. This Section
is meant to be readable by non-experts and to quickly convey the main ideas.
Section 4 contains our first application, to ${\cal N}=2$ SU(3) super Yang-
Mills, where we compare the topological features of the BPS spectrum in the
weak coupling chamber with those in a wild chamber. The interplay between
persistent homology and wall-crossing is also the focus of Section 6, which
discusses the case of the conifold in detail, using some approximation schemes
introduced in Section 5. Section 7 takes a different approach; here we compute
the Donaldson-Thomas invariants for a few distinct one parameter compact
Calabi-Yaus, compare their distributions and discuss the implications for
black hole physics. Section 8 is about a different class of black holes, in
${\cal N}=4$ string compactifications. In this case the relevant partition
functions have modular properties and we discuss the interplay between
modularity and topology. Section 9 takes a similar approach, now in the
context of elliptic genera in the Mathieu moonshine correspondence. In this
case, the technical details of the topological analysis are postponed to the
Appendix A. We summarize our finding in Section 10.

##  2 BPS states and wall-crossing

In this note we will consider certain field and string theories with extended
supersymmetry. In this Section we will quickly review some general properties
and postpone a more detailed description to later Sections on a case by case
basis. The theories we shall consider all have moduli spaces of quantum vacua
${\cal M}$. These moduli spaces often have a direct geometrical
interpretation, for example parametrizing deformation of a compactification
manifold or solutions of certain differential equations. Determining ${\cal
M}$ captures the vacuum structure of the theory. In theories with extended
supersymmetry one can often give a remarkably precise local description of the
moduli spaces ${\cal M}$, in the form of an answer determined at weak coupling
plus a series of quantum corrections.

On top of the geometry of ${\cal M}$, there is other physical information
which can be computed exactly. In this note we will be interested in the
spectrum of BPS states, which is very closely related to the series of quantum
corrections which determine the moduli space of quantum vacua ${\cal M}$.
These quantities are particularly important because due to the amount of
supersymmetry preserved, they allow for the extrapolation of weakly coupled
computations to strong coupling. In other words they are one of the few
available sources of non-perturbative information in quantum theories.

Supersymmetric theories have a Hilbert space of states ${\cal H}$ upon which
the supersymmetry generators act as operators. States in ${\cal H}$ can be
organized according to the representation theory of the supersymmetry algebra.
BPS states are characterized by the fact that a certain number of
supersymmetry generators are represented trivially. The fact that a BPS state
is annihilated by certain operators is a rather strong constraint, in many
cases strong enough to reduce quantum corrections to a computable form.

To be concrete consider ${\cal N}=2$ theories in four dimensions. We denote by
$\Gamma$ the lattice of electric and magnetic conserved charges, as measured
at spatial infinity and at a point in the moduli space of vacua. For example
in a string theory compactification or engineering, this lattice can be
realized as the homology of a certain variety. The lattice of charges is
endowed with the antisymmetric Dirac pairing

$\langle\ \ ,\ \ \rangle\,:\,\Gamma\times\Gamma\longrightarrow{\mathbb{Z}}\ .$
(2.1)

This pairing vanishes identically on the charges of particles which are
mutually local. The conserved charges of $\Gamma$ divide the Hilbert space of
states into superselection sectors. The BPS degeneracies $\Omega(\gamma;u)$
count with signs the number of BPS states with charge $\gamma\in\Gamma$. They
are defined as traces over the single particle BPS Hilbert space ${\cal
H}^{BPS}_{u}=\bigoplus_{\gamma\in\Gamma}{\cal H}^{BPS}_{\gamma;u}$, filtered
by the charge measured at spatial infinity.

The single particle Hilbert spaces ${\cal H}_{\gamma;u}$ and the BPS
degeneracies $\Omega(\gamma;u)$ depend explicitly on a point $u\in{\cal M}$.
The constraints arising from supersymmetry are such that $\Omega(\gamma;u)$
has a very specific dependence on $u\in{\cal M}$: it is a piecewise constant
function, almost independent on the physical parameters except for certain
codimension one walls in ${\cal M}$, at which it jumps suddenly. This is the
wall-crossing phenomenon. At walls of marginal stability the change in the BPS
degeneracies $\Omega(\gamma;u)$ describe physical processes of fusion or
fission of BPS particles from or into elementary constituents. The wall-
crossing of the BPS degeneracies is a very strong constraint on the
consistency of a theory at the quantum level [1]. The moduli space of vacua
${\cal M}$ is divided by the walls of marginal stability into chambers
$\mathscr{C}$. Solving the BPS spectrum of a theory amounts in finding the
$\Omega(\gamma;u)$ in each chamber.

Walls of marginal stability $\mathsf{MS}$ are defined as the loci in moduli
space where the central charges of two or more BPS particles become parallel.
In theories with extended supersymmetry the central charge is realized as an
holomorphic function over the moduli space ${\cal M}$,

${\cal Z}\,:\,{\cal M}\longrightarrow\mathrm{Hom}(\Gamma;{\mathbb{C}})\ .$
(2.2)

For example, a two body decay of a state with charge $\gamma$ into two
elementary constituents $\gamma_{1}$ and $\gamma_{2}$ is kinematically allowed
at the locus

$\mathsf{MS}(\gamma_{1},\gamma_{2})=\left\\{u\in{\cal M}\ |\ \arg{\cal
Z}_{\gamma_{1}}(u)=\arg{\cal Z}_{\gamma_{2}}(u)\right\\}\ .$ (2.3)

In many cases the central charge function has a very explicit description: in
four dimensional quantum field theories is given by the integral of the
Seiberg-Witten differential $\lambda$ over a cycle of the Seiberg-Witten curve
whose homology class correspond to a charge $\gamma\in\Gamma$. Similarly in
Calabi-Yau compactifications of the type II string it is given by an integral
of the holomorphic $(3,0)$-form over 3-cycles.

More formally we can usually describe BPS states as objects in some abelian
category ${\mathsf{A}}$. In concrete examples ${\mathsf{A}}$ could be the
category of representations of a certain quiver with potential
$\mathsf{rep}\left(Q,{\cal W}\right)$, or the category of coherent sheaves on
a Calabi-Yau threefold $X$, $\mathsf{coh}(X)$. This is not completely correct
as a more precise account would require objects in the bounded derived
category $\mathsf{D(A)}$, but for the purpose of this Section we will neglect
these issues. In these cases there is an isomorphism which identifies the
lattice of conserved charges $\Gamma$ with the topological Grothendieck group
$K(\mathsf{A})$. In the above cases the isomorphism is given by the Chern
character in the case of $\mathsf{coh}(X)$ and by the identification of the
simple representations with a basis of BPS states in the case of
$\mathsf{rep}\left(Q,{\cal W}\right)$. In any case we can regard the central
charge as a stability function on ${\mathsf{A}}$, at fixed $u\in{\cal M}$

${\cal Z}_{u}\,:\,K(\mathsf{A})\longrightarrow{\mathbb{C}}\,,$ (2.4)

which to any object $E\in\mathsf{A}$ associates a complex number ${\cal
Z}_{u}(E)$. We say that a BPS state described by an object $E$ is ${\cal
Z}$-stable if $\arg{\cal Z}_{u}(F)<\arg{\cal Z}_{u}(E)$ for any proper sub-
object $F$ of $E$. Note that the stability condition explicitly depends on
$u\in{\cal M}$ and therefore on the parameters of the theory. As these
parameters are varied, the stability condition changes and a stable object may
become unstable.

Since all that matters is the phase of the complex number ${\cal Z}_{u}(E)$,
we will loosely speak of a BPS state as a BPS ray $\ell_{\gamma}$ associated
with the BPS state of charge $\gamma$, a vector in the complex plane
${\mathbb{C}}$ (which we will refer as the central charge plane), as is by now
common use [5, 8].

The change in $\Omega(\gamma;u)$ across a wall of marginal stability is
governed by a wall-crossing formula [6, 23, 24]. The Kontsevich-Soibelman
wall-crossing formula (KSWCF) states that a certain product of operators,
which depends on the stable BPS charges and on the BPS degeneracies, remains
invariant across walls of marginal stability as to compensate for the change
in the degeneracies $\Omega(\gamma;u)$. To describe the KSWCF we need a few
more ingredients. We introduce the torus
$\mathbb{T}_{\Gamma}=\Gamma\otimes_{\mathbb{Z}}{\mathbb{C}}^{*}$ and formal
variables $X_{\gamma}$ for each $\gamma\in\Gamma$, which enjoy the property

$X_{\gamma_{i}}\,X_{\gamma_{j}}=(-1)^{\langle\gamma_{i},\gamma_{j}\rangle}\,X_{\gamma_{i}+\gamma_{j}}\,.$
(2.5)

The operators ${\cal K}_{\gamma}$ are automorphisms of the algebra of
functions on $\mathbb{T}_{\Gamma}$ which act as

$\mathcal{K}_{\gamma}(X_{\delta})=(1-(-1)^{\langle\gamma,\delta\rangle}X_{\gamma})^{\langle\gamma,\delta\rangle}\,X_{\delta}\,.$
(2.6)

To state the KSWCF we choose an angular sector $A$ in the central charge
plane. Then the KSWCF states that the phase ordered product

$\prod_{\gamma\,:\,\arg{\cal Z}_{\gamma}(u)\in A}\ {\cal
K}_{\gamma}^{\Omega(\gamma;u)}$ (2.7)

is invariant across walls of marginal stability, under the assumptions that no
BPS state enter or leaves the sector $A$. See [25] for a more in depth review.

The situation for ${\cal N}=4$ theories is similar. One can still define a
central charge function ${\cal Z}$ as a moduli dependent function which at a
point in the moduli space associates to a state a charge dependent complex
vector. The BPS condition now depends on the amount of supersymmetry
preserved. It is customary to use the notation
$({\boldsymbol{P}},{\boldsymbol{Q}})$ to indicate the charge of a generic 1/4
BPS dyon, while 1/2 BPS states are necessarily purely electrically or
magnetically charged. The degeneracies of BPS states can be defined as certain
helicity supertraces over the Hilbert space of states. The main difference in
the wall-crossing behavior respect to the ${\cal N}=2$ case is that now only
two bodies decays are allowed, namely of a 1/4 dyon into two 1/2 BPS states.

In this Section we have review very briefly some basic properties of BPS
states in supersymmetric theories. The set of stable BPS states has clearly a
lot of structure, which has lead to deep physical insights and beautiful
mathematics. These structures have deep algebraic and geometrical origin in
the theory of generalized Donaldson-Thomas invariants and of wall-crossing
structures [6, 26]. In this note we want to investigate their features from a
rather different perspective: we will look at the set of BPS states as a
distribution of points and try to understand its topological properties, and
how these change upon crossing walls of marginal stability. But first we have
to set up the appropriate tools.

##  3 Persistent homology

In this Section we will introduce the concept of persistent homology and
explain its uses in the context of topological data analysis. The idea behind
persistence is to study topological features of a space as a function of the
length scale [10, 11]. When applied to a set of datas, the topological
analysis extract qualitative features, which are independent on any particular
metric or coordinate system used, and robust to noise. In our exposition we
will mainly follow the reviews [12, 13, 14].

The techniques of topological data analysis are increasingly common in many
fields such as biology, neuroscience, complex systems or the study of
language, see [15, 16, 17, 18, 19] for a sample of the literature. An
application of these techniques to the study of string vacua appears in [20].

###  3.1 Homology of simplicial complexes

Homology captures intrinsic topological information of a space. To a
topological space $X$ we assign a collection of abelian groups $H_{i}(X)$
whose independent elements formally correspond to topological features of $X$,
such as its number of components or holes. The computation of the homology of
a space is a standard procedure to study its topology. There are several ways
to do this, as well as several homology theories which can be defined.

A very convenient approach uses simplicial complexes. We can think of a
simplicial complex as a triangulation of a space, whose elements are vertices,
edges and faces and so on, and which can be studied with combinatorial or
algebraic techniques. A simplicial complex $S$ is a pair consisting of a
finite set $V$ of vertices and a family $\Sigma$ of non-empty subsets of $V$.
The collection $\Sigma$ is defined by the property that if $\sigma\in\Sigma$
and $\tau\subseteq\sigma$, then $\tau\in\Sigma$, which for example implies
that if a certain simplex is part of $\Sigma$, so are its faces. The
$k$-simplexes of $\Sigma$ form the subset $\Sigma_{k}$ of simplexes with
cardinality $k+1$.

For example a standard simplicial complex associated to a metric space $X$ is
the Čech complex. Let $B_{\epsilon}(x)$ be the standard ball of radius
$\epsilon$ centered at $x\in X$. Assume that we can find a set $V\subset X$ so
that $X=\bigcup_{v\in V}B_{\epsilon}(v)$. Then the Čech complex is defined as

$\textsf{\v{C}ech}_{\epsilon}(X)=\left\\{\sigma=[v_{0},\dots,v_{k}]\ |\
\bigcap_{i=0}^{k}B_{\epsilon}(v_{i})\neq\emptyset\right\\}\,.$ (3.1)

This is a particular example of the nerve construction.

In this note we will be interested in a version of this construction, applied
to a very particular case. We define a point cloud ${\mathsf{X}}$ as a
collection of points $\\{\mathsf{x}_{i}\\}_{i\in I}$ in N. To a point cloud
${\mathsf{X}}$ we associate the Vietoris-Rips complex
$\mathsf{VR}_{\epsilon}({\mathsf{X}})$ as

$\mathsf{VR}_{\epsilon}({\mathsf{X}})=\left\\{\sigma=[v_{0},\dots,v_{k}]\ |\
d(v_{i},v_{j})\leq\epsilon\ \text{for all}\ i,j\right\\}\,,$ (3.2)

where $d(\ ,\ )$ is the standard distance function on N. In other words a
simplex is identified by the pairwise intersection of radius $\epsilon/2$
balls. Note that we can generalize immediately the definition of the Čech
complex to point clouds. The fact that the Vietoris-Rips complex
$\mathsf{VR}_{\epsilon}({\mathsf{X}})$ is defined only in terms of pairwise
intersection makes it much more amenable to algorithmic computations than the
Čech complex $\textsf{\v{C}ech}_{\epsilon}({\mathsf{X}})$. These complexes are
related by the inclusions

$\mathsf{\check{C}ech}_{\epsilon}({\mathsf{X}})\subseteq\mathsf{VR}_{2\epsilon}({\mathsf{X}})\subseteq\mathsf{\check{C}ech}_{2\epsilon}({\mathsf{X}})$
(3.3)

which imply that the Vietoris-Rips complex is a good approximation to the Čech
complex. Note also that the two complexes have a natural orientation which
follows by declaring that a $k$-simplex $[v_{0},\cdots,v_{k}]$ changes sign
under an odd permutation.

To a simplicial complex $S$ we can associate its homology as follows. We
define the group of $k$-chains $C_{k}$ by taking formal linear combinations of
$k$-simplices as $c=\sum_{i}a_{i}\sigma_{i}$ where $a_{i}\in{\mathbb{Z}}_{p}$
for a prime number $p$. On $C_{k}$ we define the boundary operator
$\partial_{k}\,:\,C_{k}\longrightarrow C_{k-1}$

$\partial_{k}\left([v_{0},v_{1},\cdots,v_{k}]\right)=\sum_{i=0}^{k}(-1)^{i}[v_{0},\cdots,\hat{v}_{i},\cdots,v_{k}]\
,$ (3.4)

where the element $\hat{v}_{i}$ is omitted in the right-hand side. This
operator can be used to define the chain complex

$\cdots\longrightarrow C_{k+1}\longrightarrow C_{k}\longrightarrow
C_{k-1}\longrightarrow\cdots$ (3.5)

and out of this the homology of $S$. To this end we introduce the spaces of
$k$-cycles $Z_{k}(S)=\ker\,{\partial_{k}}$ and of $k$-boundaries
$B_{k}(S)={\rm Im\,}\,{\partial_{k+1}}$. Then the $k$-th homology group
$H_{k}(S)$ is defined as the quotient $Z_{k}(S)/B_{k}(S)$. The Betti numbers
are the ranks of the homology groups, $b_{k}=\dim H_{k}(S)=\dim Z_{k}(S)-\dim
B_{k}(S)$, and measure the number of $k$-cycles which are not $k$-boundaries.

A very important feature of this construction is its functoriality. A
simplicial map $f$ between two simplicial complexes $S_{1}$ and $S_{2}$, is a
map between the corresponding vertex sets such that a simplex $\sigma$ of
$S_{1}$ is mapped into a simplex $f(\sigma)$ of $S_{2}$. A simplicial map
takes a $p$-simplex into a $k$-simplex, with $k\leq p$. A simplicial map
$f:S_{1}\longrightarrow S_{2}$ induces a map between the vector spaces of
$p$-chains, $C_{p}(f):C_{p}(S_{1})\longrightarrow C_{p}(S_{2})$. Collecting
all the induced maps we form the chain map $C_{\bullet}(f)$, that is a
collection of maps

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{p}(S_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{p}^{S_{1}}}$$\scriptstyle{C_{p}(f)}$$\textstyle{C_{p-1}(S_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{C_{p-1}(f)}$$\textstyle{\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{p}(S_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{p}^{S_{2}}}$$\textstyle{C_{p-1}(S_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$
(3.6)

such that

$C_{p-1}(f)\circ\partial_{p}^{S_{1}}=\partial_{p}^{S_{2}}\circ C_{p}(f)\,.$
(3.7)

In particular a chain map $C_{\bullet}(f)$ induced by the simplicial map $f$,
induces a map between homology groups

$f_{\star}:H_{i}(S_{1})\longrightarrow H_{i}(S_{2})\,.$ (3.8)

It will be important for us the case when the simplicial map $f$ is the
inclusion. Then by functoriality $f_{\star}$ keeps track of the individual
homology classes of $H_{i}(S_{1})$ inside $H_{i}(S_{2})$, that is it contains
the information whether an homology class remains non-trivial or not.

###  3.2 Persistent homology and barcodes

In order to define persistent homology we need to introduce a few preliminary
notions. Consider a field $\mathbb{K}$. An $\mathbb{N}$-persistence
$\mathbb{K}$-vector space (or module) is a collection of vector spaces
$\\{V_{n}\\}_{n\in\mathbb{N}}$ over $\mathbb{K}$ indexed by a natural number
$n\in\mathbb{N}$ together with a collection of morphisms
$\rho_{i,j}\,:\,V_{i}\longrightarrow V_{j}$ for every $i$ and $j$ so that
$i\leq j$. We further require a compatibility condition, that is
$\rho_{i,k}\cdot\rho_{k,j}=\rho_{i,j}$ whenever $i\leq k\leq j$. Morphisms
between $\mathbb{N}$-persistence vector spaces $\\{V_{n}\\}$ and $\\{W_{m}\\}$
are naturally defined as a collection of maps $f_{i}\,:\,V_{i}\longrightarrow
W_{i}$ such that the diagrams

$\textstyle{V_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{i}}$$\scriptstyle{\rho_{i,j}}$$\textstyle{V_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{j}}$$\textstyle{W_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{i,j}}$$\textstyle{W_{j}}$
(3.9)

commute. The same construction can be generalized to abelian groups,
simplicial complexes, chain complexes and so on. More formally a version of
these arguments can be applied to any category $\mathsf{Cat}$ to define the
category of $\mathbb{N}$-persistence objects
$\mathbb{N}_{pers}(\mathsf{Cat})$.

There is a particular class of $\mathbb{N}$-persistence modules which are of
interest in topological data analysis. A persistence module
$\\{V_{i}\\}_{i\in\mathbb{N}}$ is called tame if: i) each $V_{i}$ is finite
dimensional, and ii) $\rho_{n,n+1}:V_{n}\longrightarrow V_{n+1}$ is an
isomorphism for large enough $n$. The reason such modules are interesting is
that $\mathbb{N}$-persistence tame $\mathbb{K}$-modules are in one to one
correspondence with finitely generated modules over the graded ring
$\mathbb{K}[t]$. The fact that the latter are finitely generated allows for a
classification theorem for persistence modules, in terms of their barcodes
[11].

In order to explain this classification theorem, we define the
$\mathbb{N}$-persistence module $\mathbb{K}\,(m,n)$ as

$\mathbb{K}\,(m,n)=\begin{cases}&0,\qquad$if$\ i<m\ $or$\ i>n\\\
&\mathbb{K}\qquad$otherwise$\end{cases}$ (3.10)

where $m\leq n$ are two integers, $m$ is non-negative and $n$ can be infinity.
The morphism $\rho$ is simply $\rho_{i,j}=\mathrm{id}_{\mathbb{K}}$ for $m\leq
i<j\leq n$. $\mathbb{K}\,(m,n)$ is also known as an interval module, which
assigns a non-trivial vector space only to a certain interval.

The classification theorem then states that a given tame
$\mathbb{N}$-persistence $\mathbb{K}$-module admits the unique (up to ordering
of factors) decomposition

$\\{V_{i}\\}_{i}\simeq\bigoplus_{j=0}^{N}\mathbb{K}\,(m_{j},n_{j})\,.$ (3.11)

A tame persistence module is therefore completely specified by a collection of
$N$ intervals, for a certain $N\in\mathbb{N}$, to which we assign a non-
trivial vector space. An important consequence of this theorem is that we can
completely specify a persistence module by its barcode. A barcode is simply
the collection of non-negative integers $(m_{i},n_{i})$, where $0\leq m\leq n$
and eventually $n$ can be $+\infty$, which specify when the persistence module
is non-trivial. This classification result is a generalization of the well
known fact from elementary algebra that ordinary vector spaces are classified
up to isomorphisms by their dimension. In a similar fashion persistence
modules are characterized by a sequence of intervals. We will represent
graphically such a collection of intervals by drawing a series of bars (hence
the name barcode).

The reason these facts are important for us is that the persistent homology of
a point cloud (or of any topological space) gives a persistent module. Then a
barcode becomes a very effective tool to summarize and visualize homological
features.

Consider a point cloud ${\mathsf{X}}$, a collection of points in N. We denote
by ${\mathsf{X}}_{\epsilon}$ the point cloud ${\mathsf{X}}$ where every point
$\mathsf{x}\in{\mathsf{X}}$ has been replaced by a ball
$B_{\epsilon}(\mathsf{x})$ of radius $\epsilon$ centered at $\mathsf{x}$. We
regard ${\mathsf{X}}_{\epsilon}$ as a continuous family of topological spaces
indexed by the real variable $\epsilon\in_{\geq 0}$, with
${\mathsf{X}}_{0}={\mathsf{X}}$. For any fixed collection of values
$0=\epsilon_{0}<\epsilon_{1}<\epsilon_{2}<\cdots$, we have the sequence of
inclusions

${\mathsf{X}}_{\epsilon_{0}}\hookrightarrow{\mathsf{X}}_{\epsilon_{1}}\hookrightarrow{\mathsf{X}}_{\epsilon_{2}}\hookrightarrow\cdots\,.$
(3.12)

Similarly for each ${\mathsf{X}}_{\epsilon}$ we construct the associated
Vietoris-Rips complex $\mathsf{VR}_{\epsilon/2}({\mathsf{X}})$. Again this is
a continuous family of simplicial complexes parametrized by $\epsilon$. On the
other hand only for certain values of $\epsilon$ the simplicial complexes will
be distinct, and again for those parameters we have a sequence of inclusions

$\mathsf{VR}_{\epsilon_{0}}({\mathsf{X}})\hookrightarrow\mathsf{VR}_{\epsilon_{1}}({\mathsf{X}})\hookrightarrow\mathsf{VR}_{\epsilon_{2}}({\mathsf{X}})\hookrightarrow\cdots\,.$
(3.13)

For each $\mathsf{VR}_{\epsilon}({\mathsf{X}})$ we construct the associated
chain complex with coefficients in ${\mathbb{Z}}_{p}$, with $p$ a prime. Then
passing to the $i$-th homology gives the $\mathbb{N}$-persistence module

$H_{i}(\mathsf{VR}_{\epsilon_{0}}({\mathsf{X}});{\mathbb{Z}}_{p})\hookrightarrow
H_{i}(\mathsf{VR}_{\epsilon_{1}}({\mathsf{X}});{\mathbb{Z}}_{p})\hookrightarrow
H_{i}(\mathsf{VR}_{\epsilon_{2}}({\mathsf{X}});{\mathbb{Z}}_{p})\hookrightarrow\cdots\,.$
(3.14)

Technically both $\mathsf{VR}_{\epsilon}({\mathsf{X}})$ and the
$H_{i}(\mathsf{VR}_{\epsilon}({\mathsf{X}});{\mathbb{Z}}_{p})$ are really
-persistence modules, indexed by the real variable $\epsilon$. However only
for a finite number of $\epsilon$’s these complexes are really distinct, and
we can therefore talk of $\mathbb{N}$-persistence modules. More formally we
pick any order preserving map $\mathbb{N}\longrightarrow$ and construct the
$\mathbb{N}$-persistence modules out of the -persistence modules we have just
defined, as explained more in detail in [12].

Note that all this construction relies essentially on the functoriality of
homology. The maps in (3.14) are those induced by the inclusions of (3.13).
Without these maps (3.14) would just be a collection of vector spaces. It is
immediate to see, and follows just from the definition of homology and from
its functoriality, that these maps obey all the required properties to define
a persistence module.

In particular this means that for every $i$ the $\mathbb{N}$-persistence
module in (3.14) is completely characterized by a collection of barcodes.
These barcodes capture topological features of the point cloud ${\mathsf{X}}$.

In this case we can also give a perhaps more direct description of the
barcodes. The inclusions between topological spaces lift to maps between the
homology groups; we call this map

$\mathsf{f}^{a,b}_{i}\,:\,H_{i}(\mathsf{VR}_{\epsilon_{a}}({\mathsf{X}});{\mathbb{Z}}_{p})\longrightarrow
H_{i}(\mathsf{VR}_{\epsilon_{b}}({\mathsf{X}});{\mathbb{Z}}_{p})$ (3.15)

for $\epsilon_{a}\leq\epsilon_{b}$. Then we define the i-th persistent
homology group $\mathsf{H}^{a,b}_{i}={\rm Im\,}\,\mathsf{f}^{a,b}_{i}$, or
more explicitly

$\mathsf{H}^{a,b}_{i}=\frac{Z_{i}(\mathsf{VR}_{\epsilon_{a}}({\mathsf{X}}))}{B_{i}(\mathsf{VR}_{\epsilon_{b}}({\mathsf{X}}))\cap
Z_{i}(\mathsf{VR}_{\epsilon_{a}}({\mathsf{X}}))}\,,$ (3.16)

where for simplicity we have not written down the inclusions. The i-th
persistent Betti number is naturally defined as
$\beta_{i}^{a,b}=\mathrm{rank}\,\mathsf{H}^{a,b}_{i}$. The persistent Betti
number $\beta_{i}^{a,b}$ is given by the number of barcodes of
$H_{i}(\mathsf{VR}_{\epsilon}({\mathsf{X}});{\mathbb{Z}}_{p})$ which span the
whole interval $[\epsilon_{a},\epsilon_{b}]$.

###  3.3 Barcodes and topological features

Let us expand a bit on the interpretation of the barcodes. Barcodes are a
visual device which represent the number of persistent generators in the
$i$-th homology group
$H_{i}(\mathsf{VR}_{\epsilon}({\mathsf{X}});{\mathbb{Z}}_{p})$. Given two
values of the proximity parameter $\epsilon_{1}<\epsilon_{2}$, a persistent
homology class along the interval $[\epsilon_{1},\epsilon_{2}]$ is a non-
trivial homology class in
$H_{i}(\mathsf{VR}_{\epsilon_{1}}({\mathsf{X}});{\mathbb{Z}}_{p})$ which is
mapped into a non-trivial homology class in
$H_{i}(\mathsf{VR}_{\epsilon_{2}}({\mathsf{X}});{\mathbb{Z}}_{p})$.

Consider the barcodes at a fixed value of $\epsilon$. This means that we are
looking at the point cloud ${\mathsf{X}}$ at a certain characteristic scale
given by $\epsilon$. At this scale the generators of
$H_{i}(\mathsf{VR}_{\epsilon}({\mathsf{X}});{\mathbb{Z}}_{p})$ capture
topological features of the data set: they represent $i$-dimensional
configurations of points shaped as cycles which are not boundaries, meaning
delimiting “holes” which are not filled up by other points. The persistence of
these generators is a measure of how long these holes last as the value of the
proximity parameter $\epsilon$ increases. Intuitively the clearer is the
topological feature, for example if a certain hole contains none or very few
points in its interior, the longer the persistent homology class lasts. A non-
trivial long-lived persistent class in
$H_{i}(\mathsf{VR}_{\epsilon}({\mathsf{X}});{\mathbb{Z}}_{p})$ indicates that
the points in the data set cluster around a $i$-cycle without “filling it up”.

For example persistent homology classes in
$H_{0}(\mathsf{VR}_{\epsilon}({\mathsf{X}});{\mathbb{Z}}_{p})$ measure the
number of connected components in the point cloud as a function of the length
scale. At very small values of $\epsilon$ each barcode correspond to a point
in the original point cloud ${\mathsf{X}}$. As $\epsilon$ increases, neighbor
points will form a single connected component. Long-lived barcodes are
evidence for clustering of data into different regions. This could indicate
for example that the data have a tendency to accumulate towards a certain
point or area. A clear division into connected areas with similar behavior,
for example the repetition of the same pattern in the barcodes, could also be
regarded as evidence of an existing symmetry in the underlying physical
problem.

A similar reasoning holds for higher homology groups. A persistent generator
of $H_{1}(\mathsf{VR}_{\epsilon}({\mathsf{X}});{\mathbb{Z}}_{p})$ which is
long-lived implies that at different length scales a certain area of the point
cloud ${\mathsf{X}}$ is naturally well approximated by a one-dimensional
manifold with the topology of an empty circle $S^{1}$. Similarly non-trivial
elements of $H_{i}(\mathsf{VR}_{\epsilon}({\mathsf{X}});{\mathbb{Z}}_{p})$
will generically suggest that a certain region of the point cloud
${\mathsf{X}}$ can be approximated by a higher dimensional manifold with a
given topology. The fact that data live on a certain shape can for example
suggest a good coordinate system to approximate the point cloud; in general
the presence of non-trivial persistent topologies in ${\mathsf{X}}$ is a hint
of the existence of correlations between the data points, for example in the
form of a set of equations which constrain regions of ${\mathsf{X}}$.

When discussing the barcode distribution for
$H_{i}(\mathsf{VR}_{\epsilon}({\mathsf{X}});{\mathbb{Z}}_{p})$, we will
loosely use the terminology: barcodes at Betti number $i$, barcodes in degree
$i$, or barcodes for $H_{i}$.

###  3.4 An example

Before we proceed to apply these techniques to the study of BPS states, let us
go through a simple example. We take for our point cloud ${\mathsf{X}}$ the
simple configuration of points shown in Figure 1. We want to understand its
topological features using persistent homology, and in the process explain how
to apply the relevant techniques step by step.

The configuration of points in Figure 1 has a clear hexagonal shape. From a
topological perspective this is equivalent to say that the points are
distributed along a circle. While this is clear just by looking at the Figure,
we would like to abstract this information in a collection of barcodes. What
we gain in this abstraction will be clear in the following Sections, where we
will have to confront higher dimensional point clouds where no simple
visualization tool is available.

Figure 1: Left. A configuration of points which form the vertices of an
hexagon. Right. The corresponding barcodes computed from the Vietoris-Rips
complex. The non-trivial barcode at Betti number 1 makes precise the statement
that the six points look like an hexagon, which is topologically a circle.

We draw the relevant barcodes in Figure 1 on the right. As with all the
persistent homology computations in this paper, to obtain the barcodes we
wrote a matlab program available in [21]. Let us follow the formation and
demise of persistent homology classes in “time” $\epsilon$.

We draw different stages of the $\epsilon$ evolution in Figure 2. At
$\epsilon=0$ we just have the original six points and there is nothing worth
noticing: the homology obviously gives six distinct connected components and
no further feature. At $\epsilon=1$ two edges form, as shown in Figure 2. As a
consequence the first homology of the Vietoris-Rips complex should capture the
four connected components of ${\mathsf{X}}_{\epsilon=1}$. Indeed we see that
precisely at $\epsilon=1$ two homology classes disappear, corresponding to the
formation of two edges connecting the respective vertices. Therefore
immediately before $\epsilon=1$ there are six persistent homology classes in
degree 0 corresponding to six barcodes; immediately after two of these classes
have become trivial in homology (two vertices are the boundary of an edge) and
only four barcodes are left.

At $\epsilon=1.5$ we see that a 1-cycle form has formed. Any point has an edge
connecting it to its two nearest neighbors. By now each individual zeroth
homology class but one has died by merging into a single class, and the only
barcode left for $H_{0}$ signals the only connected component of the simplex.
On the other hand we see a non trivial barcode for $H_{1}$, corresponding to
the non-trivial cycle in the Vietoris-Rips complex.

Figure 2: The Vietoris-Rips complex (in blue) at various values of the
proximity parameter $\epsilon$ as shown by the red line in the barcodes’ plot.
In yellow the balls around the points of the point cloud ${\mathsf{X}}$, whose
radius $\epsilon/2$ is determined by the proximity parameter. At $\epsilon=1$
we see the formation of two edges and at $\epsilon=1.5$ a complete one-cycle.
As we increase $\epsilon$ new simplexes form and the one-cycle becomes the
boundary of a face in the Vietoris-Rips complex, thus disappearing from the
homology.

When we reach $\epsilon=2.5$ each one of the six original points of the
hexagon is now at a distance less then $\epsilon$ from each other point. As a
consequence a face of the simplex has formed. The persistent homology class of
$H_{1}$ has already disappeared, corresponding to the fact that the non-
trivial one cycle we saw at $\epsilon=1.5$ is now the boundary of the face
shown in Figure 2.

In this way the persistent homology of the point cloud ${\mathsf{X}}$ captures
its essential topological features: the number of points is determined by the
$H_{0}$ barcodes at small values of the proximity parameters; the existence of
a long-lived persistent homology class at $H_{1}$ is tantamount to the
statement that the point cloud ${\mathsf{X}}$ has “the shape of a circle”;
finally the number of $H_{0}$ barcodes present at large values of $\epsilon$,
in this case just one, is an information on the number of clusters in the
distribution of points.

Note that whether we call a class long-lived or short-lived depends somewhat
on the context. In this case we could identify the barcode at $H_{1}$ directly
as an interesting topological feature of the hexagonal point cloud
${\mathsf{X}}$ because we knew of its shape. In general one needs some
physical input from the problem at hand, as we will see repeatedly in the
following Sections.

##  4 BPS spectra in theories of class ${\cal S}[A_{K}]$

We have discussed techniques to extract topological information out of a
distribution of points. In this Section we will apply these techniques to BPS
states in supersymmetric quantum field theory. We will consider ${\cal N}=2$
SU(3) super Yang-Mills and construct two point clouds ${\mathsf{X}}$ out of
the distribution of BPS states in two different chambers. Then we will use the
tools of persistent homology to compare their features. We will also learn how
to extract physical information out of the barcodes.

Most of the recent progress in understanding BPS spectra has been focussed on
theories of class ${\cal S}[A_{K}]$. These theories can be seen as the low
energy limit of the compactification on ${}^{3,1}\times{\cal C}$ of the six
dimensional ${\cal N}=(2,0)$ superconformal theory. This perspective provides
an alternative description of many physical quantities in terms of the
geometry of the curve ${\cal C}$.

In this context the moduli space of quantum vacua is the Coulomb branch ${\cal
B}$ which parametrizes tuples $u=(\phi_{2},\cdots,\phi_{K})$ of meromorphic
differentials with prescribed singularities. The Wilsonian effective action of
the theory in the Coulomb branch is completely determined by a family of
curves $\Sigma_{u}$, which are $K$-fold branched coverings of ${\cal C}$.
$\Sigma_{u}$ inherits a natural holomorphic one form $\lambda_{u}$ which
descends from the Liouville one-form on ${\cal C}$.

The lattice of electric and magnetic charges $\Gamma$ is identified with a
quotient of $H_{1}(\Sigma_{u};{\mathbb{Z}})$ by the lattice of flavor charges.
For any state of charge $\gamma\in\Gamma$, the central charge is given by

${\cal Z}_{\gamma}(u)=\frac{1}{\pi}\,\int_{\gamma}\,\lambda_{u}\,,$ (4.1)

where we have identified the charge $\gamma$ with an homology class in
$H_{1}(\Sigma_{u};{\mathbb{Z}})$. Given the pair $(\Sigma_{u},\lambda_{u})$
the central charge is in principle known at any point $u\in{\cal B}$.

The BPS spectra of theories of class ${\cal S}[A_{K}]$ have striking and
unexpected features for $K>2$. Firstly these theories will generically have
higher spin BPS supermultiplets at generic points in their Coulomb branch.
Secondly, and more surprisingly, they have wild spectra: chambers where the
number of BPS states with mass less or equal to a given mass $M$ grows
exponentially with $M$. These features were demonstrated explicitly for
$\mathrm{SU}(3)$ super Yang-Mills with ${\cal N}=2$ in [27] and are believed
to hold generically. This phenomenon gives a striking example of the type of
understanding of quantum field theory that we gain by studying the wall-
crossing behavior of BPS states. With these techniques available to perform
controlled computations at various values of the coupling constant, even a
relatively simple quantum field theory such as $\mathrm{SU}(3)$ Yang-Mills
with ${\cal N}=2$ is full of surprises.

We will now apply the formalism outlined in Section 3 to this theory. In
particular we will discuss the topological features of the BPS spectrum in a
weak coupling chamber and study their behavior as we cross a wall of marginal
stability into a wild chamber. We will see that the differences are quite
striking, even at the topological level.

In this particular case, the Seiberg-Witten curve $\Sigma$ is a three sheeted
covering of ${\cal C}$, which has the topology of a cylinder, with six
ramification points:

$\lambda^{3}-\frac{u_{2}}{z^{2}}\,\lambda+\left(\frac{1}{z^{2}}+\frac{u_{3}}{z^{3}}+\frac{1}{z^{4}}\right)=0\,.$
(4.2)

One way to compute the BPS spectra is to start from a chamber where the
spectrum is known and then apply the wall-crossing formula (2.7). The strong
coupling region is characterized by small values of the moduli $u_{2}$ and
$u_{3}$ and has a finite spectrum consisting of six hypermultiplets. The
spectrum generator decomposes as

${\cal K}_{\gamma_{4}}\,{\cal K}_{\gamma_{3}}\,{\cal
K}_{\gamma_{2}+\gamma_{4}}\,{\cal K}_{\gamma_{1}+\gamma_{3}}\,{\cal
K}_{\gamma_{2}}\,{\cal K}_{\gamma_{1}}\,.$ (4.3)

For this theory the rank of the charge lattice $\Gamma$ is four and we have
picked a basis $\\{\gamma_{i}\\}$, with $i=1,2,3,4$, so that
$\langle\gamma_{1},\gamma_{2}\rangle=\langle\gamma_{2},\gamma_{3}\rangle=\langle\gamma_{3},\gamma_{4}\rangle=-2$
and
$\langle\gamma_{1},\gamma_{3}\rangle=\langle\gamma_{2},\gamma_{4}\rangle=1$
and all the other parings are vanishing.

From a mathematical perspective the study of the BPS spectrum can be recast in
terms of quiver representation theory [8]. In this case the relevant quiver
has the form

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with superpotential

$\mathcal{W}=\left(B_{2}\,\psi-\phi\,B_{1}\right)\,C_{1}+\left(A_{2}\,\psi-\phi\,A_{1}\right)\,C_{2}\,.$
(4.5)

We have labelled the nodes of the quiver such that the simple representation
at a node $\bullet_{i}$ corresponds to a BPS hypermultiplet of charge
$\gamma_{i}$. The signed number of arrows connecting the node $\bullet_{i}$
with the node $\bullet_{j}$ is the pairing
$\langle\gamma_{i},\gamma_{j}\rangle$.

We can be a bit more precise with the enumerative problem we are considering.
Given a certain quiver ${\mathsf{Q}}$, we consider the path algebra of the
quiver ${\mathbb{C}}{\mathsf{Q}}$ over ${\mathbb{C}}$ as the algebra of paths,
where the product is defined as concatenation of paths in ${\mathsf{Q}}$
whenever possible, or zero otherwise. The Jacobian algebra $\mathscr{J}_{{\cal
W}}={\mathbb{C}}{\mathsf{Q}}/\langle\partial\,{\cal W}\rangle$ arise by
considering the quotient by the equations of motion which follow from the
superpotential ${\cal W}$. Then physical states are described in terms of the
category of left modules $\mathscr{J}_{\cal W}-\mathsf{mod}$, which is
equivalent to the category of quiver representations with relations
$\mathsf{rep}({\mathsf{Q}},{\cal W})$.

We can use the central charge ${\cal Z}_{\gamma}(u)$ at a point $u\in{\cal B}$
to impose stability conditions on the physical states. More precisely
identifying the simple representations of ${\mathsf{Q}}$ with a basis of
charges gives an isomorphism between $\Gamma$ and
$K(\mathsf{rep}({\mathsf{Q}}))$ so that we can let the central charge act on
representations of ${\mathsf{Q}}$. Since the central charge is linear, it is
enough to specify its action on a the objects of $\mathsf{rep}({\mathsf{Q}})$
corresponding to a basis of $\Gamma$. In particular at a point $u\in{\cal B}$,
the central charge function ${\cal Z}_{\gamma}(u)$ specifies an ordering for
the phases of the objects corresponding to a basis of charges of $\Gamma$.
Since what defines the walls of marginal stability are the phases of the
central charge for the physical states, this identifies a chamber
$\mathscr{C}$ in the moduli space ${\cal B}$.

Therefore at any point in $u\in{\cal B}$ we know, in principle, the value of
the central charge and we can identify stable BPS states with ${\cal
Z}$-stable representations of ${\mathsf{Q}}$, modulo isomorphisms classes of
representations. We denote by $\mathscr{M}_{\mathbf{d}}^{{\cal
Z}}({\mathsf{Q}})$ the moduli space which parametrizes isomorphism classes of
${\cal Z}$-stable representations of ${\mathsf{Q}}$ with dimension vector
$\mathbf{d}$. The dimension vector determines the charge of a BPS state as
$\gamma=\sum_{i}\,d_{i}\,\gamma_{i}$, where the sum runs over a basis of
$\Gamma$. Then we obtain the physical moduli space of BPS states
$\mathscr{M}_{\mathbf{d}}^{{\cal Z}}({\mathsf{Q}},{\cal W})$ as the scheme-
theoretical critical locus of the superpotential ${\cal W}$.

Finally we identify the BPS degeneracies with the Donaldson-Thomas invariants
of $\mathscr{M}_{\mathbf{d}}^{{\cal Z}}({\mathsf{Q}},{\cal W})$

$\Omega(\gamma;u)={\tt DT}_{\mathbf{d}}({\mathsf{Q}},{\cal
W})=\chi_{\mathrm{vir}}(\mathscr{M}_{\mathbf{d}}^{{\cal Z}}({\mathsf{Q}},{\cal
W}))\,,$ (4.6)

defined as the virtual Euler characteristics of the moduli space of stable BPS
states [28]. The virtual Euler characteristic of a scheme $X$ is defined as

$\chi_{\mathrm{vir}}(X)=\int_{X}\,\nu_{X}\,{\rm
d}\chi=\sum_{n\in{\mathbb{Z}}}n\,\chi(\nu_{X}^{-1}(n))\,,$ (4.7)

as an integral against the Euler characteristic of Behrend’s constructible
function $\nu_{X}\,:\,X\longrightarrow{\mathbb{Z}}$. At a smooth point of $X$,
$\nu_{X}$ is just a sign. The Donaldson-Thomas invariant (4.6) is basically an
Euler characteristic weighted with signs, similarly to the physical definition
of degeneracies as a trace over an Hilbert space weighted with signs.

We will now apply this discussion to the quiver (4.4). For a point $u\in{\cal
B}$ the central charge function ${\cal Z}_{\gamma}(u)$ identifies a chamber
and a stable BPS spectrum. We have shown the spectrum in the strong coupling
chamber in (4.3). By moving in ${\cal B}$ we cross walls of marginal stability
and the BPS spectrum changes accordingly. Each time we can compute the
spectrum with the wall-crossing formula (2.7). This allows us to determine the
BPS degeneracies in the weak coupling chamber and study the persistent
homology of its distribution. From the weak coupling chamber we will move in
the wild chamber, where again the enumerative invariants can be computed
(albeit not in closed form) with the wall-crossing formula. We shall see that
the distribution of Donaldson-Thomas invariants in the wild chamber has
striking features which are visible already at the level of the barcodes. From
a more mathematical viewpoint our results concern the topological variations
in the distribution of virtual Euler characteristics of moduli spaces of
stable quiver representations, as the stability condition changes.

###  4.1 Weak coupling chamber

From the strong coupling chamber we follow a path $p$ in the Coulomb branch
${\cal B}$ towards the region of weak coupling. This path crosses four walls
of marginal stability:
$\mathsf{MS}(\gamma_{1}+\gamma_{3},\gamma_{2}+\gamma_{4})$,
$\mathsf{MS}(\gamma_{1},\gamma_{2})$, $\mathsf{MS}(\gamma_{3},\gamma_{4})$ and
$\mathsf{MS}(\gamma_{1},\gamma_{2}+\gamma_{4})$, in this order, ending within
a chamber $\mathscr{C}^{SU(3)}_{1}$. For each of these walls the two charges
whose central charge becomes parallel have pairing equal to two: each time the
situation is a direct analog of the transition from strong to weak coupling in
pure SU(2) super Yang-Mills, and each time a similar spectrum is generated
consisting of a vector multiplet shrouded by an infinite cloud of
hypermultiplets. In this chamber the spectrum can be written down explicitly.
Following [27] we introduce the notation

$\Pi^{n,m}(\gamma_{a},\gamma_{b})=\left(\prod_{k\nearrow n}^{\infty}{\cal
K}_{(k+1)\gamma_{a}+k\gamma_{b}}\right)\,{\cal
K}_{\gamma_{a}+\gamma_{b}}^{-2}\left(\prod_{k\searrow m}^{\infty}{\cal
K}_{k\gamma_{a}+(k+1)\gamma_{b}}\right)\,,$ (4.8)

where $k\nearrow n$ means that the product is taken in order so that the value
of $k$ increases from left to right starting from $n$, and similarly for
$k\searrow m$. Then the spectrum in $\mathscr{C}^{SU(3)}_{1}$ is

$\Pi^{(0,0)}(\gamma_{3},\gamma_{4})\,\Pi^{(0,1)}(\gamma_{1}+\gamma_{3},\gamma_{2}+\gamma_{4})\,\Pi^{(0,0)}(\gamma_{1},\gamma_{2}+\gamma_{4})\,\Pi^{(1,0)}(\gamma_{1},\gamma_{2})\,.$
(4.9)

This spectrum contains four vectormultiplets and an infinite series of dyonic
stable hypermultiplets.

Now we look at the topological structures of this spectrum, using the Rips-
Vietoris complex. That is we consider a point cloud ${\mathsf{X}}$ in 5 where
each vector has the form
$\mathsf{x}=(d_{1},d_{2},d_{3},d_{4},\Omega(\gamma;u))$, where
$\gamma=\sum_{i}^{4}d_{i}\gamma_{i}$ is the charge of a stable particle and
$u\in\mathscr{C}^{SU(3)}_{1}$.

Figure 3: Barcodes corresponding to the BPS spectrum in the weak coupling
chamber $\mathscr{C}^{SU(3)}_{1}$. Left. The point cloud ${\mathsf{X}}$ is
constructed from vectors of the form
$\mathsf{x}=(d_{1},d_{2},d_{3},d_{4},\Omega(\gamma;u))$. Right. The logarithm
of the degeneracies $\log|\Omega(\gamma;u)|$ is now used in the point cloud.

After constructing the filtered Rips-Vietoris complex as explained in Section
3, we pass to the homology over ${\mathbb{Z}}_{2}$ and compute the barcodes.
They are shown in Figure 3. The point cloud consists of 84 states and the
construction of the filtered Vietoris-Rips complex involves a total of 5348
simplices111This number is a function of the proximity parameter $\epsilon$,
as well as of how many homology groups $H_{k}$ are included in the
computation, in the sense that if one decides to truncate the computation at
some $H_{k}$, higher dimensional simplices can be neglected. In the following
we will by convention only show barcodes up to a value of the proximity
parameter $\epsilon$ and of the Betti numbers, which contain interesting
topological features, or at least the topological features we want to discuss.
We will use the terminology “number of simplices” in a similar way, referring
only to the simplices used in the homology computations under these
conditions.. Non-trivial persistent homology classes are present only for
$H_{0}$ and $H_{1}$.

For future reference we show in Figure 3 on the right the barcodes with the
logarithm of the (absolute value of the) degeneracies, that is obtained from
the point cloud $(d_{1},d_{2},d_{3},d_{4},\log|\Omega(\gamma;u)|)$. Note that
this change does not really modify the BPS point cloud substantially, since
all the non vanishing degeneracies are $1$ or $-2$ for hypermultiplets and
vector multiplets respectively. On the other hand the relative distance
between points are now different and as a consequence the total number of
simplices changes, in this case increases to 13708. The distribution of
barcodes does not deviate significantly between Figure 3 on the left and on
the right, as expected.

###  4.2 Wild chamber

Now we cross the wall
$\mathsf{MS}(2\gamma_{1}+\gamma_{2},\gamma_{2}+\gamma_{4})$ to enter into the
wild chamber $\mathscr{C}^{SU(3)}_{2}$. Note that $\langle
2\gamma_{1}+\gamma_{2},\gamma_{2}+\gamma_{4}\rangle=3$, which implies that in
this chamber the BPS quiver has a representative in its mutation class which
contains the 3-Kronecker quiver as a subquiver. In crossing the wall a
plethora of higher spin multiplets is generated with wild degeneracies. The
spectrum is given by

$\Pi^{(0,0)}(\gamma_{3},\gamma_{4})\,\Pi^{(0,1)}(\gamma_{1}+\gamma_{3},\gamma_{2}+\gamma_{4})\,\Pi^{(0,0)}(\gamma_{1},\gamma_{2}+\gamma_{4})\,\Xi(2\gamma_{1}+\gamma_{2},\gamma_{2}+\gamma_{4})\,\Pi^{(1,0)}(\gamma_{1},\gamma_{2})\
,$ (4.10)

where $\Xi(2\gamma_{1}+\gamma_{2},\gamma_{2}+\gamma_{4})$ is a so-called $3$
cohort in [27].

Figure 4: Barcodes for the BPS spectrum in the wild chamber
$\mathscr{C}^{SU(3)}_{2}$. Left. The point cloud is ${\mathsf{X}}$ constructed
out of vectors of the form $(d_{1},d_{2},d_{3},d_{4},\Omega(\gamma;u))$.
Right. Vectors in the point cloud are of the type
$(d_{1},d_{2},d_{3},d_{4},\log|\Omega(\gamma;u)|)$.

The $3$ cohort does not have a closed form expression and its BPS degeneracies
have been computed in [27] only up to a total charge $\gamma$ of $15$. From
this result we use (4.10) to construct a sample of 144 points corresponding to
as many BPS states. The Vietoris-Rips complex runs over a total of 210156
simplices and its persistent homology classes (again over ${\mathbb{Z}}_{2}$)
are shown in the left in Figure 4. The exponential growth of the degeneracies
shows up as the presence of many very long-lived barcodes in degree 0. This is
easy to understand; due to the exponential growth of the point cloud the
homology of the Vietoris-Rips complex sees many points as individual connected
components for very large values of the proximity parameter $\epsilon$.

Since the degeneracies are exponentially growing it is useful to consider also
the point cloud obtained by taking the logarithm of the modulus of the
degeneracies. This is merely a trick to simplify the computations of the
simplices. This significantly reduces the lifespan of long-lived homology
classes, also altering the number of simplices. On the other hand the features
of the barcode distribution are easier to visualize, and shown in Figure 4 on
the right. The total number of simplices for which there are interesting
topological features is now reduced to 27766.

Across the wall we see distinctively a transition at the level of the topology
of the point cloud. This is clear for example in Figure 4 (left) where the
barcodes for $H_{0}$ are very long-lived. If we compare with Figure 3 (left)
we see that the wild chamber has a much larger number of connected component
at very large scales $\epsilon$. This is indeed a consequence of the
exponential growth in the number of states: since the degeneracies
$\Omega(\gamma;u)$ grow exponentially the points are further and further apart
and therefore at the same scale $\epsilon$ at which the points in Figure 3 are
already grouped in a single cluster, the degeneracies in Figure 4 still look
like many connected components. In other words at a certain threshold the
values of the degeneracies begin to grow too far apart for the Vietoris-Rips
complex to form edges and the homology sees each point as an individual
connected component.

To have a more meaningful comparison, we turn to the logarithm of the
degeneracies, so to damp the exponential growth. We will see in the next
Sections that there are more refined methods, but for the moment this will be
enough. We now compare the results in Figures 3 and 4 (right). The transition
between the two chambers is still very clear: the barcodes in the weak
coupling chamber are very regular and the dependence on the scale $\epsilon$
rather mild. On the other hand we see in Figure 4 how persistent homology
captures the wilderness of the BPS spectrum. The homological features are very
irregular, especially for $H_{1}$ and non-trivial 1-cycles persist at every
$\epsilon$-scale. This features terminate in Figure 4 at a certain value of
$\epsilon$ only because we are using a finite sample, with roughly the same
number of points as in the weak coupling chamber, but would continue
indefinitely were we to increase the number of degeneracies computed via the
wall-crossing formula.

This is an example of the kind of information we can get on the BPS spectra
using the formalism of persistent homology. In this Section we have just
compared the topological features of the BPS spectrum in two different
chambers. Topology gives a clear meaning to the statement that these chambers
are qualitatively different. The topological features of the two distributions
are captured by the basic topological invariants of the
$\mathbb{N}$-persistence modules, namely the barcodes. It is remarkable that
upon crossing the wall of marginal stability, the transition between the two
chambers is very sharp even at the topological level.

We will now discuss a similar problem in a string theory compactification,
namely the BPS spectra in various chambers of the conifold. But before that we
need to refine our techniques in the next Section.

##  5 Witness and lazy witness

We have shown how to use the Vietoris-Rips complex $\mathsf{VR}_{\epsilon}$ to
extract topological information from the BPS spectra in the form of barcodes.
The computation of the barcodes is done exactly by evaluating the homology of
the simplicial complex for any value of the proximity parameter $\epsilon$.
However in many cases the point cloud ${\mathsf{X}}$ is too large for a direct
computation. In this cases certain approximation schemes are available, which
we will now describe. These approximation schemes are suited to deal
efficiently with large sets of data. The main idea is to use certain criteria
to select only a certain subset $\mathsf{L}$ of the point cloud
${\mathsf{X}}$, which is then used to form a simplicial complex which
approximates the Vietoris-Rips complex [12].

The operator which performs such a choice is a landmark selector. The most
used landmark selectors are:

  1. 1.

Random selector. The most straightforward way to choose a subset
$\mathsf{L}\subset{\mathsf{X}}$ is by picking a number of points at random.
This procedure is quite useful, although in practice it is better to choose
various random subsets $\mathsf{L}$ and perform homological computations for
each one separately. Indeed it is quite possible that a random selection would
miss essential features of a point cloud ${\mathsf{X}}$. The limitations of
random selection are well known from Monte Carlo algorithms and will not be
repeated here.

  2. 2.

Minmax selector. This operator selects a collection of points $\mathsf{L}$
which, in a very specific sense, is as spread out as possible. The minmax
algorithm works inductively by maximizing the distance of a point from a
previously chosen set. More in detail, the algorithm starts from a randomly
chosen point $\mathsf{x}_{0}$. Let $d(\ ,\
)\,:\,{\mathsf{X}}^{2}\longrightarrow$ be the standard distance function
between points (which is inherited from the ambient metric of N where the
point cloud is embedded). The choice of the remaining points in the set
$\mathsf{L}$ proceeds by induction. Denote by $\mathsf{L}_{i}$ the minmax
selection of $i$ points in ${\mathsf{X}}$. Then $\mathsf{L}_{i+1}$ consist of
the same set to which a single point $\mathsf{z}$ is added, in such a way that
the function $d(\mathsf{x}_{j},\mathsf{z})$ is maximized for each
$\mathsf{x}_{j}\in\mathsf{L}_{i}$. Because of this inductive definition, the
landmark selected set $\mathsf{L}$ will consists of points which are spread
apart from each other as much as possible. Therefore in general we expect this
selector to capture features of a point cloud ${\mathsf{X}}$ better than a
random selector. On the other hand one has to keep in mind that there are
cases where this expectation will fail: for example since the minmax selector
will generically pick out outlier points, a random selector might work better
with very dense sets.

Having chosen a landmark set $\mathsf{L}$, we can define a version of the
Vietoris-Rips complex which is based on it. We shall use two such simplicial
complexes, the witness complex
${\mathsf{W}}({\mathsf{X}},\mathsf{L},\epsilon)$ and the lazy witness complex
$\mathsf{LW}_{\nu}({\mathsf{X}},\mathsf{L},\epsilon)$. In both cases the
vertex set is given by $\mathsf{L}$; what changes is the definition of the
simplices. These are defined as follows:

  1. 1.

Witness complex ${\mathsf{W}}({\mathsf{X}},\mathsf{L},\epsilon)$. Let
$\mathsf{x}$ be a point in ${\mathsf{X}}$. Denote by $d_{k}(\mathsf{x})$, for
$k>0$, the distance between $\mathsf{x}$ and its $k+1$-th closest landmark
point. Now we declare that a collection of vertices
$\mathsf{l}_{i}\in\mathsf{L}$ for $i=0,\dots,k$ form the simplex
$[\mathsf{l}_{0},\dots,\mathsf{l}_{k}]$ if all of its faces are in
${\mathsf{W}}({\mathsf{X}},\mathsf{L},\epsilon)$ and there exists a witness
point $\mathsf{x}\in\mathsf{X}$ so that the following condition holds

$\max\\{d(\mathsf{l}_{0},\mathsf{x}),d(\mathsf{l}_{1},\mathsf{x}),\dots,d(\mathsf{l}_{k},\mathsf{x})\\}\leq
d_{k}(\mathsf{x})+\epsilon\,.$ (5.1)

  2. 2.

Lazy Witness complex $\mathsf{LW}_{\nu}({\mathsf{X}},\mathsf{L},\epsilon)$.
Let $\nu\in\mathbb{N}$, and let $d_{\nu}(\mathsf{x})$ be the distance between
$\mathsf{x}$ and the $\nu$-th closest landmark point (with
$d_{\nu}(\mathsf{x})\equiv 0$ if $\nu=0$). Then for $\mathsf{l}_{1}$ and
$\mathsf{l}_{2}$ points of $\mathsf{L}$, $[\mathsf{l}_{1},\mathsf{l_{2}}]$ is
an edge in $\mathsf{LW}_{\nu}({\mathsf{X}},\mathsf{L},\epsilon)$ if there
exists a witness point $\mathsf{x}\in\mathsf{X}$ so that the following
condition holds

$\max\\{d(\mathsf{l}_{1},\mathsf{x}),d(\mathsf{l}_{2},\mathsf{x})\\}\leq
d_{\nu}(\mathsf{x})+\epsilon\ .$ (5.2)

A higher dimensional simplex defines an element of
$\mathsf{LW}_{\nu}({\mathsf{X}},\mathsf{L},\epsilon)$ if and only if all of
its edges are in $\mathsf{LW}_{\nu}({\mathsf{X}},\mathsf{L},\epsilon)$

These definitions are those implemented in javaplex. It is easy to see that if
$\epsilon_{1}<\epsilon_{2}$, then we have
${\mathsf{W}}({\mathsf{X}},\mathsf{L},\epsilon_{1})\subset{\mathsf{W}}({\mathsf{X}},\mathsf{L},\epsilon_{2})$
and
$\mathsf{LW}_{\nu}({\mathsf{X}},\mathsf{L},\epsilon_{1})\subset\mathsf{LW}_{\nu}({\mathsf{X}},\mathsf{L},\epsilon_{2})$.

We can now use the same arguments of Section 3 to argue that by letting
$\epsilon$ vary we induce filtrations of witness and lazy witness complexes
and that taking the $i$-th homology of any such sequence as a function of
$\epsilon$ defines a $\mathbb{N}$-persistence module. Therefore we can easily
define persistent homology groups and barcodes and use the complexes
$\mathsf{W}({\mathsf{X}},\mathsf{L},\epsilon)$ and
$\mathsf{LW}_{\nu}({\mathsf{X}},\mathsf{L},\epsilon)$ to study persistent
topological features of distributions of BPS states. Unless specified
otherwise, when using the lazy witness complex we will always set $\nu=1$ and
omit it from the notation.

##  6 BPS invariants on the conifold

In our second example we consider BPS states in a local string theory
compactification, the resolved conifold. The geometry is given by the total
space of the fibration $X={\cal O}(-1)\oplus{\cal
O}(-1)\longrightarrow{\mathbb{P}}^{1}$ and has one Kähler modulus $t$, the
complexified size of the base ${\mathbb{P}}^{1}$. We are interested in a
particular class of BPS states which are given by bound states of a gas of
light D0 branes and D2 branes wrapping the ${\mathbb{P}}^{1}$, with a single
D6 brane wrapping the full non compact total space. Such invariants have been
computed in quite some detail at any point of the moduli space [29, 30, 31,
32].

Since the geometry is non-compact, to properly define the charges it is
necessary to embed it into a compact Calabi-Yau and then take a local limit.
This is done by considering a compact Calabi-Yau $\tilde{X}$ and taking the
limit where the Kähler moduli of all the homology classes become large, with
the sole exception of the Kähler class of a rigid rational curve.

Since we are taking a local limit, it is enough to consider the compact
Calabi-Yau $\tilde{X}$ in the large radius approximation. BPS states are
labeled by charge vectors $\gamma\in\Gamma$, where

$\gamma\ \in\ \Gamma=\Gamma^{\rm m}\oplus\Gamma^{\rm
e}=\left(H^{0}(\tilde{X},{\mathbb{Z}})\oplus
H^{2}(\tilde{X},{\mathbb{Z}})\right)\ \oplus\
\left(H^{4}(\tilde{X},{\mathbb{Z}})\oplus
H^{6}(\tilde{X},{\mathbb{Z}})\right)\ .$ (6.1)

By Poincaré duality D-branes wrapping $p$-cycles correspond to charges as

${\rm D}p\ \longleftrightarrow\
H^{6-p}(\tilde{X},{\mathbb{Z}})=H_{p}(\tilde{X},{\mathbb{Z}})\ ,\qquad
p=0,2,4,6\ .$ (6.2)

The Dirac-Schwinger-Zwanziger pairing has the geometric definition

$\big{\langle}\gamma\,,\,\gamma^{\prime}\,\big{\rangle}_{\Gamma}=\int_{\tilde{X}}\,\gamma\wedge(-1)^{{\rm
deg}/2}\,\gamma^{\prime}\ .$ (6.3)

In the large radius limit the central charge of a BPS state with charge
$\gamma$ is given by the integral

${\cal
Z}_{\tilde{X}}\left(\gamma;\tilde{t}\,\right)=-\int_{\tilde{X}}\,\gamma\wedge{\,\rm
e}\,^{-\tilde{t}}$ (6.4)

where $\tilde{t}=B+{\,{\rm i}\,}J$ is the complexified Kähler form given by
the background supergravity two-form $B$-field and the Kähler $(1,1)$-form $J$
of $\tilde{X}$.

Now we take the local limit, following [29]. We parametrize the Kähler form as

$\tilde{t}=t_{{\mathbb{P}}^{1}}+L\,{\,\rm e}\,^{{\,{\rm
i}\,}\varphi}\,t_{\perp}$ (6.5)

where $t_{{\mathbb{P}}^{1}}$ and $t_{\perp}$ are classes in
$H^{2}(\tilde{X};{\mathbb{C}})$ such that

$\int_{{\mathbb{P}}^{1}}\,t_{{\mathbb{P}}^{1}}=z\ ,\
\qquad\int_{{\mathbb{P}}^{1}}\,t_{\perp}=0\ .$ (6.6)

Now we take the local limit by sending $L\longrightarrow+\infty$.

In this limit the relevant BPS configurations are multi-centered solutions
with core charge $\gamma_{c}=(1,0,0,0)$ and halo charge
$\gamma_{h}=(0,0,-\beta,n)$. Conventionally we write the latter as
$\gamma_{h}=(0,0,-m,n)$, where $m$ denotes the number of times the class
$\beta$ wraps222We use a different notation from [29]: we call $\beta$ the
homology class of a curve, while in [29] $\beta$ is the 4-form Poincaré dual
to the ${\mathbb{P}}^{1}$. the ${\mathbb{P}}^{1}$,
$\beta=m\,[{\mathbb{P}}^{1}]$. Walls of marginal stability can be computed
explicitly as a function of $m$ and $n$. In the local limit the central
charges retain some dependence on the parameter $\varphi$. Therefore the walls
of marginal stability can be parametrized on the space $(z,\varphi)$, where
the Kähler moduli space is extended by the parameter $\varphi$ due to the
local limit. Physically $\varphi$ represent the density of the components of
the B-field normal to the exceptional locus.

The walls of marginal stability are [29, 30, 32]

$\displaystyle\mathsf{MS}_{n}^{m}$
$\displaystyle=\left\\{(z,\varphi)\,:\,\varphi=\frac{1}{3}\arg(z+\frac{n}{m})+\frac{\pi}{3}\,\right\\}\,,$
(6.7) $\displaystyle\mathsf{MS}_{n}^{-m}$
$\displaystyle=\left\\{(z,\varphi)\,:\,\varphi=\frac{1}{3}\arg(z-\frac{n}{m})\,\right\\}\,,$
(6.8) $\displaystyle\mathsf{MS}_{-n}^{-m}$
$\displaystyle=\left\\{(z,\varphi)\,:\,\varphi=\frac{1}{3}\arg(z+\frac{n}{m})\,\right\\}\,,$
(6.9)

with $n\geq 0$ and $m\geq 0$. We will denote a chamber between two walls as
$\mathscr{C}[\mathsf{MS}_{n_{1}}^{m_{1}},\mathsf{MS}_{n_{2}}^{m_{2}}]$.

Now we will study the topological features of the BPS spectra in various
chambers and compare them. From a physical perspective this is similar to what
we have done in Section 4, except that the BPS spectra count states in string
theory and not in quantum field theory (albeit we have rendered gravity non-
dynamical by taking a local limit).

This problem is also interesting from a mathematical perspective. Indeed a
proper treatment of BPS particles in string theory requires to work at the
level of a derived category, in this case ${\cal D}(\mathsf{coh}(X))$. However
for practical computations one typically chooses to work in a simpler abelian
category. Indeed choosing a stability condition on ${\cal D}(\mathsf{coh}(X))$
corresponds to the choice of an abelian category $\mathsf{A}$ and a stability
condition on $\mathsf{A}$. In this case we will discuss, for example, BPS
states computed from the moduli space of ideal sheaves at large radius, or
from the moduli space of cyclic modules of a quiver in a certain chamber.
These objects are mathematically quite different, yet describe the same
physical setting up to wall-crossings. We will provide a new viewpoint using
persistent topology.

###  6.1 Chambers $\mathscr{C}[\mathsf{MS}_{n+1}^{1},\mathsf{MS}_{n}^{1}]$

We begin by considering the chambers of the form
$\mathscr{C}[\mathsf{MS}_{n+1}^{1},\mathsf{MS}_{n}^{1}]$. These chambers are
directly connected to the core region, where only the D6 brane exists as a
stable state, situated near the wall with $\phi=\frac{1}{3}\arg
z+\frac{\pi}{3}$. The partition function in each chamber can be obtained by
applying the KSWCF (2.7). Because of the particularly simple charge
configuration of the bound states, the wall-crossing formula reduces to the
semi-primitive version [33], which amounts in multiplying the partition
function by the partition function of a halo of D0 and D2 branes. After
crossing the appropriate walls of marginal stability, the core D6 brane will
be bound to a halo of charge $\gamma_{h}=(0,0,-m,n)$. In decreasing $\phi$
from the core region, the first walls encountered are of the form
$\mathsf{MS}_{n}^{1}$.

We introduce the standard notation $q={\,\rm e}\,^{\lambda}$ where $\lambda$
is the topological string coupling, and $Q={\,\rm e}\,^{-z}$. The resulting
partition function in the chambers
$\mathscr{C}[\mathsf{MS}_{n+1}^{1},\mathsf{MS}_{n}^{1}]$ is

$Z_{n}(q,Q)=\prod_{k=1}^{n}\left(1-(-q)^{k}Q\right)^{k}\ .$ (6.10)

In these chambers the physical BPS spectrum is particularly simple and can be
obtained simply by expanding (6.10) at any fixed $n\geq 0$.

###  6.2 Large radius

At large radius the effective dynamics of the bound states is captured by a
six dimensional topological field theory, which arise as the topological twist
of the ${\cal N}=2$ supersymmetric abelian theory obtained as low energy
approximation to the DBI action on the D6 brane worldvolume [34, 35]. To reach
this chamber one has to cross all the chambers of the form
$\mathscr{C}[\mathsf{MS}_{n+1}^{-1},\mathsf{MS}_{n}^{-1}]$ by taking smaller
values for $\phi$, for $n\longrightarrow\infty$. A notable feature of the
large radius chamber, as well as other chambers in local threefolds, is that
walls of marginal stability accumulate towards it, leaving a certain ambiguity
to the statement “crossing all the walls”. The behavior is similar to what
happens in quantum field theory with BPS rays, which accumulate to higher spin
fields. Luckily in this situation the limit can be taken analytically. The
computation of the BPS spectrum can also be done directly by other means,
confirming the wall crossing prediction.

Indeed BPS bound states in this chamber can be realized as singular instanton
configurations [34]. More precisely the relevant physical configurations
correspond to ideal sheaves, torsion free sheaves of rank one with trivial
determinant. The partition function of the topological theory can be computed
explicitly using equivariant localization with respect to the natural toric
action on the moduli space of ideal sheaves, induced by the toric action on
the resolved conifold. An equivalent but more geometric perspective is
obtained by recalling that an ideal sheaf ${\cal I}$ sits in the short exact
sequence

$\textstyle{0\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\
\mathcal{I}\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\
\mathcal{O}_{X}\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\
\mathcal{O}_{Y}\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ 0\ ,}$
(6.11)

where $Y$ is a subscheme of $X$. The sequence (6.11) identifies the moduli
space of ideal sheaves with fixed characteristic classes with the Hilbert
scheme of points and curves $\mathrm{Hilb}_{n,\beta}(X)$ parametrizing
subschemes $Y\subset X$ with no component of codimension one and such that

$\displaystyle n=\chi(\mathcal{O}_{Y})\qquad\mbox{and}\qquad\beta=[Y]\ \in\
H_{2}(X,\mathbb{Z})\ ,$ (6.12)

where with $\chi$ we denote the holomorphic Euler characteristic.

The BPS degeneracies are given by the (ordinary) Donaldson-Thomas invariants
of these moduli spaces. In this case they can be computed directly via virtual
integration

$\Omega_{X}(n,\beta)={\tt
DT}_{n,\beta}(X):=\int_{[\mathrm{Hilb}_{n,\beta}(X)]^{\mathrm{vir}}}\,1\ .$
(6.13)

The gauge theory partition function

$Z^{X}_{\mathrm{gauge}}(q,Q)=\sum_{k\,\beta}\ q^{k}\,Q^{\beta}\ {\tt
DT}_{k,\beta}(X)$ (6.14)

can be evaluated explicitly by equivariant localization. In our notation
$Q^{\beta}={\,\rm e}\,^{-m\,z}$. Such a partition function receives
contributions from point like instantons located at the toric fixed points,
say at the north and south pole of the base ${\mathbb{P}}^{1}$, as well as
extended instantons fibered over the ${\mathbb{P}}^{1}$. In the case of the
conifold the result can be written in closed form as [34]

$Z^{X}_{\mathrm{gauge}}(q,Q)=M(-q)^{2}\,\prod_{k=1}^{\infty}(1-(-q)^{k}Q)^{k}\,.$
(6.15)

We have used the partition function (6.15) to compute a large number of
enumerative invariants and used them to construct a point cloud.

###  6.3 Noncommutative crepant resolution

In the noncommutative crepant resolution (NCCR) chamber, geometrical concepts
are replaced by algebraic ones. We reach this chamber by taking smaller and
smaller values of $\phi$, crossing all walls of the form
$\mathsf{MS}_{n}^{-1}$ until we reach the region $\frac{1}{3}\arg
z<\phi<\frac{1}{3}\arg(z-1)$. The conifold point lies at the boundary of this
region. Indeed in a noncommutative crepant resolution ordinary geometrical
notions don’t hold anymore and geometry is replaced by a certain algebra. Such
a phenomenon is generic in toric threefolds. This algebra is the Jacobian
algebra of an appropriate quiver. The center of this algebra is the equation
of the singular limit of the threefold; in this sense the Jacobian algebra is
a “resolution” of the singularity. The physics of such resolutions can be
understood in terms of a certain deformation of a cohomological TQFT, called
“stacky gauge theory”, which has been discussed at length in [35, 36, 37]. We
refer the interested reader to the review [38] for more details.

In the case of the conifold the relevant BPS states can be understood in terms
of the representation theory of the framed quiver [31]

$\textstyle{\star\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\
a_{\star}\ }$$\textstyle{\ \circ\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\
a_{1}\ }$$\scriptstyle{\ a_{2}\ }$$\textstyle{\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
\bullet}$$\scriptstyle{\ b_{1}\ }$$\scriptstyle{\ b_{2}\ }$ (6.16)

with superpotential

${\cal W}=a_{1}\,b_{1}\,a_{2}\,b_{2}-a_{1}\,b_{2}\,a_{2}\,b_{1}\ .$ (6.17)

The path algebra is given explicitly by

${\sf A}=\mathbb{C}[e_{\circ},e_{\bullet}]\langle
a_{1},a_{2},b_{1},b_{2}\rangle\,\big{/}\,\langle
b_{1}\,a_{i}\,b_{2}-b_{2}\,a_{i}\,b_{1}\,,\,a_{1}\,b_{i}\,a_{2}-a_{2}\,b_{i}\,a_{1}\
|\ i=1,2\rangle\ .$ (6.18)

where $e_{\circ}$ and $e_{\bullet}$ are the trivial paths of length zero at
the nodes $\circ$ and $\bullet$. In particular it is easy to see that the
center of the algebra is the equation of the conifold singularity. In this
sense the full path algebra is a noncommutative resolution of the singularity.

Again we can define BPS invariants which correspond to the degeneracies of
bound states of D-branes, the noncommutative Donaldson-Thomas invariants.
Physically we can describe bound states of D2/D0 branes with a D6 brane as
ground states of a supersymmetric quiver quantum mechanics with superpotential
derived from (6.16). The framing node represents the infinitely massive non-
compact D6 brane. From a representation theory perspective the simple
representations at the unframed nodes correspond geometrically to the
generators of $K_{c}(X)$, the K-theory group of sheaves on the conifold with
compact support (equivalently sequences of vector bundles which are exact
outside the exceptional locus). For more details of the correspondence between
representation theory of quivers and large radius geometry in the context of
BPS state counting, see [37].

The ground states of the supersymmetric quiver quantum mechanics are described
in representation theory by cyclic modules. More precisely, consider the
framed representation space

$\mathrm{Rep}({\mathsf{Q}},\circ)=\bigoplus_{(v\longrightarrow
w)\in{\mathsf{Q}}_{1}}\,\mathrm{Hom}_{\mathbb{C}}(V_{v},V_{w})\ \oplus\
\mathrm{Hom}_{\mathbb{C}}(V_{\circ},\mathbb{C})\ ,$ (6.19)

which explicitly depends on the choice of the framed node $\circ$. Let
$\mathrm{Rep}({\mathsf{Q}},\circ;{\cal W})$ be the subscheme of
$\mathrm{Rep}({\mathsf{Q}},\circ)$ cut out by the equations derived from the
superpotential $\partial_{a}{\cal W}=0$. The moduli space of cyclic modules is
the smooth Artin stack

${\mathscr{M}}_{n_{0},n_{1}}({\mathsf{Q}})=\big{[}\mathrm{Rep}({\mathsf{Q}},\circ;{\cal
W})\,\big{/}\,GL(n_{0},\mathbb{C})\times GL(n_{1},\mathbb{C})\big{]}\ .$
(6.20)

This is precisely the situation described in Section 2. The BPS degeneracies
are now defined as (weighted) Euler characteristic of the moduli spaces
${\mathscr{M}}_{n_{0},n_{1}}({\mathsf{Q}})$. In particular they can be
computed explicitly using equivariant localization at the level of the quiver
quantum mechanics, with respect to the natural toric action which rescales all
the field associated with the arrows, while keeping the F-term equations
$\partial_{a}{\cal W}=0$ invariant. The result can be summarized in the
partition function [31]

$Z_{NCCR}(q,Q)=M(-q)^{2}\prod_{k=1}^{\infty}\left(1-(-q)^{k}\,Q\right)^{k}\left(1-(-q)^{k}\,Q^{-1}\right)^{k}$
(6.21)

Again we can compute the BPS degeneracies by a direct expansion and we will
momentarily use persistent homology to study their distribution.

###  6.4 Beyond the NCCR chamber

Now we move to the study of the chambers
$\mathscr{C}[\mathsf{MS}_{-n}^{-1},\mathsf{MS}_{-n-1}^{-1}]$. In those
chambers, a different qualitative phenomenon happens, namely the crossing of
certain conjugation walls $\mathscr{S}$ after which the core charge of the
bound states, identified with the D6 brane at large radius, changes [30].
These conjugation walls are associated with particles becoming massless. As a
consequence there is a monodromy around the locus where the state becomes
massless in the moduli space. Upon crossing such walls the core charge changes
according to the monodromy, while an extra halo of particles appears to ensure
charge conservation.

The partition function can be computed using the wall-crossing formula, giving
[30]

${\cal
Z}_{n}(q,Q)=\left(\prod_{k=1}^{n}(q^{k}\,Q)^{k}\right)M(-q)^{2}\prod_{k=1}^{\infty}\left(1-(-q)^{k}\,Q^{-1}\right)^{k}\prod_{k=n+1}^{\infty}\left(1-(-q)^{k}\,Q\right)^{k}\,.$
(6.22)

Here the presence of the factor $\left(\prod_{k=1}^{n}(q^{k}Q)^{k}\right)$ is
a consequence of the monodromy

$\Gamma_{0}\longrightarrow\Gamma_{0}-\sum_{l=1}^{n}\langle\Gamma_{0},\gamma_{1,l}\rangle\,\gamma_{1,l}\,,$
(6.23)

which a core charge $\Gamma_{0}$ undergoes after crossing the conjugation
walls $\mathscr{S}(\gamma_{1,l},\Gamma_{0})$ for $l=1,\dots,n$. Each time a
conjugation wall $\mathscr{S}(\gamma_{1,l},\Gamma_{0})$ is crossed, the core
charge changes accordingly to the above monodromy, and an halo appears, made
of a filled Fermi sea of particles with charge $\gamma_{1,l}$. Therefore the
nature of the bound states changes in a qualitative fashion, and we are
interested in seeing how does this change affects the distribution of BPS
invariants.

###  6.5 Topological analysis

As we have seen the structure of BPS states on the conifold has very
distinctive features in each chamber. We want now to understand at the
qualitative level offered by the topological analysis how the nature of the
BPS bound states differs from one chamber to another. We use the partition
functions we have described above to generate a rather large number of BPS
invariants, as the coefficients of the expansion

$Z_{\mathscr{C}}(q,Q)=\sum_{k,m\in{\mathbb{Z}}}\,\Omega\left(\gamma=(1,0,-m,k);\mathscr{C}=(z,\phi)\right)\,q^{k}\,Q^{m}\,.$
(6.24)

In our case this partition function is given by (6.10), (6.15), (6.21) and
(6.22) respectively. We have shown explicitly the chamber dependence as well
as the charge vector of a state.

Figure 5: Spectra of BPS states for the conifold. The barcodes have been
obtained using the lazy witness complex
$\mathsf{LW}({\mathsf{X}},\mathsf{L},\epsilon)$ on a point cloud of the form
$\mathsf{x}=(k,m,\log|\Omega(\gamma;\mathscr{C})|)$. Left. Chambers of the
form $\mathscr{C}[\mathsf{MS}_{n+1}^{1},\mathsf{MS}_{n}^{1}]$ for $n=90$. The
point cloud is made out of 1093 BPS invariants and the filtered homology
computation involves 3457 simplices. Right. The large radius chamber. The
point cloud is constructed with 1183 invariants and the filtered homology
computation involves 3632 simplices.

Since for Calabi-Yau manifolds the degeneracies of BPS states grow rather
fast, we take the log of $\Omega(\gamma;\mathscr{C})$. Therefore in all the
cases our point cloud ${\mathsf{X}}$ contains vectors of the form
$\mathsf{x}=(k,m,\log|\Omega(\gamma;\mathscr{C})|)$, where the chamber
$\mathscr{C}$ is specified by the value of the moduli $(z,\phi)$.

For each of the chambers under consideration we take around a thousands of BPS
invariants. For those partition functions in chambers $\mathscr{C}$ which
depend on a choice of an integer $n$, namely (6.10) and (6.22), we chose this
integer high enough to provide enough invariants. Due to the large number of
states, we employ the lazy witness complex
$\mathsf{LW}({\mathsf{X}},\mathsf{L},\epsilon)$, with 200 landmark points. The
computation of persistent homology leads to the barcode distributions shown in
Figures 5, 6 and 7.

Let us understand what kind of topological information we can infer. Consider
firstly Figure 5. On the left we have the chambers close to the core region,
while on the right the large radius chamber. The transition is very clear and
by looking at the homology in degree one, we see that the large radius chamber
presents many more topological features. In a sense moving towards large
radius there is an increase in complexity as more cycles form in an irregular
pattern. We interpret this as evidence that the stability condition which
identifies the large radius chamber offers more possibilities to construct BPS
states, while the enumerative problem in chambers nearby the core region is
comparatively poorer. Interestingly the length of the long-lived persistent
classes is roughly comparable. Looking at the zeroth Betti number in both
cases most homology classes have died by $\epsilon\simeq 1.5$.

Figure 6: Spectra of BPS states for the conifold. The barcodes have been
obtained using the lazy witness complex
$\mathsf{LW}({\mathsf{X}},\mathsf{L},\epsilon)$ on a point cloud of the form
$\mathsf{x}=(k,m,\log|\Omega(\gamma;\mathscr{C})|)$. Left. The noncommutative
crepant resolution chamber. The point cloud is made of 1165 invariants and the
number of simplices is 4361. Right. Chambers of the kind
$\mathscr{C}[\mathsf{MS}_{-n}^{-1},\mathsf{MS}_{-n-1}^{-1}]$ for $n=45$. The
point cloud has 1230 points and the number of simplices is 5095.

Also quite interesting are the similarities between the large radius chamber,
Figure 5 on the right, where BPS invariants count ideal sheaves on the
resolved conifold, and the noncommutative crepant resolution chamber, Figure 6
on the left, where BPS invariants count cyclic modules over the path algebra
of the framed conifold quiver. Despite the difference between the enumerative
problems the shapes of the barcode diagrams are roughly comparable, the main
dissimilarity being the presence of more homology classes in degree zero which
persist over $\epsilon\simeq 3$ in the large radius region. Of course while we
are discussing the distribution of barcodes, the actual homology classes have
a different behavior in both cases, and it would be very interesting to try to
interpret each interval, especially in degree one. The abundance of non-
trivial 1-cycles in both cases points out that, if we try to interpret the
point clouds as a discretization of an underlying geometry, at a generic point
within this geometry there should exist a canonical set of circle
coordinates333Canonical in the sense of being associated to the 1-form dual to
the 1-cycle. More precisely, assuming there is no torsion, one can lift
cohomology from ${\mathbb{Z}}_{2}$ to , and then pick the harmonic
representative of the 1-form, which locally integrates to a smooth coordinate
function..

Figure 7: Chambers of the kind
$\mathscr{C}[\mathsf{MS}_{-n}^{-1},\mathsf{MS}_{-n-1}^{-1}]$ for $n=90$. The
point cloud has 1183 points and the number of simplices is 3632.

Another intriguing phenomenon happens when continuing beyond the
noncommutative crepant resolution chamber, to chambers of the form
$\mathscr{C}[\mathsf{MS}_{-n}^{-1},\mathsf{MS}_{-n-1}^{-1}]$. We have computed
the persistent homology in Figure 6 on the right and Figure 7 for two values
of the chamber label $n$. As $n$ increases the barcode distribution
interpolates between the noncommutative chamber and the large radius chamber.
In Figure 7 the similarities with Figure 5 on the right, are striking, despite
the actual generating functions being distinct. A physical interpretation of
these chambers is non straightforward; as explained in [30], proceeding by
keeping $z$ fixed while sending $\phi$ to zero, makes the large radius
approximation to the periods used to compute the central charges, less
reliable. What we seem to be finding is that getting further and further from
the noncommutative chamber, one reproduces a similar structure to large
radius. Indeed this is reminiscent of the result of [31] which relates the
partition function of large radius Donaldson-Thomas invariants, and the
partition function of noncommutative Donaldson-Thomas invariants, with the
large radius partition function of the flipped conifold, where a topology
changing transition has replaced the exceptional divisor with a topologically
distinct ${\mathbb{P}}^{1}$. According to this interpretation, the shape of
the barcode distribution in Figure 7 seem to imply that the only difference
with the large radius BPS states is the core charge, which has changed due to
crossing a large number of conjugation walls $\mathscr{S}$. It would be very
interesting to have an independent check in the low energy effective theory.

##  7 Quantum geometry of compact threefolds

Now we want to discuss BPS invariants for a few compact Calabi-Yau varieties.
So far we have fixed an underlying compactification and we have studied how
the BPS spectra are affected by crossing walls of marginal stability. We will
now take a somewhat different perspective, by comparing BPS spectra in
different compactifications but in the “same” chamber, at large radius. What
we mean by the same chamber is as follows. The topological string on a compact
Calabi-Yau $X$ captures a particular collection of BPS states which play a
role in the microscopic description of certain five dimensional black holes
[3], the Gopakumar-Vafa invariants [39, 40]. It is generically believed, and
in some cases proven, that the generating function of Gopakumar-Vafa
invariants coincides with the generating function of Donaldson-Thomas
invariants in a certain chamber, upon a change of variables. This chamber
captures physically BPS spectra of bound states of D0/D2 branes with a single
D6 brane. Having a single D6 brane means that the relevant moduli space is the
moduli space of ideal sheaves on $X$. Implicit in the definition of an ideal
sheaf, is a notion of a chamber, for which ideal sheaves represent stable
objects. We will compare BPS spectra in such a chamber for different Calabi-
Yaus.

We have chosen certain one parameter varieties for which a great deal is known
about Gopakumar-Vafa invariants, and the topological string explicitly solved
up to a certain genus [41]. Thanks to Castelnuovo’s theory of curves, these
results allow for the computation of several Donaldson-Thomas invariants [42].

###  7.1 Topological strings on one parameter CY

We will discuss the distributions of BPS invariants on certain one-parameters
models where the topological string amplitudes were computed explicitly in
[41] to high genus. These models have all the form of complete intersection
Calabi-Yaus in weighted projective spaces. Complete intersection manifolds are
constructed as the zero locus of a finite number of homogeneous polynomials in
a product of projective spaces. If we denote a weighted projective space by
${\mathbb{P}}^{n-1}(w_{1},\dots,w_{n})$, then a complete intersection CY of
degree $(d_{1},\dots,d_{k})$ will be denoted by
$X_{d_{1},\dots,d_{k}}(w_{1},\dots,w_{n})$ and $w^{m}$ will denote the $m$
times repetition of the weight $w$. For example the quintic threefold
$X_{5}(1)$ is obtained as the zero locus of a single degree 5 polynomial in
${\mathbb{P}}^{4}$.

A collection of 13 such threefolds was discussed in [41]. All these one-
parameter models have the property that their mirror has a Picard-Fuchs system
with three regular singular points. The moduli spaces have the form ${\cal
M}(X)={\mathbb{P}}^{1}\setminus\\{0,1,\infty\\}$ where the point denoted by
$\infty$ denotes the point of maximal monodromy at large radius, where we will
discuss BPS spectra of D-branes bound states. The other two points correspond
to a conifold point and a point with rational branching, for example an
orbifold point corresponding to a Gepner model. The constraints imposed by
modularity on the topological string amplitudes implies that these can be
written as polynomials over a ring generated by certain modular objects.

In particular the holomorphic ambiguity in the topological amplitude is fixed
by the specific form of the amplitude near the conifold and orbifold points;
for example at the conifold points, the “gap condition” determines the form of
the poles in the amplitude via knowledge of the light BPS states, and this
structure is universal [41].

###  7.2 BPS invariants

To the Calabi-Yaus $X$ under consideration one can associate the partition
function

$Z_{{\tt
GV}}(X;q,t)=\prod_{d=1}^{\infty}\left[\prod_{r=1}^{\infty}(1-q^{r}\,{\,\rm
e}\,^{-dt})^{r\,{\tt GV}_{0,d}}\
\prod_{g=1}^{\infty}\prod_{l=0}^{2g-2}\left(1-q^{g-l-1}{\,\rm
e}\,^{-dt}\right)^{(-1)^{g+l}\left(\begin{matrix}2g-2\\\
l\end{matrix}\right){\tt GV}_{g,d}}\right]\,.$ (7.1)

The topological string free energy $F(X;q,t)=\log Z_{{\tt GV}}(X;q,t)$ was
computed in [41] using the techniques outline above, up to high genus. Here
$t$ represents the Kähler modulus of the Calabi-Yau, while $q={\,\rm
e}\,^{{\,{\rm i}\,}\lambda}$, with $\lambda$ the topological string coupling.
The integers ${\tt GV}_{g,d}$ are the Gopakumar-Vafa invariants of [39, 40].
The free energy $F(X;q,t)$ has an interpretation as a certain coupling in the
low energy ${\cal N}=2$ supergravity. This term can be obtained by a one-loop
computation in a constant graviphoton background, where it receives
contribution from BPS particles engineered by an M2 brane wrapping a degree
$d$ curve, in the M-theory limit of the type IIA string. The Gopakumar-Vafa
invariant ${\tt GV}_{g,d}$ is a twisted supersymmetric index which captures
the contribution of these particles to the effective coupling. The quantum
numbers of these particles can be labelled by the representation of the little
group $\rm SO(4)\sim SU(2)_{L}\otimes SU(2)_{R}$. It turns out that only
supersymmetric particles with quantum number $g$ associated with
representations of $\rm SU(2)_{L}$ of the form

$\left[2(\mathbf{0})+\left(\mathbf{\frac{1}{2}}\right)\right]^{\otimes g}$
(7.2)

contribute to the index.

These couplings and the invariants ${\tt GV}_{g,d}$ play a very important role
in the microscopical description of black holes in quantum gravity. A five
dimensional black hole can be engineered via an M-theory compactification on a
Calabi-Yau, by an M2 brane wrapping a curve $\beta=d\,t\in
H_{2}(X,{\mathbb{Z}})$. The microscopic degeneracies of such a black hole,
with charge $\beta$ and angular momentum $m$ in $\rm SU(2)_{L}$ can be
computed as functions of the invariants ${\tt GV}_{g,d}$, and reproduce
precisely the macroscopic entropy [3].

In this Section we will be more interested in certain D-brane bound states
which are dual to the above description. We will study numerical Donaldson-
Thomas invariants of the moduli space of ideal sheaves of $X$ which physically
represent the degeneracies of BPS bound states of a D6 brane wrapping $X$ with
a gas of D0/D2 branes. Conjecturally the duality has the form

$Z_{{\tt DT}}(X;q,t)=\sum_{n,d}\,{\tt DT}_{n,d}\,q^{n}\,{\,\rm
e}\,^{-dt}=Z_{{\tt GV}}(X;q,t)\,M(-q)^{\chi(X)}\,,$ (7.3)

where

$M(q)=\prod_{n=1}^{\infty}\left(1-q^{n}\right)^{-n}$ (7.4)

is the MacMahon function. Using the knowledge of the topological string
amplitude computed in [41] we will follow the approach of [42] to compute the
BPS invariants ${\tt DT}_{n,d}$ up to a certain $d$. By expanding (7.3) one
sees that the invariant ${\tt DT}_{n,d}$ with fixed $d$, receives
contributions from all the invariants ${\tt GV}_{g,d^{\prime}}$ with
$d^{\prime}\leq d$, but arbitrary $g$. Therefore in principle one should know
the full right hand side of (7.3) to compute the invariants ${\tt DT}_{g,d}$.
However for the models at hand there is an enormous simplification coming from
Castelnuovo theory. Roughly speaking Castelnuovo’s theory provides a bound on
the genus $g$ of a degree $d$ curve in a projective space, and certain
generalizations thereof. For the models we are studying, the Castelnuovo bound
is either known or estimated in [42]. Due to the Castelnuovo bound, given a
degree $d$, there is a $g_{max}$ so that for $g>g_{max}$ all the ${\tt
GV}_{g,d}$ are vanishing. For example for the quintic $X_{5}(1)$

$g_{max}\leq\frac{1}{10}\left(10+5\,d+d^{2}\right)\,.$ (7.5)

Taking as inputs the invariants ${\tt GV}_{g,d}$ of [41, 42] and using
Castelnuovo’s theory, we have computed all the BPS degeneracies ${\tt
DT}_{n,d}$ up to $d=9$ and $n=9$ for the complete intersection varieties
$X_{5}(1)$, $X_{3,3}(1^{6})$, $X_{4,2}(1^{6})$, $X_{3,2,2}(1^{7})$,
$X_{2,2,2,2}(1^{8})$ and $X_{4,3}(1^{5},2)$, to have an homogenous set of data
to use as a point cloud. We have similar computations for the remaining of the
13 one parameter models, but with less data; the results are in line with what
we will discuss momentarily but we will not include them here.

###  7.3 Topological analysis

We collect and discuss here our results. Out of our physical settings we
construct a number of point clouds ${\mathsf{X}}$ which have the form
$\mathsf{x}=(d,n,\log|{\tt DT}_{n,d}|)$ for $\mathsf{x}\in{\mathsf{X}}$. Again
we are taking the logarithm of the degeneracies. As we have already stressed
this should be handled with care, since the logarithm is likely to wash out
more subtle topological features in the BPS spectra. However as already
explained this is not a problem in our case, since we are not using persistent
homology to determine the properties of a BPS spectrum on its own, but to
compare various spectra. From a purely computational perspective, taking the
logarithm is almost a necessity. The BPS degeneracies that we are discussing
correspond to black hole microstates and on general grounds in quantum gravity
these numbers grow exponentially.

Let us apply our formalism. We construct the point cloud ${\mathsf{X}}$ with
BPS spectra consisting of 108-135 states, depending on the geometry. These are
all the non-vanishing Donaldson-Thomas invariants ${\tt DT}_{n,d}$ of the
compact threefold up to degree $d=9$, and $n=9$ (chosen in order to have point
clouds of roughly the same order). Out of ${\mathsf{X}}$ we construct the
family of topological spaces ${\mathsf{X}}_{\epsilon}$ and the Vietoris-Rips
complex $\mathsf{VR}_{\epsilon}({\mathsf{X}})$. Then we pass to the homology
$H_{i}(\mathsf{VR}_{\epsilon}({\mathsf{X}});{\mathbb{Z}}_{2})$ and look at the
topological features of the $\mathbb{N}$-persistence modules.

Figure 8: Barcodes for the distribution of Donaldson-Thomas invariants on
compact Calabi-Yaus. The computation uses the Vietoris-Rips complex
$\mathsf{VR}_{\epsilon}({\mathsf{X}})$ build on a point cloud ${\mathsf{X}}$
whose generic element is $\mathsf{x}=(d,n,\log|{\tt DT}_{n,d}|)$. We give
explicitly the number of simplices used in the computation. Left. The quintic
$X_{5}(1)$, with 135 BPS invariants, from 2662 simplices. Right. The variety
$X_{3,3}(1^{6})$, with 121 BPS invariants, from 3472 simplices.

One immediate feature of Figures 8-10 is that they all look rather similar.
The differences between the distributions of barcodes are minimal. For certain
geometries the $H_{0}$ barcodes are slightly more long-lived and for others
the $H_{1}$ distribution is a bit more regular. However these differences,
despite the damping due to the logarithmic scale, are still very small and it
is natural to think that there is a physical mechanism behind this.

Of course there is a natural candidate for the problem at hand: the
universality expected for black hole physics. The degeneracies of black hole
microstates must be such as to recover the macroscopic entropy, given by the
area law: for the large charges the entropy scales with the area of the black
hole horizon. Furthermore the area law receives an infinite series of higher
derivative corrections [43] which are expected to be determined by the
topological string [44, 33, 45, 46].

Figure 9: Barcode computation, same as in Figure 8. Left. The variety
$X_{4,2}(1^{6})$, with 126 BPS invariants, from 3200 simplices. Right. The
variety $X_{3,2,2}(1^{7})$, with 115 BPS invariants, from 3795 simplices.

What we are seeing here is that also the topological features of the BPS
spectra (but not the actual values of the BPS invariants, which fluctuate
wildly between different geometries) appear to be universal. Note that the
usual ideas about black hole universality refers to the emergence of universal
features in the large charges expansion. Here we are dealing with the BPS
spectrum without any limit (although it should be noted that from a similar
partial knowledge, the authors of [42] could extrapolate a large order result
which agrees with the macroscopic predictions).

Figure 10: Barcode computation, same as in Figure 8. Left. The variety
$X_{2,2,2,2}(1^{8})$, with 108 BPS invariants, from 4163 simplices. Right. The
variety $X_{4,3}(1^{5},2)$, with 133 BPS invariants, from 2988 simplices.

It is natural to believe that what we are seeing is how universal aspects of
black hole physics impact the topological features of the distribution of BPS
states. In particular it is not just the shape of the barcode distribution,
but the actual lengths and positions of the barcodes which are comparable,
both in degree zero and one. We stress again that the values of the BPS
invariants and the corresponding point clouds are very different from case to
case. It is natural to conjecture that this phenomenon would become more
pronounced were we to increase the number of BPS invariants in the point
clouds. To be more precise we put forward the following conjecture, which we
expect to be valid at least for one-parameter Calabi-Yaus: as the number of
BPS invariants increases, approaching the large charge limit, the
$\mathbb{N}$-persistence modules associated with $H_{0}$ and $H_{1}$ tend to
universal $\mathbb{N}$-persistence modules. Note that if we interpret the
point clouds as a discretization of an underlying surface, we can rephrase the
above statement in terms of a single universal homotopy class of surfaces,
governing the distribution of BPS states.

Assuming that such a conjecture is true, it would seem natural to imagine that
such a homotopy class of surfaces is determined by the attractor mechanism.
Unfortunately we do not now how such an explicit link could be realized.

It is clear that we cannot draw any conclusive statement from the limited
spectra we have analyzed here. To check our conjecture directly would require
a better knowledge of the BPS spectra to higher values of the curves’ degree,
and possibly for a larger sample of Calabi-Yaus.

##  8 ${\cal N}=4$ dyons and quantum black holes

As next example we now consider dyons in an ${\cal N}=4$ compactification on
$\rm{K3}\times{\mathbb{T}}^{2}$. The reason to consider this example is that
contrary to the cases discussed in the previous section, the BPS invariants
are directly constrained by modularity and appear as coefficients of certain
number theoretical functions. Indeed in this case the high degree of
supersymmetry and the action of the modular group $\rm
SL(2;{\mathbb{Z}})\times SO(22,6;{\mathbb{Z}})$ of S- and T-dualities allows
for explicit counting functions for BPS states [47]. The number of $1/4$ BPS
states can be written as coefficients of the Fourier expansion of
$1/\Phi_{10}$, where $\Phi_{10}$ is the Igusa cusp form of weight $10$. These
coefficients represent indices of BPS bound states computed at weak string
coupling; as the string coupling grows the geometry backreacts and a black
hole can form. Certain Fourier coefficients of this expansion then represent
degeneracies of black holes microstates which can then be compared with the
macroscopic thermodynamical quantities [48, 45, 49, 50]. We will be interested
in the interplay between such counting functions and modularity; the relevant
counting functions are meromorphic Jacobi forms and the associated mock Jacobi
forms are interpreted as partition functions of single-centered black holes
[51]. We want to discuss how (mock) modularity of the microscopic degeneracies
impacts the topological features of the distribution of BPS states. For a
macroscopic supergravity perspective on how modularity affects the counting
functions, see [52, 53].

The charges of the BPS states take values in the lattice of electric-magnetic
charges $\Gamma_{e}^{6,22}\oplus\Gamma_{m}^{6,22}$, where each factor is
isomorphic to the lattice $H_{2}(\rm{K3};{\mathbb{Z}})\oplus 3\,\Gamma^{1,1}$,
and $\Gamma^{1,1}$ is an hyperbolic lattice. The microscopic degeneracies can
be labelled by three T-duality invariants $(n,l,m)$. They can be extracted
from the expansion of the Siegel modular form of weight 10,
$\Phi_{10}(\tau,z,v)$, which is itself a function of the chemical potentials
for the T-duality invariants [47]. By writing $w={\,\rm e}\,^{2\pi{\,{\rm
i}\,}v}$, we can expand

$\frac{1}{\Phi_{10}}=\sum_{m\geq 1}\ \psi_{m}(\tau,z)\,w^{m}\,,$ (8.1)

where the functions $\psi_{m}(\tau,z)$ are meromorphic Jacobi forms of index
$m$ [51]. Similarly we will also introduce the notation $q={\,\rm
e}\,^{2\pi{\,{\rm i}\,}\tau}$ and $\zeta={\,\rm e}\,^{2\pi{\,{\rm i}\,}z}$. A
meromorphic Jacobi form $\psi(\tau,z)$ of weight $k$ and index $m$ is
holomorphic in $\tau$ and meromorphic in $z$ and transforms as

$\psi\left(\frac{a\,\tau+b}{c\,\tau+d},\frac{z}{c\,\tau+d}\right)=(c\,\tau+d)^{k}\,{\,\rm
e}\,^{\frac{2\pi{\,{\rm i}\,}mcz^{2}}{c\,\tau+d}}\psi(\tau,z)\,,\
\qquad\text{with}\ \left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)\in\mathrm{SL}(2,{\mathbb{Z}})\,,$ (8.2)

under the modular group, and enjoys the elliptic transformation

$\psi(\tau,z+\lambda\,\tau+\mu)={\,\rm e}\,^{-2\pi{\,{\rm
i}\,}m(\lambda^{2}\,\tau+2\lambda\,z)}\ \psi(\tau,z)\,,$ (8.3)

under translations of $z$ by ${\mathbb{Z}}\,\tau+{\mathbb{Z}}$, where
$\lambda\,,\,\mu\in{\mathbb{Z}}$.

All the expansions we shall consider in this Section are meaningful for a
particular choice of the compactification moduli which correspond to the
attractor region. Explicitly we can write the meromorphic Jacobi forms
$\psi_{m}(\tau,z)$ in terms of the elliptic genus of a symmetric product of K3
surfaces [51]

$\psi_{m}(\tau,z)=\frac{1}{A(\tau,z)}\frac{1}{\eta(\tau)^{24}}\mathscr{E}(\tau,z;\mathrm{Sym}^{m+1}(\rm{K3}))\,.$
(8.4)

We have introduced the standard notation

$\displaystyle A(\tau,z)$
$\displaystyle=\frac{\theta_{1}^{2}(\tau,z)}{\eta^{6}(\tau)}\,,$ (8.5)
$\displaystyle\eta(\tau)$ $\displaystyle=q^{\frac{1}{24}}\,\prod_{n\geq
1}(1-q^{n})$ (8.6) $\displaystyle B(\tau,z)$
$\displaystyle=4\left(\frac{\theta_{2}^{2}(\tau,z)}{\theta_{2}^{2}(\tau)}+\frac{\theta_{3}^{2}(\tau,z)}{\theta_{3}^{2}(\tau)}+\frac{\theta_{4}^{2}(\tau,z)}{\theta_{4}^{2}(\tau)}\right)\,.$
(8.7)

and the last function will be used momentarily. The first few $\psi_{m}$
functions can be written down explicitly [51]

$\displaystyle\psi_{1}$
$\displaystyle=\frac{1}{4\,\eta^{24}}\left(9A^{-1}\,B^{2}+3E_{4}\,A\right)\,,$
(8.8) $\displaystyle\psi_{2}$
$\displaystyle=\frac{1}{27\,\eta^{24}}\left(50A^{-1}\,B^{3}+48E_{4}\,A\,B+10\,E_{6}\,A^{2}\right)\,,$
(8.9) $\displaystyle\psi_{3}$
$\displaystyle=\frac{1}{384\,\eta^{24}}\left(475\,A^{-1}\,B^{4}+886\,E_{4}\,A\,B^{2}+360\,E_{6}\,A^{2}\,B+199\,E_{4}^{2}A^{3}\right)\,,$
(8.10) $\displaystyle\psi_{4}$
$\displaystyle=\frac{1}{72\,\eta^{24}}\left(51\,A^{-1}\,B^{5}+155\,E_{4}\,A\,B^{3}+93\,E_{6}\,A^{2}\,B^{2}+102\,E_{4}^{2}\,A^{3}\,B+31\,E_{4}\,E_{6}\,A^{4}\right)\,.$
(8.11)

We have introduced the Eisenstein series $E_{k}$ of weight $k$, with $k\geq
2$,

$E_{k}(\tau)=1-\frac{2\,k}{B_{k}}\sum_{n\geq 1}\sigma_{k-1}(n)\,q^{n}$ (8.12)

with $\sigma_{k-1}(n)=\sum_{d|n}\,d^{k-1}$ and $B_{k}$ the $k$-th Bernoulli
number. We will use the explicit form of these functions to compute the
degeneracies of single centered black holes and study their distributions.

Meromorphic Jacobi forms are associated to mock Jacobi forms. This in practice
means that the functions $\psi_{m}$ can be written as a sum of a finite part
and a polar part

$\psi_{m}=\psi_{m}^{F}+\psi_{m}^{P}\,.$ (8.13)

Both are holomorphic in $\tau$, but while the finite part is holomorphic also
in $z$ the polar part is not and is indeed completely determined by the poles
of $\psi_{m}$. The finite part $\psi_{m}^{F}$ does not transform as a Jacobi
form, but (8.2) is recovered upon adding the polar part $\psi_{m}^{P}$. Both
$\psi_{m}^{F}$ and $\psi_{m}^{P}$ still enjoy the elliptic transformation
(8.3).

Physically this decomposition corresponds to the fact that the Fourier
coefficients of $\psi_{m}^{F}$ capture the degeneracies of single centered
black holes, while the polar part $\psi_{m}^{P}$ determines the jump in the
degeneracies due to decay into two centered black holes at walls of marginal
stability [51]. More complicated decays are forbidden by ${\cal N}=4$
supersymmetry (in the sense that too many supercharges realized non-linearly
produce too many fermionic zero modes to contribute to the relevant indices).
The presence of the polar part is necessary by consistency with wall-crossing,
but its contribution has to be subtracted to determine the degeneracies of
single centered black holes [51]. In particular the polar part of $\psi_{m}$
is known for all $m$

$\psi_{m}^{P}(\tau,z)=\frac{p_{24}(m+1)}{\eta^{24}(\tau)}\sum_{s\in{\mathbb{Z}}}\
\frac{q^{ms^{2}+s}\zeta^{2ms+1}}{(1-\zeta q^{s})^{2}}\,.$ (8.14)

The function $p_{n}(m)$ counts integer partitions of $m$ with $n$ available
slots. The finite part $\psi_{m}^{F}$ is a mock Jacobi form of index $m$,
which means that it can be completed by $\psi_{m}^{P}$ to give the Jacobi form
$\psi_{m}$. If we expand

$\psi_{m}^{F}=\sum_{n,l}\ c(n,r)q^{n}\,\zeta^{l}\,,$ (8.15)

then the precise physical statement is that the microscopic degeneracies
$d(n,l,m)$ corresponding to single centered black holes are related to the
coefficients of $\psi_{m}$ as $d(n,l,m)=(-1)^{l+1}c(n,l)$ for $n\geq m$.
Furthermore on can restrict to the range $0\leq l\leq m$ due to elliptic
invariance of $\psi_{m}$.

We have computed the barcodes for the coefficients of $\psi_{m}^{F}$ and
$\psi_{m}$ for $m=1,2,3,4$, up to a certain value of $n$. We want to
understand the impact on the degeneracies of subtracting the two centered
contribution. We have also computed the persistent homology for $\psi_{m}^{F}$
relaxing the condition $0\leq l\leq m$ to see explicitly the constraint
imposed by elliptic invariance on the data set. The results are presented in
Figures 11 and 12. Let us now discuss them.

Figure 11: Barcodes for the spectrum of single-centered BPS ${\cal N}=4$
dyons. The point cloud is constructed out of vectors of the form
$\mathsf{x}=(\log|d(n,l,m)|,m,l,n)$. The degeneracies are obtained from the
coefficients of $\psi_{m}^{F}$, having subtracted the polar part from
$\psi_{m}$. To compute the persistent homology we use the Vietoris-Rips
complex $\mathsf{VR}_{\epsilon}(\mathsf{X})$ with 254 BPS states, running over
254073 simplices.

Consider Figure 11. The point cloud is constructed out of degeneracies
$d(n,l,m)$ of microstates of single centered black holes, obtained from
subtracting the polar part from $\psi_{m}$. Differently from all the cases we
have seen in the previous Sections, the persistent homology of ${\cal N}=4$
single centered black hole degeneracies has non-trivial classes in degree 2.
These are not much long-lived, albeit their lifespan is comparable with those
we see in the classes for the first Betti number. These classes represent
2-cycles regularly distributed within the point cloud. One interesting feature
is that their lifespans are mostly distributed around the same scale
$\epsilon\sim 3$. Around this scale a number of 2-cycles appear simultaneously
an with the same size, since all the lifespans have comparable lengths. This
pattern is somewhat reminiscent of the “voids” found in the distribution of
flux vacua in rigid Calabi-Yaus in [54]. In those class of type IIB string
flux compactification the only complex structure parameter in the flux
superpotential is the axion-dilaton $\tau_{\rm ad}$. When one studies the
distribution of minima of the flux superpotential in a fundamental domain of
$\mathrm{SL}(2;{\mathbb{Z}})$ in the $\tau_{\rm ad}$-plane, one discovers
empty regions of the form of circles of various sizes, with a big degeneracy
of vacua at the center. This pattern is due to the
$\mathrm{SL}(2;{\mathbb{Z}})$ symmetry. It is natural to suspect that the
similar structure we see in Figure 11 is due to the (mock) modularity of the
degeneracy partition function. Partial evidence comes from a persistent
homology study of the flux vacua of [54], discussed in [20], where a similar
pattern in the persistent homology is reproduced.

Figure 12: Barcodes for the spectrum of ${\cal N}=4$ dyons, the point cloud
has the same form as in Figure 11. Left. The degeneracies include the
contribution of two-centered states, and are obtained from the expansion of
$\psi_{m}$. The point cloud contains 254 BPS states and the persistent
homology computation runs over a total of 255260 simplices. Right. Single
centered degeneracies, but now relaxing the condition $0\leq l\leq m$, that is
including modular ellitpic images. The point cloud contains 707 states and the
homological computation runs over 1318999 simplices.

In Figure 12 on the left, we show the results of the same computation, now
done including also two-centered black holes. The generating function is now
the full $\psi_{m}$ and not $\psi^{F}_{m}$. The barcode distributions are
essentially identical to those in Figure 11 (again we stress that the actual
numbers in the point cloud are in general different). In other words the
presence of two-centered solutions has no effect whatsoever at the level of
persistent homology. One should note however that the expansions we are using
are valid in a region of the moduli space corresponding to the attractor
region. It is therefore natural that the contribution of single centered black
holes is dominating. The situation in other regions of the moduli space might
be rather different.

Finally Figure 12 on the right, shows again the persistent homology of single
centered black holes, but relaxing the condition $0\leq l\leq m$. We are
therefore computing the coefficients of $\psi_{m}^{F}$ this time including the
images under elliptic transformations. Perhaps unsurprisingly the barcode
distribution is again basically the same, where however every feature is more
pronounced444In the comparison one should take into account that this point
cloud contains many more BPS invariants that in the previous cases, which
alters the precise position at which features appear and disappear. Also since
the Figure has to accommodate all the degree zero homology classes provided by
the initial points, many features are pushed up. For example the bump at
around $\epsilon\sim 2$ of Figure 11 in degree zero, is now visible in Figure
12 in the first upper third of the frame.. Here we see at the level of
topology the structure of a (mock) modular form: the barcodes give a visual
representation of the mechanism which connects various coefficients via
elliptic transformations.

##  9 Mathieu moonshine

As a final example we will apply our techniques to study the coefficients of
certain number theoretical functions which play an important role in string
theory compactifications on K3 geometries. Mathieu moonshine is based on the
observation of Eguchi, Ooguri and Tachikawa [55] that the coefficients of the
Fourier expansion of the elliptic genus of K3 can be written as sum of
dimensions of the irreducible representation of the Mathieu group
$\mathbb{M}_{24}$. This observation is a generalization of the monstrous
moonshine, where the fact that sums of the dimensions of the irreducible
representations of the monster group appear as coefficients in the
$J$-function, has an explicit realization via certain modules in bosonic
conformal field theory. These facts have been reviewed in [56, 57, 58], to
which we refer the reader for further references, and whose presentation we
will follow. We shall also assume that the relevant conjecture formulated in
the moonshine literature are true.

In the case at hand the monster group, the largest sporadic group, is replaced
by the Mathieu group $\mathbb{M}_{24}$, which is a subgroup of the group of
permutations with 24 elements which preserves a certain fixed set, known as
the extended binary Golay code. The conformal field theory is a ${\cal N}=4$
superconformal sigma model with target space K3. The spectrum of
supersymmetric string states is encoded in the elliptic genus

$\mathscr{E}(\tau,z)={\rm tr}_{{\cal
H}_{RR}}\left((-1)^{J_{0}+\overline{J}_{0}}\,y^{J_{0}}\,q^{L_{0}-\frac{c}{24}}\,\overline{q}^{\overline{L}_{0}-\frac{\overline{c}}{24}}\right)$
(9.1)

where $q={\,\rm e}\,^{2\pi{\,{\rm i}\,}\tau}$ and $y={\,\rm e}\,^{2\pi{\,{\rm
i}\,}z}$, and $L_{0}$, $\overline{L}_{0}$, $J_{0}$ and $\overline{J}_{0}$ are
the zero modes of the corresponding operators in the superconformal algebra.
The trace is taken over the Ramond-Ramond sector and only the right moving
ground states contribute. As a result the elliptic genus is independent of
$\overline{\tau}$. Such states form a subspace of ${\cal H}_{RR}$ which
decomposes according to the representations of the ${\cal N}=4$ supersymmetry
algebra. As a consequence the elliptic genus can be decomposed as

$\mathscr{E}(\tau,z)=20\,{\rm ch}_{\frac{1}{4},0}(\tau,z)-2\,{\rm
ch}_{\frac{1}{4},\frac{1}{2}}+\sum_{n=1}^{\infty}\,A_{n}\,{\rm
ch}_{\frac{1}{4}+n,\frac{1}{2}}(\tau,z)\ ,$ (9.2)

where

${\rm ch}_{h,j}(\tau,z)={\rm tr}_{{\cal
H}_{h,j}}\left((-1)^{J_{0}}\,y^{J_{0}}\,q^{L_{0}-\frac{c}{24}}\right)$ (9.3)

are the characters over the irreducible representation spaces ${\cal H}_{h,j}$
labeled by the conformal dimension $h$ and the $\mathfrak{su}(2)$ spin $j$. In
particular the $A_{n}$ are dimension of representations $R_{n}$ of
$\mathbb{M}_{24}$ (the same is also true for the first two coefficients
interpreted as virtual representations).