††institutetext: 1School of Theoretical Physics, Dublin Institute of Advanced
Studies, Dublin 4, Ireland 2School of Mathematics, Trinity College, Dublin 2,
Ireland
3Hamilton Mathematical Institute, Trinity College, Dublin 2, Ireland
4School of Natural Sciences, Institute for Advanced Study, 1 Einstein Drive,
Princeton, NJ 08540 USA

# Numerical experiments on coefficients of instanton partition functions

Aradhita Chattopadhyaya1, Jan Manschot2,3,4<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We analyze the coefficients of partition functions of Vafa-Witten theory for
the complex projective plane $\mathbb{CP}^{2}$. We experimentally study the
growth of the coefficients for gauge group $SU(2)$ and $SU(3)$, which are
examples of mock modular forms of depth $1$ and 2 respectively. We also
introduce the notion of “mock cusp form”, and study an example of weight 3
related to the $SU(3)$ partition function. Numerical experiments on the first
200 coefficients suggest that the coefficients of a mock modular form of
weight $k$ grow as the coefficients of a modular form of weight $k$, that is
to say as $n^{k-1}$. On the other hand the coefficients of the mock cusp form
appear to grow as $n^{3/2}$, which exceeds the growth of classical cusp forms
of weight 3. We provide bounds using saddle point analysis, which however
largely exceed the experimental observation.

##  1 Introduction

Instantons are (anti-)self-dual solutions of the Yang-Mills equations of
motion. These non-perturbative solutions are central in theoretical physics
tHooft:1974kcl ; Belavin1975 ; Losev:1997tp ; Dorey:2002ik ; Nekrasov:2002qd ;
Vandoren:2008xg , and connect to many subjects including geometry
Uhlenbeck1986 ; DONALDSON1990 ; Donaldson90 ; Klyachko:91 ; Yoshioka94 ;
Kool_2014  and analytic number theory Gottsche1990 ; Vafa:1994tf ;
Gottsche:1996aoa ; Bringmann:2010sd ; Coskun2022 . Of particular interest are
moduli spaces of instanton solutions, and their topological invariants. This
article considers generating functions of Euler numbers of instanton moduli
spaces for the four-manifold $\mathbb{P}^{2}$. Rather strikingly, these
partition functions are examples of (mock) modular forms as a consequence of
electric-magnetic duality MONTONEN1977117 ; Vafa:1994tf ; Verlinde:1995mz ,
and even give rise to new functions of this type Manschot:2010nc ;
Manschot:2014cca ; Alexandrov:2016enp ; Manschot:2017xcr .

Physically, these generating functions are the partition functions of a
specific topological twist of $\mathcal{N}=4$ Yang-Mills theory Witten:1988ze
; Vafa:1994tf  on a four-manifold $X$. The supersymmetry makes an explicit
analysis feasible for many physical quantities of interest. If $X$ is Ricci
flat, for example $X=K3$, the partition functions of this theory are known to
give rise to classical modular forms. On other four-manifolds, such as the
rational surfaces, the partition functions are instances of mock modular forms
Vafa:1994tf ; Manschot:2011dj ; Dabholkar:2020fde , and even mock modular
forms of higher depth Alexandrov:2016enp ; Manschot:2017xcr . More precisely,
the functions are examples of mixed mock modular forms, and their coefficients
grow exponentially. An exact formula of Rademacher type was derived for these
coefficients for gauge group $SU(2)$ in Bringmann:2010sd , and gauge group
$SU(3)$ in Bringmann:2018cov .

This article concerns the factor of the partition function which are pure mock
modular forms. These are attributed to the smooth instanton solutions, or the
locally free sheaves. For gauge group $SU(2)$ these coefficients are famously
Hurwitz class numbers Klyachko:91 ; zagier:75 . This article explores the
coefficients for gauge group $SU(3)$. We find intriguing patterns for the
coefficients of these partition functions, while we furthermore determine an
upperbound on the asymptotic growth of these coefficients. In our analysis, we
introduce the notion of a “mock cusp form”. For our specific example of a mock
cusp form of weight $k=3$, we find experimentally that the coefficients of the
function accidentally grow as $\sim n^{3/2}$. We are unable to prove this
growth. Using the saddle point method, we are able to bind the growth of the
coefficients by $n^{5/2}$, and using a more heuristic argument based on
lattice sums by $n^{2}$. To put these bounds into context, we note that the
saddle point method applied to classical cusp forms of weight $k$ (or Hecke
bound) gives $n^{k/2}$, which is notably larger than the sharper Deligne-
Petersson-Ramanujan bound $n^{(k-1)/2}$. It would be interesting to explore
the growth of similar functions, such as the coefficients of VW partition
functions for $N>3$, and those of generating functions of bound states of
black holes Chattopadhyaya:2021rdi .

The outline of this article is as follows. We briefly review VW theory in
section 2.1, followed by a discussion of the partition functions for gauge
groups $SU(2)$ and $SU(3)$ in section 2.2. In section 3.1, we define modular
forms and their transformation properties. In section 3.2, we define the mock
modular forms, state their transformation properties and extend the definition
to mock cusp forms. Section 4 provides detailed numerical results and plots
obtained for the partition functions of $SU(3)$ VW theory and how these
functions grow for $p$-th coefficients for prime $p$. Finally in section 5, we
analyse the asymptotics of the Fourier coefficients of the partition functions
of VW theory for gauge group $SU(3)$. In section 5.1, we review the rough
bound on the growth of coefficients of modular and cusp forms. We extend this
in section 5.2 to the growth of coefficients of mock cusp forms associated
with the partition functions of VW theory. Section 5.3 provides an heuristic
argument for the growth of the coefficients of theta series.

##  2 Yang-Mills theory and mock modularity

This section briefly reviews ${\cal N}=4$ super Yang-Mills theory and its
topologically twisted version, Vafa-Witten theory Vafa:1994tf . For more
detailed expositions in mathematics and physics we suggest Klyachko:91 ;
Nakajima:1993jg ; Nakajima:1994nid ; Yoshioka94 ; Yoshioka:k95 , and
Dorey:1996hu ; Dorey:1996bf ; Dorey:1999pd ; Dorey:2002ik .

###  2.1 Instanton solutions

For a simply connected gauge group $G$, we let $F_{ij}$ be the field strength
for the gauge potential $A_{i}$ with $i,j\in\\{1,\dots,4\\}$,

$\displaystyle F_{ij}$ $\displaystyle=$
$\displaystyle\partial_{i}A_{j}-\partial_{j}A_{i}+[A_{i},A_{j}].$ (1)

The Yang-Mills action reads

$S=\frac{1}{4g^{2}}\int_{X}d^{4}x\,{\rm Tr}\;F_{ij}F^{ij},$ (2)

where the trace is over the indices of the representation of the Lie algebra
$G$. The action $S$ is left invariant by the gauge symmetry, whic acts on the
covariant derivative $D_{i}=\partial_{i}+A_{i}$ as,

$\displaystyle D_{i}\rightarrow h^{-1}D_{i}h,\;{\rm for}\;h(\vec{x})\in G.$
(3)

The action is bounded below by the instanton number

$k=-\frac{1}{8\pi^{2}}\int_{X}{\rm Tr}\;F\wedge F,$ (4)

which is a topological invariant of a solution to the Yang-Mills equations of
motion. The instanton number gives a lower bound on the action,

$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{1}{4g^{2}}\int_{X}d^{4}x\,{\rm Tr}\;F_{ij}F^{ij}$
$\displaystyle=$ $\displaystyle\frac{1}{8g^{2}}\int_{X}d^{4}x\,{\rm
Tr}\;\left((F^{\pm})^{2}\mp 2F\wedge F\right)$ $\displaystyle\geq$
$\displaystyle\pm\frac{2\pi^{2}}{g^{2}}\,k,$

where $F^{\pm}$ are the self dual and anti-self dual decomposition of
$F_{ij}$:

$\displaystyle 2F_{ij}^{+}$ $\displaystyle=$ $\displaystyle
F_{ij}+\frac{1}{2}\epsilon_{ijkl}F^{kl},\quad
2F_{ij}^{-}=F_{ij}-\frac{1}{2}\epsilon_{ijkl}F^{kl}.$ (6)

Solutions which saturate the bound (2.1), those with with $F^{+}=0$ or
$F^{-}=0$, are called “instantons” or “anti-instantons”.

Instanton solutions were studied by several physicists as well as
mathematicians since 1970s. On the physics side, instantons are related to
magnetic monopole solutions by dimensional reduction tHooft:1974kcl ;
tHooft:1976snw ; tHooft:1976rip ; tHooft:1986ooh . A complete construction
using linear algebra alone for the Yang-Mills self dual instantons in
Euclidean $S^{4}$ was first given in Atiyah:1978ri ; Drinfeld:1978xr .

Modulo gauge transformations, the moduli space $\mathcal{M}_{k}$ of instanton
solutions with instanton number $k$ is finite dimensional. The dimension
corresponds physically to the number of fermionic zero modes (ground states),
or more precisely the index of the Dirac operator. The sequence of topological
invariants of instanton moduli spaces as function of the Euler number is
naturally combined to a generating function. For example, the generating
function of Euler numbers of moduli spaces of instantons on $\mathbb{R}^{4}$
with boundary conditions for the gauge potential,
$A|_{r\rightarrow\infty}\sim\frac{1}{r^{2}}$ and
$F|_{r\rightarrow\infty}\sim\frac{1}{r^{3}}$ for $r\to\infty$ were computed in
Nakajima:1993jg ; Nakajima:1994nid . Such generating functions are often
realized physically as a statistical partition function of a topologically
twisted supersymmetric Yang-Mills theory. Depending on the field content,
different topological observables can be realized Losev:1997tp . The subject
of this paper is the Vafa-Witten twist of $\mathcal{N}=4$ Yang-Mills theory,
whose partition function is a generating function of Euler numbers of
instanton moduli spaces.

We recall a few aspects from Vafa:1994tf  in what follows. The action of the
bosonic fields of ${\cal N}=4$ super Yang-Mills theory reads

$\begin{split}&S_{b}(A_{i},v_{a})=\\\
&\quad\frac{1}{2g^{2}}\int_{X}d^{4}x\,{\rm
Tr}\;\left(\frac{1}{2}F_{ij}F^{ij}+\sum_{a,b=1}^{6}[v_{a},v_{b}]+\sum_{a=1}^{6}(D_{i}v_{a})^{2}+\frac{ig^{2}\theta}{4\pi^{2}}F\wedge
F\right),\end{split}$ (7)

where $v_{a}$, $a=1,\dots,6$, are scalar fields, and $g$ is the gauge
coupling, $X$ is a 4-manifold. The theory has a global $R$-symmetry group
$SU(4)$. The four scalars $v_{a}$ transform under the 6-dimensional
representation of $SU(4)$, whereas the gauge field $A_{i}$ is a singlet. The
four supercharges transform under the four-dimensional representation ${\bf
4}$ of $SU(4)$.

A topological twist of this $\mathcal{N}=4$ theory on a compact four-manifold
$X$ identifies a principal $SU(4)$ R-symmetry bundle. The identification
follows by the action on the associated bundle with fiber the four-dimensional
representation ${\bf 4}$ of $SU(4)$. The Vafa-Witten twist is the twist for
which the ${\bf 4}$ is identified with the representation $({\bf 1},{\bf
2})\oplus({\bf 1},{\bf 2})$ of the local frame group $Spin(4)\simeq
SU(2)_{+}\times SU(2)_{-}$. The identification of the principal $SU(4)$
R-symmetry bundle thus follows from the chiral spin bundle $W^{-}$ on $X$,
i.e. the bundle associated to the $({\bf 1},{\bf 2})$ representation of
$Spin(4)$. Crucially, the supercharges of the topologically twisted theory
includes two scalar supercharges. The fermionic fields of the theory consists
of two self-dual two-forms, two vectors and two scalars, whereas the bosonic
field content of the twisted theory is:

  1. 1.

Gauge field $A_{i}$,

  2. 2.

Self-dual two-form $B_{ij}$. For $i=0$, we abbreviate $B_{0j}=B_{j}$,

  3. 3.

Real scalar $C$, and complex scalar $\phi$.

The bosonic part of the action with $\phi,\bar{\phi}$ set to zero is given by,

$\begin{split}S(A_{i},B_{i},C)&=\frac{1}{2g^{2}}\int_{X}\,d^{4}x\,{\rm
Tr}\left(F_{ij}^{2}+(D_{i}B_{j})^{2}+(D_{i}C)^{2}\right.\\\
&\quad\left.+\sum_{i<j}[B_{i},B_{j}]^{2}+\sum_{i}[C,B_{i}]^{2}+\frac{ig^{2}\theta}{4\pi^{2}}F\wedge
F\right).\end{split}$ (8)

The instanton solutions are given by

$\displaystyle B_{ij}=C=F_{ij}^{+}=0\quad{\forall}\;i,j.$ (9)

In addition, there is a monopole or Abelian branch Vafa:1994tf . This latter
branch is absent on manifolds with positive scalar curvature such as
$\mathbb{CP}^{2}$.

###  2.2 Partition functions of Vafa-Witten theory

We review the partition functions of the Vafa-Witten theory on gauge groups
$SU(N)$ for the four manifolds $K3$ and $\mathbb{CP}^{2}$. The partition
function is given schematically by the path integral formalism as

$\displaystyle Z(\tau,\bar{\tau})$ $\displaystyle=$ $\displaystyle\int{\cal
D}\vec{\Phi}\,e^{-S(\vec{\Phi})},$ (10)

where $\vec{\Phi}$ represents the field content of the theory, which includes
the fermionic super-partners, ghost anti-ghost pairs and auxiliary fields and

$\displaystyle\tau=\frac{\theta}{2\pi}+\frac{4\pi i}{g^{2}},$ (11)

is the complexified coupling constant.

The partition functions of Vafa-Witten theory on $K3$ and gauge group $U(1)$
were given in Vafa:1994tf

$Z^{K3}_{1}(\tau)=\frac{1}{\eta^{24}(\tau)}=q^{-1}\sum_{k\geq
0}p_{24}(n)q^{k},$ (12)

where $q=e^{2\pi i\tau}$, $p_{24}(k)$ is the partition of a positive integer
$k$ in 24 colors, and $\eta(\tau)$ is the Dedekind eta function given by,

$\eta(\tau)=q^{1/24}\prod_{n=1}^{\infty}(1-q^{n}).$ (13)

The function in (12) is also the partition function of 24 free scalar fields
in two dimensions. In the twisted $SU(2)$ theory the partition function is
given by,

$\displaystyle Z^{K3}_{2}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{8}Z_{K3}(2\tau)+\frac{1}{4}Z_{K3}(\frac{\tau}{2})+\frac{1}{4}Z_{K3}(\frac{\tau+1}{2})$
(14)

More generally, for gauge group $SU(N)$ with prime $N$ the result was proposed
in Vafa:1994tf ,

$\displaystyle Z^{K3}_{N}$ $\displaystyle=$
$\displaystyle\frac{1}{2N^{2}}Z_{K3}(N\tau)+\frac{1}{2N}\sum_{b=0}^{N-1}Z_{K3}(\frac{\tau+b}{N}).$
(15)

The partition function for other ’t Hooft fluxes can be expressed in a similar
form Vafa:1994tf . Explicit partition functions for other algebraic surfaces
with $b_{2}^{+}>1$ are expressed in terms of Seiberg-Witten invariants
Dijkgraaf:1997ce ; Tanaka:2017jom ; Gottsche:2017vxs .

We continue with the four-manifold $\mathbb{CP}^{2}$. While the partition
functions for K3 are given in terms of modular forms for any $N$, we will see
that the partition functions for $\mathbb{CP}^{2}$ give rise to new functions.
The result for gauge group $U(1)$ is still given in terms of the Dedekind eta
function,

$\displaystyle Z^{\mathbb{CP}^{2}}_{U(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{\eta^{3}(\tau)}.$ (16)

However for $SU(2)$ with ’t Hooft flux $\mu$, the partition functions read
Vafa:1994tf :

$Z^{\mathbb{CP}^{2}}_{2,\mu}(\tau,\bar{\tau})=\frac{\widehat{f}_{2,\mu}(\tau,\bar{\tau})}{\eta^{6}(\tau)},\qquad\mu=0,1,$
(17)

with the $\widehat{f}_{2,\mu}$, explicitly given as

$\displaystyle\widehat{f}_{2,\mu}(\tau,\bar{\tau})$ $\displaystyle=$
$\displaystyle
f_{2,\mu}(\tau)-\frac{3i}{4\sqrt{2}\pi}\int_{-\bar{\tau}}^{i\infty}\frac{\Theta_{\mu/2}(w)}{(-i(w+\tau))^{3/2}}dw.$
(18)

The holomorphic parts $f_{2,\mu}$ are a multiple of the generating functions
$G_{\mu}$ of Hurwitz class numbers $H(n)$ Yoshioka94 ; Yoshioka:k95 ;
Klyachko:91 ; zagier2 ; zagier:75 ,

$f_{2,\mu}(\tau)=3G_{\mu}(\tau),\qquad
G_{\mu}(\tau)=\sum_{n=0}^{\infty}H(4n-\mu)\,q^{n-\mu/4}.$ (19)

Moreover, $\Theta_{\mu/2}$ in (18) is the theta series defined by

$\Theta_{\alpha}(\tau)=\sum_{n\in\mathbb{Z}+\alpha}q^{n^{2}/2}.$ (20)

See equation (76) for explicit generating series for the $f_{2,\mu}(\tau)$.
Table LABEL:coefclass in Section 4 lists the first few Hurwitz class numbers.

The holomorphic part $f_{2,\mu}$ transforms as

$\begin{split}f_{2,\mu}\\!\left(-\frac{1}{\tau}\right)&=-\frac{\tau^{3/2}}{\sqrt{2}}\sum_{\nu=0}^{1}(-1)^{\mu\nu}\left(f_{2,\nu}(\tau)-\frac{3i}{4\sqrt{2}\pi}\int_{0}^{i\infty}\frac{\Theta_{\nu/2}(w)}{(-i(w+\tau))^{3/2}}\;dw\right),\\\
f_{2,\mu}(\tau+1)&=e^{-\pi i\mu^{2}/2}f_{2,\mu}(\tau).\end{split}$ (21)

For the non-holomorphic function $\widehat{f}_{2,\mu}$, the shift by the
period integral is absorbed by the non-holomorphic integral such that
$\widehat{f}_{2,\mu}$ transforms as a modular form of weight 3/2. The full
partition function is thus a non-holomorphic modular form of weight $-3/2$.
Physically one can derive the holomorphic anomaly using localization
techniques on the Coulomb branch of the effective field theory
Dabholkar:2020fde ; Bonelli:2020xps ; Manschot:2021qqe .

For $X=\mathbb{CP}^{2}$, the partition functions of $SU(N)$ Vafa-Witten theory
are determined for arbitrary $N$ in Manschot:2014cca . These expressions give
rise to higher dimensional analogues of Appell functions. For $SU(3)$, the
complete non-holomorphic partition function has a similar form to (17), it
reads Manschot:2010nc ; Manschot:2011ym ; Manschot:2017xcr :

$\displaystyle Z^{\mathbb{CP}^{2}}_{3,\mu}(\tau,\bar{\tau})$ $\displaystyle=$
$\displaystyle\frac{\widehat{f}_{3,\mu}(\tau,\bar{\tau})}{\eta(\tau)^{9}},$
(22)

where $\widehat{f}_{3,\mu}$ reads

$\begin{split}\widehat{f}_{3,\mu}(\tau,\bar{\tau})&=f_{3,\mu}(\tau)-\frac{i\left(3/2\right)^{3/2}}{\pi}\sum_{\nu=0,1}\int_{-\bar{\tau}}^{i\infty}\frac{\widehat{f}_{2,\nu}(\tau,-v)\,\Theta_{\frac{\mu}{3}+\frac{\nu}{2}}(3v)}{(-i(v+\tau))^{3/2}}\,dv,\end{split}$
(23)

with $\widehat{f}_{2,\nu}$ as in (18). The first few terms of the
$q$-expansions of $f_{3,\mu}(\tau)$ are:

$\begin{split}f_{3,0}(\tau)&=\frac{1}{9}-q+3q^{2}+\dots,\\\
f_{3,1}(\tau)&=f_{3,2}(\tau)=3q^{5/3}+15q^{8/3}+36q^{11/3}+\dots.\end{split}$
(24)

See Table LABEL:coef in Section 4 for a longer list. The holomorphic parts of
$\widehat{f}_{3,\mu}$ transform as:

$\begin{split}f_{3,\mu}(-1/\tau)&=\frac{i\tau^{3}}{\sqrt{3}}\sum_{\nu=0}^{2}e^{-2\pi
i\mu\nu/3}\\\
&\times\left(f_{3,\nu}(\tau)-\frac{i\left(3/2\right)^{3/2}}{\pi}\sum_{\alpha=0,1}\int_{0}^{i\infty}\frac{\widehat{f}_{2,\alpha}(\tau,-w)\,\Theta_{\frac{2\nu+3\alpha}{6}}(3w)}{(-i(w+\tau))^{3/2}}\;dw\right),\\\
f_{3,\mu}(\tau+1)&=(-1)^{\mu}e^{\pi i\mu^{2}/3}f_{3,\mu}(\tau).\end{split}$
(25)

The non-holomorphic parts satisfy a compact holomorphic anomaly equation
Minahan:1998vr ; Alexandrov:2016tnf ; Alexandrov:2019rth ; Bonelli:2020xps ;
Dabholkar:2020fde ; Manschot:2021qqe .

In the next section, we will introduce various notions of modular forms and
mock modular forms to characterize the various functions, which appeared
above.

##  3 Modular forms and mock modular forms

In this section, we introduce the definitions of modular and mock modular
forms for $SL_{2}(\mathbb{Z})$ and their vector-valued counterparts. We also
introduce the notion of mock cusp forms.

###  3.1 Modular, mock modular and mock cusp forms

We start with the basic definition of a modular form:

Definition: A modular form of weight $k$ is a holomorphic function
$f(\tau):\mathbb{H}\to\mathbb{C}$, which

  1. 1.

transforms under an $SL_{2}(\mathbb{Z})$ matrix $\begin{pmatrix}a&b\\\
c&d\end{pmatrix}$ as follows:

$\displaystyle f\left(\frac{a\tau+b}{c\tau+d}\right)=(c\tau+d)^{k}f(\tau),$
(26)

  2. 2.

and whose growth for $\tau\to i\infty$ is such that

$\lim_{\tau\to i\infty}(c\tau+d)^{-k}f\left(\frac{a\tau+b}{c\tau+d}\right)$
(27)

is bounded for all $\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in
SL_{2}(\mathbb{Z})$.

There are two generators of $SL_{2}(\mathbb{Z})$, namely
$T=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}$ and $S=\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}$. Under these transformations

$\displaystyle f(\tau+1)=f(\tau),\quad
f\left(-\frac{1}{\tau}\right)=\tau^{k}f(\tau).$ (28)

Due to the symmetry under the $T$-transformation, $f(\tau)$ can be expanded as
a Fourier series. This series starts with a constant term $a_{0}$ as a result
of the growth condition,

$f(\tau)=\sum_{n=0}^{\infty}a(n)\,q^{n},\qquad q=e^{2\pi i\tau}.$

One can allow for phases in the transformations

$\begin{split}&f(\tau+1)=\varepsilon(T)\,f(\tau),\\\
&f\left(\frac{-1}{\tau}\right)=\varepsilon(S)\,\tau^{k}f(\tau).\end{split}$

Since $S^{2}=(ST)^{3}=-I$ we have $\varepsilon(S)=\varepsilon(T)^{-3}$.

We next introduce the notion of a cusp form:

Definition: A modular form $f(\tau)$ for $SL_{2}(\mathbb{Z})$ is a cusp form
for which the combination (27) vanishes for $\tau\to i\infty$. As a result,
the constant term $a_{0}$ of its Fourier series vanishes.

The space of modular forms is well-known to be finite dimensional for fixed
weight $k$. The Petterson innerproduct

$\left<f,g\right>=\int_{\mathbb{H}/SL_{2}(\mathbb{Z})}\frac{d\tau\wedge
d\bar{\tau}}{y^{2-k}}f(\tau)\,\overline{g(\tau)},$ (29)

forms a natural innerproduct on the space of cusp forms. Moreover,
$\left<f,g\right>$ vanishes for $f$ an Eisenstein series and $g$ a cusp form.

The definitions of a modular form and cusp are readily extended from the full
modular group $SL_{2}(\mathbb{Z})$ to a congruence subgroup $\Gamma\subset
SL_{2}(\mathbb{Z})$. Another useful notion is a vector-valued modular form:
Definition: A $d$-dimensional vector-valued modular form of weight $k$ under
$SL_{2}(\mathbb{Z})$ is a vector of holomorphic functions

$\vec{f}=\left[\begin{array}[]{c}f_{0}\\\ \vdots\\\
f_{d-1}\end{array}\right]:\mathbb{H}\to\mathbb{C}^{d},$ (30)

with the following properties:

  1. 1.

The elements of the vector transform under the $S$ and $T$ transformations as

$\displaystyle S:\qquad\vec{f}\left(\frac{-1}{\tau}\right)$ $\displaystyle=$
$\displaystyle\tau^{k}\,{\bf M}(S)\vec{f}(\tau),$ (31) $\displaystyle
T:\qquad\vec{f}(\tau+1)$ $\displaystyle=$ $\displaystyle{\bf
M}(T)\vec{f}(\tau).$ (32)

with ${\bf M}(S)$ and ${\bf M}(T)$ as $d\times d$ matrices. ${\bf M}(T)$ is
diagonal in many cases.

  2. 2.

For each element $f_{\mu}$, $\mu=0,\dots,d-1$, the combination (27) is bounded
for all matrices $\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)\in
SL_{2}(\mathbb{Z})$.

This notion is readily extended to a vector-valued cusp form. The elements
$f_{\mu}$ of the vector $\vec{f}$ are modular forms for a congruence subgroup
$\Gamma\subset SL_{2}(\mathbb{Z})$.

We next continue with the definition of mock modular form zagier:75 ;
ZwegersThesis ; zagier2 . We first introduce a map on non-holomorphic
functions. For a function $g:\mathbb{H}\times\mathbb{\bar{H}}\to\mathbb{C}$,
we define the function $g^{*}:\mathbb{H}\times\mathbb{\bar{H}}\to\mathbb{C}$
by:

$\displaystyle
g^{*}(\tau,\bar{\tau})=-2^{1-k}i\int_{-\bar{\tau}}^{i\infty}\frac{g(\tau,-v)}{(-i(v+\tau))^{k-\ell}}\,dv\;.$
(33)

Assuming that $g$ transforms under an element of $\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in SL_{2}(\mathbb{Z})$ as

$g\left(\frac{a\tau+b}{c\tau+d},\frac{au+b}{cu+d}\right)=(c\tau+d)^{k}(cu+d)^{2-k+\ell}g(\tau,u),$
(34)

for some $k$ and $\ell$, the function $g^{*}$ transforms as,

$g^{*}\left(\frac{a\tau+b}{c\tau+d},\frac{a\bar{\tau}+b}{c\bar{\tau}+d}\right)=(c\tau+d)^{k}\left(g^{*}(\tau,\bar{\tau})+2^{1-k}i\int_{d/c}^{i\infty}\frac{g(\tau,-v)}{(-i(v+\tau))^{k-\ell}}\,dv\right).$
(35)

Definition: A mock modular form of weight $k$ is a holomorphic $q$-series
$f:\mathbb{H}\to\mathbb{C}$, such that its completion

$\widehat{f}(\tau,\bar{\tau})=f(\tau)+g^{*}(\tau,\bar{\tau})$ (36)

  1. 1.

transforms as a modular form of weight $k$,

  2. 2.

$g^{*}$ is the image under the map (33) of the complex conjugate of a modular
form with weight $2-k$,

  3. 3.

The combination (27) is bounded for $f$ and all matrices
$\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)\in SL_{2}(\mathbb{Z})$.

There are many variations to the above definitions. A mixed mock modular form
if a function $f$ as above but with $g$ a product of a modular form of weight
$k$ and the complex conjugate of a modular form of weight $2-k+\ell$ for some
$k$ and $\ell$. Similarly, one can consider sums of products.

Clearly, the function $g$ is crucial information to characterize the mock
modular form $f$. It is called the shadow of $f$, and can be obtained by
taking a non-holomorphic derivative of $\widehat{f}$,

$g=y^{k-\ell}\partial_{\bar{\tau}}\widehat{f},$ (37)

with $y={\rm Im}(\tau)$. Thus the shadow is an element of
$\overline{{M}}_{2-k+\ell}(\Gamma)$ for some group $\Gamma\subset
SL_{2}(\mathbb{Z})$, that is to say the shadow is the complex conjugate of a
modular form of weight $2-k+\ell$.

A further interesting extension of mock modular forms are those of depth
$r\geq 1$. Mock modular forms of depth 1 are the functions defined in the
previous definition and equation (36). The notion of mock modular forms of
depth $r$ is defined iteratively for $r\geq 1$ Nazaroglu:2016lmr ; Larry ;
Manschot:2017xcr . To explain the definition, we introduce the vector spaces
$\mathbb{M}_{k}^{r}(\Gamma)$ of mock modular forms of depth $r\geq 1$, with
weight $k$ for the group $\Gamma\subset SL_{2}(\mathbb{Z})$. We furthermore
let $\mathbb{M}_{k}^{-1}(\Gamma)=\emptyset$, and
$\mathbb{M}_{k}^{0}(\Gamma)=M_{k}(\Gamma)$, i.e. the space of classical
modular forms of weight $k$. We furthermore introduce the vector spaces
$\widehat{\mathbb{M}}_{k}^{r}$ of the completed, non-holomorphic functions of
$\mathbb{M}_{k}^{r}(\Gamma)$. We then have the following definition:
Definition: A mock modular form of depth $r$ and with weight $k$ for the group
$\Gamma\subset SL_{2}(\mathbb{Z})$ is defined by the property that its shadow
is an element of
$\widehat{\mathbb{M}}_{k}^{r-1}\otimes\overline{{M}}_{2-k+\ell}$.
We introduce moreover the notion of a mock modular cusp form.
Definition: A mock cusp form of depth $r$ is a mock modular form $f(\tau)$ of
depth $r$ such that the combination (27) vanishes for all elements of
$SL_{2}(\mathbb{Z})$. Thus in particular the constant term of the Fourier
series of $f$ vanishes.

The notion of mock modular form and mock cusp form are readily extended to
vector-valued mock modular forms and vector-valued mock cusp forms.

###  3.2 VW partition functions and (mock) modular forms

With the terminology developed in above, we can characterise the functions
appearing Section 2.2. Clearly, the Dedeking eta function (13) is a modular
form of weight 1/2 with $\varepsilon(T)=e^{2\pi i/24}$. Furthermore,
$\Theta_{\mu}$, $\mu=0,1$, are modular forms of weight 1/2 for the congruence
subgroup $\Gamma_{0}(4)$. The vector $\vec{\Theta}$ of the two functions,

$\vec{\Theta}(\tau)=\left[\begin{array}[]{c}\Theta_{0}(\tau)\\\
\Theta_{1}(\tau)\end{array}\right]$ (38)

is a vector-valued modular form for $SL_{2}(\mathbb{Z})$.

The vector $\vec{f}_{2}$ of the functions $f_{2,\mu}$ (19),

$\vec{f}_{2}(\tau)=\left[\begin{array}[]{c}f_{2,0}(\tau)\\\
f_{2,1}(\tau)\end{array}\right],$ (39)

is a vector-valued mock modular form of weight 3/2 and depth 1. Moreover, the
vector $\vec{f}_{3}$ of the functions $f_{3,\mu}$ (19) is a vector-valued mock
modular form of weight 3 and depth 2 Manschot:2017xcr .

To further study these functions, we first introduce the theta series

$b_{3,j}(\tau)=\sum_{k_{1},k_{2}\in\mathbb{Z}+j/3}q^{k_{1}^{2}+k_{2}^{2}+k_{1}k_{2}}.$
(40)

The modular transformations of these functions are:

$\displaystyle b_{3,j}(-\frac{1}{\tau})$ $\displaystyle=$
$\displaystyle-\frac{i\tau}{\sqrt{3}}\sum_{\ell=0}^{2}e^{-2\pi
ij\ell/3}b_{3,\ell}(\tau),$ (41) $\displaystyle b_{3,j}(\tau+1)$
$\displaystyle=$ $\displaystyle e^{2\pi ij^{2}/3}b_{3,j}(\tau).$

We then form the 3-dimensional vector-valued modular form
$\vec{m}:\mathbb{H}\to\mathbb{C}^{3}$,

$\displaystyle\vec{m}=\begin{bmatrix}m_{0}\\\ m_{1}\\\
m_{2}\end{bmatrix}=\begin{bmatrix}\frac{b_{3,0}^{3}+2b_{3,1}^{3}}{9}\\\
\frac{b_{3,0}b_{3,1}^{2}}{3}\\\ \frac{b_{3,0}b_{3,1}^{2}}{3}\end{bmatrix},$
(42)

The transformations of $m_{\mu}$ under the generators of $SL_{2}(\mathbb{Z})$
are given by,

$\begin{split}m_{\mu}(\tau+1)&=(-1)^{\mu}e^{\pi i\mu^{2}/3}m_{\mu}(\tau),\\\
m_{\mu}\\!\left(-\frac{1}{\tau}\right)&=\frac{i\tau^{3}}{\sqrt{3}}\sum_{\nu=0}^{2}e^{-2\pi
i\nu\mu/3}m_{\nu}(\tau).\end{split}$ (43)

These transformations are identical to those of the completed mock modular
form $\widehat{f}_{3,\mu}$ (23) obtained from $SU(3)$ Vafa-Witten theory.
There is no analog of this type of purely holomorphic functions for $SU(2)$,
since a holomorphic modular form with the same transformations as
$\widehat{f}_{2,\mu}$ does not exist Manschot:2008zb .

The first few terms in the series expansion of $m_{\mu}$ can be given by,

$\displaystyle m_{0}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{9}+8q+30q^{2}+\dots$ (44) $\displaystyle m_{1}(\tau)$
$\displaystyle=$ $\displaystyle 3q^{2/3}+24q^{5/3}+51q^{8/3}+\dots$

We can in fact explicitly write the series for $m_{0}(\tau)$, and
$m_{1}(\tau)$ as follows:

$\begin{split}m_{0}(\tau)&=\frac{1}{9}+\sum_{n=0\;{\rm mod}\;3\atop
n>0}\sum_{d|n}\chi_{n,d}\;d^{2}q^{n/3},\\\ m_{1}(\tau)&=\sum_{n=2\;{\rm
mod}\;3}\sum_{\begin{smallmatrix}\frac{n}{d}=1\;{\rm mod}\;3\\\
d\in\mathbb{Z}\end{smallmatrix}}{\rm sgn}(d)\;d^{2}q^{n/3},\end{split}$ (45)

where, the character $\chi_{n,d}$ is given for $n>0$ by,

$\chi_{n,d}=\begin{cases}-1&\text{if $d=1$ mod 3, $n/d=0$ mod $3$}\\\
1&\text{if $d=2$ mod 3, $n/d=0$ mod $3$}\\\ 1&\text{if $d=0$ mod 3, $n/d=1$
mod $3$}\\\ -1&\text{if $d=0$ mod 3, $n/d=2$ mod $3$}\\\
0&\text{otherwise}.\end{cases}$ (46)

As a result, the coefficient of $q^{p/3}$ of $m_{1}(\tau)$ with $p$ a prime is
given by $p^{2}-1$. The coefficients $d_{0}(p)$ of $q^{p}$ of $m_{0}(\tau)$
with $p$ prime are of two types,

  1. 1.

For $p=2\;{\rm mod}\;3$,

$d_{0}(p)=10p^{2}-10,$ (47)

  2. 2.

For $p=1\;{\rm mod}\;3$,

$d_{0}(p)=8p^{2}+8.$ (48)

For example $d_{0}(3)=80$.

We observe that coefficients of $q^{p}$ for prime $p$ in $m_{0}$ are bounded
by $10p^{2}$.

Now having described $\vec{m}$, we can obtain an example $\vec{S}$ of a
vector-valued mock cusp form of depth 2. Namely, we define $\vec{S}$ as

$\vec{S}=\frac{1}{3}(\vec{m}-\vec{f}).$ (49)

The first terms in the $q$-series of $S_{0}$ and $S_{1}$ are:

$\displaystyle S_{0}(\tau)$ $\displaystyle=$ $\displaystyle
3q+9q^{2}+21q^{3}+\dots,$ (50) $\displaystyle S_{1}(\tau)$ $\displaystyle=$
$\displaystyle S_{2}(\tau)=q^{2/3}+7q^{5/3}+\dots.$

The constant terms of $S_{\mu}$ thus vanish as for the classical cusp forms.
More terms are listed in Table LABEL:coef in Section 4.

Moreover with Eq. (25), we see that the elements $S_{\mu}$ of $\vec{S}$
transform as

$\begin{split}S_{\mu}(-1/\tau)&=\frac{i\tau^{3}}{\sqrt{3}}\sum_{\nu=0}^{2}e^{-2\pi
i\mu\nu/3}\\\
&\times\left(S_{\nu}(\tau)+\frac{\sqrt{3}i}{2\sqrt{2}\pi}\sum_{\alpha=0,1}\int_{0}^{i\infty}\frac{\widehat{f}_{2,\alpha}(\tau,-w)\,\Theta_{\frac{2\nu+3\alpha}{6}}(3w)}{(-i(w+\tau))^{3/2}}\;dw\right),\\\
S_{\mu}(\tau+1)&=(-1)^{\mu}e^{\pi i\mu^{2}/3}S_{\mu}(\tau).\end{split}$ (51)

Since

$\lim_{\tau\to\infty}\int_{0}^{i\infty}\frac{\widehat{f}_{2,\alpha}(\tau,w)\,\Theta_{\frac{2\nu+3\alpha}{6}}(3w)}{(-i(w+\tau)^{3/2})}\;dw=0,$
(52)

the combination (27) vanishes for $S_{\mu}$. We thus confirm that $S_{\mu}$
are mock cusp forms for the congruence subgroup $\Gamma(3)\subset
SL_{2}(\mathbb{Z})$. It would be interesting to understand better the spaces
of such functions, and for example define a suitable inner product. The
standard innerproduct (29) diverges for functions such as
$\widehat{S}_{\mu}(\tau,\bar{\tau})$, and a suitable regularization will need
to be defined. We leave this for future work.

We have some interesting observations for the growth of the Fourier
coefficients of these functions. These are shown as plots in the next section.
In section 5 we briefly discuss the behavior of the growth of coefficients of
cusp forms and how they might change when there is a non-holomorphic piece in
the modular transformation.

##  4 Numerical experiments for the coefficients of VW partition functions

This section carries out varies numerical experiments on the coefficients of
$f_{2,\mu}$, $f_{3,\mu}$, $m_{\mu}$ and $S_{\mu}$. Especially for $S_{\mu}$ we
find some intriguing pattern, namely the prime coefficients appear to be well
approximated by a constant times $p^{3/2}$.

###  4.1 Coefficients for $N=2$

We study in this subsection the coefficients of $f_{2,\mu}=3G_{2,\mu}$. Since
the coefficients of $G_{2,\mu}$ are the well-known Hurwitz class numbers, we
will focus on these. We define for $\mu=0,1$:

$G_{2,\mu}(\tau)=\sum_{n\in\mathbb{Z}-\frac{\mu}{4}}a_{\mu}(n)q^{n}.$ (53)

We first tabulate the coefficients for of $G_{2,\mu}$ up to $n=45$ in Table
LABEL:coefclass.

Table 1: First few coefficients $a_{\mu}(n)=H(4n-\mu)$ of the class number generating functions $G_{\mu}$ (76). $n$ |  $a_{0}(n)$ |  $a_{1}(n-1/4)$
---|---|---
0 |  $-\frac{1}{12}$ |  0
1 |  $\frac{1}{2}$ |  $\frac{1}{3}$
2 |  1 |  1
3 |  $\frac{4}{3}$ |  1
4 |  $\frac{3}{2}$ |  2
5 |  2 |  1
6 |  2 |  3
7 |  2 |  $\frac{4}{3}$
8 |  3 |  3
9 |  $\frac{5}{2}$ |  2
10 |  2 |  4
11 |  4 |  1
12 |  $\frac{10}{3}$ |  5
13 |  2 |  2
14 |  4 |  4
15 |  4 |  3
16 |  $\frac{7}{2}$ |  5
17 |  4 |  1
18 |  3 |  7
19 |  4 |  $\frac{7}{3}$
20 |  6 |  5
21 |  4 |  3
22 |  2 |  6
23 |  6 |  2
24 |  6 |  8
25 |  $\frac{5}{2}$ |  3
26 |  6 |  5
27 |  $\frac{16}{3}$ |  3
28 |  4 |  8
29 |  6 |  2
30 |  4 |  10
31 |  6 |  2
32 |  7 |  5
33 |  4 |  5
34 |  4 |  8
35 |  8 |  3
36 |  $\frac{15}{2}$ |  10
37 |  2 |  $\frac{7}{3}$
38 |  6 |  7
39 |  8 |  4
40 |  6 |  10
41 |  8 |  1
42 |  4 |  11
43 |  4 |  5
44 |  10 |  7
45 |  6 |  5

We observe from the table that the behavior of the coefficients as function of
$n$ is not monotonic, but that the coefficients grow on average. To get a
better impression, we plot the coefficients up to $n=400$ in Figure 1. We
observe that the coefficients are highly scattered, but on average appear to
grow as a power law.

Figure 1: Plot of the coefficients $a_{\mu}(n)=H(4n-\mu)$ of $G_{\mu}$ as
function of $n$. The red dots represent the coefficients $a_{0}(n)$, while the
green ones represent the coefficients $a_{1}(n-1/4)$.

Now in many cases, such as Eisenstein series of integer weight, the growth of
the coefficients is more regular for prime numbers. This we for example
encountered for the functions $m_{\mu}$ (45). We therefore present plots for
$a_{\mu}(p)$ as function of prime $p$ in Figures 2 and 3.

Figure 2: Diagram with the coefficients $a_{0}(p)$ as function of prime $p$.
Figure 3: Diagram with the coefficients $a_{1}(p/4)$ as function for prime $p$
of type $4n+3$.

###  4.2 Coefficients for $N=3$

We proceed in this subsection with partition functions for $SU(3)$, and
compare our findings with those for $N=2$. We define the coefficients
$c_{\mu}(n)$, $d_{\mu}(n)$ and $s_{\mu}(n)$ through

$\displaystyle f_{3,\mu}(\tau)$ $\displaystyle=$
$\displaystyle\sum_{n\in\mathbb{Z}-\frac{\mu}{3}}c_{\mu}(n)q^{n},$ (54)
$\displaystyle m_{\mu}(\tau)$ $\displaystyle=$
$\displaystyle\sum_{n\in\mathbb{Z}-\frac{\mu}{3}}d_{\mu}(n)q^{n},$
$\displaystyle S_{\mu}(\tau)$ $\displaystyle=$
$\displaystyle\sum_{n\in\mathbb{Z}-\frac{\mu}{3}}s_{\mu}(n)q^{n}.$

First we list the coefficients of $f_{3,\mu}$ and $S_{\mu}$ in Table
LABEL:coef. Since $S_{\mu}$ is a mock cusp form, we expect that the
coefficients of are smaller than those for $f_{\mu}$ for $n\gg 1$. We observe
from the table that this is indeed the case.

Table 2: First few coefficients of the mock modular forms $f_{3,\mu}$ and the mock cusp forms $S_{\mu}$ $n$ |  $c_{0}(n)$ |  $c_{1}(n-\frac{1}{3})$ |  $s_{0}(n)$ |  $s_{1}(n-\frac{1}{3})$
---|---|---|---|---
0 |  $\frac{1}{9}$ |  0 |  0 |  0
1 |  $-1$ |  0 |  3 |  1
2 |  3 |  3 |  9 |  7
3 |  17 |  15 |  21 |  12
4 |  41 |  36 |  21 |  28
5 |  78 |  69 |  54 |  27
6 |  120 |  114 |  42 |  58
7 |  193 |  165 |  69 |  49
8 |  240 |  246 |  90 |  94
9 |  359 |  303 |  123 |  69
10 |  414 |  432 |  54 |  136
11 |  579 |  492 |  207 |  109
12 |  626 |  669 |  138 |  177
13 |  856 |  726 |  168 |  120
14 |  906 |  975 |  198 |  235
15 |  1194 |  999 |  258 |  187
16 |  1172 |  1332 |  156 |  292
17 |  1638 |  1338 |  414 |  155
18 |  1569 |  1743 |  207 |  355
19 |  1987 |  1716 |  303 |  278
20 |  2040 |  2226 |  360 |  418
21 |  2578 |  2130 |  474 |  252
22 |  2340 |  2775 |  180 |  435
23 |  3255 |  2625 |  675 |  373
24 |  2940 |  3354 |  414 |  562
25 |  3665 |  3129 |  381 |  327
26 |  3642 |  4041 |  486 |  653
27 |  4490 |  3735 |  690 |  395
28 |  3940 |  4752 |  420 |  712
29 |  5484 |  4317 |  972 |  411
30 |  4734 |  5532 |  342 |  796
31 |  5815 |  5070 |  627 |  598
32 |  5814 |  6393 |  792 |  765
33 |  7014 |  5694 |  942 |  553
34 |  5832 |  7317 |  360 |  961
35 |  8274 |  6582 |  1242 |  696
36 |  7115 |  8277 |  783 |  1057
37 |  8566 |  7272 |  798 |  456
38 |  8322 |  9345 |  846 |  1141
39 |  10018 |  8325 |  1194 |  865
40 |  8334 |  10425 |  486 |  1325
41 |  11778 |  9087 |  1674 |  693
42 |  9708 |  11541 |  864 |  1161
43 |  11785 |  10281 |  1005 |  942
44 |  11604 |  12855 |  1332 |  1435
45 |  13614 |  11058 |  1302 |  804
46 |  10998 |  14175 |  558 |  1531
47 |  15843 |  12327 |  2079 |  1091
48 |  13178 |  15486 |  1074 |  1638
49 |  15531 |  13263 |  1359 |  909
50 |  14817 |  16959 |  1071 |  1747

To get a better impression of the growth, we plot the coefficients of the
functions for a larger range, starting with $f_{0}$ in Figure 4. We observe
from the table that the magnitude of the coefficients alternate between even
and odd $n$. We therefore distinguish the even and odd coefficients with red
and green respectively. We observe from the plot in Figure 4 that the even and
odd coefficients remain separate. This behavior is not unique to $f_{0}$.
Figure 5 demonstrates a similar behavior for the coefficients of $m_{0}$.

Figure 4: Green dots represent the coefficients $c_{0}(n)$ for odd $n$ while
the red ones represent the coefficients $c_{0}(n)$ for even $n$. Figure 5:
Green dots represent the coefficients $d_{0}(n)$ for odd $n$ while the red
ones represent the coefficients $d_{0}(n)$ for even $n$.

We have seen in Section 3.2 that the prime coefficients of $m_{\mu}$ have a
very simple expression. See Eqs (47) and (48). This is a general property of
Eisenstein series. Let us therefore plot the prime coefficients $d_{0}(p)$ for
$m_{0}$, and $c_{0}(p)$ of $f_{3,0}$. We observe that the plot is far less
scattered than the original plots for both $m_{0}$ and $f_{3,0}$. We also
observe that the growth in this range is roughly comparable. We will derive an
upperbound for the growth in the next section, as well as discuss an average.

Figure 6: Blue dots represent the coefficients $d_{0}(p)$ of $m_{0}(\tau)$,
and the pink ones represent the coefficients $c_{0}(p)$ of $f_{3,0}(\tau)$ for
prime $p$.

Next we consider the coefficients of the mock cusp form,
$S_{0}(\tau)=\frac{1}{3}(m_{0}-f_{3,0})$, which are possibly of most interest.
Its coefficients $s_{\mu}(n)$ are plotted in Figure 7, again distinguishing
even and odd $n$. We observe that coefficients are very scattered, and that
the two sets of coefficients are not separated as was the case for $f_{3,0}$
or $m_{0}$. Still, the coefficients of odd powers of $q$ are typically larger
than those of even powers of $q$.

Figure 7: Green dots represent the coefficients $s_{0}(n)$ for odd $n$ while
the red ones represent the coefficients $s_{0}(n)$ for even $n$.

As mentioned above, the behavior of the coefficients is in general better for
the prime coefficients. We plot the coefficients $s_{0}(p)$ for prime $p$ in
Figure 8. Here we do see a striking behavior, namely that these points appear
to lie on a smooth curve. Such a regular curve is not generally the case for
classical cusp forms, such as $\eta^{24}$.

We make a curve fit in Figure 9. The least square fit in the last plot (except
for $p=3$), suggests that the coefficients grow as $n\sim n^{3/2}$. Naturally,
it is desirable to prove this growth. In Section 5, we give an upperbound,
which is unfortunately much weaker.

Figure 8: The dots $s_{0}(p)$ for prime $p$. To note that from 11 upto 199 all
twin primes show the following behavior, if $p$ and $p+2$ are twin primes then
$s_{0}(p)>s_{0}(p+2)$. Figure 9: Least square fits for $s_{0}(p)$ with primes
$p$ of the form $3n\pm 1$. The red dots represent the coefficient $s_{0}(p)$
for prime $p=3n-1$ and the grey line is given by $6.75467\,x^{3/2}$, the green
dots represent $s_{0}(p)$ for prime $p=3n+1$ and the dashed line is given by
$3.57843\,x^{3/2}$.

Next we repeat the above plots for the functions with $\mu=1$. Figure 10 plots
the coefficient $c_{1}(n+2/3)$ of $f_{3,1}$. We again observe different bands
as in Figure 4 for $f_{3,0}$. The plot here appears less scattered though.

Figure 10: Plot of the coefficients $c_{1}(n+2/3)$ of $f_{3,1}$. The numbers
on the horizontal axis are $3n+2$ for integer $n\leq 200$.

We continue with the plot for the coefficients $d_{1}(n+2/3)$ of $m_{1}$ in
Figure 11. The plot is quite similar to Figures 5 and 10. The curves are
quadratic since for $3n+2$ equal to a prime $p$, $d_{1}(p/3)=p^{2}-1$. The
prime coefficients of $c_{1}(p/3)$ of $f_{3,1}$ are also well approximated by
a parabolic curve.

Figure 11: Plot of the coefficients $d_{1}(n+2/3)$ of $m_{1}$. The numbers on
the horizontal axis are $3n+2$ for $n\leq 200$.

Lastly, we consider the coefficients $s_{1}(n+2/3)$ of the mock cusp form
$S_{1}$. We plot in Figure 12 the coefficients $s_{1}(n+2/3)$, and the least
square fit for the prime coefficients $s_{1}(p/3)$. Similar to the $S_{0}$,
the least square fit suggests that the coefficients grow as $\sim n^{3/2}$.
This is notably larger than Deligne’s bound $\sim n^{1}$ for the coefficients
of proper cusp forms of weight 3\. Thus the modified transformation of
$S_{\mu}$ involving the period integral must have an important impact on the
growth of the coefficients.

Figure 12: Plot of coefficients $s_{1}(n+2/3)$ with the numbers $3n+2$ on the
horizontal axis. The least square fit for the prime coefficients $s_{1}(p/3)$
is given by $0.993519\,x^{3/2}$.

##  5 Asymptotics of Fourier coefficients

This section reviews the calculation of upper bounds on the growth of Fourier
coefficients of modular forms and cusp forms using the saddle point method. We
will then generalize this analysis to mock modular forms and cusp forms, in
particular to the function $f_{2,\mu}$, $f_{3,\mu}$ and $S_{\mu}$.

It is well known that the saddle point method (or Hecke bound) typically gives
a very large upperbound, that is to say order $n^{k}$ for modular forms,
whereas the correct bound is order $n^{k-1}$. For cusp forms the saddle point
method gives $n^{k/2}$. On the other hand, the sharper Ramanujan-Petersson
conjecture, proved by Deligne in deligne:74 ; deligne:80 , states that the
Fourier coefficients $a(n)$ of a holomorphic cusp forms for
$SL_{2}(\mathbb{Z})$ bounded by

$|a(n)|<\sigma_{0}(n)n^{\frac{k-1}{2}},$

where $\sigma_{0}(n)$ is the number of divisors of $n$. Thus for prime numbers
the bound is given by,

$|a(p)|<2p^{\frac{k-1}{2}},$

where $k$ is the weight of the cusp form.

Similarly, we do find that the saddle point analysis for mock modular and mock
cusp forms gives rise to larger exponents than those observed in the previous
section. We observe furthermore that the modified modular transformations
involving the iterated period integral increases the exponent in the bound.

###  5.1 Bound for coefficients of modular forms and cusp forms

We review in this section the elementary saddle point analysis to determine a
rough bound for modular forms. Since we know the bound is not sharp, we will
not be careful with errors to the bound.
Modular forms
First we consider a modular form of weight $k$, $f(\tau)=\sum_{n\geq
0}a(n)q^{n}$ and $n\in\mathbb{Z}$ and $a(0)\neq 0$. Its transformation
property is given by,

$\displaystyle f\\!\left(-\frac{1}{\tau}\right)=C\tau^{k}f(\tau),$ (55)

with $C$ a phase. The $n$-th Fourier coefficient is given by,

$\displaystyle a(n)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}f(\tau)\,e^{-2\pi in\tau}\,d\tau\;,$ (56)
$\displaystyle|a(n)|$ $\displaystyle\leq$
$\displaystyle\int_{0}^{1}\left|f\\!\left(-\frac{1}{\tau}\right)\,\tau^{-k}e^{-2\pi
in\tau}\right|\,d\tau\;.$

We assume that ${\rm Im}(-1/\tau)\gg 0$ at the saddle point, such that the
saddle point can be determined from the first non-vanishing term in the
$q$-expansion, $a(0)$. This gives for the saddle point $\tau_{s}$,

$\displaystyle-\frac{k}{\tau_{s}}=2\pi in,\quad\tau_{s}=\frac{ik}{2\pi n}.$
(57)

Thus the assumption is indeed satisfied for sufficiently large $n$. The upper
bound on $a(n)$ is given by,

$\displaystyle|a(n)|<c\,n^{k}\;.$ (58)

with for the constant $c=|a(0)|\left(\frac{2\pi e}{k}\right)^{k}$. This is
clearly not a sharp bound, since we know from the discussion on Eisenstein
series that the bound for modular forms is proportional to $p^{k-1}$ for prime
coefficients $a(p)$.
Cusp forms
Now we show that the same method applied to a cusp form

$S(\tau)=\sum_{n\geq\Delta>0}a(n)q^{n}$

of weight $k$, gives $c\,n^{k/2}$ for some constant $c$ for the bound of its
Fourier coefficients. We assume the following transformation property for
$S(\tau)$

$\displaystyle S\left(-\frac{1}{\tau}\right)$ $\displaystyle=$ $\displaystyle
C\tau^{k}S(\tau),$ (59)

for a phase $C$. Assuming again that ${\rm Im}(-1/\tau)\gg 0$ at the saddle
point, we can approximate $S(-1/\tau)$ by its first non-vanishing term
$a(\Delta)\,e^{-2\pi i\Delta/\tau}$. We then have

$\displaystyle|a(n)|$ $\displaystyle\leq$
$\displaystyle\int_{0}^{1}\left|f(\tau)e^{-2\pi
in\tau}\right|\,d\tau\approx\int_{0}^{1}\left|\tau^{-k}a(\Delta)\,e^{-2\pi
i\Delta/\tau}e^{-2\pi in\tau}\right|\,d\tau.$ (60)

Extremization of $-k\log(\tau)-2\pi i\Delta/\tau-2\pi in\tau$, gives for the
saddle point,

$\displaystyle\tau_{s}$ $\displaystyle=$
$\displaystyle\frac{-k\pm\sqrt{k^{2}-16\pi^{2}\Delta n}}{4\pi in}=\frac{-k\pm
i\sqrt{\delta_{n}}}{4\pi in},$ (61)

where we introduced $\delta_{n}=16\pi^{2}\Delta n-k^{2}$. In the regime of
interest, large $n$, $\delta_{n}>0$, and the assumption ${\rm
Im}(-1/\tau_{s})\gg 0$ is satisfied. We have for the magnitude

$|\tau_{s}|^{2}=\frac{\Delta}{n}.$ (62)

Substitution of $\tau_{s}$ in the exponents, gives

$e^{-2\pi i\Delta/\tau_{s}}e^{-2\pi in\tau_{s}}=e^{\mp i\delta_{n}}.$ (63)

This has unit magnitude and thus does not contribute to the magnitude of the
integrand. We therefore find for the bound,

$|a(n)|\leq c\,n^{k/2},$ (64)

with $c=\Delta^{-k/2}|a(\Delta)|$. This bound is known as the Hecke bound. We
note that this bound is less accurate than Deligne’s theorem ie, for a cusp
form of weight $k$ we expect the growth of the coefficients to grow maximally
as $\sigma_{0}(n)n^{\frac{k-1}{2}}$, where $\sigma_{0}(n)$ is the number of
divisors of $n$.

###  5.2 Bound for coefficients of mock cusp forms

Next we proceed with the mock cusp form $S_{\mu}$ (49). From Eq. (51), we have
the the following transformation

$\begin{split}S_{0}(\tau)&=\frac{-i\tau^{-3}}{\sqrt{3}}\left(S_{0}(-1/\tau)+2S_{1}(-1/\tau)+{\cal
J}_{0}(-1/\tau)\right),\\\
S_{1}(\tau)&=\frac{-i\tau^{-3}}{\sqrt{3}}\left(S_{0}(-1/\tau)-S_{1}(-1/\tau)+{\cal
J}_{1}(-1/\tau)\right),\end{split}$ (65)

where,

$\displaystyle{\cal J}_{0}(\tau)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3}i}{2\sqrt{2}\pi}\sum_{\nu=0}^{2}\sum_{\alpha=0}^{1}\int_{0}^{i\infty}\frac{\widehat{f}_{2,\alpha}(\tau,w)\Theta_{\frac{2\nu+3\alpha}{6}}(3w)}{(-i(w+\tau))^{3/2}}\;dw,$
(66) $\displaystyle{\cal J}_{1}(\tau)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3}i}{2\sqrt{2}\pi}\sum_{\nu=0}^{2}\sum_{\alpha=0}^{1}e^{-2\pi
i\nu/3}\int_{0}^{i\infty}\frac{\widehat{f}_{2,\alpha}(\tau,w)\Theta_{\frac{2\nu+3\alpha}{6}}(3w)}{(-i(w+\tau))^{3/2}}\;dw.$

Similarly to the discussion above, we have

$\begin{split}|s_{\mu}(n)|&\leq\int_{0}^{1}\left|S_{\mu}(\tau)\,e^{-2\pi
in\tau}\right|d\tau.\end{split}$ (67)

We then have to determine the leading contribution among the terms on the rhs
of (65). For large ${\rm Im}(-1/\tau)$, the leading terms of
$S_{\mu}(-1/\tau)$ follow from (50). These are exponentially decreasing. To
compare these with ${\cal J}_{\mu}$, we recall Lemma 3.1 in Bringmann:2010sd ,
(see also Bringmann2007 )

$\displaystyle\int_{0}^{\infty}\frac{\Theta_{\alpha/2}(iz)}{(z+x)^{3/2}}dz=\frac{2}{\sqrt{x}}\delta_{\alpha,0}+O(x^{-3/2}),$
(68)

with the leading term coming from the constant term of $\Theta_{0}$. As a
result, we have

$\begin{split}\mathcal{J}_{\mu}(\tau)=-\frac{\sqrt{3}i}{2\sqrt{2}\pi}\frac{1}{4}\frac{2}{\sqrt{-i\tau}}+O(\tau^{-3/2}).\end{split}$
(69)

Thus the contribution from $\mathcal{J}_{\mu}(-1/\tau)$ provides the leading
term to the rhs in (65). We thus have for the coefficients

$|s_{\mu}(n)|\leq\frac{1}{4\sqrt{2}\pi}\int_{0}^{1}\left|\tau^{-5/2}e^{-2\pi
in\tau}\right|d\tau.$ (70)

Using the saddle point method, we then arrive at

$|s_{\mu}(n)|\leq c\,n^{5/2},$ (71)

with $c=\frac{1}{4\sqrt{2}\pi}(4\pi e/5)^{5/2}$. This upperbound agrees with
the numerical experiments, in the sense that the numerics suggest a growth
proportional to $n^{3/2}$, which is clearly much smaller than $n^{5/2}$. We
note that for a classical cusp form terms like ${\cal J}_{\mu}$ are absent in
the modular transformation, and the saddle point method (64) gives $n^{3/2}$
for the growth.

###  5.3 Average growth for theta series

If more is known about the arithmetic nature of the coefficients, sharper
bounds can often be obtained than the saddle point method. This is for example
the case of Eisenstein series and cusp forms. Often the modular forms can also
be expressed as lattice sum or theta series.

We give a heuristic argument (probably well-known to many) that the leading
term of the average growth of the coefficients of a theta series equals that
of Eisenstein series. This should of course be the case since all theta series
which transform as modular forms can be expressed as linear combinations of
Eisenstein series. To make the argument, let us consider a $d$-dimensional
positive definite lattice $L$ with integral quadratic form $Q$. The associated
theta series $\Theta_{L}(\tau)$ and coefficients $d(n)$ are defined through,

$\begin{split}\Theta_{L}(\tau)&=\sum_{k\in L}q^{Q(k)/2}\\\ &=\sum_{n\geq
0}d(n)\,q^{n}.\end{split}$ (72)

We also introduce the cumulative sum $D(N)$ as,

$D(N)=\sum_{0\leq n\leq N}d(n),$ (73)

such that

$d(N)=D(N)-D(N-1).$ (74)

For a theta series, $D(N)$ is a count of lattice points and thus scales as the
volume of the domain in $L$, whose lattice points are enumerated by $D(N)$.
The volume scales on average as $|k|^{d}=|Q(k)|^{d/2}=N^{d/2}$, such that
$d(N)=D(N)-D(N-1)$ scales as $N^{(d-2)/2}$. Since the weight $k$ of
$\Theta_{L}$ is $k=d/2$, we find for the leading term

$d(n)\approx C\,n^{k-1}.$ (75)

We can use the same approximation for indefinite theta series, which involve a
sum over a positive definite cone in an indefinite lattice. The class number
generating function $G_{2,\mu}$ can be expressed in this form. Moreover, the
expression of Kool Kool_2014  for $f_{3,1}$ are of this form. This matches our
observation that the coefficients of $f_{3,\mu}$ grow as $n^{2}$, similar to
the Eisenstein series of weight 3. This rough estimate is for the coefficients
of $f_{3,\mu}$ is even sharper than the bound for the coefficients of
$S_{\mu}$ using the saddle point method.

## Acknowledgements

The majority of this work was carried out while AC was a postdoctoral fellow
in the School of Mathematics, Trinity College Dublin. During this time, AC and
JM were supported by the Laureate Award 15175 “Modularity in Quantum Field
Theory and Gravity” of the Irish Research Council. JM is also supported by the
Ambrose Monell Foundation. AC is presently funded by fellowship from DIAS. We
thank Prof. Werner Nahm for useful discussions. We also thank the participants
of “Modular Forms in Number Theory and Beyond”, Bielefeld especially Olivia
Beckwith, Jan Vonk, Caner Nazaroglu and Michael Mertens for useful insights.

##  Appendix A Explicit expressions for $f_{N,\mu}$ for $N=2,3$

The explicit expressions for the class number generating functions $G_{\mu}$
as $q$-series were given in Bringmann:2010sd , as:

$\displaystyle G_{0}(\tau)$ $\displaystyle=$
$\displaystyle-\frac{1}{2\Theta_{0}(\tau+1/2)}\sum_{n\in\mathbb{Z}}\frac{n(-1)^{n}q^{n^{2}}}{1+q^{2n}}-\frac{1}{12}\Theta_{0}^{3}(\tau),$
(76) $\displaystyle G_{1}(\tau)$ $\displaystyle=$
$\displaystyle-\frac{q^{-1/4}}{2\Theta_{0}(\tau)}\sum_{n\in\mathbb{Z}}\frac{(2n-1)q^{n^{2}}}{(1-q^{2n-1})}+\frac{1}{6}\Theta_{1}^{3}(\tau),$

where,

$\displaystyle\Theta_{j}(\tau)$ $\displaystyle=$
$\displaystyle\sum_{n\in\mathbb{Z}}q^{\frac{(2n+j)^{2}}{4}}.$ (77)

The explicit expressions for $f_{3,\mu}$ as expansions in $q$ series were
given in Manschot:2017xcr , which are quoted as follows:

$\displaystyle b_{3,0}(\tau)f_{3,0}(\tau)$ $\displaystyle=$
$\displaystyle\frac{13}{240}+\frac{E_{2}(\tau)}{24}+\frac{E_{2}(\tau)^{2}}{72}+\frac{E_{4}(\tau)}{720}$
$\displaystyle-\frac{9}{2}\sum_{k\in\mathbb{Z}}k^{2}q^{3k^{2}}+\sum_{k_{1},k_{2}\in\mathbb{Z}}(k_{1}+2k_{2})^{2}q^{k_{1}^{2}+k_{2}^{2}+k_{1}k_{2}}$
$\displaystyle+\sum_{\begin{smallmatrix}k\in\mathbb{Z}\\\ k\neq
0\end{smallmatrix}}S_{1,0}(k,q)q^{3k^{2}}$
$\displaystyle+\sum_{\begin{smallmatrix}k_{1},k_{2}\in\mathbb{Z}\\\
2k_{1}+k_{2}\neq 0\\\ k_{2}-k_{1}\neq
0\end{smallmatrix}}S_{2}(2k_{1}+k_{2},k_{2}-k_{1},q)q^{k_{1}^{2}+k_{2}^{2}+k_{1}k_{2}+2k_{1}+k_{2}}$
$\displaystyle b_{3,0}(\tau)f_{3,1}(\tau)$ $\displaystyle=$
$\displaystyle\sum_{\begin{smallmatrix}k\in\mathbb{Z}\end{smallmatrix}}S_{1,1}(k,q)q^{3k^{2}-1/3}$
$\displaystyle+\sum_{\begin{smallmatrix}k_{1},k_{2}\in\mathbb{Z}\\\
2k_{1}+k_{2}\neq 1\\\ k_{2}-k_{1}\neq
0\end{smallmatrix}}S_{2}(2k_{1}+k_{2}-1,k_{2}-k_{1},q)q^{k_{1}^{2}+k_{2}^{2}+k_{1}k_{2}+2k_{1}+k_{2}-1/3}$

where we have,

$\displaystyle S_{1,\mu}(k,q)$ $\displaystyle=$
$\displaystyle\frac{(E_{2}(\tau)-1)(k-\mu+1)}{2(1-q^{3k-\mu})}+\frac{9(k-\mu)^{2}+33(k-\mu)+31-E_{2}(\tau)}{2(1-q^{3k-\mu})^{2}}$
(80)
$\displaystyle-\frac{15(k-\mu)+34}{(1-q^{k-\mu})^{3}}+\frac{19}{(1-q^{3k-\mu})^{4}}$
$\displaystyle S_{2}(a,b,q)$ $\displaystyle=$
$\displaystyle\frac{4q^{b}}{(1-q^{a})(1-q^{b})^{3}}+\frac{4q^{a}}{(1-q^{b})(1-q^{a})^{3}}+\frac{4}{(1-q^{a})^{2}(1-q^{b})^{2}}$
$\displaystyle-\frac{2q^{b}(a+b+1)}{(1-q^{a})(1-q^{b})^{2}}-\frac{2q^{a}(a+b+1)}{(1-q^{b})(1-q^{a})^{2}}+\frac{(a+b-2)^{2}-8}{(1-q^{a})(1-q^{b})}$

The blow-up formula provides relations among different representations for the
$q$-series of $f_{3,\mu}$ Bringmann_2016 .

The Eisenstein series $E_{2}$ and $E_{4}$ are given by,

$\displaystyle E_{2}(\tau)$ $\displaystyle=$ $\displaystyle
1-24\sum_{n=1}^{\infty}\sigma_{1}(n)q^{n},\quad
E_{4}(\tau)=1+240\sum_{n=1}^{\infty}\sigma_{3}(n)q^{n}$ (81)

for $\sigma_{k}(n)=\sum_{d|n}d^{k}$.

## References

  * (1) G. ’t Hooft, Magnetic Monopoles in Unified Gauge Theories, Nucl. Phys. B 79 (1974) 276–284.
  * (2) A. Belavin, A. Polyakov, A. Schwartz, and Y. Tyupkin, Pseudoparticle solutions of the yang-mills equations, Physics Letters B 59 (1975), no. 1 85–87.
  * (3) A. Losev, N. Nekrasov, and S. L. Shatashvili, Issues in topological gauge theory, Nucl. Phys. B 534 (1998) 549–611, [hep-th/9711108].
  * (4) N. Dorey, T. J. Hollowood, V. V. Khoze, and M. P. Mattis, The Calculus of many instantons, Phys. Rept. 371 (2002) 231–459, [hep-th/0206063].
  * (5) N. A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003), no. 5 831–864, [hep-th/0206161].
  * (6) S. Vandoren and P. van Nieuwenhuizen, Lectures on instantons, arXiv:0802.1862.
  * (7) K. Uhlenbeck and S. T. Yau, On the existence of hermitian-yang-mills connections in stable vector bundles, Communications on Pure and Applied Mathematics 39 (1986), no. S1 S257–S293, [https://onlinelibrary.wiley.com/doi/pdf/10.1002/cpa.3160390714].
  * (8) S. Donaldson, Polynomial invariants for smooth four-manifolds, Topology 29 (1990), no. 3 257–315.
  * (9) S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds / S.K. Donaldson and P.B. Kronheimer.  Clarendon Press ; Oxford University Press Oxford : New York, 1990.
  * (10) A. Klyachko, Moduli of vector bundles and numbers of classes., Functional Analysis Its Applications 25 (1991) 67–69.
  * (11) K. Yoshioka, The betti numbers of the moduli space of stable sheaves of rank 2 on p2., preprint, Kyoto University 1994 (1994), no. 453 193–220.
  * (12) M. Kool, Euler characteristics of moduli spaces of torsion free sheaves on toric surfaces, Geometriae Dedicata 176 (feb, 2014) 241–269.
  * (13) L. Göttsche, The Betti numbers of the Hilbert scheme of points on a smooth projective surface., Mathematische Annalen 286 (1990), no. 1-3 193–208.
  * (14) C. Vafa and E. Witten, A Strong coupling test of S duality, Nucl. Phys. B 431 (1994) 3–77, [hep-th/9408074].
  * (15) L. Göttsche and D. Zagier, Jacobi forms and the structure of Donaldson invariants for 4-manifolds with $b_{2}^{+}=1$, alg-geom/9612020.
  * (16) K. Bringmann and J. Manschot, From sheaves on $P^{2}$ to a generalization of the Rademacher expansion, Am. J. Math. 135 (2013), no. 4 1039–1065, [arXiv:1006.0915].
  * (17) I. Coskun and M. Woolf, The stable cohomology of moduli spaces of sheaves on surfaces, Journal of Differential Geometry 121 (2022), no. 2 291 – 340.
  * (18) C. Montonen and D. Olive, Magnetic monopoles as gauge particles?, Physics Letters B 72 (1977), no. 1 117–120.
  * (19) E. P. Verlinde, Global aspects of electric - magnetic duality, Nucl. Phys. B 455 (1995) 211–228, [hep-th/9506011].
  * (20) J. Manschot, The Betti numbers of the moduli space of stable sheaves of rank 3 on $P^{2}$, Lett. Math. Phys. 98 (2011) 65–78, [arXiv:1009.1775].
  * (21) J. Manschot, Sheaves on $\mathbb{P}^{2}$ and generalized Appell functions, Adv. Theor. Math. Phys. 21 (2017) 655–681, [arXiv:1407.7785].
  * (22) S. Alexandrov, S. Banerjee, J. Manschot, and B. Pioline, Indefinite theta series and generalized error functions, Selecta Math. 24 (2018) 3927, [arXiv:1606.0549].
  * (23) J. Manschot, Vafa–Witten Theory and Iterated Integrals of Modular Forms, Commun. Math. Phys. 371 (2019), no. 2 787–831, [arXiv:1709.1009].
  * (24) E. Witten, Topological Quantum Field Theory, Commun. Math. Phys. 117 (1988) 353.
  * (25) J. Manschot, BPS invariants of $\mathcal{N}=4$ gauge theory on Hirzebruch surfaces, Commun. Num. Theor. Phys. 6 (2012) 497–516, [arXiv:1103.0012].
  * (26) A. Dabholkar, P. Putrov, and E. Witten, Duality and Mock Modularity, SciPost Phys. 9 (2020), no. 5 072, [arXiv:2004.1438].
  * (27) K. Bringmann and C. Nazaroglu, An exact formula for $\mathrm{U}(3)$ Vafa-Witten invariants on $\mathbb{P}^{2}$, Trans. Am. Math. Soc. 372 (2019), no. 9 6135–6159, [arXiv:1803.0927].
  * (28) D. Zagier, Nombres de classes et formes modulaires de poids 3/2, C.R. Acad. Sci. Paris (A) 281 (1975) 883–886.
  * (29) A. Chattopadhyaya, J. Manschot, and S. Mondal, Scaling black holes and modularity, JHEP 03 (2022) 001, [arXiv:2110.0550].
  * (30) H. Nakajima, Resolution of moduli spaces of ideal instantons on R**4, 1993\.
  * (31) H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994), no. 2 365–416.
  * (32) K. Yoshioka, The betti numbers of the moduli space of stable sheaves of rank 2 on a ruled surface., Math Ann. 302 (1995) 519–540.
  * (33) N. Dorey, V. V. Khoze, and M. P. Mattis, Multi - instanton calculus in N=2 supersymmetric gauge theory, Phys. Rev. D 54 (1996) 2921–2943, [hep-th/9603136].
  * (34) N. Dorey, V. V. Khoze, and M. P. Mattis, Multi instanton calculus in N=2 supersymmetric gauge theory. 2. Coupling to matter, Phys. Rev. D 54 (1996) 7832–7848, [hep-th/9607202].
  * (35) N. Dorey, T. J. Hollowood, V. V. Khoze, M. P. Mattis, and S. Vandoren, Multi-instanton calculus and the AdS / CFT correspondence in N=4 superconformal field theory, Nucl. Phys. B 552 (1999) 88–168, [hep-th/9901128].
  * (36) G. ’t Hooft, Computation of the Quantum Effects Due to a Four-Dimensional Pseudoparticle, Phys. Rev. D 14 (1976) 3432–3450. [Erratum: Phys.Rev.D 18, 2199 (1978)].
  * (37) G. ’t Hooft, Symmetry Breaking Through Bell-Jackiw Anomalies, Phys. Rev. Lett. 37 (1976) 8–11.
  * (38) G. ’t Hooft, How Instantons Solve the U(1) Problem, Phys. Rept. 142 (1986) 357–387.
  * (39) M. F. Atiyah, N. J. Hitchin, V. G. Drinfeld, and Y. I. Manin, Construction of Instantons, Phys. Lett. A 65 (1978) 185–187.
  * (40) V. G. Drinfeld and Y. I. Manin, A DESCRIPTION OF INSTANTONS, Commun. Math. Phys. 63 (1978) 177.
  * (41) R. Dijkgraaf, J.-S. Park, and B. J. Schroers, N=4 supersymmetric Yang-Mills theory on a Kahler surface, hep-th/9801066.
  * (42) Y. Tanaka and R. P. Thomas, Vafa-Witten invariants for projective surfaces I: stable case, J. Alg. Geom. 29 (2020), no. 4 603–668, [arXiv:1702.0848].
  * (43) L. Göttsche and M. Kool, Virtual Refinements of the Vafa–Witten Formula, Commun. Math. Phys. 376 (2020), no. 1 1–49, [arXiv:1703.0719].
  * (44) D. Zagier, Ramanujan’s Mock Theta Functions and Their Applications, 2007\.  https://people.mpim-bonn.mpg.de/zagier/files/aster/326/fulltext.pdf.
  * (45) G. Bonelli, F. Fucito, J. F. Morales, M. Ronzani, E. Sysoeva, and A. Tanzini, Gauge theories on compact toric manifolds, Lett. Math. Phys. 111 (2021) 77, [arXiv:2007.1546].
  * (46) J. Manschot and G. W. Moore, Topological correlators of $SU(2)$, $\mathcal{N}=2^{*}$ SYM on four-manifolds, arXiv:2104.0649.
  * (47) J. Manschot, BPS invariants of semi-stable sheaves on rational surfaces, Lett. Math. Phys. 103 (2013) 895–918, [arXiv:1109.4861].
  * (48) J. A. Minahan, D. Nemeschansky, C. Vafa, and N. P. Warner, E strings and N=4 topological Yang-Mills theories, Nucl. Phys. B 527 (1998) 581–623, [hep-th/9802168].
  * (49) S. Alexandrov, S. Banerjee, J. Manschot, and B. Pioline, Multiple D3-instantons and mock modular forms I, Commun. Math. Phys. 353 (2017), no. 1 379–411, [arXiv:1605.0594].
  * (50) S. Alexandrov, J. Manschot, and B. Pioline, S-duality and refined BPS indices, Commun. Math. Phys. 380 (2020), no. 2 755–810, [arXiv:1910.0309].
  * (51) S. P. Zwegers, Mock Theta Functions.  PhD thesis, Utrecht University, 2008.
  * (52) C. Nazaroglu, $r$-Tuple Error Functions and Indefinite Theta Series of Higher-Depth, Commun. Num. Theor. Phys. 12 (2018) 581–608, [arXiv:1609.0122].
  * (53) K. Bringmann, A. Folsom, K. Ono, and L. Rolen, Harmonic Maass forms and mock modular forms: theory and applications, vol. 64 of American Mathematical Society Colloquium Publications.  American Mathematical Society, Providence, RI, 2017.
  * (54) J. Manschot, On the space of elliptic genera, Commun. Num. Theor. Phys. 2 (2008) 803–833, [arXiv:0805.4333].
  * (55) P. Deligne, The weil conjecture: I, Publications Mathématiques de l’IHÉS 43 (1974) 273–307.
  * (56) P. Deligne, The Weil conjecture: II, Publications Mathématiques de l’IHÉS 52 (1980) 137–252.
  * (57) K. Bringmann and J. Lovejoy, Dyson’s Rank, Overpartitions, and Weak Maass Forms, International Mathematics Research Notices 2007 (01, 2007).
  * (58) K. Bringmann, J. Manschot, and L. Rolen, Identities for generalized appell functions and the blow-up formula, Letters in Mathematical Physics 106 (jul, 2016) 1379–1395.


* [162] Wang L, Yang Z-X, Liu Y-M, Bai C-H, Wang D-Y, Zhang S and Wang H-F 2020 Magnon blockade in a $\mathcal{P}$$\mathcal{T}$-symmetric-like cavity magnomechanical system Ann. Phys. 532 2000028
  * [163] Wang M, Zhang D, Li X-H, Wu Y-Y and Sun Z-Y 2019 Magnon chaos in $\mathcal{P}$$\mathcal{T}$-symmetric cavity magnomechanics IEEE Photonics J. 11 5300108
  * [164] Jing H, Özdemir S K, Lü X-Y, Zhang J, Yang L and Nori F 2014 $\mathcal{PT}$-symmetric phonon laser Phys. Rev. Lett. 113 053604
  * [165] Zhao J, Liu Y, Wu L, Duan C-K, Liu Y-x and Du J 2020 Observation of anti-$\mathcal{P}$$\mathcal{T}$-symmetry phase transition in the magnon-cavity-magnon coupled system Phys. Rev. Appl. 13 014053
  * [166] Nair J M P, Mukhopadhyay D and Agarwal G S 2021 Enhanced sensing of weak anharmonicities through coherences in dissipatively coupled anti-PT symmetric systems Phys. Rev. Lett. 126 180401
  * [167] Yang Y, Wang Y-P, Rao J W, Gui Y S,Yao B M, Lu W and Hu C-M 2020 Unconventional singularity in anti-parity-time symmetric cavity magnonics Phys. Rev. Lett. 125 147202
  * [168] Mukhopadhyay D, Nair J M P and Agarwal G S 2022 Anti-$\mathcal{P}$$\mathcal{T}$ symmetry enhanced interconversion between microwave and optical fields Phys. Rev. B 105 064405
  * [169] Lu T-X, Zhang H, Zhang Q and Jing H 2021 Exceptional-point-engineered cavity magnomechanics Phys. Rev. A 103 063708
  * [170] Harder M, Yang Y, Yao B M, Yu C H, Rao J W, Gui Y S, Stamps R L and Hu C-M 2018 Level attraction due to dissipative magnon-photon coupling Phys. Rev. Lett. 121 137203
  * [171] Xu P-C, J W, Gui Y S, Jin X and Hu C-M 2019 Cavity-mediated dissipative coupling of distant magnetic moments: Theory and experiment Phys. Rev. B 100 094415
  * [172] Li W, Cheng J, Gong W and Li J 2023 Nonlinear self-sustaining dynamics in cavity magnomechanics Phys. Rev. A 108 033518
  * [173] Cheng J, Li W and Li J 2023 Synchronization by magnetostriction (arXiv:2306.13942)
  * [174] Li Y, Zhao C, Zhang W, Hoffmann A and Novosad V 2021 Advances in coherent coupling between magnons and acoustic phonons APL Mater. 9 060902
  * [175] Gonzalez-Ballestero C, Gieseler J and Romero-Isart O 2020 Quantum acoustomechanics with a micromagnet Phys. Rev. Lett. 124 093602
  * [176] Kani A, Sarma B and Twamley J 2022 Intensive cavity-magnomechanical cooling of a levitated macromagnet Phys. Rev. Lett. 128 013602
  * [177] Hei X-L, Li P-B, Pan X-F and Nori F 2023 Enhanced tripartite interactions in spin-magnon-mechanical hybrid systems Phys. Rev. Lett. 130 073602
  * [178] Wang Y, Zhang H-L, Wu J-L, Song J, Yang K, Qin W, Jing H and Kuang L-M 2023 Quantum parametric amplification of phonon-mediated magnon-spin interaction Sci. China Phys. Mech. Astron. 66 110311
  * [179] Li J, Wang Y-P, Wu W-J, Zhu S-Y and You J Q 2021 Quantum network with magnonic and mechanical nodes PRX Quantum 2 040344
  * [180] Tan H and Li J 2021 Einstein-Podolsky-Rosen entanglement and asymmetric steering between distant macroscopic mechanical and magnonic systems Phys. Rev. Res. 3 013192
  * [181] Tan H 2023 Measurement-based control of quantum entanglement and steering in a distant magnomechanical system Photonics 10 1081
  * [182] Weiler M, Dreher L, Heeg C, Huebl H, Gross R, Brandt M S and Goennenwein S T B 2011 Elastically driven ferromagnetic resonance in nickel thin films Phys. Rev. Lett. 106 117601
  * [183] Weiler M, Huebl H, Goerg F S, Czeschka F D, Gross R and Goennenwein S T B 2012 Spin pumping with coherent elastic waves Phys. Rev. Lett. 108 176601
  * [184] Dreher L, Weiler M, Pernpeintner M, Huebl H, Gross R, Brandt M S and Goennenwein S T B 2012 Surface acoustic wave driven ferromagnetic resonance in nickel thin films: Theory and experiment Phys. Rev. B 86 134415
  * [185] Hatanaka D, Asano M, Okamoto H, Kunihashi Y, Sanada H and Yamaguchi H 2022 On-chip coherent transduction between magnons and acoustic phonons in cavity magnomechanics Phys. Rev. Appl. 17 034024
  * [186] Hatanaka D, Asano M, Okamoto H and Yamaguchi H 2023 Phononic crystal cavity magnomechanics Phys. Rev. Appl. 19 054071