††institutetext: School of Theoretical Physics
Dublin Institute for Advanced Studies
Dublin 4, Ireland

# Sign of BPS index for ${\cal N}=4$ dyons

Aradhita Chattopadhyaya<EMAIL_ADDRESS>
###### Abstract

In this paper we argue how the sign changes on an average for the positive
weight mock modular forms associated with the ${\cal N}=4$ type II string
black holes compactified on orbifolds of $K3\times T^{2}$. The orbifolds of
order $N$ act with $g^{\prime}\in[M_{23}]$ an order $N$ symplectic orbifold on
$K3$ and a $1/N$ shift in one of the circles of the torus $T^{2}$. We expand
the inverse Siegel modular forms of subgroups of $Sp_{2}(\mathbb{Z})$ for the
magnetic charge $P^{2}=2$ in terms of mock Jacobi forms and Appell Lerch sums.
We analyze the average growth of the coefficients of these mock modular forms
after theta decomposition and removing inverse eta products. In particular we
remove the contribution of the fundamental string which rightfully dominates
the growth of the positive weight modular forms after the first few
coefficients and ensures the positivity of the helicity trace index $-B_{6}$.
Using numerics and limits of divisor sum function we predict the sign of these
mock modular forms. We also observe that the cusp forms associated with the
non-geometric orbifolds of $K3$ can only contribute for sign changes up to the
first few terms hence their contribution can be neglected for large electric
charges.

##  1 Introduction

Black hole microstates counting and extracting the Bekenstein Hawking entropy
is a subject of great interest Strominger:1996sh ; Dijkgraaf:1996it .
Partition function of ${\cal N}=4$ supersymmetric black holes in type II
string theory were studied in Dijkgraaf:1996it . In later works partition
function for these black holes for geometric orbifolds on $K3$ (CHL orbifolds)
were studied Jatkar:2005bh ; LopesCardoso:2004law ; David:2006yn ; Sen:2007qy
; Dabholkar:2012zz . The calculation is done by computing the helicity trace
index $B_{2n}$, where $4n$ is the number of broken supersymmetry generators
for these dyons Sen:2010mz . The helicity trace index $B_{6}$ computes the
degeneracy of the 1/4-th BPS dyons in these theories.

$B_{2n}=\frac{1}{(2n)!}{\rm Tr}\left(e^{2\pi ih}(2h)^{2n}\right)$ (1)

where $h$ is the third component of the angular of the black hole in its rest
frame. The trace is over all states with a given set of electric and magnetic
charges.

In ${\cal N}=4$ supersymmetric theories for the $1/4$-th BPS dyons the number
of broken supersymmetry generators is 12. These correspond to hair modes for
the black holes Sen:2010mz . In Sen:2010mz  it was assumed that there is a
frame of reference where the only angular momentum in the rest frame comes
from these broken supersymmetry generators. The number of microstates of the
black hole can be counted from $-B_{6}$

$B_{6}=-d_{\rm horizon}\times d_{\rm hair}$ (2)

where $d_{\rm horizon}$ counts the number of degrees of freedom for a given
set of charges while $d_{\rm hair}$ counts the hair degrees of freedom for the
BPS black holes. In Banerjee:2009uk ; Jatkar:2009yd  it was observed that
these hair modes were frame dependent and in one reference frame this was
computed for the canonical case of $K3\times T^{2}$. In Chattopadhyaya:2020yzl
it was found that not in all ${\cal N}=4$ type II black holes these fermionic
zero modes are only hairs. Explicit computation of the hair modes were done in
Chakrabarti:2020ugm ; Chattopadhyaya:2020yzl . The interesting feature we
found is that these hairs are always given by partition functions of some
bosonic degrees of freedom.

In this paper we argue the positivity of the generating function of BPS
indices of single centered ${\cal N}=4$ supersymmetric black holes in type II
theories. These generating functions are given in terms of inverse Siegel
modular forms of subgroups of $Sp_{2}(\mathbb{Z})$ Dabholkar:2005by ;
David:2006ji ; David:2006ru ; David:2006ud ; Govindarajan:2009qt ; Sen:2007vb
; Persson:2013xpa ; Chattopadhyaya:2017ews . At fixed magnetic charges we can
expand these inverse Siegel forms in terms of Jacobi forms. This was done for
$K3$ case in Dabholkar:2012nd  at small values of magnetic charge. A list of
these Jacobi forms were obtained for the orbifold compactifications for
magnetic charge $P^{2}=2$ in Chattopadhyaya:2018xvg . These Jacobi forms
contain poles which can be written as Appell Lerch sums and a finite piece. In
this paper we use the results of Dabholkar:2012nd ; Chattopadhyaya:2018xvg  to
see the behavior of the relevant Jacobi forms after removing their poles and
also extracting out the contribution of the fundamental string. In particular,
for most of our analysis we take the corresponding weight $k=9/2$ mock modular
form which is obtained after theta decomposition and extracting out an extra
factor of $\frac{1}{\eta^{6}}$ such that the resultant function has no effect
from any bosonic partition function whose growth is $\sim e^{\sqrt{n}}$ while
the coefficients of modular forms with positive weight can only grow
polynomially.

This paper is organized in the following manner. In section 2 we give a brief
review of computing $1/4$-th BPS dyons. In section 3.1 we give the definitions
of modular, Jacobi, mock modular, mock Jacobi forms. In section 3.2 we propose
two statements based on numerical evidence and assuming a similarity with
Prime number theorem. We also review some theorems on growth of the sum of
divisors of $n$. In section 4 we review the results of the canonical case of
$K3\times T^{2}$, in section 4.1 we extend the results to the 2A orbifold
based on numerical evidence and the results of section 3 and in section 4.2 we
further extend the results for orbifolds of order $N>2$ odd.

##  2 Counting $1/4$-th BPS dyons

In this section we give a brief introduction of the computation of the 6th
helicity trace index, which computes the number of $1/4$th BPS dyons in ${\cal
N}=4$ supersymmetric type II theories. We consider the compactifications on
symplectic orbifolds of $K3$ by an orbifold of $g^{\prime}$ of order $N$
corresponding to conjugacy classes of Mathieu group $M_{23}$. In the attractor
chamber only single centered solutions survive and since the black hole is
spherically symmetric and regular at the horizon and assuming there is a frame
of reference where the only hair modes were the 12 broken supersymmetry
generators the sign of $B_{6}$ was predicted to be negative Sen:2010mz . We
can obtain the Fourier coefficient of the black hole partition functions for
given electromagnetic charges in the attractor chamber. This can be written as
follows:

$\displaystyle d(Q,P)=\frac{1}{N}(-1)^{Q\cdot P+1}\int_{{\cal
C}}d\tilde{\rho}d\tilde{\sigma}d\tilde{v}\;e^{-\pi
i(N\tilde{\rho}Q^{2}+\tilde{\sigma}P^{2}/N+2\tilde{v}Q\cdot
P)}\frac{1}{\tilde{\Phi}_{k}(\tilde{\rho},\tilde{\sigma},\tilde{v})}.$ (3)

The contour ${\cal C}$ is defined over a 3 dimensional subspace of
$\mathbb{C}^{3}$ where,

$\tilde{\rho}=\tilde{\rho}_{1}+i\tilde{\rho}_{2},\tilde{\sigma}=\tilde{\sigma}_{1}+i\tilde{\sigma}_{2},\tilde{v}=\tilde{v}_{1}+i\tilde{v}_{2}.$
$\displaystyle\tilde{\rho}_{2}=M_{1},\qquad\tilde{\sigma}_{2}=M_{2},\qquad\tilde{v}_{2}=-M_{3},$
(4) $\displaystyle 0\leq\tilde{\rho}_{1}\leq 1,\qquad
0\leq\tilde{\sigma}_{1}\leq N,\qquad 0\leq\tilde{v}_{1}\leq 1.$

Here $M_{1},M_{2},M_{3}$ are positive numbers, which are fixed and large and
$M_{3}<<M_{1},M_{2}$. For CHL models there is a symmetry in the electric and
magnetic charges which essentially render the contour as the Fourier
expansions are done in the order $e^{2\pi i\tilde{\sigma}}$ and $e^{2\pi
i\tilde{\rho}}$ and then in $e^{-2\pi i\tilde{v}}$. The function
$\tilde{\Phi}$, which is present in the integrand is a Siegel modular form of
weight $k$ transforming under a subgroup of $Sp(2,\mathbb{Z})$. We can write
this explicitly as:

$\displaystyle\tilde{\Phi}_{k}(\rho,\sigma,v)$ $\displaystyle=$ $\displaystyle
e^{2\pi i(\tilde{\rho}+\tilde{\sigma}/N+\tilde{v})}$
$\displaystyle\prod_{r=0}^{N-1}\prod_{\begin{smallmatrix}k^{\prime}\in\mathbb{Z}+\frac{r}{N},l\in\mathcal{Z},\\\
j\in 2\mathbb{Z}\\\ k^{\prime},l\geq
0,\;j<0\;k^{\prime}=l=0\end{smallmatrix}}(1-e^{2\pi
i(k^{\prime}\sigma+l\rho+jv)})^{\sum_{s=0}^{N-1}e^{2\pi
isl/N}c^{r,s}(4k^{\prime}l-j^{2})}.$

Here $c^{(r,s)}$ can be obtained from the Fourier coefficients of the twisted
elliptic genus of $K3$ given as $F^{(r,s)}(\tau,z)$

$\displaystyle F^{(r,s)}(\tau,z)$ $\displaystyle=$
$\displaystyle\sum_{n,j}c^{(r,s)}(4n-j^{2})q^{n}y^{j}$ (6)

A list of $F^{(r,s)}$ can be found in (Chattopadhyaya:2017ews, ). In general
we can write $F^{(r,s)}$ as,

$\displaystyle F^{(r,s)}(\tau,z)$ $\displaystyle=$
$\displaystyle\alpha_{N}^{(r,s)}A(\tau,z)+\beta_{N}^{(r,s)}(\tau)B(\tau,z)$
(7)

where $\alpha_{N}^{(r,s)}$ is a positive number for the orbifolds in question.
$\beta_{N}^{(r,s)}$ is generally a function of weight two holomorphic
Eisenstein series and cusp forms involving various products of eta functions.
We can extract the Eisenstein series and cusp forms from
$\beta_{N}^{(r,s)}(\tau)$

$\beta_{N}^{(r,s)}(\tau)=\sum_{d|N}f_{d}{\cal
E}_{d}(\frac{a\tau+b}{d})+b_{N}\eta_{N}(\tau)$

where $f_{d},b_{N}$ are numbers, $\eta_{N}$ is a weight two cusp form and
$a,b$ depend on $N$ and $(a,d)=1$.

We note that the Fourier expansion in terms of Jacobi forms after removing the
poles in $v$-plane we get an overlap with the attractor chamber mentioned in
Sen:2010mz . In this region we have the classical area of the black hole
horizon to be positive and the electromagnetic charges are positive

$P^{2}Q^{2}>(Q\cdot P)^{2},\quad P^{2}>0,\quad Q^{2}>0,\quad Q\cdot P\geq 0.$

For orbifolds of $K3$ the resultant attractor chamber is smaller than obtained
from the above constraints alone, however for this analysis we require only
the above mentioned constraints which is also obtained by removing the poles
in $v$-plane.

Before moving into the mathematical results and numerical estimates let us
briefly state the results of the contributions from the two forms in the
heterotic counting for $K3$ and the orbifold models. This was done in
Chattopadhyaya:2020yzl . In the CHL models the 2-forms compute the
contribution of hair that comes from the bosonic modes. These are given by the
contribution of the fundamental string however without the shift in zero point
energy

$\displaystyle Z_{{\rm hair}:{\rm CHL}}^{\rm bosonic}$ $\displaystyle=$
$\displaystyle\prod_{l=1}^{\infty}(1-e^{2\pi i(Nl\rho)})^{-4}(1-e^{2\pi
i(l\rho)})^{-\sum_{s=0}^{N-1}e^{-2\pi isl/N}c^{(0,s)}(0)}.$ (8)

In our analysis we will remove all these states as the growth of these objects
are as that of partitions which remain far higher than the growth of ordinary
positive weight modular forms.

##  3 A brief review of some results in number theory

Before moving into the details of the calculations we first introduce the
notion of modular forms and its associated objects. These are a subject of
interest for black hole thermodynamics in string theory as well as counting
different kind of partitions in number theory.

###  3.1 Modular, Jacobi, mock modular forms

In this section we briefly review the definitions of modular forms, Jacobi
forms, mock modular forms and mock Jacobi forms. For more rigorous definitions
see Eichler:1985 ; MR2605321 ; ZwegersThesis ; Dabholkar:2012nd ;
Manschot:2017xcr ; Chattopadhyaya:2023aua .

Definition: A modular form $f(\tau)$ is a holomorphic function of
$\tau\in\mathbb{H}$ of weight $k$ with the following

  1. 1.

Transformation property:

$f(\frac{a\tau+b}{c\tau+d})=(c\tau+d)^{k}f(\tau)$ (9)

where $\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\Gamma\subseteq SL_{2}(\mathbb{Z})$.

  2. 2.

Boundedness criteria:

$\lim_{\tau\to i\infty}(c\tau+d)^{-k}f\left(\frac{a\tau+b}{c\tau+d}\right)$
(10)

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

The modular group is generated by $S$ and $T$ transformations given by

$S=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix},\qquad T=\begin{pmatrix}1&1\\\
0&1\end{pmatrix}.$ (11)

Due to the $T$ transformations we can write these in terms of Fourier series
with $q=e^{2\pi i\tau}$

$f(\tau)=\sum_{n=0}^{\infty}a_{n}q^{n}.$ (12)

The above definition can be easily generalized to vector valued modular forms
and modular forms of subgroups of $SL_{2}(\mathbb{Z})$. An important subgroup
of $SL_{2}(\mathbb{Z})$ is $\Gamma_{0}(N)$ given by

$\begin{pmatrix}a&b\\\ Nc&d\end{pmatrix},\quad{\rm
with}\;(a,b,c,d)\in\mathbb{Z}^{4}$

.

Definition: Cusp form $f(\tau)$ is a modular forms of positive weight $k$
given by the additional growth constraint

$\lim_{\tau\to i\infty}(c\tau+d)^{-k}f\left(\frac{a\tau+b}{c\tau+d}\right)=0$
(13)

for all $\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in SL_{2}(\mathbb{Z})$. The
implies that the zeroth order Fourier coefficient of a cusp form is always
zero.

The space of modular forms are spanned by Eisenstein series and cusp forms.

$\displaystyle E_{4}(\tau)$ $\displaystyle=$ $\displaystyle
1+240\sum_{n}\sigma_{3}(n)q^{n},\quad\sigma_{\ell}(n)=\sum_{d|n}d^{\ell},\quad
q=e^{2\pi i\tau}$ (14) $\displaystyle{\cal E}_{N}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{N-1}(NE_{2}(N\tau)-E_{2}(\tau))$

where $E_{2}(\tau)=1-24\sum_{n=1}^{\infty}\sigma(n)q^{n}$. The weight two
Eisenstein series is mock modular (or quasi-modular) but ${\cal E}_{N}$ is a
holomorphic modular form of $\Gamma_{0}(N)$.

Modular forms are generally defined with a boundedness criteria, however those
do not hold everywhere in the upper half of the complex plane if the weight
$k$ is negative.

Definition: A weight $k$ index $m$ Jacobi form $J(\tau,z)$ is a holomorphic
function of $\tau\in\mathbb{H}$ and $z\in\mathbb{C}$ with the following
transformation properties:

  1. 1.

For $\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right)\in
SL_{2}(\mathbb{Z})$

$J\left(\frac{a\tau+b}{c\tau+d},\frac{z}{c\tau+d}\right)=(c\tau+d)^{k}e^{\frac{2\pi
imcz^{2}}{c\tau+d}}J(\tau,z)$

  2. 2.

For $(\lambda,\mu)\in\mathbb{Z}^{2}$

$J(\tau,z+\lambda\tau+\mu)=e^{2\pi im(\lambda^{2}\tau+2\lambda z)}J(\tau,z)$

  3. 3.

$J(\tau,z)$ has a Fourier expansion:

$J(\tau,z)=\sum_{n\geq 0}\sum_{\ell^{2}\leq 4mn}c_{n,\ell}q^{n}e^{2\pi
iz\ell}$

Jacobi forms of index 1 and weight $k\in\mathbb{Z}$ $J(\tau,z)$ can be
decomposed with respect to theta functions also known as theta decomposition.

$J(\tau,z)=\theta_{3}(2\tau,2z)j_{0}(\tau)+\theta_{2}(2\tau,2z)j_{1}(\tau)$

where $j_{0}$ and $j_{1}$ are weight $k-1/2$ modular forms. In the case of
weakly holomorphic Jacobi forms of index $1$ as the twisted elliptic genus of
$K3$ the Fourier expansion of $4n-\ell^{2}\geq-1$ instead of $4n\geq 0$.

$J(\tau,z)=\sum_{n\geq 0}\sum_{\ell^{2}\leq 4n+1}c_{n,\ell}\;q^{n}e^{2\pi
iz\ell}.$

The theta decomposition is still possible for the weakly holomorphic cases.

Definition: A mock modular form $f(\tau)$ of weight $k$ is a holomorphic
function in $\tau\in\mathbb{H}$ such that its completion
$\hat{f}(\tau,\bar{\tau})=f(\tau)+g^{*}(\tau,\bar{\tau})$ transforms like a
modular form of weight $k$ where $g^{*}$ is given as follows:

$\displaystyle
g^{*}(\tau,\bar{\tau})=-2^{1-k}i\int_{-\bar{\tau}}^{i\infty}\frac{g(-v)}{(-i(v+\tau))^{k-\ell}}\,dv\;.$
(15)

and $g(\tau)$ is a modular form of weight $2-k$.

A mock Jacobi form $J(\tau,z)$ of weight $k$ and index $1$ is given by a
similar theta decomposition property as a Jacobi form of index 1, however in
this case,

$J(\tau,\nu)=j_{0}(\tau)\theta_{3}(2\tau,2z)+j_{1}(\tau)\theta_{2}(2\tau,2z)$

where $j_{0},j_{1}$ are mock modular in nature.

###  3.2 Growth of $\sigma(n)$\- theorems and numerical evidence

Our argument involves two theorems from number theory and one assumption.
Before stating the assumption we first write the statements of the theorems we
use. The first one is a statement on the maximal growth of the divisor sum
function ie, $\sigma(n)=\sum_{d|n}d$ for positive $d,n$ in $\mathbb{Z}$. The
upper limit on $\sigma(n)$ was given by Ramanujan assuming Riemann hypothesis.
The upper bound can be written as

$\displaystyle\sigma(n)<e^{\gamma}n\log\log(n)$ (16)

where $\gamma=2.718$ is the Euler Mascheroni constant. Robin showed that
considering Riemann hypothesis being false this limit has some modifications
robin . We state Robin’s theorem.

######  Theorem 1.

(Robin’s Theorem) Considering Riemann hypothesis to be false, the upper limit
of divisor sum $\sigma(n)$ for $n>3$ can be given by,

$\displaystyle\sigma(n)<e^{\gamma}n\log\log(n)+\frac{0.6483n}{\log\log(n)}<e^{\gamma}n\log\log(n)+O(n)$
(17)

Most of our analysis revolves around the average growth of the divisor sum and
we need to estimate the average growth of $\sigma(n)=\sum_{d|n}d$.

######  Theorem 2.

Average growth of $\sigma(n)\sim\frac{\pi^{2}n}{6}+O(\log(n)).$ It is
important to note that the average growth of the $\sigma(n)$ function is
linear and close to $1.645n<2n$ hardy-wright  111see theorem 324.

Next we state the prime number theorem. This theorem will not be used in the
paper but we will motivate a proposition based on numerical evidence which
relates closely to prime number theorem.

######  Theorem 3.

(Prime Number Theorem) Let $\pi(n)$ denote the number of primes below $n$,
then as $n\rightarrow\infty$,

$\pi(n)=\frac{n}{\log(n)}.$

A simple proof of the prime number theorem given by Newman can be found in
zagier-prime .

Motivated by the above results we propose the following based on numerical
evidences. We introduce two propositions for the growth of the following ratio
$\frac{\sigma(4n+k}{2\sigma(2n)-4\sigma(n)}$.

######  Proposition 1.

For a positive integer $n>10^{5}$ the following ratios

$\displaystyle\frac{\sigma(4n+1)}{2\sigma(2n)-4\sigma(n)}<1,\quad\frac{\sigma(4n+3)}{2\sigma(2n)-4\sigma(n)}<1.$
(18)

hold for a maximum of $\frac{\kappa n}{\log(n)}$ terms and $\kappa\leq 2$.

Numerically this is checked to be correct for $10^{8}$ terms. We shall not
attempt to prove this in this paper. A heuristic argument can be given as
follows: The possibility of finding $\sigma(4n+1)<2\sigma(2n)-4\sigma(n)$
occurs most often when $4n+1$ is a prime number (or a product of very few
terms) and $n$ contains several factors such that $2\sigma(2n)-4\sigma(n)>4n$.
A similar possibility also arises for $4n+3$ primes, however as $n$ cannot
have the divisor 3, it is even harder to find such numbers for small $n$. Upto
$10^{8}$ there is only one term where $\sigma(4n+3)<2\sigma(2n)-4\sigma(n)$
222We checked upto $n=10^{8}$ for the ratios
$\frac{\sigma(4n+1)}{2\sigma(2n)-4\sigma(n)}$. There are $1.4\times 10^{6}$
terms where the ratio is below 1 and number of these terms are below less than
the number primes below $10^{8}$ which is approximately $5.4\times 10^{6}$.
which is clearly far below the proposed bound $\frac{\kappa n}{\log(n)}$ with
$\kappa\leq 2$.

######  Proposition 2.

For a positive integer $n>10^{5}$ the following also holds

$\displaystyle\sigma(4n+1)-(2\sigma(2n)-4\sigma(n))<n,\quad\sigma(4n+3)-(2\sigma(2n)-4\sigma(n))<n.$
(19)

for a maximum of $\frac{\kappa n}{\log(n)}$ terms upto $n$ and $\kappa\leq 2$.

Numerically this is also checked to be correct for $10^{8}$ terms. We shall
not attempt to prove this in this paper. In the following sections we will see
the need for the above propositions and the theorem on average growth of
$\sigma(n)$.

##  4 Sign changes for compactifications on orbifolds of $K3\times T^{2}$

In this section we introduce the behavior of the Fourier coefficients of the
inverse Siegel modular forms associated with the $1/4$-th BPS dyons of ${\cal
N}=4$ type IIB theories compactified on $K3\times T^{2}$ and its orbifolds. We
then extract out the hair modes as obtained in Chakrabarti:2020ugm ;
Chattopadhyaya:2020yzl  and theta decompose to observe the positivity of the
Fourier coefficients of the resultant half integral weight modular forms for
the magnetic charge $P^{2}=2$. The inverse Siegel modular form of weight $k$
can be expanded in terms of weak Jacobi forms $A$ and $B$ given as:

$\displaystyle\frac{1}{\Phi_{k}(q,p,y)}$ $\displaystyle=$
$\displaystyle\sum_{m=-1}^{\infty}\psi^{(N)}_{m}(q,y)p^{m/N}$ (20)

where the order of $K3$ orbifold is given by $N$. The Jacobi forms associated
to the canonical case for the compactification $K3\times T^{2}$ at magnetic
charge $P^{2}=2$ can be given as follows:

$\displaystyle\psi_{1}^{(1)}$ $\displaystyle=$
$\displaystyle-48\frac{A^{2}(\tau,z)}{B(\tau,z)}$ (21)

where, the weak Jacobi form of weight zero and weight $-2$ given as $A,B$
respectively (index 1 for both cases) are defined as follows:

$\displaystyle A(\tau,z)$ $\displaystyle=$
$\displaystyle\frac{1}{\eta^{6}}\left[6\theta_{3}^{2}(\tau,z)\theta_{2}^{2}\theta_{4}^{2}-2\theta_{1}^{2}(\tau,z)(\theta_{4}^{4}-\theta_{2}^{4}))\right],\quad
B(\tau,z)=\frac{\theta_{1}^{2}(\tau,z)}{\eta^{6}}$

$\psi_{1}^{(N)}$ can be written in terms of an Appell Lerch sum which is a sum
over poles in the $z$ plane and a mock modular form which remains finite
written as follows:

$\displaystyle\psi_{1}^{(N)}$ $\displaystyle=$
$\displaystyle\psi_{1}^{(N)}|_{\rm polar}+\psi_{1}^{(N)}|_{\rm finite}$ (22)

For the canonical case of $K3$ we get,

$\displaystyle\psi_{1}^{(1)}|_{\rm
polar}=\frac{48}{\eta^{24}}\sum_{n\in\mathbb{Z}}\frac{q^{n^{2}+n}y^{2n+1}}{(1-q^{n}y)^{2}},\quad
q=e^{2\pi i\tau},\;\;y=e^{2\pi iz}.$ (23)

The finite peice can be given in terms of class number generating function and
the Eisenstein series $E_{4}$

$\displaystyle\psi_{1}^{(1)}|_{\rm
finite}=-\frac{3}{\eta^{24}}(E_{4}(\tau)B(\tau,z)+216{\cal H}(\tau,z)$ (24)

We can write ${\cal H}(\tau,z)$ in terms of Hurwitz class number generating
functions. It is possible to theta decompose $B(\tau,z)$ and ${\cal
H}(\tau,z)$ which separate the even and odd powers of $y$, given as

$\displaystyle
B(\tau,z)=\frac{\theta_{1}^{2}(\tau,z)}{\eta^{6}}=b_{0}(\tau)\theta_{3}(2\tau,2z)+b_{1}(\tau)\theta_{2}(2\tau,2z)$
(25) $\displaystyle{\cal
H}(\tau,z)=h_{0}(\tau)\theta_{3}(2\tau,2z)+h_{1}(\tau)\theta_{2}(2\tau,2z)$

where we have,

$\displaystyle b_{0}(\tau)=\frac{1}{\eta^{6}}(\theta_{2}(2\tau)),\quad
b_{1}(\tau)=\frac{1}{\eta^{6}}(-\theta_{3}(2\tau))$ (26)

One can obtain an extensive list of the Fourier coefficients of $h_{0}(\tau)$
and $h_{1}(\tau)$ referred to as $f_{2,0}$ and $f_{2,1}$ respectively in
Chattopadhyaya:2023aua  table 1. It was shown in Bringmann:2012zr ;
Chattopadhyaya:2020yzl  that the Fourier coefficients of $\psi_{1}^{K3}$ after
theta decomposition can only change sign at $q^{0}$ and the sign remains
correct as predicted from black hole physics even after removing the
contribution of the hair modes. Since $\eta^{6}(\tau)$ is a cusp form one can
similarly argue that the weight $9/2$ mock modular forms obtained as
$h_{\mu}\eta^{6}$ will have coefficients growing
$<a_{\Delta}(\Delta)^{-9/4}n^{9/4}$ and thus,

$\displaystyle-3\eta^{6}(\tau)(E_{4}(\tau)b_{0}(\tau)+216h_{0}(\tau),\quad
3\eta^{6}(\tau)(E_{4}(\tau)b_{1}(\tau)-216h_{1}(\tau)$ (27)

can only have a finite number of sign changes (upto $q^{2}$). In the rest of
this paper we shall mostly ignore cusp forms as their saddle point growth
remains $a_{\Delta}\Delta^{-k/2}n^{k/2}\sim O(n^{k/2})$ and from Deligne bound
deligne:74 ; deligne:80  these have growth $O(n^{\frac{k-1}{2}})$ while the
modular forms have the saddle point growth as $O(n^{k})$ and average growth
$O(n^{k-1})$. So if there is a sign change this will only affect the first few
terms which can be easily computed numerically.

###  4.1 2A orbifold

Let us now write the explicit forms of $\psi_{1}^{(2)}$ for the 2A orbifold of
$K3$ where the orbifold is a $g^{\prime}$ action corresponding to $[M_{23}]$,
where $M_{23}\subset M_{24}$ Mathieu moonshine group. In this case we need the
definition of the holomorphic Eisenstein series of weight 2 given by ${\cal
E}_{N}$

${\cal E}_{N}(\tau)=\frac{1}{N-1}\left(NE_{2}(N\tau)-E_{2}(\tau)\right).$ (28)

We can write the finite part as $g^{8}(\tau)\psi_{1}^{(2)}|_{finite}$ as:

$-\frac{20}{9}A(2\tau,z){\cal
E}_{2}(\tau)-B(2\tau,z)\left(\frac{1}{16}E_{4}(\tau)+\frac{13}{36}E_{4}(2\tau)+\frac{19}{144}{\cal
E}_{2}^{2}(\tau)\right)-104{\cal H}(2\tau,z)$ (29)

where $g^{8}(\tau)=\eta^{8}(\tau)\eta^{8}(2\tau)$. Simplifying using ${\cal
E}_{2}^{2}(\tau)=\frac{4}{5}(E_{4}(2\tau)+\frac{1}{4}E_{4}(\tau))$ we get,

$-\frac{20}{9}A(2\tau,z){\cal
E}_{2}(\tau)-B(2\tau,z)\left(\frac{4}{45}E_{4}(\tau)+\frac{21}{45}E_{4}(2\tau)\right)-104{\cal
H}(2\tau,z)$ (30)

The components $\frac{4}{45}E_{4}(\tau)+\frac{21}{45}E_{4}(2\tau)$ will have
always positive Fourier coefficients. So does ${\cal E}_{2}(\tau)$ has only
positive coefficients in its Fourier expansion as $\sigma(2n)>2\sigma(n)$.

The last two terms are similar to the results obtained in the previous section
and the only term which we need to deal with in this part is
$\phi^{2A}_{A}=-A(2\tau,z){\cal E}_{2}(\tau)\eta^{6}$. The theta decomposition
for $A(\tau,z)\eta^{6}$ can be given as follows:

$\displaystyle A(\tau,z)\eta^{6}$ $\displaystyle=$ $\displaystyle
6\theta_{3}^{2}(\tau,z)\theta_{2}^{2}\theta_{4}^{2}-2\theta_{1}^{2}(\tau,z)(\theta_{4}^{4}-\theta_{2}^{4})=A_{0}(\tau)\theta_{3}(2\tau,2z)+A_{1}(\tau)\theta_{2}(2\tau,2z)$

where we have,

$\displaystyle A_{0}(\tau)$ $\displaystyle=$ $\displaystyle
6\,\theta_{2}^{2}\theta_{4}^{2}\theta_{3}(2\tau)-2(\theta_{4}^{4}-\theta_{2}^{4})\theta_{2}(2\tau),\quad
A_{1}(\tau)=6\,\theta_{2}^{2}\theta_{4}^{2}\theta_{2}(2\tau)+2(\theta_{4}^{4}-\theta_{2}^{4})\theta_{3}(2\tau)$

We used the double angle identities to do the theta decomposition above,

$\displaystyle\theta_{1}^{2}(\tau,z)$ $\displaystyle=$
$\displaystyle\theta_{2}(2\tau)\theta_{3}(2\tau,2z)-\theta_{3}(2\tau)\theta_{2}(2\tau,2z)$
(32) $\displaystyle\theta_{3}^{2}(\tau,z)$ $\displaystyle=$
$\displaystyle\theta_{3}(2\tau)\theta_{3}(2\tau,2z)+\theta_{2}(2\tau)\theta_{2}(2\tau,2z)$

Before moving into the relations between the divisor sums and the propositions
let us first check the asymptotic signs from saddle point analysis. For this
we first need the result of various S transformations involving theta
functions:

$\displaystyle\theta_{2}(-\frac{1}{\tau})$ $\displaystyle=$
$\displaystyle\sqrt{-i\tau}\theta_{4}(\tau)$ (33)
$\displaystyle\theta_{3}(-\frac{1}{\tau})$ $\displaystyle=$
$\displaystyle\sqrt{-i\tau}\theta_{3}(\tau)$
$\displaystyle\theta_{4}(-\frac{1}{\tau})$ $\displaystyle=$
$\displaystyle\sqrt{-i\tau}\theta_{2}(\tau)$

Also for the multiple angled theta functions we can write,

$\displaystyle\theta_{2}(-\frac{2}{\tau})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\sqrt{-i\tau}(\theta_{3}(2\tau)-\theta_{2}(2\tau))$
(34) $\displaystyle\theta_{3}(-\frac{2}{\tau})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\sqrt{-i\tau}(\theta_{3}(2\tau)+\theta_{2}(2\tau))$

Now using these we can write the transformation properties of the even and odd
parts of theta decomposition of $\eta^{6}A(\tau,z)$ given by $A_{0}$ and
$A_{1}$. This is given by

$\displaystyle A_{0}(\frac{1}{\tau})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(-i\tau)^{5/2}(A_{0}(\tau)+A_{1}(\tau))$ (35)
$\displaystyle A_{1}(\frac{1}{\tau})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(-i\tau)^{5/2}(A_{0}(\tau)-A_{1}(\tau))$

for $A_{0},A_{1}$ as in (LABEL:a0a1). Therefore we can write

$\displaystyle A_{0}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(-i\tau)^{-5/2}\left(A_{0}(-\frac{1}{\tau})+A_{1}(-\frac{1}{\tau})\right)$
(36) $\displaystyle A_{1}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(-i\tau)^{-5/2}\left(A_{0}(-\frac{1}{\tau})-A_{1}(-\frac{1}{\tau})\right)$

Now we do a saddle point analysis for $A_{j}$, consider $\tau_{2}\rightarrow
0$ and $j$ corresponds to 0 or 1. Let us denote the coefficient of the Fourier
expansion of $A_{j}(\tau)=\sum_{n}a_{j}(n)q^{n}$ for $n\in\mathbb{Z}+1/4-j/4$

$\displaystyle a_{j}(n)$ $\displaystyle=$ $\displaystyle\int_{0}^{1}e^{-2\pi
in\tau}A_{j}(\tau)d\tau$ (37)

Now we have,

$a_{j}(n)=\frac{(-i)^{5/2}}{\sqrt{2}}\int_{0}^{1}e^{-2\pi
in\tau}\tau^{-5/2}\sum_{\ell=0}^{1}\chi_{\ell,j}A_{\ell}(-\frac{1}{\tau})d\tau$
(38)

for the characters $\chi_{ij}$ as follows:

$\chi_{0,0}=1,\quad\chi_{0,1}=1,\quad\chi_{1,0}=1,\quad\chi_{1,1}=-1$

. The $q^{0}$ term appears only in $A_{1}$ whose coefficient is 2 which is the
major contribution from the inverted forms evaluated as
$\frac{1}{\tau}\rightarrow\infty$. At saddle point obtained from the
extremization of the exponent $\tau_{s}=\frac{ik}{2\pi n}$ for a weight $k$
modular form the rough growth estimates as

$\displaystyle a_{0}(n)\sim\sqrt{2}n^{5/2}(4\pi e/5)^{5/2}$ (39)
$\displaystyle a_{1}(n)\sim-\sqrt{2}n^{5/2}(4\pi e/5)^{5/2}$

Now $a_{0}$ corresponds to even $Q\cdot P$ while $a_{1}$ corresponds to odd
values of $Q\cdot P$. Hence the sign asymptotically looks as expected from
$-B_{6}$ considering the overall sign of the function $A(\tau,z){\cal
E}_{2}(\tau)$.

Next we attempt to use the propositions for the growth of the divisor sum.
Using the weight 2 Eisenstein series
$E_{2}=1-24\sum_{n=1}^{\infty}\sigma(n)q^{n}$ we can write $A_{0},A_{1}$ as
follows:

$\displaystyle A_{0}$ $\displaystyle=$ $\displaystyle
6\theta_{3}(2\tau)\left[-\frac{i}{6}E_{2}(\frac{\tau-1}{4})+\frac{i}{2}E_{2}(\frac{\tau-1}{2})-\frac{i}{3}E_{2}(\tau)\right]-2\theta_{2}(2\tau)\left[2E_{2}(\tau)-E_{2}(\frac{\tau-1}{2})\right]$
$\displaystyle A_{1}$ $\displaystyle=$ $\displaystyle
6\theta_{2}(2\tau)\left[-\frac{i}{6}E_{2}(\frac{\tau-1}{4})+\frac{i}{2}E_{2}(\frac{\tau-1}{2})-\frac{i}{3}E_{2}(\tau)\right]+2\theta_{3}(2\tau)\left[2E_{2}(\tau)-E_{2}(\frac{\tau-1}{2})\right]$

We can also write the above in terms of holomorphic Eisenstein series:

$\displaystyle A_{0}$ $\displaystyle=$ $\displaystyle
i\theta_{3}(2\tau)\left[{\cal E}_{2}(\frac{\tau-1}{4})-{\cal
E}_{2}(\frac{\tau-1}{2})\right]-2\theta_{2}(2\tau){\cal
E}_{2}(\frac{\tau-1}{2})$ $\displaystyle A_{1}$ $\displaystyle=$
$\displaystyle i\theta_{2}(2\tau)\left[{\cal E}_{2}(\frac{\tau-1}{4})-{\cal
E}_{2}(\frac{\tau-1}{2})\right]+2\theta_{3}(2\tau){\cal
E}_{2}(\frac{\tau-1}{2})$

Since $A(\tau,z)$ is a Jacobi form of index 1 the question of sign changes
reduces to a question on sign changes on the following two expressions for the
2A orbifold of $K3$ at magnetic charge $P^{2}=2$.

$\displaystyle\theta_{3}(2\tau)\sum_{n=0}^{\infty}24\sigma(4n+1)q^{n+1/4}-2\theta_{2}(2\tau)\left(1+\sum_{n=1}^{\infty}q^{n}\left(24\sigma(2n)-48\sigma(n)\right)\right)$
(41)
$\displaystyle-\theta_{2}(2\tau)\sum_{n=0}^{\infty}24\sigma(4n+3)q^{n+3/4}+2\theta_{3}(2\tau)\left(1+\sum_{n=1}^{\infty}q^{n}\left(24\sigma(2n)-48\sigma(n)\right)\right)$

We observe numerically that the only sign change appears in the second
expression at $q^{0}$ upto a Fourier expansion of the first $1.6\times 10^{5}$
terms. The first few Fourier coefficients of $a_{0}$ and $a_{1}$ are listed
below,

$\displaystyle A_{0}(\tau)$ $\displaystyle=$ $\displaystyle
20{q}^{1/4}+96q^{5/4}+500q^{9/4}+480q^{13/4}+960q^{17/4}+960q^{21/4}+O(q^{25/4})$
$\displaystyle A_{1}(\tau)$ $\displaystyle=$ $\displaystyle
2-140q-240q^{2}-480q^{3}-1100q^{4}-1056q^{5}-1440q^{6}-O(q^{7})$ (42)

Using Propositions 1 and 2, we argue that the average growth of the
coefficients of $a_{0}$ and $a_{1}$ remain positive. Let $C(x)$ be the average
of the maximum growth of the coefficients of $\sigma(4x+k)$ such that
$\sigma(4x+k)>2(\sigma(2x)-2\sigma(x))$ considering a frequency of occurrence
$\frac{\kappa}{\log(x)}$ given by,

$\displaystyle C(M)=\kappa
e^{\gamma}\frac{d}{dM}\int_{M_{0}}^{M}\frac{x\log\log(x)}{\log(x)}dx$ (43)

and $D(x)$ be the growth of the rest of the coefficients which are of $O(x)$,
we can write,

$\displaystyle
D(M)=\frac{d}{dM}\int_{M_{0}}^{M}x\left(1-\frac{\kappa}{\log(x)}\right)dx$
(44)

Then for $\kappa<2$ we have $C(M)<D(M)$. Since the growth of theta functions
which multiply $\sigma(4x+k)$ and $2(\sigma(2x)-2\sigma(x))$ are similar we
expect the resultant average growth of these products to remain positive.

###  4.2 Odd $N>2$

When we look at $N>2$ orbifolds there are in general more terms than the $N=2$
counterpart for the same magnetic charge $P^{2}=2$ whose positivity and growth
we need to check individually. In this part we describe the growth of
coefficients corresponding to the mock Jacobi form $\psi_{1}^{(N)}$ where, $N$
is odd. These were evaluated in Chattopadhyaya:2018xvg . We can write
$\psi_{1}^{(N)}$ in terms of the coefficients of the Jacobi forms in the
twisted elliptic genus of $K3$.

$\displaystyle-g^{(k+2)}\psi_{1}^{(N)}=\frac{N}{2}\alpha^{(1,0)}_{N}(\frac{N}{2}\alpha^{(1,0)}_{N}+1)\frac{A^{2}(N\tau,z)}{B(N\tau,z)}+\frac{N}{2}\alpha^{(1,0)}\frac{3}{16}E_{4}(N\tau)B(N\tau,z)$
(45)
$\displaystyle\qquad\qquad+N^{2}\alpha^{(1,0)}_{N}A(N\tau,z)\beta^{(1,0)}(N\tau)+\frac{N^{2}}{2}\beta^{(1,0)}_{N}(N\tau)^{2}B(N\tau,z)$
$\displaystyle+\frac{N}{2}\left[4\frac{\theta_{2}(N\tau,z)^{2}}{\theta_{2}^{2}(N\tau)}\beta^{(1,0)}_{N}(2N\tau)+\frac{\theta_{3}(N\tau,z)^{2}}{\theta_{3}^{2}(N\tau)}\beta^{(1,0)}_{N}(\frac{N(\tau+1)}{2})+\frac{\theta_{4}(N\tau,z)^{2}}{\theta_{4}^{2}(N\tau)}\beta^{(1,0)}_{N}(\frac{N\tau}{2})\right].$

We use the results obtained in Dabholkar:2012nd  to separate the Appell Lerch
sum and the finite component

$\frac{A^{2}(\tau,z)}{B(\tau,z)}=-9\sum_{n\in\mathbb{Z}}\frac{q^{n^{2}+n}y^{2n+1}}{(1-q^{n}y)^{2}}+\frac{1}{16}E_{4}(\tau)B(\tau,z)+18{\cal
H}(\tau,z).$ (46)

Hence we can write the finite part after removing the fundamental string as
follows:

$\displaystyle-g^{(k+2)}\psi_{1}^{(N)}=\left(\frac{N^{2}}{4}(\alpha^{(1,0)}_{N})^{2}+(\frac{N}{8}\alpha^{(1,0)}_{N})\right)\left(\frac{1}{16}E_{4}(N\tau)B(N\tau,z)\right)$
(47) $\displaystyle+9N\alpha^{(1,0)}_{N}(\frac{N}{2}\alpha^{(1,0)}_{N}+1){\cal
H}(N\tau,z)+N^{2}\alpha^{(1,0)}_{N}A(N\tau,z)\beta^{(1,0)}(N\tau)+\frac{N^{2}}{2}\beta^{(1,0)}_{N}(N\tau)^{2}B(N\tau,z)$
$\displaystyle+\frac{N}{2}\left[4\frac{\theta_{2}(N\tau,z)^{2}}{\theta_{2}^{2}(N\tau)}\beta^{(1,0)}_{N}(2N\tau)+\frac{\theta_{3}(N\tau,z)^{2}}{\theta_{3}^{2}(N\tau)}\beta^{(1,0)}_{N}(\frac{N(\tau+1)}{2})+\frac{\theta_{4}(N\tau,z)^{2}}{\theta_{4}^{2}(N\tau)}\beta^{(1,0)}_{N}(\frac{N\tau}{2})\right].$

We can now use the following results relating the theta functions and the weak
Jacobi forms $A,B$

$\displaystyle 4\frac{\theta_{2}(\tau,z)^{2}}{\theta_{2}^{2}(\tau)}$
$\displaystyle=$ $\displaystyle\frac{4}{3}A(\tau,z)-\frac{2}{3}B(\tau,z){\cal
E}_{2}(\tau),$ (48) $\displaystyle
4\frac{\theta_{4}(\tau,z)^{2}}{\theta_{4}^{2}(\tau)}$ $\displaystyle=$
$\displaystyle\frac{4}{3}A(\tau,z)+\frac{1}{3}B(\tau,z){\cal E}_{2}(\tau/2),$
$\displaystyle 4\frac{\theta_{3}(\tau,z)^{2}}{\theta_{3}^{2}(\tau)}$
$\displaystyle=$ $\displaystyle\frac{4}{3}A(\tau,z)+\frac{1}{3}B(\tau,z){\cal
E}_{2}((\tau+1)/2).$

Using propositions 1 and 2 and the analysis of the section 4.1 we observe that
the only inconvenient piece may appear from the third line as follows:

$-\frac{N}{3}B(\tau,z){\cal E}_{2}(\tau)\beta^{(1,0)}_{N}(2N\tau)$

Next we observe that the following lemma holds:

######  Lemma 1.

The coefficients of the Fourier expansion of

$\sum_{j=0}^{1}{\cal E}_{N}(\frac{\tau+j}{2}){\cal
E}_{2}(\frac{\tau+j}{2})-16{\cal E}_{N}(2\tau){\cal E}_{2}(N\tau)$

are always positive for positive powers of $q$ for odd $N$.

###### Proof.

This can be proved using $\sigma(nk)>k\sigma(n)\forall k,n$. ∎

Hence the potential effect of the coefficients of the term
$-\frac{N}{3}B(\tau,z){\cal E}_{2}(\tau)\beta^{(1,0)}_{N}(2N\tau)$ which could
violate the sign conjecture are cancelled by the terms:

$\displaystyle\frac{N}{24}\left[{\cal
E}_{2}\left(\frac{N(\tau+1)}{2}\right)\beta^{(1,0)}_{N}\left(\frac{N(\tau+1)}{2}\right)+{\cal
E}_{2}\left(\frac{N\tau}{2}\right)\beta^{(1,0)}_{N}\left(\frac{N\tau}{2}\right)\right]$
(49)

for the Eisenstein series components for prime $N$. This is also true for
composite odd $N$ the relevant case being the orbifold 15A provided the
Fourier coefficient of the Eisenstein series component of $\beta_{15}^{(1,0)}$
remain positive.

For $N=11,15,23$ there are additional cusp forms in $\beta_{N}^{(1,0)}$ which
grow as $O(n^{9/4})$ while the growth of weight 2 Eisnenstein series remain
$n^{3}$, hence we can neglect the contributions after first few terms which
were checked to be positive in Chattopadhyaya:2017ews . This shows that the
sign of the coefficients remain correct as expected from computing the
helicity trace index $-B_{6}$.

##  5 Open questions

In this paper we only dealt with the cases of $N=2$ and odd $N$ at magnetic
charge $P^{2}=2$. With the increasing magnetic charge the space of the modular
forms also increase. For the cases with even $N>2$ one needs to look at the
terms case by case as the generic result for $\psi^{(N)}_{1}$ involves several
Eisenstein series of level 2. Individually looking at each term it is not
clear whether they remain positive as in the case for $N$ odd. We leave
further analysis for a future project.

For a complete proof of growth and signs of the Fourier coefficients of half
integral weight Eisenstein series one requires more understanding on the
behavior of the characters shurman ; koblitz1984  and the behavior of $L$
functions ben-lola  associated with these objects. Unlike where the modular
forms have negative weight where the behavior of the generating function of
partitions have the dominating effect, the growth of coefficients of positive
weight modular forms are understood less. However growth, sign changes are
more well studied in the context of cusp forms kohnen13 333See also several
other papers by the authors.. There are interesting congruences on the
behavior of coefficients of these forms and also for the class number
generating function which is mock modular. As the level $N$ increases the
basis of weight $k$ modular forms also increases which increases the
complexity of the problem regarding frequency of possible sign changes. It is
worth noting that not only we need $\sigma(4n+1)$ (or $\sigma(4n+3)$) to have
few factors we simultaneously need $\sigma(n)$ to have a large number of
factors such that the ratio in (18) holds. So the contest is between “highly
composite numbers” and numbers which have relatively small number of factors.
However we need to modify the definition of highly composite numbers through
the statement of Proposition 1 instead of its usual definition ramanujan  as
we observe numerically that the frequency of such terms we get is higher than
the usual definition of “highly composite”. We leave to study these in future
work.

###### Acknowledgements.

The author thanks Kathrin Bringmann, Soumya Das for useful insights and
references. The author also thanks Justin David, Jan Manschot and Ashoke Sen
for useful discussions. The author is funded by fellowship from Dublin
Institute for Advanced Studies. We thank International Center for Theoretical
Sciences (ICTS) Bangalore for hospitality where part of the project was
completed during an academic visit.

##  Appendix A Notations and identities

In this section we write the notations used in this paper and some identities.
The following conventions are used in the paper for theta and eta functions.

$\displaystyle\theta_{1}(\tau,z)$ $\displaystyle=$
$\displaystyle\sum_{n}(-1)^{n-1/2}q^{\frac{(n-1/2)^{2}}{2}}y^{n-1/2},\quad\theta_{2}(\tau,z)=\sum_{n}q^{\frac{(n-1/2)^{2}}{2}}y^{n-1/2},$
$\displaystyle\theta_{3}(\tau,z)$ $\displaystyle=$
$\displaystyle\sum_{n}q^{\frac{n^{2}}{2}}y^{n},\quad\theta_{4}(\tau,z)=\sum_{n}(-1)^{n}q^{\frac{n^{2}}{2}}y^{n},$
$\displaystyle\eta(\tau)$ $\displaystyle=$ $\displaystyle
q^{1/24}\prod_{n=1}^{\infty}(1-q^{n}),\qquad q=e^{2\pi i\tau},\;y=e^{2\pi
iz}.$

For $z=0$ we write $\theta_{j}(\tau,z)=\theta_{j}(\tau)$.

Jacobi triple product identity used to extract the positive weight modular
forms from the black hole partition functions is given as:

$\displaystyle\theta_{2}(\tau)\theta_{3}(\tau)\theta_{4}(\tau)=2\eta^{3}(\tau).$
(50)

Theta decomposition of Jacobi forms $A,B$ are done using the following double
angle relations:

$\displaystyle\theta_{1}^{2}(\tau,z)$ $\displaystyle=$
$\displaystyle\theta_{2}(2\tau)\theta_{3}(2\tau,2z)-\theta_{3}(2\tau)\theta_{2}(2\tau,2z)$
(51) $\displaystyle\theta_{2}^{2}(\tau,z)$ $\displaystyle=$
$\displaystyle\theta_{2}(2\tau)\theta_{3}(2\tau,2z)+\theta_{3}(2\tau)\theta_{2}(2\tau,2z)$
$\displaystyle\theta_{3}^{2}(\tau,z)$ $\displaystyle=$
$\displaystyle\theta_{3}(2\tau)\theta_{3}(2\tau,2z)+\theta_{2}(2\tau)\theta_{2}(2\tau,2z)$
$\displaystyle\theta_{4}^{2}(\tau,z)$ $\displaystyle=$
$\displaystyle\theta_{3}(2\tau)\theta_{3}(2\tau,2z)-\theta_{2}(2\tau)\theta_{2}(2\tau,2z).$

Relations between theta functions and Eisenstein series used in this paper can
be given by,

$\displaystyle\theta_{2}^{2}(\tau)\theta_{4}^{2}(\tau)$ $\displaystyle=$
$\displaystyle\left[-\frac{i}{6}E_{2}(\frac{\tau-1}{4})+\frac{i}{2}E_{2}(\frac{\tau-1}{2})-\frac{i}{3}E_{2}(\tau)\right]$
(52) $\displaystyle\theta_{4}^{4}(\tau)-\theta_{2}^{4}(\tau)$ $\displaystyle=$
$\displaystyle 2E_{2}(\tau)-E_{2}(\frac{\tau-1}{2})$

## References

  * (1) R. Dijkgraaf, E. P. Verlinde, and H. L. Verlinde, Counting dyons in N=4 string theory, Nucl. Phys. B 484 (1997) 543–561, [hep-th/9607026].
  * (2) A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99–104, [hep-th/9601029].
  * (3) J. R. David and A. Sen, CHL Dyons and Statistical Entropy Function from D1-D5 System, JHEP 11 (2006) 072, [hep-th/0605210].
  * (4) D. P. Jatkar and A. Sen, Dyon spectrum in CHL models, JHEP 04 (2006) 018, [hep-th/0510147].
  * (5) G. Lopes Cardoso, B. de Wit, J. Kappeli, and T. Mohaupt, Asymptotic degeneracy of dyonic N = 4 string states and black hole entropy, JHEP 12 (2004) 075, [hep-th/0412287].
  * (6) A. Sen, Black Hole Entropy Function, Attractors and Precision Counting of Microstates, Gen. Rel. Grav. 40 (2008) 2249–2431, [arXiv:0708.1270].
  * (7) A. Dabholkar and S. Nampuri, Quantum black holes, Lect. Notes Phys. 851 (2012) 165–232, [arXiv:1208.4814].
  * (8) A. Sen, How Do Black Holes Predict the Sign of the Fourier Coefficients of Siegel Modular Forms?, Gen. Rel. Grav. 43 (2011) 2171–2183, [arXiv:1008.4209].
  * (9) N. Banerjee, I. Mandal, and A. Sen, Black Hole Hair Removal, JHEP 07 (2009) 091, [arXiv:0901.0359].
  * (10) D. P. Jatkar, A. Sen, and Y. K. Srivastava, Black Hole Hair Removal: Non-linear Analysis, JHEP 02 (2010) 038, [arXiv:0907.0593].
  * (11) A. Chattopadhyaya and J. R. David, Horizon states and the sign of their index in ${\cal N}=4$ dyons, JHEP 03 (2021) 106, [arXiv:2010.0896].
  * (12) S. Chakrabarti, S. Govindarajan, P. Shanmugapriya, Y. K. Srivastava, and A. Virmani, Black Hole Hair Removal For N=4 CHL Models, arXiv:2010.0224.
  * (13) A. Dabholkar, F. Denef, G. W. Moore, and B. Pioline, Exact and asymptotic degeneracies of small black holes, JHEP 08 (2005) 021, [hep-th/0502157].
  * (14) J. R. David, D. P. Jatkar, and A. Sen, Dyon Spectrum in N=4 Supersymmetric Type II String Theories, JHEP 11 (2006) 073, [hep-th/0607155].
  * (15) J. R. David, D. P. Jatkar, and A. Sen, Product representation of Dyon partition function in CHL models, JHEP 06 (2006) 064, [hep-th/0602254].
  * (16) J. R. David, D. P. Jatkar, and A. Sen, Dyon spectrum in generic N=4 supersymmetric Z(N) orbifolds, JHEP 01 (2007) 016, [hep-th/0609109].
  * (17) S. Govindarajan and K. Gopala Krishna, BKM Lie superalgebras from dyon spectra in Z(N) CHL orbifolds for composite N, JHEP 05 (2010) 014, [arXiv:0907.1410].
  * (18) A. Sen, Walls of Marginal Stability and Dyon Spectrum in N=4 Supersymmetric String Theories, JHEP 05 (2007) 039, [hep-th/0702141].
  * (19) A. Chattopadhyaya and J. R. David, Dyon degeneracies from Mathieu moonshine symmetry, Phys. Rev. D 96 (2017), no. 8 086020, [arXiv:1704.0043].
  * (20) D. Persson and R. Volpato, Second Quantized Mathieu Moonshine, Commun. Num. Theor. Phys. 08 (2014) 403–509, [arXiv:1312.0622].
  * (21) A. Dabholkar, S. Murthy, and D. Zagier, Quantum Black Holes, Wall Crossing, and Mock Modular Forms, arXiv:1208.4074.
  * (22) A. Chattopadhyaya and J. R. David, Properties of dyons in $\mathcal{N}$ = 4 theories at small charges, JHEP 05 (2019) 005, [arXiv:1810.1206].
  * (23) M. M. Eichler and D. Zagier, The theory of Jacobi forms.  Progress in mathematics v. 55. Birkhauser, Boston, 1st ed. 1985. ed., 1985\.
  * (24) D. Zagier, Ramanujan’s mock theta functions and their applications (after Zwegers and Ono-Bringmann), Astérisque (2009), no. 326 Exp. No. 986, vii–viii, 143–164 (2010). Séminaire Bourbaki. Vol. 2007/2008.
  * (25) S. P. Zwegers, Mock Theta Functions.  PhD thesis, Utrecht University, 2008.
  * (26) J. Manschot, Vafa–Witten Theory and Iterated Integrals of Modular Forms, Commun. Math. Phys. 371 (2019), no. 2 787–831, [arXiv:1709.1009].
  * (27) A. Chattopadhyaya and J. Manschot, Numerical experiments on coefficients of instanton partition functions, arXiv:2301.0671.
  * (28) G.Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de riemann, Journal de Mathématiques Pures et Appliquées. Neuvième Série 63 (2) (1984) 187–213.
  * (29) G. Hardy and E. Wright, An Introduction to the Theory of Numbers, 5th ed. (1980).
  * (30) D. Zagier, Newman’s short proof of the prime number theorem, The American Mathematical Monthly 104 (1997), no. 8 705–708, [https://doi.org/10.1080/00029890.1997.11990704].
  * (31) K. Bringmann and S. Murthy, On the positivity of black hole degeneracies in string theory, Commun. Num. Theor Phys. 07 (2013) 15–56, [arXiv:1208.3476].
  * (32) P. Deligne, The weil conjecture: I, Publications Mathématiques de l’IHÉS 43 (1974) 273–307.
  * (33) P. Deligne, The Weil conjecture: II, Publications Mathématiques de l’IHÉS 52 (1980) 137–252.
  * (34) F. Diamond and J. Shurman, A First Course in Modular Forms (Graduate Texts in Mathematics, vol. 228.  Springer, Berlin, 2005.
  * (35) N. Koblitz, Introduction to Elliptic Curves and Modular Forms.  3Island Press, 1984.
  * (36) B. Linowitz and L. Thompson, The sign changes of fourier coefficients of eisenstein series, The Ramanujan Journal 37 (05, 2013).
  * (37) W. Kohnen and Y. Chloe, A short remark on sign changes of fourier coefficients of half-integral weight modular forms, Archiv der Mathematik 100 (2013), no. 4.
  * (38) S. Ramanujan Proceedings of the London Mathematical Society s2 14, Issue 1 (1915) 347–409.


###  E.6 Proof of Equation A10

######  Proposition 6.

Consider a regression model $\hat{f}(\mathbf{x})$ including only one feature
$x_{1}$ such as $\mathbf{x}=x_{1}$. We can show that for the performance
metric
$G(y_{i};\hat{f}(\mathbf{x}_{i});\delta_{0})=-(y_{i}-\hat{f}(\mathbf{x}_{i}))^{2}$,
if we assume that $\mathbb{E}(\hat{f}(x_{i,1}))=0$, then the corresponding
individual XPER value $\phi_{i,1}$ is equal to:

$\phi_{i,1}=2\hat{\varepsilon}_{i}\phi_{i,1}^{SHAP}+\left(\phi_{i,1}^{SHAP}\right)^{2}+\mathbb{V}\left(\hat{f}(x_{i,1})\right),$
(A77)

where $\hat{\varepsilon}_{i}=y_{i}-\hat{f}(x_{i,1})$ is the prediction error
of the model for individual $i$, and $\phi_{i,1}^{SHAP}$ refers to the SHAP
value of the feature $x_{1}$ for this individual.

###### Proof.

Consider a regression model $\hat{f}(\mathbf{x})$ including only one feature
$x_{1}$ such as $\mathbf{x}=x_{1}$, and the performance metric
$G(y_{i};\hat{f}(\mathbf{x}_{i});\delta_{0})=-(y_{i}-\hat{f}(\mathbf{x}_{i}))^{2}$.

According to Equation 7, the XPER value $\phi_{i,1}$ decomposes
$G(y_{i};\hat{f}(\mathbf{x}_{i});\delta_{0})$ such as:

$\displaystyle
G(y_{i};\hat{f}(\mathbf{x}_{i});\delta_{0})=\phi_{i,0}+\phi_{i,1},$ (A78)

with $\phi_{i,0}$ the benchmark value of the performance metric. Therefore, if
we replace $G(y_{i};\hat{f}(\mathbf{x}_{i});\delta_{0})$ by its expression in
the previous equation, we can see that:

$\phi_{i,1}=-y_{i}^{2}-\hat{f}(x_{i,1})^{2}+2y_{i}\hat{f}(x_{i,1})-\phi_{i,0}.$
(A79)

Moreover, the benchmark value $\phi_{i,0}$ is equal to:

$\phi_{i,0}=\mathbb{E}_{x_{1}}\left(G(y_{i};\hat{f}(\mathbf{x}_{i});\delta_{0})\right)=-y_{i}^{2}-\mathbb{E}\left(\hat{f}(x_{i,1})^{2}\right)+2y_{i}\mathbb{E}\left(\hat{f}(x_{i,1})\right).$
(A80)

As we assume that $\mathbb{E}\left(\hat{f}(x_{i,1})\right)=0$, we obtain:

$\phi_{i,0}=-y_{i}^{2}-\mathbb{V}\left(\hat{f}(x_{i,1})\right).$ (A81)

Replacing the expression of $\phi_{i,0}$ in Equation A79 we obtain:

$\phi_{i,1}=2y_{i}\hat{f}(x_{i,1})+\mathbb{V}\left(\hat{f}(x_{i,1})\right)-\hat{f}(x_{i,1})^{2}.$
(A82)

As the prediction error $\varepsilon_{i}$ can be expressed as the difference
between the target variable $y_{i}$ and the prediction $\hat{f}(x_{i,1})$,
i.e., $\hat{\varepsilon}_{i}=y_{i}-\hat{f}(x_{i,1})$, $\phi_{i,1}$ is then
equal to:

$\phi_{i,1}=2\hat{\varepsilon}_{i}\hat{f}(x_{i,1})+\hat{f}(x_{i,1})^{2}+\mathbb{V}\left(\hat{f}(x_{i,1})\right).$
(A83)

Now, according to Equation A8 and A9, as
$\mathbb{E}\left(\hat{f}(x_{i,1})\right)=0$, we can see that:

$\hat{f}(x_{i,1})=\phi_{i,1}^{SHAP}.$ (A84)

with $\phi_{i,1}^{SHAP}$ the SHAP value associated to feature $x_{1}$ for
individual $i$.

Finally, from Equation A83 and A84, we obtain:

$\phi_{i,1}=2\hat{\varepsilon}_{i}\phi_{i,1}^{SHAP}+\left(\phi_{i,1}^{SHAP}\right)^{2}+\mathbb{V}\left(\hat{f}(x_{i,1})\right).$
(A85)

∎

###  E.7 Proof of Proposition 1

See 1

###### Proof.

According to Definition 4, the individual XPER value $\phi_{j}$ associated to
the performance metric
$G\left(y_{i};\mathbf{x}_{i};\delta_{0}\right)=\hat{f}(\mathbf{x}_{i})$ is
equal to:

$\phi_{i,j}=\sum_{S\subseteq\mathcal{P}(\\{\mathbf{x}\\}\setminus\left\\{x_{j}\right\\})}w_{S}\left[\mathbb{E}_{\mathbf{x}^{\overline{S}}}\left(\hat{f}\left(x_{i,j},\mathbf{x}_{i}^{S},\mathbf{x}^{\overline{S}}\right)\right)-\mathbb{E}_{x_{j},\mathbf{x}^{\overline{S}}}\left(\hat{f}\left(x_{j},\mathbf{x}_{i}^{S},\mathbf{x}^{\overline{S}}\right)\right)\right].$
(A86)

Moreover, according Equation A7, the SHAP value $\phi_{j}^{SHAP}$ associated
to $\hat{f}(\mathbf{x}_{i})$ is equal to:

$\displaystyle\phi_{i,j}^{SHAP}=\sum_{S\subseteq\mathcal{P}(\\{\mathbf{x}\\}\setminus\left\\{x_{j}\right\\})}w_{S}\left[\mathbb{E}_{\mathbf{x}^{\overline{S}}}\left(\hat{f}(x_{i,j},\mathbf{x}_{i}^{S},\mathbf{x}^{\overline{S}})\right)-\mathbb{E}_{x_{j},\mathbf{x}^{\overline{S}}}\left(\hat{f}(x_{j},\mathbf{x}_{i}^{S},\mathbf{x}^{\overline{S}})\right)\right].$
(A87)

Therefore, in the particular case where
$G\left(y_{i};\mathbf{x}_{i};\delta_{0}\right)=\hat{f}(\mathbf{x}_{i})$,
according to Equation A86 and A87, the individual XPER value $\phi_{j}$ is
equal to the SHAP value $\phi_{j}^{SHAP}$. ∎

##  Appendix F Simulations: Explaining Overfitting

In Appendix F.1 and F.2, we propose two additional Monte Carlo simulation
experiments illustrating how XPER values can be used to detect the origin of
overfitting. The latter can arise for at least two reasons: (1) an improper
control of the bias-variance trade-off through model hyperparameters, or (2) a
shift of the feature distributions between the training and test samples. We
illustrate each of these two cases in the following subsections.

###  F.1 Case 1: Improper control of the bias-variance trade-off

Consider a DGP given by $y_{i}=\mathbf{1}(y^{*}_{i}>0)$ with
$y_{i}^{*}=\mathbf{\omega}_{i}\beta+\varepsilon_{i}$ a latent variable,
$\mathbf{\omega}_{i}=(1:\mathbf{x}^{{}^{\prime}}_{i})$, and $\varepsilon_{i}$
an i.i.d. error term with $\varepsilon_{i}\sim\mathcal{N}(0,1)$. We consider
three independent features such that
$\mathbf{x}_{i}\sim\mathcal{N}(\mathbf{0},\mathbf{\Sigma})$ with
$diag(\Sigma)=(1.3,1.2,1.1)$. The true vector of parameters is
$\beta=(\beta_{0},\beta_{1},\beta_{2},\beta_{3})^{{}^{\prime}}=(0.05,0.5,0.5,0.5)^{{}^{\prime}}$
with $\beta_{0}$ the intercept. We generate $K=5,000$ pseudo-samples
$\\{y_{i}^{s},\mathbf{x}_{i}^{s}\\}_{i=1}^{T+n}$ of size $1,00$0 using this
DGP. Then, we estimate a decision tree using 5-fold cross validation on the
first $T=700$ observations of each pseudo-sample and we use the remaining
$n=300$ observations as a test sample. In order to intentionally generate
overfitting, we impose a minimum tree-depth of 6 nodes for only three features
in the model. For each trained model, we implement XPER to decompose the
effect of the features on the AUC of the training and the test samples. We
display in Figure 1(a) the empirical distributions of the AUC. As expected,
the trained tree models are overfitting the data, illustrated by the
relatively low AUC values obtained on the test samples compared to the
training samples. The empirical distributions of the XPER values reported in
other panels of Figure A1 show that this drop in performance does not come
from a particular feature. Indeed, the XPER contributions to the AUC are
relatively close between the training and the test sample for all features.
Thus, when overfitting is due to an improper control of the bias-variance
trade-off, we observe a large decrease of the performance metric along with a
stability of XPER values between the training and the test sample. Therefore,
XPER can be used as a reverse engineering tool to detect wrong settings of
hyperparameters.

(a) AUC

(b) $\phi_{0}$

(c) $\phi_{1}$

(d) $\phi_{2}$

(e) $\phi_{3}$

Figure A1: Empirical distributions of AUC and XPER values in case of
overfitting due to improper control of the bias-variance trade-off

Note: This figure displays the empirical distributions of the AUC and XPER
values on the training (dark color) and test (light color) sample according to
the framework detailed in Illustration 2, case 1. XPER values are divided by
the difference between the AUC of the model and the benchmark value to be
comparable between the training and the test sample. The solid lines refer to
kernel density estimations.

###  F.2 Case 2: Shift of the feature distribution

Overfitting can also arise from a shift of the feature distributions between
the training and the test sample. To illustrate this origin of overfitting, we
consider two distinct DGPs for the training and the test sample. For the
former, we keep the same DGP as in the first case. For the test sample, we
assume an increase in the variance of the first feature while keeping other
parameters unchanged, such that $diag(\mathbf{\tilde{\Sigma}})=(3,1.2,1.1)$.
In the context of time series, such shift in the variance can come from a
structural change. See Perron and Yamamoto, (2021) on how to detect
forecasting performance changes with structural change tests. As in case 1, we
generate $K=5,000$ pseudo-samples
$\\{y_{i}^{s},\mathbf{x}_{i}^{s}\\}_{i=1}^{T+n}$ of size 1,000 ($T=700$ and
$n=300$). For each pseudo-sample, we estimate a decision tree with a depth
between 1 to 5 using 5-fold cross validation. Setting a relatively low tree
depth avoids overfitting due to an improper control of the bias-variance
trade-off.

(a) AUC

(b) $\phi_{0}$

(c) $\phi_{1}$

(d) $\phi_{2}$

(e) $\phi_{3}$

Figure A2: Empirical distributions of AUC and XPER values in case of
overfitting due to a shift of the distribution of the features

Note: This figure displays the empirical distributions of the AUC and XPER
values on the training (dark color) and test (light color) sample according to
the framework detailed in Illustration 2, case 2. XPER values are divided by
the difference between the AUC of the model and the benchmark value to be
comparable between the training and the test sample. The solid lines refer to
kernel density estimations.

In Figure 2(a), we observe a decrease in AUC between the training and test
samples. Contrary to the previous case, this decrease is due to the shift of
the distribution of $x_{1}$ which has also an impact on XPER values. More
precisely, in Figure A2, we observe that the contribution of feature $x_{1}$
to the AUC increases from the training to the test sample whereas the
contribution of the other features decreases. Thus, observing both a drop in
the performance of the model and some variations in the XPER values from the
training to the test sample can indicate a change in the data structure. Such
change is not captured by the model and not related to hyperparameter
settings.

##  Appendix G Empirical Application: Some additional results

###  G.1 Summary statistics and features distribution

Table A4: Summary Statistics

| Count | Mean | Std. | Minimum | 25% | 50% | 75% | Maximum
---|---|---|---|---|---|---|---|---
Job tenure | 7,440 | 9.3298 | 9.9787 | 0 | 2 | 5 | 15 | 58
Age | 7,440 | 45.1691 | 14.7965 | 18 | 33 | 46 | 55 | 89
Car price | 7,440 | 12,935 | 6,204 | 546 | 8,149 | 11,950 | 16,500 | 47,051
Funding amount | 7,440 | 11,461 | 6,019 | 546 | 6,846 | 10,382 | 15,000 | 30,000
Loan duration | 7,440 | 56.2176 | 19.3833 | 6 | 48 | 60 | 72 | 96
Monthly payment | 7,440 | 0.1051 | 0.0611 | 0.0051 | 0.0690 | 0.0947 | 0.1304 | 2.6300
Downpayment | 7,440 | 0.0897 |  | 0 |  |  |  | 1
Credit event | 7,440 | 0.0220 |  | 0 |  |  |  | 1
Married | 7,440 | 0.5347 |  | 0 |  |  |  | 1
Homeowner | 7,440 | 0.3848 |  | 0 |  |  |  | 1
Default | 7,440 | 0.2000 |  | 0 |  |  |  | 1

Note: This table displays summary statistics for each feature used in the
XGBoost model as well as the target variable. For each categorical feature the
standard deviation (Std.) and the quartiles (25%, 50% and 75%) are not
displayed.

Figure A3: Features distribution by default class

Note: This figure displays the distribution of the features by default class
on the training sample, using kernel density estimation for continuous
features. Dark red refers to defaulting borrowers and light red to non-
defaulting borrowers.

###  G.2 XPER decomposition

Figure A4: XPER decomposition of the AUC

Note: This figure displays the XPER values of the AUC of the XGBoost model
estimated on the training sample.

###  G.3 Individual XPER decomposition and data visualization

We analyse the impact of the various features on the performance metric but we
now do it for each borrower individually. We start by analysing in Figure A5
the XPER decomposition for two sample borrowers. These force plots enable us
to decompose the individual performance of each borrower, as defined in
Equation 7. By doing so, they allow us to understand why some individuals
contribute more to the AUC of the model than others. In each panel of Figure
A5, Performance refers to the contribution of the borrower to the AUC of the
model and Benchmark to their benchmark value, i.e., $\phi_{i,0}$ in Equation
7. For each borrower, the features increasing (respectively decreasing) the
performance appear in red (blue). Borrower #3 has a relatively high individual
AUC compared to borrower #28 (both have theoretically the same benchmark). The
over-performance of borrower #3 is mainly due to the large positive XPER
values for funding amount, job tenure, and car price. It also comes from the
small negative XPER values for the marital status (married) and the share of
the monthly payment in the borrower’s income (monthly payment).

(a) Borrower #3

(b) Borrower #28

Figure A5: Force plots of individual XPER values

Note: This figure displays the XPER decomposition (for the AUC) of two loan
borrowers (see Equation 4). Borrower #3 did not default on his loan and has a
probability of default of 8% according to the XGBoost ((Panel (a)). Borrower
#28 did not default on his loan and has a probability of default of 57%
according to the XGBoost (Panel (b)). Performance refers to the individual
level of the AUC whereas Benchmark represents the individual contribution to
the AUC associated to a population where the target variable $y_{i}$ is
independent from the features $\mathbf{x}_{i}$. The red color refers to
positive XPER values, i.e., features increasing performance. The blue color
refers to negative XPER values, i.e., features decreasing performance.

To better understand the relative influence of each feature for the two
borrowers, we analyse their risk-profiles and probabilities of default
predicted by the model. Let us start with borrower #3. He is 41 years old,
homeowner, has a stable job, and applied for a loan to buy a moderately-priced
car. He provided a down payment greater than 50% of the car value and
experienced no past credit event. Intuitively, we would naturally classify
this borrower as low-risk and this is confirmed by the 8% default probability
estimated by the XGBoost model. Thus, as borrower #3 eventually did not
default on his loan, his contribution to the AUC is high. The situation of
borrower #28 is quite different as he exhibits a higher risk profile (young,
jobless, not married, relatively large credit amount, no down payment). Yet,
the model remains quite undecided about his capacity to pay back the loan with
a 57% estimated default probability. As the AUC measures the discriminatory
ability of the model, this uncertainty leads to a low individual contribution,
and even lower than the benchmark value.

Figure A6: Summary plots of individual XPER values

Note: This figure displays the individual XPER values for each feature used in
the XGBoost model. Each dot represents the value for a given borrower (see
Equation 4). We display the results for the borrowers who paid back their
loans (left graphic) and those who do not (right graphic).

We then consider the entire sample of borrowers. In Figure A6, we display the
XPER values for each feature as a function of the feature value. We analyse
these results according to two types of borrowers: non-defaulting borrowers
(y=0) and defaulting borrowers (y=1). We clearly see that depending on the
value of the feature and the type of borrower, we know if this feature
contributes to increase or decrease the performance of the model. For
instance, for a non-defaulting borrower (left panel), a relatively high job
tenure is associated to a positive XPER value. This result is due to the fact
that a relatively long job tenure tends to lower the probability of default in
the model. Hence, this increases the ability of the model to distinguish him
from the defaulting borrowers and boosts the XPER value. On the opposite, for
a defaulting borrower (right panel), a relatively high job tenure leads to a
negative XPER value and thus decreases his contribution to the AUC of the
model.

###  G.4 Boosting model performance

(a) Features distribution by default class in group 1

(b) Features distribution by default class in group 2

Figure A7: Features distribution by default class for each group

Note: This figure displays the distribution of the features by default class
on the training sample, for the first (Panel (a)) and second group (Panel (b))
created from individual XPER values using the KMedoids methodology. For
continuous features, we use a kernel density estimation. Dark red refers to
defaulting borrowers and light red to non-defaulting borrowers.

Figure A8: XPER decomposition of the AUC by group

Note: This figure displays the XPER values of the AUC of the XGBoost model
estimated on the training sample by group creating with the KMedoid method.