USTC-ICTS/PCFT-23-33

KIAS-Q23022

Schur indices for $\mathcal{N}=4$ super-Yang-Mills
with more general gauge groups

Bao-ning<EMAIL_ADDRESS>Min-xin<EMAIL_ADDRESS>Xin<EMAIL_ADDRESS>

∗ Max Planck Institute for Mathematics,

Vivatsgasse 7, D-53111 Bonn, Germany

† Interdisciplinary Center for Theoretical Study,

University of Science and Technology of China, Hefei, Anhui 230026, China

† Peng Huanwu Center for Fundamental Theory,

Hefei, Anhui 230026, China

‡ Quantum Universe Center, Korea Institute for Advanced Study

Hoegiro 85, Seoul 02455, Korea

We study the unflavored Schur indices in the $\mathcal{N}=4$ super-Yang-Mills
theory for the $B_{n},C_{n},D_{n},G_{2}$ gauge groups. We explore two methods,
namely the character expansion method and the Fermi gas method, to efficiently
compute the $q$-series expansion of the Schur indices to some high orders.
Using the available data and the modular properties, we are able to fix the
exact formulas for the general gauge groups up to some high ranks and discover
some interesting new features. We also identify some empirical modular anomaly
equations, but unlike the case of $A_{n}$ groups, they are quite complicated
and not sufficiently useful to fix exact formulas for gauge groups of
arbitrary rank.

###### Contents

  1. 1 Introduction
  2. 2 The Schur indices: some general properties
  3. 3 Character expansion method
  4. 4 Fermi gas method
    1. 4.1 Schur indices for $BCD$-type gauge groups
      1. 4.1.1 Even ranks
      2. 4.1.2 Odd ranks
  5. 5 Fixing the exact modular formulas
    1. 5.1 $BC$-type gauge groups
    2. 5.2 $D$-type gauge groups
  6. 6 Discussions
  7. A Elliptic functions and modular forms
  8. B Ring of modular forms under congruence subgroups of $\mathrm{SL}_{2}(\mathbb{Z})$
  9. C Proof of Proposition 1
  10. D Schur Indices for $SO(N)$ Groups up to $N=17$

##  1 Introduction

As a type of topological invariants, Witten index is a very powerful non-
perturbative tool for supersymmetric theories [1]. In the context of
superconformal quantum field theories, such indices were first constructed in
[2, 3]. For theories with a Lagrangian description, the $d$-dimensional
superconformal index can be computed by path integral formalism as the
supersymmetric partition function on $S^{1}\times S^{d-1}$. If an effective
Lagrangian flows to a superconformal fixed point in the UV or IR, we can also
compute the superconformal index at the fixed point from the Lagrangian
theory. There are many important applications, e.g. providing quantitative
tests of field theory dualities, and counting holographic dual black hole
microstates. We will focus on the 4d case, where the superconformal index can
be computed with an integral over the Cartan generators of the gauge algebra,
counting gauge invariant operators. For reviews see e.g. [4, 5].

For theories with extended supersymmetry, a particular specialization of the
4d superconformal index, known as the Schur index [6], has some further nice
mathematical properties. For example, in some cases it can be computed from
the $q$-deformed 2d Yang-Mills [7], or the vacuum character of a corresponding
chiral algebra [8, 9]. The Schur index has been studied extensively, e.g. [10,
11, 12]. For the case of $\mathcal{N}=4$ supersymmetry, besides a universal
fugacity parameter denoted as $q$, the Schur index may have an extra flavor
fugacity from the symmetry $SU(2)_{F}\subset SU(4)_{R}$. We will continue the
study of unflavored Schur index in the $\mathcal{N}=4$ super-Yang-Mills theory
by one of the authors in [13], generalizing the results for $A_{n}$ gauge
groups to more gauge groups $B_{n},C_{n},D_{n},G_{2}$. A modular anomaly
equation was proposed and proved in [13], inspired by similar equations in
topological string theories [14, 15]. We also explore the modular anomaly
equations in this paper and find that they are much more complicated comparing
to the case of $A$-type gauge groups. Some other recent studies relating to
Schur index are [16, 17, 18, 19, 20, 21, 22]. The AdS/CFT correspondence for
$BCD$ types of gauge groups was described in [23].

The paper is organized as follows. In Section 2 we introduce the general
formulas for the Schur index and discuss the modular properties for various
gauge groups. We discuss how to compute the $q$-series expansion and use the
modular property to fix the exact formulas for the case of $G_{2}$ gauge
group. In Section 3 we explore the character expansion method to compute the
$q$-series expansion for the $BCD$-type gauge groups. In Section 4 we explore
the Fermi gas method. We find that the method is more efficient for $D$-type
gauge groups than $BC$-type gauge groups. In Section 5 we use the perturbative
results as well as some empirical patterns to fix the exact modular formulas
up to some high ranks, and discuss the modular anomaly equations. Finally, in
Section 6 we discuss the main results and potential future research. In
Appendix A we provide our convention for elliptic functions and modular forms.
In Appendix B we discuss the generators of modular forms of some relevant
congruence subgroups. In Appendix D we list the main results of the exact
modular formulas.

##  2 The Schur indices: some general properties

The 4d $\mathcal{N}=2$ superconformal index is a type of Witten index [3], as
a trace over the Hilbert space on $S^{3}$ in the radial quantization. For a
theory with a weakly coupled Lagrangian description, the index is computed
explicitly as a matrix integral [7]. In this paper, we consider the Schur
limit of the superconformal index, known as the Schur index, for the 4d
$\mathcal{N}=4$ super-Yang-Mills theories. Alternatively, the Schur index can
be derived by the supersymmetric localization of $\mathcal{N}=2$ SCFT theory
on $S^{3}\times S^{1}$. For the $\mathcal{N}=4$ case, it can be regarded as a
$\mathcal{N}=2^{*}$ theory with massless adjoint hypermultiplet, and we have
111The expression of the Schur index appears in different forms in the
literature up to a factor $q^{c(G)}$. Here we use a convention such that the
result has nice modular properties. See Appendix A for notations of Jacobi
theta functions $\theta_{i}(z)$ and other elliptic functions.

$\begin{split}\mathcal{I}_{G}(q)&=\frac{(-\mathrm{i})^{\,\mathrm{rank}\,G-\dim
G}\,\eta(\tau)^{3\,\mathrm{rank}\,{G}}}{|W|\,\theta_{4}^{\,\mathrm{rank}\,{G}}}\oint\prod_{j=1}^{\,\mathrm{rank}\,{G}}da_{j}\prod_{\rho\in\mathcal{R}_{G}^{*}}\frac{\theta_{1}(\rho(a_{j}))}{\theta_{4}(\rho(a_{j}))}\\\
&=\frac{1}{|W|}\left(\frac{\eta(\tau)^{3}}{\theta_{4}}\right)^{\,\mathrm{rank}\,{G}}\oint\prod_{j=1}^{\,\mathrm{rank}\,{G}}da_{j}\prod_{\rho\in\mathcal{R}_{G}^{+}}\frac{\theta_{1}(\rho(a_{j}))^{2}}{\theta_{4}(\rho(a_{j}))^{2}},\end{split}$
(2.1)

where $|W|$ stands for the dimension of the Weyl group of the gauge group $G$,
$\dim G$ is the dimension of the adjoint representation, $\,\mathrm{rank}\,G$
is the rank of $G$, $\mathcal{R}_{G}^{*}$ is the set of non-zero roots, and
$\mathcal{R}_{G}^{+}$ is the set of positive roots. The integrals $\oint$ are
performed for $a_{i}$ from $0$ to $2\pi\mathrm{i}$, effectively eliminating
the terms $e^{ma}$ with $m\in\mathbb{Z}-\\{0\\}$, leaving only the constant
term, similar as a residue calculation.

The Schur index can be calculated by explicitly performing the integral (2.1).
It is usually difficult to directly compute the integrals of theta functions,
see e.g. some examples of such elaborate efforts in [10, 11]. Instead, we will
expand the integrand in terms of $q=e^{\pi\mathrm{i}\tau}$ to some high order,
and integrate over the gauge fugacities $a_{i}$, then we can use the ring of
quasi-modular forms in the congruence subgroups listed in Table 1, to fix the
exact modular expressions of the Schur index. More precisely, the Schur
indices can be expressed as

$\displaystyle\mathcal{I}_{G}(q)=s_{0}^{b(G)}\cdot\widetilde{\mathcal{I}}_{G}(q),$
(2.2)

where

$\displaystyle
s_{0}=\frac{\theta_{4}(\tau)}{\eta(\tau)^{3}}=q^{-\frac{1}{4}}(1-2q+3q^{2}-6q^{3}+\cdots),$
(2.3)

and $\widetilde{\mathcal{I}}_{G}(q)$ can be written in terms with the second
Eisenstein series and the modular forms of the congruence subgroup
$\Gamma\subset\mathrm{SL}_{2}(\mathbb{Z})$. The subgroups $\Gamma$ and the
values of $b(G)$ are summarized in Table 1. Some properties and generators of
the relevant congruence subgroups are explained in Appendix B.

Group | $A_{2N-1}$ | $A_{2N}$ | $B_{N},C_{N}$ | $D_{2N-1}$ | $D_{2N}$ | $G_{2}$
---|---|---|---|---|---|---
Modular group $\Gamma$ | $\Gamma^{0}(2)$ | $\mathrm{SL}_{2}(\mathbb{Z})$ | $\Gamma(2)$ | $\Gamma_{0}(6)\cap\Gamma^{0}(2)$
$b(G)$ | 0 | 1 | $N$ | $2N-3$ | $2N$ | 2
Table 1: The modular groups for $\mathcal{N}=4$ Schur indices. The modular
group of $G_{2}$ can be also written as $\Gamma_{0}(3)\cap\Gamma(2)$.

For example, for the $U(N)$ group, we have

$\displaystyle\mathcal{I}_{U(N)}(q)=\frac{\eta(\tau)^{3N}}{N!\,\theta_{4}^{N}}\oint\prod_{j=1}^{N}da_{j}\prod_{1\leq
i<j\leq N}\frac{\theta_{1}(a_{i}-a_{j})^{2}}{\theta_{4}(a_{i}-a_{j})^{2}},$
(2.4)

from which we can solve

$\mathcal{I}_{U(1)}=s_{0}^{-1}=\frac{\eta^{3}}{\theta_{4}}.$ (2.5)

The $SU(N)$ Schur index can be solved from $U(N)$ Schur index, by factoring
out the $U(1)$ contribution, we have

$\displaystyle\mathcal{I}_{SU(N)}(q)=\mathcal{I}_{U(N)}(q)/\mathcal{I}_{U(1)}(q)=\frac{\eta(\tau)^{3(N-1)}}{N!\,\theta_{4}^{N-1}}\oint\prod_{j=1}^{N}da_{j}\prod_{1\leq
i<j\leq N}\frac{\theta_{1}(a_{i}-a_{j})^{2}}{\theta_{4}(a_{i}-a_{j})^{2}}.$
(2.6)

One of the integrals in $\oint\prod_{j=1}^{N}da_{i}$ is actually trivial so we
have indeed $N-1$ integrals corresponding to the rank. We also use
$\mathcal{I}_{A_{N-1}}$ to denote the Schur index for the $SU(N)$ gauge group.
For example for the low ranks, we have

$\displaystyle\mathcal{I}_{A_{1}}$
$\displaystyle=\frac{1}{2}E_{2}+\frac{1}{24}(\theta_{2}^{4}+\theta_{3}^{4}),$
(2.7) $\displaystyle\widetilde{\mathcal{I}}_{A_{2}}$
$\displaystyle=\frac{1}{2}E_{2}+\frac{1}{24}.$ (2.8)

The formulas for $A_{N}$ of arbitrary rank have been determined [12, 11, 13].

Before focusing on the main examples of $BCD$-type gauge groups, we consider
an isolated but still non-trivial example of the $G_{2}$ gauge group. Based on
the symmetry of the Dynkin diagram, we propose the congruence subgroup of
Schur index to be $\Gamma_{0}(6)\cap\Gamma^{0}(2)$, as shown in Table 1. In
this case, the rank is quite small, and there are only two integrals to
perform. We can compute up to a very high order in the $q$-series expansion
and fix the exact formula in terms of the generators of the congruence
subgroup with some redundant checks. The formula is

$\displaystyle\mathcal{I}_{G_{2}}(q)$
$\displaystyle=\frac{\eta(\tau)^{6}}{12\,\theta_{4}^{2}}\oint\prod_{j=1}^{2}da_{j}\cdot\frac{\theta_{1}(a_{1})^{2}\theta_{1}(a_{2})^{2}\theta_{1}(3a_{1}-a_{2})^{2}}{\theta_{4}(a_{1})^{2}\theta_{4}(a_{2})^{2}\theta_{4}(3a_{1}-a_{2})^{2}}$
(2.9)
$\displaystyle=\left(\frac{\theta_{4}}{\eta^{3}}\right)^{2}\left(\mathcal{I}_{A_{1}}^{2}-\frac{1}{288}(8E_{2}^{(3)}+\theta_{2}^{4}-8\theta_{3}^{4}+3\theta_{2}^{4}\theta_{3}^{4})\right),$
(2.10)

where $E_{2}^{(3)}$ is a weight two modular form defined in (B.10).

As expected from the famous S-duality of $\mathcal{N}=4$ super-Yang-Mils
theory, we can check that the Schur indices for the $B_{N}$ and $C_{N}$ groups
are the same up to some low orders in the $q$-series expansion. The tests can
be pushed to much higher orders with the more efficient methods in the next
two sections. It is well known that the $B_{N}$ and $C_{N}$ root systems have
the same Weyl group and similar structures [24]. For low ranks we may find an
explicit isomorphic transformation between the roots of these two groups. For
example for the simple case of $N=2$, the Dynkin diagrams of $B_{2}$ and
$C_{2}$ are actually the same. In our parametrization explained in more detail
later (4.1), their roots are expressed in terms of simple roots as follows

$\displaystyle B_{2}$ $\displaystyle:~{}~{}\mathcal{R}_{B_{2}}^{*}=\\{\pm
a_{1},\pm a_{2},\pm(a_{1}\pm a_{2})\\},$ (2.11) $\displaystyle C_{2}$
$\displaystyle:~{}~{}\mathcal{R}_{C_{2}}^{*}=\\{\pm 2a^{\prime}_{1},\pm
2a^{\prime}_{2},\pm(a^{\prime}_{1}\pm a^{\prime}_{2})\\}.$

We can set a relation between simple roots
$a_{1}=a^{\prime}_{1}+a^{\prime}_{2},a_{2}=a^{\prime}_{1}-a^{\prime}_{2}$,
transforming the roots of $B_{2}$ to $C_{2}$. Since the computations of the
Schur index pick out the constant terms of the simple roots, they are the same
$\mathcal{I}_{B_{2}}=\mathcal{I}_{C_{2}}$ in this case. However, for general
rank $N$, we are not aware of a universal mathematical proof of the non-
trivial identity $\mathcal{{I}}_{B_{N}}=\mathcal{{I}}_{C_{N}}$.

##  3 Character expansion method

From the definition of the Witten index, the Schur index can be written as an
integral of plethystic exponential function [6], which in our context can be
obtained by rewritings of the Jacobi theta functions in (2.1). They can then
be expanded as characters of the gauge groups, see e.g. some early works [25,
26]. The formalism is nicely applied to the $BCD$ types of gauge groups in a
recent paper [27], which provides explicit formulas in terms of sums over 2D
Young diagrams of integer partitions.

The Schur index for the (mass deformed) 4d $\mathcal{N}=4$ super-Yang-Mills
theory with gauge group $G$ has the expression

$\mathcal{I}_{G}(q;m)=q^{c(G)}\int[da]\mathrm{PE}\left[\chi^{G}_{\mathrm{\mathbf{Adj}}}(e^{\alpha})\left(i_{V}(q)+i_{H}(q)(e^{m}+e^{-m})\right)\right],$
(3.1)

where PE denotes the plethystic exponential function, $i_{V}(q)$ and
$i_{H}(q)$ are the single letter indices for the vector multiplet and
hypermultiplet that are defined as

$i_{V}(q)=\frac{-2q^{2}}{1-q^{2}},\quad\quad i_{H}(q)=\frac{q}{1-q^{2}}.$
(3.2)

Here $\chi^{G}_{\mathrm{\mathbf{Adj}}}(e^{\alpha})$ denotes the character of
the gauge group $G$ in the adjoint representation, it is defined as the sum
over the roots of $G$

$\displaystyle\chi^{G}_{\mathrm{\mathbf{Adj}}}(e^{\alpha})=\sum_{\rho\in\mathcal{R}_{G}}e^{\rho(a_{i})}.$
(3.3)

$[da]$ is the Haar measure defined as

$\displaystyle[da]=\frac{1}{|W|}\prod_{i}\frac{da_{i}}{2\pi\mathrm{i}}\cdot\prod_{\rho\in\mathcal{R}_{G}^{+}}\left(1-e^{\rho(a)}\right)^{2},$
(3.4)

where we used $\mathcal{R}_{G}$ and $\mathcal{R}_{G}^{+}$ denote the roots and
positive roots of the lie algebra corresponds to $G$.

For the massless case, the total single letter index is
$i_{\mathcal{N}=4}(q)=\frac{2q}{1+q}$. The integral in (3.1) can be expanded a
series of $q$, starting from a constant term as $1+\mathcal{O}(q)$. The factor
$q^{c(G)}$ can be determined by comparing with the formula (2.1), counting
such powers of $q$ from the $\eta,\theta_{1}$ functions.

The formula (3.1) can be computed using the properties of character for the
$BCD$ types of groups [27, 19]. They can be expressed by the following
formulas respectively

$\displaystyle\mathcal{I}_{B_{N}}$
$\displaystyle=q^{c(G)}\sum_{\lambda}\Big{[}\frac{f_{\lambda}(q)}{z_{\lambda}}\frac{1}{2^{l(\lambda)}}\sum_{\tilde{\lambda}\in\text{Ev}(\lambda)}(-1)^{l(\tilde{\lambda})}\sum_{\mu\in
R^{r}_{2N+1}(2|\lambda|)}\chi^{S}_{\mu}(\tilde{\lambda})\Big{]},$ (3.5)
$\displaystyle\mathcal{I}_{C_{N}}$
$\displaystyle=q^{c(G)}\sum_{\lambda}\Big{[}\frac{f_{\lambda}(q)}{z_{\lambda}}\frac{1}{2^{l(\lambda)}}\sum_{\tilde{\lambda}\in\text{Ev}(\lambda)}\sum_{\mu\in
R^{c}_{2N}(2|\lambda|)}\chi^{S}_{\mu}(\tilde{\lambda})\Big{]},$ (3.6)
$\displaystyle\mathcal{I}_{D_{N}}$
$\displaystyle=q^{c(G)}\sum_{\lambda}\Big{[}\frac{f_{\lambda}(q)}{z_{\lambda}}\frac{1}{2^{l(\lambda)}}\sum_{\tilde{\lambda}\in\text{Ev}(\lambda)}(-1)^{l(\tilde{\lambda})}\sum_{\mu\in
R^{r}_{2N}(2|\lambda|)\cup
W^{r}_{2N}(2|\lambda|)}\chi^{S}_{\mu}(\tilde{\lambda})\Big{]},$ (3.7)
$\displaystyle f_{\lambda}(q)$
$\displaystyle:=\prod_{i=1}^{l(\lambda)}i_{\mathcal{N}=4}(q^{\lambda_{i}}),~{}~{}~{}~{}z_{\lambda}:=\prod_{i=1}^{\infty}k_{i}!i^{k_{i}}.$
(3.8)

We use the notations in [27], explained in the followings. The sums are over
integer partitions $\lambda$, depicted by 2D Young tableaux as non-increasing
sequences $\lambda_{1}\geq\lambda_{2}\geq\cdots$. The usual notations
$l(\lambda),|\lambda|$ denote the number of rows and boxes and $\lambda^{T}$
denote the transpose of the Young tableau.
$k_{m}(\lambda)=\lambda^{T}_{m}-\lambda^{T}_{m+1}$ is the number of $m$’s in
the partition. $\text{Ev}(\lambda),R^{r}_{n}(|\lambda|),W^{r}_{n}(|\lambda|)$
and $R^{c}_{n}(|\lambda|)$ are the set of partitions defined by:

  * •

$\text{Ev}(\lambda)$: a set of partitions that is obtained by replacing
$\lambda_{i}$ with $2\lambda_{i}$ or $\lambda_{i},\lambda_{i}$ for all $i$’s,

  * •

$R^{r}_{n}(|\lambda|):=\\{\mu|~{}l(\mu)\leq n,|\mu|=|\lambda|,\forall
i(\mu_{i}~{}\text{is even})\\}$,

  * •

$W^{r}_{n}(|\lambda|):=\\{\mu|~{}l(\mu)=n,|\mu|=|\lambda|,\forall
i(\mu_{i}~{}\text{is odd})\\}$,

  * •

$R^{c}_{n}(|\lambda|):=\\{\mu|~{}l(\mu)\leq n,|\mu|=|\lambda|,\forall
i(\mu^{T}_{i}~{}\text{is even})\\}$ .

Finally, $\chi_{\mu}^{S}(\lambda)$ is the character of the symmetric group
$S_{|\lambda|}$, defined for $|\mu|=|\lambda|$. It can be computed by the
Frobenius method, as the coefficient of
$\prod_{i=1}^{l(\mu)}x_{i}^{\mu_{i}+l(\mu)-i}$ in the following expression

$\prod_{1\leq i<j\leq
l(\mu)}(x_{i}-x_{j})~{}\cdot~{}\prod_{j=1}^{\lambda_{1}}(\sum_{i=1}^{l(\mu)}x_{i}^{j})^{k_{j}(\lambda)}.$
(3.9)

The character expansion method is much more efficient than directly computing
(2.1). For example, we can check the S-duality relation
$\mathcal{{I}}_{B_{N}}=\mathcal{{I}}_{C_{N}}$ to much higher orders in
$q$-series expansion.

When $N$ is large, we can compute to further high orders by comparing with the
$N\sim\infty$ result, similar as in the more familiar $A_{N}$ case. It was
pointed out in [27] that for the adjoint representation, the large $N$ results
are the same for $BCD$ types of gauge groups (omitting the rather trivial
factor $q^{c(G)}$ in our convention), and a simple expression was proposed
there

$\mathcal{I}_{\infty}=\sum_{\lambda}\Big{[}\frac{f_{\lambda}(q)}{z_{\lambda}}\frac{1}{2^{l(\lambda)}}\sum_{\tilde{\lambda}\in\text{Ev}(\lambda)}(-1)^{l(\tilde{\lambda})}\prod_{m=1}^{\infty}a_{m,k_{m}(\tilde{\lambda})}\Big{]},$
(3.10)

where the coefficients are determined by the initial values and recursion
relations

$\displaystyle
a_{m,n}=m(n-1)a_{m,n-2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}a_{m,0}=1,a_{m,1}=0,~{}~{}~{}\text{if}~{}m~{}\text{is
odd};$ (3.11) $\displaystyle
a_{m,n}=a_{m,n-1}+m(n-1)a_{m,n-2},~{}~{}a_{m,0}=a_{m,1}=1,~{}~{}~{}\text{if}~{}m~{}\text{is
even}.$

The calculation of $\mathcal{I}_{\infty}$ is much more efficient than the
finite $N$ formulas (3.5, 3.6, 3.7), and one can easily compute to much higher
order. It is easy to see when $n\geq|\lambda|$, the sets are the same
$R^{r}_{n}(2|\lambda|)=R^{r}_{\infty}(2|\lambda|)$, so $\mathcal{{I}}_{B_{N}}$
agrees with $\mathcal{I}_{\infty}$ up to the order $q^{2N+1}$, i.e.

$\mathcal{{I}}_{B_{N}}=\mathcal{{I}}_{C_{N}}=q^{c(G)}[\mathcal{I}_{\infty}+\mathcal{O}(q^{2N+2})].$
(3.12)

On the other hand, since $W^{r}_{n}(|\lambda|)=W^{r}_{\infty}(|\lambda|)=\O$
for $n\geq|\lambda|+1$ but $W^{r}_{|\lambda|}(|\lambda|)\neq\O$, the available
data from $\mathcal{I}_{\infty}$ for $D$-type group is much fewer, only up to
the order $q^{N-1}$, i.e.

$\mathcal{{I}}_{D_{N}}=q^{c(G)}[\mathcal{I}_{\infty}+\mathcal{O}(q^{N})].$
(3.13)

Some examples are the followings. Omitting the factor $q^{c(G)}$, we have

$\displaystyle\mathcal{I}_{\infty}(q)$
$\displaystyle=1+3q^{2}-4q^{3}+15q^{4}-24q^{5}+62q^{6}-120q^{7}+270q^{8}+\mathcal{O}(q^{9}),$
(3.14) $\displaystyle\mathcal{I}_{B_{4}}(q)=\mathcal{I}_{C_{4}}(q)$
$\displaystyle\sim\underline{1+3q^{2}-4q^{3}+15q^{4}-24q^{5}+62q^{6}-120q^{7}+270q^{8}}+\mathcal{O}(q^{9}),$
$\displaystyle\mathcal{I}_{B_{3}}(q)=\mathcal{I}_{C_{3}}(q)$
$\displaystyle\sim\underline{1+3q^{2}-4q^{3}+15q^{4}-24q^{5}+62q^{6}-120q^{7}}+255q^{8}+\mathcal{O}(q^{9}),$
$\displaystyle\mathcal{I}_{D_{4}}(q)$
$\displaystyle\sim\underline{1+3q^{2}-4q^{3}}+20q^{4}-32q^{5}+86q^{6}-176q^{7}+415q^{8}+\mathcal{O}(q^{9}),$
$\displaystyle\mathcal{I}_{D_{3}}(q)$
$\displaystyle\sim\underline{1+3q^{2}}+9q^{4}-6q^{5}+22q^{6}-18q^{7}+51q^{8}+\mathcal{O}(q^{9}),$

where the underlines denote the agreements with $\mathcal{I}_{\infty}(q)$.

##  4 Fermi gas method

The Fermi gas approach provides a powerful method to calculate the matrix
integrals in various contexts [28, 29, 30, 31, 32]. It was first used in [10]
to calculate the $\mathcal{N}=4$ Schur indices of unflavored $SU(N)$ cases and
was later generalized to the flavored cases [18] and cases with line defects
[33, 34, 19]. The purpose of this section is to generalize the Fermi gas
approach to the $\mathcal{N}=4$ Schur indices with the $SO(N)$ and $Sp(N)$
gauge groups.

The key ingredient of the Fermi gas approach is the elliptic generalization of
the _Cauchy determinant formula_ , that the integrand of the integral can be
expressed as a determinant of a density matrix $\rho(a_{i},a_{j})$, for the
$U(N)$ theory, we have [10]

$\begin{split}\prod_{1\leq i<j\leq
N}\frac{\theta_{1}(a_{i}-a_{j})^{2}}{\theta_{4}(a_{i}-a_{j})^{2}}&=f_{N}(\tau)\det_{1\leq
i,j\leq N}\rho(a_{i},a_{j}),\end{split}$ (4.1)

where

$\rho(a_{i},a_{j})=\frac{\theta_{2}(a_{i}-a_{j})}{\theta_{4}(a_{i}-a_{j})}.$
(4.2)

The coefficient $f_{N}(\tau)$ is a simple function defined by

$f_{N}(\tau)=\begin{cases}\frac{\theta_{4}^{N}}{\theta_{3}^{N}},\quad&\text{if
$N$ is even},\\\
\frac{\theta_{4}^{N}}{\theta_{2}\theta_{3}^{N-1}},\quad&\text{if $N$ is
odd}.\end{cases}$ (4.3)

Equation (4.1) is a special case of the Frobenius’s determinant formula [35,
36, 37], the $\theta_{2}(a_{i}-a_{j})$ in the numerator of $\rho(a_{i},a_{j})$
can be replaced by either $\theta_{1}(a_{i}-a_{j})$ or
$\theta_{3}(a_{i}-a_{j})$ with a different prefactor $f_{N}(\tau)$, due to
different specializations of the Frobenius’s determinant formula.

By using the determinant formula (4.1) and the Leibniz formula for the
determinant, the index can be written as

$\mathcal{I}_{U(N)}(q)\sim Z(N),\qquad Z(N)=\frac{1}{N!}\sum_{\sigma\in
S_{N}}(-1)^{\epsilon(\sigma)}\int\prod_{i}\frac{\theta_{2}\left(a_{i}-a_{\sigma(j)}\right)}{\theta_{4}\left(a_{i}-a_{\sigma(j)}\right)}da_{i},$
(4.4)

where $S_{N}$ is the permutation group of $N$ elements, and $\epsilon(\sigma)$
is the signature of the permutation $\sigma$. Equation (4.4) can be treated as
the canonical partition function of an ideal Fermi gas, it can be written as a
sum over conjugacy classes of the permutation group. Define the spectral trace

$\displaystyle Z_{\ell}=\int_{0}^{2\pi\mathrm{i}}da_{1}\cdots
da_{\ell}\rho(a_{1},a_{2})\rho(a_{2},a_{3})\cdots\rho(a_{\ell-1},a_{\ell})\rho(a_{\ell},a_{1}),$
(4.5)

then the partition function is given by

$Z(N)=\sum_{m_{\ell}}^{\prime}\prod_{\ell}\frac{Z_{\ell}^{m_{\ell}}(-1)^{m_{\ell}+1}}{m_{\ell}!l^{m_{\ell}}},$
(4.6)

where the prime denotes a sum over the conjugacy class, specified by a set of
non-negative integers $m_{\ell}$ that satisfy $\sum_{\ell=1}^{N}\ell
m_{\ell}=N.$ The great benefit of the Fermi gas approach is that the integral
representation of $Z_{\ell}$ is relatively easy to compute, it has a close
form expression

$Z_{\ell}=\sum_{n\in\mathbb{Z}}\left(\frac{1}{q^{n-\frac{1}{2}}+q^{-(n-\frac{1}{2})}}\right)^{\ell}.$
(4.7)

So, the close-form expressions of the partition function $Z(N)$, and then the
Schur index $\mathcal{I}_{U(N)}(q)$ can be written down according to (4.6) and
(4.4) respectively.

###  4.1 Schur indices for $BCD$-type gauge groups

In this subsection, we generalize the Fermi gas approach to $BCD$-type gauge
groups. The nontrivial roots for $B_{N}$, $C_{N}$ and $D_{N}$ groups are

$\displaystyle\mathcal{R}_{B_{N}}^{*}$ $\displaystyle=\\{\pm a_{j}|1\leq j\leq
N\\}\cup\\{\pm a_{i}\pm a_{j}|1\leq i<j\leq N\\},$
$\displaystyle\mathcal{R}_{C_{N}}^{*}$ $\displaystyle=\\{\pm 2a_{j}|1\leq
j\leq N\\}\cup\\{\pm a_{i}\pm a_{j}|1\leq i<j\leq N\\},$ (4.8)
$\displaystyle\mathcal{R}_{D_{N}}^{*}$ $\displaystyle=\\{\pm a_{i}\pm
a_{j}|1\leq i<j\leq N\\},$

respectively. Their roots are almost the same, but for $B$\- and $C$-type
groups, there are additional “diagonal roots” $\\{\pm\delta a_{j}|1\leq j\leq
N\\}$ where $\delta=1$ when the gauge group is $B_{N}$ and $\delta=2$ when the
gauge group is $C_{N}$. In the spirit of [36, 37], we propose two propositions
that will be shown to be useful for later calculations:

######  Proposition 1.

For the $D_{2N}$ group, we have the determinant formula:

$\displaystyle\prod_{1\leq i<j\leq 2N}\frac{\theta_{1}(a_{j}\pm
a_{i})^{2}}{\theta_{4}(a_{j}\pm a_{i})^{2}}=\det_{1\leq i,j\leq
2N}\left(\frac{\theta_{1}(a_{j}\pm a_{i})}{\theta_{4}(a_{j}\pm
a_{i})}\right).$ (4.9)

For the $D_{2N-1}$ group, we have the determinant formula:

$\displaystyle\prod_{1\leq i<j\leq 2N-1}\frac{\theta_{1}(a_{j}\pm
a_{i})^{2}}{\theta_{4}(a_{j}\pm a_{i})^{2}}=\det_{1\leq i,j\leq
2N}\left(\rho_{ij}\right),$ (4.10)

where $\rho_{ij}$ is an anti-symmetric matrix, with
$\rho_{ii}=0,\,\rho_{ij}=-\rho_{ji}$ and when $i<j$,

$\rho_{ij}=\begin{cases}\frac{\theta_{1}(a_{j}\pm a_{i})}{\theta_{4}(a_{j}\pm
a_{i})},\quad&\text{if }i<j\leq 2N-1,\\\ -1,\quad&\text{if
}i=2N,j<2N.\end{cases}$ (4.11)

The proof of Proposition (1) can be found in Appendix C.

As a direct consequence of Proposition 1, the determinant formulas for
$BC$-type groups can be written from the determinant formulas for $D$-type
groups, we have:

######  Proposition 2.

For the $B_{2N}$ or $C_{2N}$ group, we have the determinant formula

$\displaystyle\prod_{1\leq i<j\leq 2N}\frac{\theta_{1}(\delta
a_{j})^{2}}{\theta_{4}(\delta a_{j})^{2}}\cdot\frac{\theta_{1}(a_{j}\pm
a_{i})^{2}}{\theta_{4}(a_{j}\pm a_{i})^{2}}=\det_{1\leq i,j\leq
2N}\left(\frac{\theta_{1}(\delta a_{j})^{2}}{\theta_{4}(\delta
a_{j})^{2}}\cdot\frac{\theta_{1}(a_{j}\pm a_{i})}{\theta_{4}(a_{j}\pm
a_{i})}\right).$ (4.12)

where $\delta=1$ when the gauge group is $B$-type and $\delta=2$ when the
gauge group is $C$-type. For the $B_{2N-1}$ or $C_{2N-1}$ group, we have the
determinant formula

$\displaystyle\prod_{1\leq i<j\leq 2N-1}\frac{\theta_{1}(\delta
a_{j})^{2}}{\theta_{4}(\delta a_{j})^{2}}\cdot\frac{\theta_{1}(a_{j}\pm
a_{i})^{2}}{\theta_{4}(a_{j}\pm a_{i})^{2}}=\det_{1\leq i,j\leq
2N}\left(\rho_{ij}\right),$ (4.13)

where $\rho_{ij}$ is an anti-symmetric matrix, with $\rho_{ii}=0$ and when
$i\neq j$,

$\rho_{ij}=\begin{cases}\frac{\theta_{1}(\delta a_{j})^{2}}{\theta_{4}(\delta
a_{j})^{2}}\cdot\frac{\theta_{1}(a_{j}\pm a_{i})}{\theta_{4}(a_{j}\pm
a_{i})},\quad&\text{if }i<j\leq 2N-1,\\\ -\rho_{ji},&\text{if }j<i\leq
2N-1,\\\ -1,\quad&\text{if }i<2N,j=2N,\\\ \frac{\theta_{1}(\delta
a_{j})^{2}}{\theta_{4}(\delta a_{j})^{2}},\quad&\text{if
}i=2N,j<2N.\end{cases}$ (4.14)

In the following subsections, we use these determinant formulas to perform the
calculations on the integral representations of Schur indices.

####  4.1.1 Even ranks

##### $D_{2N}$

We want to use the Fermi gas approach to compute the Schur index for $D_{2N}$
theory, the key part is to get the result for $Z_{\ell}$ defined in (4.5). By
using the identities of Jacobi elliptic functions:

$\displaystyle\frac{\theta_{1}(z)}{\theta_{4}(z)}=\frac{\theta_{2}}{\theta_{3}}\,\mathrm{sn}\,(z\theta_{3}^{2})=\frac{2i}{\theta_{2}\theta_{3}}\sum_{n\in\mathbb{Z}}\frac{q^{n+\frac{1}{2}}e^{(n+\frac{1}{2})z}}{1-q^{2n+1}}=\frac{2i}{\theta_{2}\theta_{3}}\sum_{n=0}^{\infty}\frac{q^{n+\frac{1}{2}}}{1-q^{2n+1}}\left(e^{(n+\frac{1}{2})z}-e^{-(n+\frac{1}{2})z}\right)\,,$
(4.15)

the density matrix for $D_{2N}$ gauge group of the Fermi gas is

$\displaystyle\rho(a,a^{\prime})\equiv\frac{\theta_{1}(a+a^{\prime})\theta_{1}(a-a^{\prime})}{\theta_{4}(a+a^{\prime})\theta_{4}(a-a^{\prime})}=-\frac{4}{(\theta_{2}\theta_{3})^{2}}\sum_{n,n^{\prime}\in\mathbb{Z}}\frac{q^{n+n^{\prime}+1}e^{(n+n^{\prime}+1)a+(n-n^{\prime})a^{\prime}}}{(1-q^{2n+1})(1-q^{2n^{\prime}+1})}.$
(4.16)

By changing the parameters, we can rewrite the summation in a form that is
convenient for later calculations

$\displaystyle\rho(a,a^{\prime})=-\frac{4}{(\theta_{2}\theta_{3})^{2}}\left(\sum_{n,n^{\prime}\in
2\mathbb{Z}}\frac{q^{n+1}e^{(n+1)a+n^{\prime}a^{\prime}}}{(1-q^{n+n^{\prime}+1})(1-q^{n-n^{\prime}+1})}+\sum_{n,n^{\prime}\in
2\mathbb{Z}+1}\frac{q^{n+1}e^{(n+1)a+n^{\prime}a^{\prime}}}{(1-q^{n+n^{\prime}+1})(1-q^{n-n^{\prime}+1})}\right).$
(4.17)

Define

$\displaystyle Z_{\ell}=\int_{0}^{2\pi\mathrm{i}}da_{1}\cdots
da_{l}\rho(a_{1},a_{2})\cdots\rho(a_{\ell},a_{1}),$ (4.18)

only the zero order part of $a_{j}$ contributes to the integral. One may then
find a surprisingly easy solution

$\displaystyle Z_{2\ell}$
$\displaystyle=\frac{2^{4\ell+1}}{(\theta_{2}\theta_{3})^{4\ell}}\left(\sum_{n_{j}\in\mathbb{Z}}\frac{q^{2(n_{1}+\cdots
n_{2\ell})+\ell}}{\prod_{j=1}^{\ell}(1-q^{2n_{2j-1}\pm
2n_{2j}+1})(1-q^{2n_{2j}\pm(2n_{2j+1}+1)})}\right)$
$\displaystyle=2^{\ell+1}\left(\frac{\eta^{3}}{\theta_{4}}\right)^{-4\ell}\left(\sum_{n_{j}\in\mathbb{Z}_{\geq
0}}\frac{q^{2(n_{1}+\cdots
n_{2\ell})+\ell}\prod_{j=1}^{\ell}2^{1-\delta_{0,n_{2j}}}}{\prod_{j=1}^{\ell}(1-q^{2n_{2j-1}\pm
2n_{2j}+1})(1-q^{2n_{2j}\pm(2n_{2j+1}+1)})}\right),$ (4.19)

with the notation $n_{2\ell+1}=n_{1}$, and

$\displaystyle Z_{2\ell+1}=0.$ (4.20)

One can then compute the Schur index of $D_{2N}$ gauge group

$\displaystyle\mathcal{I}_{D_{2N}}=\frac{1}{2^{2N-1}}\left(\frac{\eta^{3}}{\theta_{4}}\right)^{2N}\sum_{m_{\ell}}^{\prime}\prod_{\ell}\frac{Z_{2\ell}^{m_{\ell}}(-1)^{m_{\ell}}}{m_{\ell}!(2l)^{m_{\ell}}},$
(4.21)

where the prime denotes a sum over non-negative integers $m_{\ell}$ that
satisfy $\sum_{\ell=1}^{N}\ell m_{\ell}=N.$222In the $A$-type case, the grand
canonical ensemble
$\displaystyle\Xi(\kappa)=1+\sum_{N=1}^{\infty}Z(N)\kappa^{N}=\exp\left(-\sum_{\ell=1}^{\infty}\frac{(-\kappa)^{\ell}}{\ell}Z_{\ell}\right)$
has a closed form expression and was used in [10] to further simplify the
result. For the $D_{2N}$-type and all other types we derive in the paper, we
couldn’t find a closed form expression for it, so we don’t use the grand
canonical ensemble calculations in the paper.

##### $B_{2N}$ and $C_{2N}$

Recall that the non-zero roots for $B_{N}$ algebra are

$\displaystyle\mathcal{R}_{B_{N}}^{*}=\\{\pm a_{j}|1\leq j\leq N\\}\cup\\{\pm
a_{i}\pm a_{j}|1\leq i<j\leq N\\},$ (4.22)

and for $C_{N}$ algebra are

$\displaystyle\mathcal{R}_{C_{N}}^{*}=\\{\pm 2a_{j}|1\leq j\leq N\\}\cup\\{\pm
a_{i}\pm a_{j}|1\leq i<j\leq N\\},$ (4.23)

from which we have the density function

$\displaystyle\rho(a,a^{\prime})\equiv\frac{\theta_{1}(\delta
a)^{2}\theta_{1}(a+a^{\prime})\theta_{1}(a-a^{\prime})}{\theta_{4}(\delta
a)^{2}\theta_{4}(a+a^{\prime})\theta_{4}(a-a^{\prime})}$
$\displaystyle=\frac{16}{(\theta_{2}\theta_{3})^{4}}\sum_{m\in\mathbb{Z}}\left(\sum_{n,n^{\prime}\in
2\mathbb{Z}}\frac{c(m)q^{n+1}e^{(n+1+\delta
m)a+n^{\prime}a^{\prime}}}{(1-q^{n+n^{\prime}+1})(1-q^{n-n^{\prime}+1})}+\sum_{n,n^{\prime}\in
2\mathbb{Z}+1}\frac{c(m)q^{n+1}e^{(n+1+\delta
m)a+n^{\prime}a^{\prime}}}{(1-q^{n+n^{\prime}+1})(1-q^{n-n^{\prime}+1})}\right),$
(4.24)

where $\delta=1$ for $B$-type and $\delta=2$ for $C$-type and

$\displaystyle c(m)=\begin{cases}\frac{mq^{m}}{1-q^{2m}},&\text{if }m\neq
0,\\\ -2\mathcal{I}_{A_{1}},&\text{if }m=0.\end{cases}$ (4.25)

The last line in equation (4.1.1) can be derived by using equation (A.8) and
(A.9) in Appendix A. For the $C_{2N}$ case, $\delta m$ in (4.1.1) is an even
number, which can be absorbed by $n$, so the structure of the solution to the
integral (4.18) is similar to the $D_{2N}$ case. We have

$\displaystyle
Z_{C,2\ell}=\frac{(-1)^{\ell}2^{8\ell+1}}{(\theta_{2}\theta_{3})^{8\ell}}\left(\sum_{n_{j},n_{j}^{\prime}\in\mathbb{Z}}\frac{q^{2(n_{1}+\cdots
n_{2\ell})+2\ell}\prod_{j=1}^{\ell}c(n_{2j-1}-n_{2j})c(n_{2j}^{\prime}-n_{2j+1}^{\prime})}{\prod_{j=1}^{2\ell}(1-q^{2n_{j}\pm
2n_{j}^{\prime}+1})}\right),$ (4.26)

with the notation $n^{\prime}_{2\ell+1}=n^{\prime}_{1}$ and

$\displaystyle Z_{C,2\ell+1}=0.$ (4.27)

Finally, the Schur index for $C_{2N}$ group is

$\displaystyle\mathcal{I}_{C_{2N}}=\frac{1}{2^{2N}}\left(\frac{\eta^{3}}{\theta_{4}}\right)^{2N}\sum_{m_{\ell}}^{\prime}\prod_{\ell}\frac{Z_{C,2\ell}^{m_{\ell}}(-1)^{m_{\ell}}}{m_{\ell}!(2l)^{m_{\ell}}},$
(4.28)

where the prime denotes a sum over non-negative integers $m_{\ell}$ that
satisfy $\sum_{\ell=1}^{N}\ell m_{\ell}=N.$

Our method also applies to $B_{2N}$ case. However, since $\delta=1$, the shift
$\delta m$ in (4.1.1) changes the even/odd properties of $n$ in the summation.
So the expression is more complicated. We experimentally test the calculation
by expanding the $q$ series for the first few ranks, we find the agreement
between the $B_{2N}$ and $C_{2N}$ Schur indices.

####  4.1.2 Odd ranks

In this subsection, we compute the Schur indices for $BCD$-groups with odd
ranks. We will start with $D_{2N-1}$ case and then generalize it to $B_{2N-1}$
and $C_{2N-1}$ cases.

##### $D_{2N-1}$

According to Proposition 1, the determinant formula for $D_{2N-1}$ gauge group
is represented with a $N\times N$ matrix where the entries of the matrix are
not completely in the same pattern. To perform the integral with the help of
the Fermi gas approach, we assume that there is an additional integral
$\int_{0}^{2\pi\mathrm{i}}d{a_{2N}}$, which does not change the result. Then
if all the $a_{i}$ in the integral are not equal to $a_{2N}$, we have the
integrals that were studied in the last subsection. If one of the $a_{i}$ in
the integral is $a_{2N}$, let us suppose $a_{\ell}=a_{2N}$, we will encounter
the integral in the form

$\widetilde{Z}_{\ell}=-\int_{0}^{2\pi\mathrm{i}}da_{1}\cdots
d{a_{\ell-1}}\rho(a_{1},a_{2})\cdots\rho(a_{\ell-2},a_{\ell-1}).$ (4.29)

The result of the integral (4.29) is quite similar to the $D_{2N}$ case, we
have

$\displaystyle\widetilde{Z}_{2\ell+2}$
$\displaystyle=-\frac{2^{4\ell+1}}{(\theta_{2}\theta_{3})^{4\ell}}\left(\sum_{\begin{subarray}{c}n_{2\ell}=0\\\
n_{j<2\ell}\in\mathbb{Z}\end{subarray}}\frac{q^{2(n_{1}+\cdots
n_{2\ell-1})+\ell}}{\prod_{j=1}^{\ell}(1-q^{2n_{2j-1}\pm
2n_{2j}+1})(1-q^{2n_{2j}\pm(2n_{2j+1}+1)})}\right)$
$\displaystyle=-2^{\ell+1}\left(\frac{\eta^{3}}{\theta_{4}}\right)^{-4\ell}\left(\sum_{\begin{subarray}{c}n_{2\ell}=0\\\
n_{j<2\ell}\in\mathbb{Z}\end{subarray}}\frac{q^{2(n_{1}+\cdots
n_{2\ell})+\ell}\prod_{j=1}^{\ell}2^{1-\delta_{0,n_{2j}}}}{\prod_{j=1}^{\ell}(1-q^{2n_{2j-1}\pm
2n_{2j}+1})(1-q^{2n_{2j}\pm(2n_{2j+1}+1)})}\right),$ (4.30)

with the notation $n_{2\ell+1}=n_{1}$, and

$\displaystyle\widetilde{Z}_{2\ell+1}=0.$ (4.31)

Then the Schur index of the gauge group $D_{2N-1}$ can be calculated by using
a similar equation (4.21), but replacing one of the $Z_{2\ell}$ with
$\widetilde{Z}_{2\ell}$ in the product. We have

$\displaystyle\mathcal{I}_{D_{2N-1}}=\frac{1}{2^{2N-2}}\left(\frac{\eta^{3}}{\theta_{4}}\right)^{2N-1}\sum_{m_{\ell}}^{\prime}\prod_{\ell}\frac{Z_{2\ell}^{m_{\ell}}(-1)^{m_{\ell}}}{m_{\ell}!(2\ell)^{m_{\ell}}}\sum_{k=1}^{\ell}{km_{k}}\frac{\widetilde{Z}_{2k}}{{Z}_{2k}},$
(4.32)

where the prime denotes a sum over non-negative integers $m_{\ell}$ that
satisfy $\sum_{\ell=1}^{N}\ell m_{\ell}=N.$

##### $B_{2N-1}$ and $C_{2N-1}$

The logic to calculate the $B_{2N-1}$ and $C_{2N-1}$ cases is similar to the
$D_{2N-1}$. We assume that there is an additional integral
$\int_{0}^{2\pi\mathrm{i}}d{a_{2N}}$, which does not change the result. Then
if all the $a_{i}$ in the integral are not equal to $a_{2N}$, we have the
integrals that were studied in Section 4.1.1. If one of the $a_{i}$ in the
integral is $a_{2N}$, let us suppose $a_{\ell}=a_{2N}$, we will encounter the
integral in the form

$\widetilde{Z}_{\ell}=-\int_{0}^{2\pi\mathrm{i}}da_{1}\cdots
d{a_{\ell-1}}\rho(a_{1},a_{2})\cdots\rho(a_{\ell-2},a_{\ell-1})\cdot\frac{\theta_{1}(\delta
a_{1})^{2}}{\theta_{4}(\delta a_{1})^{2}}.$ (4.33)

Using equations (4.1.1) and (A.9), we determine that for the $C_{2N-1}$ case,

$\displaystyle\widetilde{Z}_{C,2}=-4\mathcal{I}_{A_{1}},$ (4.34)

if $\ell>0$,

$\displaystyle\widetilde{Z}_{C,2\ell+2}=\frac{(-1)^{\ell}2^{8\ell+1}}{(\theta_{2}\theta_{3})^{8\ell}}$
$\displaystyle\times\left(\sum_{n_{j},n_{j}^{\prime}\in\mathbb{Z}}\frac{q^{2(n_{1}+\cdots
n_{2\ell})+2\ell}\prod_{j=1}^{\ell}c(n_{2j-1}-n_{2j})\cdot
c(n^{\prime}_{1})c(n^{\prime}_{2\ell})\prod_{j=2}^{\ell-1}c(n_{2j}^{\prime}-n_{2j+1}^{\prime})}{\prod_{j=1}^{2\ell}(1-q^{2n_{j}\pm
2n_{j}^{\prime}+1})}\right),$ (4.35)

and

$\displaystyle\widetilde{Z}_{C,2\ell+1}=0.$ (4.36)

Then the Schur index for $C_{2N-1}$ group is

$\displaystyle\mathcal{I}_{C_{2N-1}}=\frac{1}{2^{2N-1}}\left(\frac{\eta^{3}}{\theta_{4}}\right)^{2N-1}\sum_{m_{\ell}}^{\prime}\prod_{\ell}\frac{Z_{C,2\ell}^{m_{\ell}}(-1)^{m_{\ell}}}{m_{\ell}!(2\ell)^{m_{\ell}}}\sum_{k=1}^{\ell}{km_{k}}\frac{\widetilde{Z}_{C,2k}}{{Z}_{C,2k}},$
(4.37)

where the prime denotes a sum over non-negative integers $m_{\ell}$ that
satisfy $\sum_{\ell=1}^{N}\ell m_{\ell}=N.$

Our method also applies to $B_{2N-1}$ case. However, similar to the $B_{2N}$
case, the expression is more complicated so we don’t present it here. We
experimentally test the calculation by expanding the $q$ series for the first
few ranks, and we find the agreement between the $B_{2N-1}$ and $C_{2N-1}$
Schur indices.

##  5 Fixing the exact modular formulas

We shall try to fix the exact modular formulas for the $BCD$ types of gauge
groups, using the calculations from the two methods in previous sections, as
well as the modular properties in Table 1. In this section we mostly use
$\widetilde{\mathcal{I}}_{G}$ in (2.2), which has better modular formula than
$\mathcal{I}_{G}$. The generators of the modular group $\Gamma(2)$ are
$\theta_{2}(q)^{4},\theta_{3}(q)^{4}$. Due to the quasi-modularity, we also
need to add the second Eisenstein series $E_{2}$. Similar to the $A_{N}$ case,
the maximal weight can be easily read off from the formulas (2.1, 2.2) and
grows linearly in the rank $N$. So the number of unknown coefficients of a
generic quasi-modular ansatz goes like $N^{3}$ for large $N$, though the
scaling factor here is larger since there is no universal symmetry between
$\theta_{2}(q)^{4}$ and $\theta_{3}(q)^{4}$. Also similar to the $A_{N}$ case,
the most significant constraint to fix the ansatz comes from the vanishing
conditions that the $q$-series expansion starts from a very high power,
scaling like $N^{2}$ for large rank $N$.

For the $A_{N}$ case, there is a simple modular anomaly equation [13], which
fixes the dependence of $E_{2}$, so the number of unknown coefficients of the
remaining modular ansatz goes like $N^{2}$, with a scaling factor smaller than
that from the vanishing conditions. This enables the calculations of the Schur
index for arbitrary rank in the $A_{N}$ cases. However, for the $BCD$ types of
gauge groups, we do not find a simple modular anomaly equation, so the
behaviors of the number of unknown coefficients $N^{3}$ is eventually bigger
than the available conditions at a sufficiently large rank $N$.

For the $A_{N}$ cases, the quasi-modular formulas have contributions from each
even weight up to the maximal weight, except for the absence of the constant
term when $N$ is odd. For the $BCD$ types of gauge groups, there is an
interesting new phenomenon that there is no term with weight smaller than one
half of the maximal weight, so there is a non-trivial minimal modular weight.
This constraint would reduce the number of unknown coefficients in the quasi-
modular ansatz by about $\frac{1}{8}$.

We observe an additional pattern that at each modular weight, the $q$-series
expansion does not start from the generic constant term, but also from a high
power at $q^{[\frac{N+1}{4}]}$ for the $SO(N)$ gauge group. If this is true,
it would also provide additional conditions for helping to fix the quasi-
modular ansatz at larger $N$. Furthermore, there is a pattern in the quasi-
modular formulas that $\theta_{2}$ and $\theta_{3}$ are symmetric in the
maximal weight terms, and further for the next maximal weight terms when $N$
is even. These constraints are nevertheless much less significant than the
other conditions mentioned above. Of course, the results of $q$-series
calculations from the previous two sections also provide additional conditions
and redundant checks.

We summarize the conditions at the Table 2 and the formulas in Appendix D. In
the following, we discuss some more details for the $BC$-type and $D$-type
gauge groups separately, and analyze some potential ansatz for modular anomaly
equations.

SO(N) | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14
---|---|---|---|---|---|---|---|---|---|---|---|---
maximal weight | 2 | 4 | 4 | 4 | 6 | 8 | 8 | 8 | 10 | 12 | 12 | 12
minimal weight | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6
leading order | $q$ | $q^{2}$ | $q^{3}$ | $q^{4}$ | $q^{6}$ | $q^{8}$ | $q^{10}$ | $q^{12}$ | $q^{15}$ | $q^{18}$ | $q^{21}$ | $q^{24}$
leading order of each weight | $q$ | $\underline{q^{2}}$ | $q$ | $q$ | $q^{2}$ | $q^{2}$ | $q^{2}$ | $q^{2}$ | $q^{3}$ | $q^{3}$ | $q^{3}$ | $q^{3}$
Table 2: Some properties of the Schur indices for the $SO(N)$ group. There is
a cyclic pattern when $N$ increases by 4. Because $SO(4)$ is not a simple Lie
group, its behavior is an aberration of the usual patterns.

###  5.1 $BC$-type gauge groups

Counting the power of $q$ in the formulas (2.1, 2.2), we find the leading
order in $q$-series expansion

$\widetilde{\mathcal{I}}_{B_{N}}(q)=\mathcal{O}(q^{\frac{N(N+1)}{2}}),$ (5.1)

Using the conditions mentioned above, we fix some exact formulas in the
following with more results in the Appendix D

$\displaystyle\widetilde{\mathcal{I}}_{B_{1}}$
$\displaystyle=\widetilde{\mathcal{I}}_{A_{1}}=\frac{E_{2}}{2}+\frac{1}{24}\Theta_{0,1},$
(5.2) $\displaystyle\widetilde{\mathcal{I}}_{B_{2}}$
$\displaystyle=\frac{E_{2}^{2}}{4}+\frac{E_{2}}{24}\Theta_{0,1}+\frac{1}{48}\big{(}4\Theta_{0,2}-5\Theta_{1,1}+18\theta_{2}(q)^{4}\big{)},$
$\displaystyle\widetilde{\mathcal{I}}_{B_{3}}$
$\displaystyle=\frac{E_{2}^{3}}{12}+\frac{E_{2}^{2}}{48}\Theta_{0,1}+\frac{E_{2}}{4608}\big{(}8\Theta_{0,2}-\Theta_{1,1}+10\theta_{2}(q)^{4}+64\theta_{3}(q)^{4}\big{)},$
$\displaystyle+\frac{1}{82944}\big{(}4\Theta_{0,3}-15\Theta_{1,2}+15\theta_{2}(q)^{8}+96\theta_{3}(q)^{8}-33\theta_{2}(q)^{4}\theta_{3}(q)^{4}\big{)},$

where

$\Theta_{r,s}(q)=\theta_{2}(q)^{4r}\theta_{2}(q)^{4s}+\theta_{2}(q)^{4s}\theta_{3}(q)^{4r}.$
(5.3)

is the generator of $\Gamma^{0}(2)$ modular group. The case of $N=1,2$ can be
obtained by using the formula in the literature [11] to take the limit of the
flavor parameter $b\rightarrow 0$.

Analogous to the $A_{N}$ case [13], we test the various ansatzes for the
modular anomaly equation with the available formulas. After some trials, we
observed a simplest equation with the following form

$\partial_{E_{2}}\widetilde{\mathcal{I}}_{B_{N}}=\sum_{k=1}^{N}c^{N}_{k}\widetilde{\mathcal{I}}_{B_{N-k}},$
(5.4)

where we use the convenient initial condition
$\widetilde{\mathcal{I}}_{B_{0}}=1$. In contrast to the $A_{N}$ case, the
coefficients $c^{N}_{k}$ are no longer constants, but dependent on
$\theta_{2}^{4}$ and $\theta_{3}^{4}$. Additionally, for the case
$c^{N}_{2}(E_{2},\theta_{2},\theta_{3})$ we also need to introduce a $E_{2}$
dependence. This aberration has the compensating effect of simplifying the
higher $c^{N}_{k}$ coefficients. The first few coefficients are

$\displaystyle c^{N}_{1}(\theta_{2},\theta_{3})$ $\displaystyle=\frac{1}{2},$
(5.5) $\displaystyle c^{N}_{2}(\theta_{2},\theta_{3})$
$\displaystyle=\frac{1}{2}\widetilde{\mathcal{I}}_{B_{1}}=\frac{E_{2}}{4}+\frac{1}{48}\Theta_{0,1},$
$\displaystyle c^{N}_{3}(\theta_{2}^{4},\theta_{3}^{4})$
$\displaystyle=c_{1}\theta_{2}^{4}+c_{2}\theta_{3}^{4},$ $\displaystyle
c^{N}_{4}(\theta_{2},\theta_{3})$ $\displaystyle=-\frac{\Theta_{1,1}}{1536}.$

We see that the coefficients $c^{N}_{1},c^{N}_{2},c^{N}_{4}$ are actually
independent of $N$, while $c^{N}_{3}$ is a linear combination of
$\theta_{2}^{4}$ and $\theta_{3}^{4}$ with coefficients dependent on $N$. For
example, some specific anomaly equations are:

$\displaystyle\partial_{E_{2}}\widetilde{\mathcal{I}}_{B_{2}}$
$\displaystyle=\frac{1}{2}\widetilde{\mathcal{I}}_{B_{1}}+\frac{1}{2}\widetilde{\mathcal{I}}_{B_{1}}=\widetilde{\mathcal{I}}_{B_{1}},$
(5.6) $\displaystyle\partial_{E_{2}}\widetilde{\mathcal{I}}_{B_{3}}$
$\displaystyle=\frac{1}{2}\widetilde{\mathcal{I}}_{B_{2}}+\frac{1}{2}\widetilde{\mathcal{I}}_{B_{1}}^{2}+\frac{1}{576}(-\theta_{2}^{4}+8\theta_{3}^{4}),$
$\displaystyle\partial_{E_{2}}\widetilde{\mathcal{I}}_{B_{4}}$
$\displaystyle=\frac{1}{2}\widetilde{\mathcal{I}}_{B_{3}}+\frac{1}{2}\widetilde{\mathcal{I}}_{B_{2}}\widetilde{\mathcal{I}}_{B_{1}}+\frac{1}{576}(23\theta_{2}^{4}-4\theta_{3}^{4})\widetilde{\mathcal{I}}_{B_{1}}-\frac{\Theta_{1,1}}{1536}.$

For $k\geq 4$, the coefficient $c^{N}_{k}$ is a polynomial of $\theta_{2}^{4}$
and $\theta_{3}^{4}$. Of course they are constrained by the maximal and
minimal weights in $\widetilde{\mathcal{I}}_{B_{N}}$. We find that not all
generic terms appear and many terms actually vanish. However, we do not
otherwise identity any particularly simple pattern in these higher
coefficients. Overall, although this is not as nice as the $A_{N}$ case, the
modular anomaly is not completely random and still contains some useful
information.

###  5.2 $D$-type gauge groups

Counting the power of $q$ in the formulas (2.1, 2.2), we find the leading
order in $q$-series expansion

$\displaystyle\widetilde{\mathcal{I}}_{D_{2N}}$
$\displaystyle=\mathcal{O}(q^{2N^{2}}),$ (5.7)
$\displaystyle\widetilde{\mathcal{I}}_{D_{2N+1}}$
$\displaystyle=\mathcal{O}(q^{2N(N+1)}),$

Despite the initial term of the $q$-expansion being an even number, the series
also contains odd numbers. Again, using the conditions mentioned above, we fix
some exact formulas in the following with more results in the Appendix D

$\displaystyle\widetilde{\mathcal{I}}_{D_{1}}$
$\displaystyle=\widetilde{\mathcal{I}}_{B_{1}}=\widetilde{\mathcal{I}}_{A_{1}}=\frac{E_{2}}{2}+\frac{1}{24}\Theta_{0,1},$
(5.8) $\displaystyle\widetilde{\mathcal{I}}_{D_{2}}$
$\displaystyle=\widetilde{\mathcal{I}}_{A_{1}}^{2}=\frac{E_{2}^{2}}{4}+\frac{E_{2}}{24}\Theta_{0,1}+\frac{1}{576}(\Theta_{0,2}+\Theta_{1,1}),$
$\displaystyle\widetilde{\mathcal{I}}_{D_{3}}$
$\displaystyle=\widetilde{\mathcal{I}}_{A_{3}}=\frac{E_{2}^{2}}{8}+\frac{E_{2}}{48}(2+\Theta_{0,1})+\frac{1}{1152}(\Theta_{0,2}-2\Theta_{1,1}+4\Theta_{0,1}),$
$\displaystyle\widetilde{\mathcal{I}}_{D_{4}}$
$\displaystyle=\frac{E_{2}^{4}}{64}+\frac{E_{2}^{3}}{192}\Theta_{0,1}+\frac{E_{2}^{2}}{1536}(\Theta_{0,2}+\Theta_{1,1}+8\Theta_{0,1})$
$\displaystyle+\frac{E_{2}}{27648}(\Theta_{0,3}+3\Theta_{1,2}+24\Theta_{0,2}+6\Theta_{1,1})$
$\displaystyle+\frac{1}{1327104}(\Theta_{0,4}+4\Theta_{1,3}-24\Theta_{2,2}+48\Theta_{0,3}+162\Theta_{2}^{8}).$

Since the properties of the Schur indices for the $D_{N}$ group exhibit
distinctions based on whether $N$ is even or odd, we treat them separately in
our ansatzes for the modular anomaly equation. The simplest equation from our
search is

$\displaystyle\partial_{E_{2}}\widetilde{\mathcal{I}}_{D_{2N}}$
$\displaystyle=\sum_{k=1}^{N}\big{(}d_{k}^{N}\widetilde{\mathcal{I}}_{D_{2(N-k)}}\widetilde{\mathcal{I}}_{D_{1}}+e_{k}^{N}\widetilde{\mathcal{I}}_{D_{2(N-k)}}\big{)},$
(5.9) $\displaystyle\partial_{E_{2}}\widetilde{\mathcal{I}}_{D_{2N+1}}$
$\displaystyle=\sum_{k=1}^{N}\big{(}f_{k}^{N}\widetilde{\mathcal{I}}_{D_{2(N-k)+1}}\widetilde{\mathcal{I}}_{D_{1}}+g_{k}^{N}\widetilde{\mathcal{I}}_{D_{2(N-k)+1}}\big{)},$

where we use the convention $\widetilde{\mathcal{I}}_{D_{0}}=1$. Because there
is a weight difference of 4 between $\widetilde{\mathcal{I}}_{D_{2N}}$ and
$\widetilde{\mathcal{I}}_{D_{2(N-1)}}$, we include the term
$\widetilde{\mathcal{I}}_{D_{N}}\widetilde{\mathcal{I}}_{D_{1}}$. The low
order coefficients $d,e,f,g$ are again somewhat simple

$\displaystyle d_{1}^{N}$ $\displaystyle=f_{1}^{N}=\frac{1}{2},$ (5.10)
$\displaystyle e_{1}^{N}$ $\displaystyle=0,~{}g_{1}^{N}=\frac{1}{24},$
$\displaystyle d_{2}^{N}$ $\displaystyle\sim\Theta_{0,1},$ $\displaystyle
f_{2}^{N}$ $\displaystyle=-\frac{1}{288}+\frac{1}{96}\Theta_{0,1}.$

However, similar to the previous case, there seems no simple pattern in other
higher coefficients as polynomials of $\theta_{2}^{4}$ and $\theta_{3}^{4}$
although many generic terms actually vanish.

##  6 Discussions

We develop and improve some methods for calculating the unflavored Schur index
in $\mathcal{N}=4$ super-Yang-Mills theory. The main results for $BCD$-type
groups are listed in Appendix D and the formula for the $G_{2}$ case is (2.9).

There are some remaining questions for potential future research. It would be
certainly interesting to find more constraints which would enable the complete
calculations of Schur indices for $BCD$-type groups of arbitrary rank,
possibly from some improvements of the modular anomaly equations.

It would be interesting to derive the various empirical features for the
formulas in Appendix D. In particular, the existence of a minimal modular
weight, which is one half of the maximal weight, is an intriguing new feature
which does not appear in the $A_{N}$ case, and seems deserving further study.
Some mathematical techniques e.g. in the review paper [4] may be useful to
provide a proof of the non-trivial identity
$\mathcal{I}_{B_{N}}=\mathcal{I}_{C_{N}}$ expected from S-duality.

The calculation for the $G_{2}$ case is simple due to the smallness of its
rank. Further improvements in computational techniques are needed in order to
fix the exact formulas for $F_{4},E_{6,7,8}$ gauge groups, completing the
picture for exceptional groups.

For the $A_{N}$ case, the Schur index is related to the generalized MacMahon’s
sum-of-divisors functions [38, 39, 40]. However, the congruence subgroups in
[39] do not match those of the Schur indices of $BCD$-type groups. It would be
interesting to search for more generalizations of such MacMahon’s sum-of-
divisors functions that could give the Schur indices in this paper.

Acknowledgments

We thank Sheldon Katz, Albrecht Klemm for stimulating collaborations on
related papers and Jun-Hao Li, Gao-fu Ren, Pei-xuan Zeng for helpful
discussions. XW thanks Tadashi Okazaki and Yongchao Lü for helpful
discussions. BD and MH thank Chiung Hwang, Sung-Soo Kim, Yiwen Pan, Futoshi
Yagi, Wenbin Yan for related enlightening lectures at the PCFT (Peng Huanwu
Center for Fundamental Theory) mini advanced school “Supersymmetric field
theories and related topics” in August 2023. The works of MH was supported in
parts by National Natural Science Foundation of China (Grant No. 12247103). XW
is supported by a KIAS Individual Grant QP079201.

##  Appendix A Elliptic functions and modular forms

In this appendix, we summarize some basic definitions for various elliptic
functions that we have used in the main text.

##### Jacobi theta functions

The Jacobi theta functions $\theta_{i}(z;\tau),i=1,\cdots,4,$ are defined as
follows:

$\displaystyle\theta_{1}(z;\tau)$
$\displaystyle=\mathrm{i}\sum_{n\in\mathbb{Z}}(-1)^{n}q^{(n+\frac{1}{2})^{2}}e^{(n+\frac{1}{2})z},$
(A.1) $\displaystyle\theta_{2}(z;\tau)$
$\displaystyle=\sum_{n\in\mathbb{Z}}q^{(n+\frac{1}{2})^{2}}e^{(n+\frac{1}{2})z},$
(A.2) $\displaystyle\theta_{3}(z;\tau)$
$\displaystyle=\sum_{n\in\mathbb{Z}}q^{n^{2}}e^{nz},$ (A.3)
$\displaystyle\theta_{4}(z;\tau)$
$\displaystyle=\sum_{n\in\mathbb{Z}}(-1)^{n}q^{\frac{1}{2}n^{2}}e^{nz},$ (A.4)

where $q=e^{\pi\mathrm{i}\tau}$. We denote
$\theta_{i}=\theta_{i}(0;\tau),i=2,3,4$ as theta constants.

##### Dedekind eta function

The Dedekind eta function is defined by

$\eta(\tau)=q^{\frac{1}{12}}\prod_{n=1}^{\infty}(1-q^{2n}).$ (A.5)

It is related to the theta constants by

$\eta(\tau)^{3}=\frac{1}{2}\theta_{2}\theta_{3}\theta_{4}.$ (A.6)

If there is no confusion regarding notations, we also use $\eta$ to denote the
Dedekind eta function.

##### Jacobi elliptic functions

The Jacobi elliptic functions are a set of basic elliptic functions. In this
section, we will focus on the elliptic sine function $\mathrm{sn}(z,k)$, and
review some of its properties which can also be found in the textbook [41].
The elliptic sine function can be defined from the Jacobi theta functions as
333Here we don’t use the standard notation, in order to avoid unnecessary
additional definitions.:

$\mathrm{sn}(z\theta_{3}^{2})=\frac{\theta_{3}}{\theta_{2}}\frac{\theta_{1}(z)}{\theta_{4}(z)}=\frac{2\mathrm{i}}{\theta_{2}^{2}}\sum_{n\in\mathbb{Z}}\frac{q^{n+\frac{1}{2}}e^{(n+\frac{1}{2})z}}{1-q^{2n+1}},$
(A.7)

so we can derive the expansion

$\frac{\theta_{1}(z)}{\theta_{4}(z)}=\frac{2\mathrm{i}}{\theta_{2}\theta_{3}}\sum_{n\in\mathbb{Z}}\frac{q^{n+\frac{1}{2}}e^{(n+\frac{1}{2})z}}{1-q^{2n+1}}.$
(A.8)

Similarly, we can derive the expansion

$\displaystyle\frac{\theta_{1}(z)^{2}}{\theta_{4}(z)^{2}}=-\frac{4}{{\theta_{2}^{2}\theta_{3}^{2}}}\left(-2\mathcal{I}_{A_{1}}+\sum_{n=1}^{\infty}\frac{nq^{n}}{(1-q^{2n})}(e^{nz}+e^{-nz})\right),$
(A.9)

where $\mathcal{I}_{A_{1}}$ is the $\mathcal{N}=4$ Schur index for $SU(2)$
gauge group, it has the expression

$\displaystyle\mathcal{I}_{A_{1}}=\mathcal{I}_{SU(2)}=\sum_{n=1}^{\infty}\frac{q^{2n-1}}{(1-q^{2n-1})^{2}}={\frac{1}{2}E_{2}+\frac{1}{24}(\theta_{2}^{4}+\theta_{3}^{4})}.$
(A.10)

##  Appendix B Ring of modular forms under congruence subgroups of
$\mathrm{SL}_{2}(\mathbb{Z})$

We summarize some known facts about modular forms of congruence subgroups.
Some recent references are [42, 43, 44].

##### Modular form of $\mathrm{SL}_{2}(\mathbb{Z})$

A modular form of weight $k$ is a holomorphic function
$f:\mathcal{H}\rightarrow\mathbb{C}$ on the upper-half plane that satisfying:

$\displaystyle f\left(\frac{a\tau+b}{c\tau+d}\right)=(c\tau+d)^{k}f(\tau),$
(B.1)

for all $\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\mathrm{SL}_{2}(\mathbb{Z})$. The modular group
$\mathrm{SL}_{2}(\mathbb{Z})$ is defined as

$\displaystyle\mathrm{SL}_{2}(\mathbb{Z}):=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\,\middle|\,a,b,c,d\in\mathbb{Z},ad-bc=1\right\\}.$ (B.2)

Define the Eisenstein series

$\displaystyle
E_{2k}(\tau)=-\frac{B_{2k}}{(2k)!}\left(1+\frac{4k}{B_{2k}}\sum_{n=1}^{\infty}\frac{n^{2k-1}q^{2n}}{1-q^{2n}}\right),\quad
k>0$ (B.3)

where $q=e^{\mathrm{i}\pi\tau}$ and $B_{2k}$ is Bernoulli number with the
values $B_{2}=\frac{1}{6},B_{4}=-\frac{1}{30},B_{6}=\frac{1}{42},\cdots,$ for
$k=1,2,3,\cdots$. Then the even weight modular forms
$M_{*}(\mathrm{SL}_{2}(\mathbb{Z}),\tau)=\bigoplus\limits_{k=0}^{\infty}M_{2k}(\mathrm{SL}_{2}(\mathbb{Z}),\tau)$
of $\mathrm{SL}_{2}(\mathbb{Z})$ are finitely generated by the fourth and
sixth Eisenstein series

$M_{*}(\mathrm{SL}_{2}(\mathbb{Z}),\tau)=\mathbb{C}[E_{4}(\tau),E_{6}(\tau)].$
(B.4)

The second Eisenstein series $E_{2}(\tau)$ is not a modular form; however, it
admits a non-holomorphic completion

$\widehat{E}_{2}(\tau,\bar{\tau})=E_{2}(\tau)+\frac{1}{4\pi\,\mathrm{Im}\tau},$
(B.5)

which transforms as a weight-two modular form. We call
$\widehat{E}_{2}(\tau,\bar{\tau})$ the almost-holomorphic modular form, and
the holomorphic part $E_{2}(\tau)$ is called the quasi-modular form. The ring
of quasi-modular form for a congruence subgroup is consistent of the ring of
modular form together with $E_{2}(\tau)$.

##### Modular form of congruence subgroups

A modular form of weight $k$, level $n$ is a holomorphic function
$f:\mathcal{H}\rightarrow\mathbb{C}$ satisfying:

$\displaystyle f\left(\frac{a\tau+b}{c\tau+d}\right)=(c\tau+d)^{k}f(\tau),$
(B.6)

for all $\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\Gamma_{0}(n)$. The subgroup
$\Gamma_{0}(n)\in\mathrm{SL}_{2}(\mathbb{Z})$, which are called the Hecke
congruence subgroup of level $n$, is defined as

$\displaystyle\Gamma_{0}(n):=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\mathrm{SL}_{2}(\mathbb{Z}):c\equiv
0\quad(\mathrm{mod}\,\,n)\right\\}.$ (B.7)
$\displaystyle\Gamma^{0}(n):=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\mathrm{SL}_{2}(\mathbb{Z}):b\equiv
0\quad(\mathrm{mod}\,\,n)\right\\}.$ (B.8)

We also encounter the intersection of the congruence subgroups

$\displaystyle\Gamma_{0}(n_{1})\cap\Gamma^{0}(n_{2}):=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\mathrm{SL}_{2}(\mathbb{Z}):c\equiv
0\quad(\mathrm{mod}\,\,n_{1})\,\,\&\,\,b\equiv
0\quad(\mathrm{mod}\,\,n_{2})\right\\}.$ (B.9)

It is clear to see, if $f(\tau)$ is a modular form of
$\Gamma_{0}(n_{1}n_{2})$, then $f(\tau/n_{2})$ is a modular form of
$\Gamma_{0}(n_{1})\cap\Gamma^{0}(n_{2})$. Define

$E_{2}^{(n)}(\tau)=-\frac{24}{n-1}q\partial_{q}\log\frac{\eta(\tau)}{\eta(n\tau)},$
(B.10)

the rings of the even weight modular forms
$M_{*}(\Gamma,\tau)=\bigoplus\limits_{k=0}^{\infty}M_{2k}(\Gamma,\tau)$
provided in SageMath [45] can be expressed as

$\displaystyle
M_{*}(\Gamma_{0}(2),\tau)=\mathbb{C}[E_{2}^{(2)}(\tau),E_{4}(\tau)],$ (B.11)
$\displaystyle
M_{*}(\Gamma_{0}(3),\tau)=\mathbb{C}[E_{2}^{(3)}(\tau),E_{4}(\tau),E_{6}(\tau)],$
(B.12) $\displaystyle
M_{*}(\Gamma_{0}(4),\tau)=\mathbb{C}[E_{2}^{(2)}(\tau),E_{2}^{(4)}(\tau)],$
(B.13) $\displaystyle
M_{*}(\Gamma_{0}(6),\tau)=\mathbb{C}[E_{2}^{(2)}(\tau),E_{2}^{(3)}(\tau),E_{2}^{(6)}(\tau)],$
(B.14) $\displaystyle
M_{*}(\Gamma_{0}(12),\tau)=\mathbb{C}[E_{2}^{(2)}(\tau),E_{2}^{(3)}(\tau),E_{2}^{(4)}(\tau),E_{2}^{(6)}(\tau),E_{2}^{(12)}(\tau)],$
(B.15)

and

$M_{*}(\Gamma_{0}(n)\cap\Gamma^{0}(2),\tau)=M_{*}(\Gamma_{0}(2n),\frac{1}{2}\tau).$
(B.16)

##  Appendix C Proof of Proposition 1

Denote the functions on the left-hand side and on the right-hand side of
equation (4.9) as $f_{\mathrm{L}}(a_{1},\cdots,a_{2N})$ and
$f_{\mathrm{R}}(a_{1},\cdots,a_{2N})$ respectively. It is easy to see that
$f_{\mathrm{L}}(a_{1},\cdots,a_{2N})$ and
$f_{\mathrm{R}}(a_{1},\cdots,a_{2N})$ are elliptic functions with the same
arguments. The denominator of $f_{\mathrm{L}}(a_{1},\cdots,a_{2N})$ is
divisible by the denominator of each term in the determinant expression of
$f_{\mathrm{R}}(a_{1},\cdots,a_{2N})$. So the poles of
$f_{\mathrm{L}}(a_{1},\cdots,a_{2N})$ contain those of
$f_{\mathrm{R}}(a_{1},\cdots,a_{2N})$ including multiplicity. We note that the
converse is not obviously true since the poles in
$f_{\mathrm{R}}(a_{1},\cdots,a_{2N})$ may cancel among different terms in the
determinant. In the followings we also prove that the zero points of
$f_{\mathrm{R}}$ contain those of $f_{\mathrm{L}}$ including multiplicity, so
$f_{\mathrm{R}}/f_{\mathrm{L}}$ would be an analytic function with no pole.
According to Liouville’s theorem on elliptic functions, it would be just a
constant.

It is easy to see that all the zero points of
$f_{\mathrm{L}}(a_{1},\cdots,a_{2N})$ are of degree two and they are located
at the points

$\displaystyle a_{i}=\pm a_{j},\quad j\neq i.$ (C.1)

Now we show that the $f_{\mathrm{R}}$ has the same zero points which are at
least of degree two. Denote the entry of the matrix in the determinant to be

$\displaystyle\rho(a_{i},a_{j})\equiv\frac{\theta_{1}(a_{j}\pm
a_{i})}{\theta_{4}(a_{j}\pm a_{i})},$ (C.2)

then $f_{\mathrm{R}}$ can be expanded as

$\displaystyle
f_{\mathrm{R}}(a_{1},\cdots,a_{2N})=\sum_{i_{1},\cdots,i_{2N}}\varepsilon_{i_{1}\cdots
i_{2N}}\rho(a_{1},a_{i_{1}})\rho(a_{2},a_{i_{2}})\cdots\rho(a_{2N},a_{i_{2N}}),$
(C.3)

where $\varepsilon_{i_{1}\cdots i_{2N}}$ is the Levi-Civita symbol. If
$a_{1}=a_{2}$, then (C.3) is zero due to the antisymmetric summation. If we
exchange $a_{1}$ and $a_{2}$, the determinant is obviously invariant. So the
zero at $a_{1}=a_{2}$ must be of even degree, implying it is at least of
degree two. By performing the reflection action $a_{2}\rightarrow-a_{2}$ and
permutation action on the variables $a_{j}$, we find that all the zero points
in (C.1) are also at least degree two zero points of $f_{\mathrm{R}}$. We can
conclude $f_{\mathrm{R}}/f_{\mathrm{L}}$ is analytic and thus is a constant.
By using the asymptotic behavior of $f_{\mathrm{R}}/f_{\mathrm{L}}$, we find
$f_{\mathrm{R}}/f_{\mathrm{L}}=1$, which finish the proof of equation (4.9).
The proof of equation (4.10) can be done in the same manner that we have done.

In contrast to the $A_{N}$ case, we can not simply change the numerator of
$\rho(a_{i},a_{j})$ to $\theta_{2,3}(a_{i}\pm a_{j})$. If we do this, it would
give additional poles $\theta_{4}(2a_{i})=0$ from the diagonal matrix elements
so that $f_{R}/f_{L}$ may not be analytic.

##  Appendix D Schur Indices for $SO(N)$ Groups up to $N=17$

This appendix presents the Schur indices $\widetilde{I}_{SO(N)}$ for the
$SO(N)$ groups up to $N=17$. This includes the $B$ and $D$ types of gauge
groups. The results for the $C$ type symplectic gauge groups are the same as
those of the $B$ types with the same ranks. Some sporadic results on low ranks
have appeared in the literature, e.g. [11, 20]. To show the symmetry of
$\theta_{2}$ and $\theta_{3}$ near the maximal weight, we also use the
notation
$\Theta_{i,j}\equiv\theta_{2}^{4i}\theta_{3}^{4j}+\theta_{3}^{4i}\theta_{2}^{4j}$.

$\displaystyle\widetilde{\mathcal{I}}_{SO(3)}$
$\displaystyle=\widetilde{\mathcal{I}}_{SU(2)}=\frac{E_{2}}{2}+\frac{1}{24}\Theta_{0,1},$
(D.1) $\displaystyle\widetilde{\mathcal{I}}_{SO(4)}$
$\displaystyle=(\widetilde{\mathcal{I}}_{SU(2)})^{2}=\frac{E_{2}^{2}}{4}+\frac{E_{2}}{24}\Theta_{0,1}+\frac{1}{576}(\Theta_{0,2}+\Theta_{1,1}),$
$\displaystyle\widetilde{\mathcal{I}}_{SO(5)}$
$\displaystyle=\frac{E_{2}^{2}}{4}+\frac{E_{2}}{24}\Theta_{0,1}+\frac{1}{1152}\big{(}2\Theta_{0,2}-5\theta_{2}^{4}\theta_{3}^{4}+9\theta_{2}^{4}\big{)},$
$\displaystyle\widetilde{\mathcal{I}}_{SO(6)}$
$\displaystyle=\widetilde{\mathcal{I}}_{SU(4)}=\frac{E_{2}^{2}}{8}+\frac{E_{2}}{48}(2+\Theta_{0,1})+\frac{1}{1152}(\Theta_{0,2}-2\Theta_{1,1}+4\Theta_{0,1}),$
$\displaystyle\widetilde{\mathcal{I}}_{SO(7)}$
$\displaystyle=\frac{E_{2}^{3}}{12}+\frac{E_{2}^{2}}{48}\Theta_{0,1}+\frac{E_{2}}{4608}\big{(}8\Theta_{0,2}-\Theta_{1,1}+10\theta_{2}^{4}+64\theta_{3}^{4}\big{)}$
$\displaystyle+\frac{1}{82944}\big{(}4\Theta_{0,3}-15\Theta_{1,2}+15\theta_{2}^{8}+96\theta_{3}^{8}-33\theta_{2}^{4}\theta_{3}^{4}\big{)},$
$\displaystyle\widetilde{\mathcal{I}}_{SO(8)}$
$\displaystyle=\frac{E_{2}^{4}}{64}+\frac{E_{2}^{3}}{192}\Theta_{0,1}+\frac{E_{2}^{2}}{1536}(\Theta_{0,2}+\Theta_{1,1}+8\Theta_{0,1})$
$\displaystyle+\frac{E_{2}}{27648}(\Theta_{0,3}+3\Theta_{1,2}+24\Theta_{0,2}+6\Theta_{1,1})$
$\displaystyle+\frac{1}{1327104}(\Theta_{0,4}+4\Theta_{1,3}-24\Theta_{2,2}+48\Theta_{0,3}+162\theta_{2}^{8}),$
$\displaystyle\widetilde{\mathcal{I}}_{SO(9)}$
$\displaystyle=\frac{5E_{2}^{4}}{192}+\frac{5E_{2}^{3}}{576}\Theta_{0,1}+\frac{E_{2}^{2}}{9216}(10\Theta_{0,2}+\Theta_{1,1}+106\theta_{2}^{4}+16\theta_{3}^{4})$
(D.2)
$\displaystyle+\frac{E_{2}}{82944}(5\Theta_{0,3}-12\Theta_{1,2}+159\theta_{2}^{8}+24\theta_{3}^{8}+3\theta_{2}^{4}\theta_{3}^{4})$
$\displaystyle+\frac{1}{15925248}(20\Theta_{0,4}-136\Theta_{1,3}+87\Theta_{2,2}+1272\theta_{2}^{12}$
$\displaystyle+192\theta_{3}^{12}-1116\theta_{2}^{8}\theta_{3}^{4}-1224\theta_{2}^{4}\theta_{3}^{8}+4374\theta_{2}^{8}),$
$\displaystyle\widetilde{\mathcal{I}}_{SO(10)}$
$\displaystyle=\frac{E_{2}^{4}}{128}+\frac{E_{2}^{3}}{384}(2+\Theta_{0,1})+\frac{E_{2}^{2}}{3072}(12\Theta_{0,1}+\Theta_{0,2})+$
$\displaystyle\frac{E_{2}}{276480}(5\Theta_{0,3}-15\Theta_{1,2}+150\Theta_{0,2}-30\Theta_{1,1}+36\theta_{2}^{4}+576\theta_{3}^{4})$
$\displaystyle+\frac{1}{13271040}(5\Theta_{0,4}-40\Theta_{1,3}+36\Theta_{2,2}+280\Theta_{0,3}-600\Theta_{1,2}-504\Theta_{1,1}+954\theta_{2}^{8}+2304\theta_{3}^{8}),$
$\displaystyle\widetilde{\mathcal{I}}_{SO(11)}$
$\displaystyle=\frac{13E_{2}^{5}}{1920}+\frac{13E_{2}^{4}}{4608}\Theta_{0,1}+\frac{E_{2}^{3}}{27648}(13\Theta_{0,2}+4\Theta_{1,1}+6\theta_{2}^{4}+168\theta_{3}^{4})$
$\displaystyle+\frac{E_{2}^{2}}{331776}(13\Theta_{0,3}-15\Theta_{1,2}+18\theta_{2}^{8}+504\theta_{3}^{8}+162\theta_{2}^{4}\theta_{3}^{4})$
$\displaystyle+\frac{E_{2}}{398131200}(650\Theta_{0,4}-2800\Theta_{1,3}-825\theta_{2}^{8}\theta_{3}^{8}+1800\theta_{2}^{12}$
$\displaystyle+50400\theta_{3}^{12}-24750\theta_{2}^{8}\theta_{3}^{4}-12600\theta_{2}^{4}\theta_{3}^{8}+459\theta_{2}^{8}+221184\theta_{3}^{8}+63936\theta_{2}^{4}\theta_{3}^{4})$
$\displaystyle+\frac{1}{4777574400}(130\Theta_{0,5}-1150\Theta_{1,4}+1975\Theta_{2,3}+600\theta_{2}^{16}+16800\theta_{3}^{16}$
$\displaystyle-24150\theta_{2}^{12}\theta_{3}^{4}-28200\theta_{2}^{4}\theta_{3}^{12}-4950\theta_{2}^{8}\theta_{3}^{8}+459\theta_{2}^{12}+221184\theta_{3}^{12}-72981\theta_{2}^{8}\theta_{3}^{4}-46656\theta_{2}^{4}\theta_{3}^{8}),$
$\displaystyle\widetilde{\mathcal{I}}_{SO(12)}$
$\displaystyle=\frac{E_{2}^{6}}{1536}+\frac{E_{2}^{5}}{3072}\Theta_{0,1}+\frac{E_{2}^{4}}{73728}(48\Theta_{0,1}+5\Theta_{0,2}+5\Theta_{1,1})$
$\displaystyle+\frac{E_{2}^{3}}{663552}(5\Theta_{0,3}+15\Theta_{1,2}+144\Theta_{0,2}+90\Theta_{1,1})$
$\displaystyle+\frac{E_{2}^{2}}{53084160}(25\Theta_{0,4}+100\Theta_{1,3}-60\Theta_{2,2}+1440\Theta_{0,3}$
$\displaystyle+2160\Theta_{1,2}+4224\Theta_{1,1}+16842\theta_{2}^{8}+3072\theta_{3}^{8})$
$\displaystyle+\frac{E_{2}}{318504960}(5\Theta_{0,5}+25\Theta_{1,4}-220\Theta_{2,3}+480\Theta_{0,4}+840\Theta_{1,3}$
$\displaystyle-720\Theta_{2,2}+16842\theta_{2}^{12}+3072\theta_{3}^{12}+1242\theta_{2}^{8}\theta_{3}^{4}+6912\theta_{2}^{4}\theta_{3}^{8})$
$\displaystyle+\frac{1}{22932357120}(5\Theta_{0,6}+30\Theta_{1,5}-735\Theta_{2,4}+1184\Theta_{3,3}+720\Theta_{0,5}+1440\Theta_{1,4}-12240\Theta_{2,3}$
$\displaystyle+50526\theta_{2}^{16}+9216\theta_{3}^{16}-17892\theta_{2}^{12}\theta_{3}^{4}+16128\theta_{2}^{4}\theta_{3}^{12}-40770\theta_{2}^{8}\theta_{3}^{8}+174960\theta_{2}^{12}),$
$\displaystyle\widetilde{\mathcal{I}}_{SO(13)}$
$\displaystyle=\frac{19E_{2}^{6}}{11520}+\frac{19E_{2}^{5}}{23040}\Theta_{0,1}+\frac{E_{2}^{4}}{221184}(38\Theta_{0,2}+31\theta_{2}^{4}\theta_{3}^{4}+589\theta_{2}^{4}-32\theta_{3}^{4})$
(D.3)
$\displaystyle+\frac{E_{2}^{3}}{1990656}(38\Theta_{0,3}-21\Theta_{1,2}+1767\theta_{2}^{8}-96\theta_{3}^{8}+735\theta_{2}^{4}\theta_{3}^{4})$
$\displaystyle+\frac{E_{2}^{2}}{796262400}(950\Theta_{0,4}-2950\Theta_{1,3}-1725\theta_{2}^{8}\theta_{3}^{8}+88350\theta_{2}^{12}$
$\displaystyle-4800\theta_{3}^{12}+6750\theta_{2}^{8}\theta_{3}^{4}-25650\theta_{2}^{4}\theta_{3}^{8}+435099\theta_{2}^{8}-24576\theta_{3}^{8}+162096\theta_{2}^{4}\theta_{3}^{4})$
$\displaystyle+\frac{E_{2}}{4777574400}(190\Theta_{0,5}-1300\Theta_{1,4}+1225\Theta_{2,3}+29450\theta_{2}^{16}-1600\theta_{3}^{16}$
$\displaystyle-31600\theta_{2}^{12}\theta_{3}^{4}-32950\theta_{2}^{4}\theta_{3}^{12}-11400\theta_{2}^{8}\theta_{3}^{8}+435099\theta_{2}^{12}-24576\theta_{3}^{12}+45027\theta_{2}^{8}\theta_{3}^{4}-74448\theta_{2}^{4}\theta_{3}^{8})$
$\displaystyle+\frac{1}{687970713600}(380\Theta_{0,6}-4470\Theta_{1,5}+15150\Theta_{2,4}-14675\Theta_{3,3}+88350\theta_{2}^{20}-4800\theta_{3}^{20}$
$\displaystyle-267900\theta_{2}^{16}\theta_{3}^{4}-182850\theta_{2}^{4}\theta_{3}^{16}+161925\theta_{2}^{12}\theta_{3}^{8}+176100\theta_{2}^{8}\theta_{3}^{12}+2610594\theta_{2}^{16}-147456\theta_{3}^{16}$
$\displaystyle-1908477\theta_{2}^{12}\theta_{3}^{4}-1865952\theta_{2}^{4}\theta_{3}^{12}-614142\theta_{2}^{8}\theta_{3}^{8}+8365275\theta_{2}^{12}),$
$\displaystyle\widetilde{\mathcal{I}}_{SO(14)}$
$\displaystyle=\frac{E_{2}^{6}}{3072}+\frac{E_{2}^{5}}{6144}(2+\Theta_{0,1})+\frac{E_{2}^{4}}{147456}(68\Theta_{0,1}+5\Theta_{0,2}+2\Theta_{1,1})+$
$\displaystyle\frac{E_{2}^{3}}{6635520}(25\Theta_{0,3}-15\Theta_{1,2}+870\Theta_{0,2}+330\Theta_{1,1}-3492\theta_{2}^{4}+6768\theta_{3}^{4})$
$\displaystyle+\frac{E_{2}^{2}}{106168320}(25\Theta_{0,4}-80\Theta_{1,3}-24\Theta_{2,2}+1640\Theta_{0,3}$
$\displaystyle-840\Theta_{1,2}+6168\Theta_{1,1}+2874\theta_{2}^{8}+30144\theta_{3}^{8})$
$\displaystyle+\frac{E_{2}}{4459069440}(35\Theta_{0,5}-245\Theta_{1,4}+224\Theta_{2,3}+3710\Theta_{0,4}-10360\Theta_{1,3}-840\Theta_{2,2}+69006\theta_{2}^{12}$
$\displaystyle+116256\theta_{3}^{12}-20664\theta_{2}^{4}\theta_{3}^{8}-67914\theta_{2}^{8}\theta_{3}^{4}-60372\theta_{2}^{8}+442368\theta_{3}^{8}+145152\theta_{2}^{4}\theta_{3}^{4}),$
$\displaystyle+\frac{1}{321052999680}(35\Theta_{0,6}-420\Theta_{1,5}+1407\Theta_{2,4}-568\Theta_{3,3}+5460\Theta_{0,5}-33180\Theta_{1,4}$
$\displaystyle+31920\Theta_{2,3}+255906\theta_{2}^{16}+254016\theta_{3}^{16}-579096\theta_{2}^{12}\theta_{3}^{4}-480816\theta_{2}^{4}\theta_{3}^{12}-16254\theta_{2}^{8}\theta_{3}^{8}$
$\displaystyle+862488\theta_{2}^{12}+2654208\theta_{3}^{12}-1295352\theta_{2}^{8}\theta_{3}^{4}-456192\theta_{2}^{4}\theta_{3}^{8}),$
$\displaystyle\widetilde{\mathcal{I}}_{SO(15)}$
$\displaystyle=\frac{29E_{2}^{7}}{80640}+\frac{29E_{2}^{6}}{138240}\Theta_{0,1}+\frac{E_{2}^{5}}{2211840}(116\Theta_{0,2}+115\theta_{2}^{4}\theta_{3}^{4}-367\theta_{2}^{4}+2144\theta_{3}^{4})$
(D.4)
$\displaystyle+\frac{E_{2}^{4}}{15925248}(116\Theta_{0,3}-3\Theta_{1,2}-1101\theta_{2}^{8}+6432\theta_{3}^{8}+2883\theta_{2}^{4}\theta_{3}^{4})$
$\displaystyle+\frac{E_{2}^{3}}{2388787200}(1450\Theta_{0,4}-2975\Theta_{1,3}-2775\theta_{2}^{8}\theta_{3}^{8}-27525\theta_{2}^{12}$
$\displaystyle+160800\theta_{3}^{12}-20700\theta_{2}^{8}\theta_{3}^{4}+58275\theta_{2}^{4}\theta_{3}^{8}-163089\theta_{2}^{8}+827136\theta_{3}^{8}+318744\theta_{2}^{4}\theta_{3}^{4})$
$\displaystyle+\frac{E_{2}^{2}}{9555148800}(290\Theta_{0,5}-1475\Theta_{1,4}+200\Theta_{2,3}-9175\theta_{2}^{16}+53600\theta_{3}^{16}$
$\displaystyle-39175\theta_{2}^{12}\theta_{3}^{4}-22975\theta_{2}^{4}\theta_{3}^{12}-25575\theta_{2}^{8}\theta_{3}^{8}-163089\theta_{2}^{12}+827136\theta_{3}^{12}$
$\displaystyle-29313\theta_{2}^{8}\theta_{3}^{4}+329112\theta_{2}^{4}\theta_{3}^{8})$
$\displaystyle+\frac{E_{2}}{67421129932800}(56840\Theta_{0,6}-518910\Theta_{1,5}+984900\Theta_{2,4}+442225\theta_{2}^{12}\theta_{3}^{12}$
$\displaystyle-2697450\theta_{2}^{20}+15758400\theta_{3}^{20}-21403200\theta_{2}^{16}\theta_{3}^{4}-30333450\theta_{2}^{4}\theta_{3}^{16}+3156825\theta_{2}^{12}\theta_{3}^{8}$
$\displaystyle-117600\theta_{2}^{8}\theta_{3}^{12}-95896332\theta_{2}^{16}+486355968\theta_{3}^{16}-222500817\theta_{2}^{12}\theta_{3}^{4}$
$\displaystyle-93012192\theta_{2}^{4}\theta_{3}^{12}-141703884\theta_{2}^{8}\theta_{3}^{8}-109024137\theta_{2}^{12}$
$\displaystyle+1719926784\theta_{3}^{12}+236475936\theta_{2}^{8}\theta_{3}^{4}+555393024\theta_{2}^{4}\theta_{3}^{8})$
$\displaystyle+\frac{1}{809053559193600}(8120\Theta_{0,7}-115150\Theta_{1,6}+501270\Theta_{2,5}$
$\displaystyle-542675\Theta_{3,4}-539490\theta_{2}^{24}+3151680\theta_{3}^{24}-6337170\theta_{2}^{20}\theta_{3}^{4}$
$\displaystyle-13005090\theta_{2}^{4}\theta_{3}^{20}+13858425\theta_{2}^{16}\theta_{3}^{8}+13560750\theta_{2}^{8}\theta_{3}^{16}+6508425\theta_{2}^{12}\theta_{3}^{12}$
$\displaystyle-31965444\theta_{2}^{20}+162118656\theta_{3}^{20}-142790949\theta_{2}^{16}\theta_{3}^{4}-224056224\theta_{2}^{4}\theta_{3}^{16}+8038107\theta_{2}^{12}\theta_{3}^{8}$
$\displaystyle-10120068\theta_{2}^{8}\theta_{3}^{12}-109024137\theta_{2}^{16}+1719926784\theta_{3}^{16}-492641433\theta_{2}^{12}\theta_{3}^{4}$
$\displaystyle-304570368\theta_{2}^{4}\theta_{3}^{12}-363706848\theta_{2}^{8}\theta_{3}^{8}),$

$\displaystyle\widetilde{\mathcal{I}}_{SO(16)}$
$\displaystyle=\frac{E_{2}^{8}}{49152}+\frac{E_{2}^{7}}{73728}\Theta_{0,1}+\frac{E_{2}^{6}}{1769472}(72\Theta_{0,1}+7\Theta_{0,2}+7\Theta_{1,1})$
(D.5)
$\displaystyle+\frac{E_{2}^{5}}{10616832}(7\Theta_{0,3}+21\Theta_{1,2}+216\Theta_{0,2}+162\Theta_{1,1})$
$\displaystyle+\frac{E_{2}^{4}}{2548039680}(175\Theta_{0,4}+700\Theta_{1,3}+120\Theta_{2,2}+10800\Theta_{0,3}$
$\displaystyle+21600\Theta_{1,2}+85248\theta_{2}^{4}\theta_{3}^{4}+219582\theta_{2}^{8}-67968\theta_{3}^{8})$
$\displaystyle+\frac{E_{2}^{3}}{7644119040}(35\Theta_{0,5}+175\Theta_{1,4}-460\Theta_{2,3}+3600\Theta_{0,4}+9000\Theta_{1,3}$
$\displaystyle+2160\Theta_{2,2}+219582\theta_{2}^{12}-67968\theta_{3}^{12}+133056\theta_{2}^{4}\theta_{3}^{8}+51246\theta_{2}^{8}\theta_{3}^{4}),$
$\displaystyle+\frac{E_{2}^{2}}{321052999680}(245\Theta_{0,6}+1470\Theta_{1,5}-13335\Theta_{2,4}-1904\Theta_{3,3}+37800\Theta_{0,5}+113400\Theta_{1,4}$
$\displaystyle-257040\Theta_{2,3}+4611222\theta_{2}^{16}-1427328\theta_{3}^{16}+362124\theta_{2}^{12}\theta_{3}^{4}+3798144\theta_{2}^{4}\theta_{3}^{12}+1121526\theta_{2}^{8}\theta_{3}^{8}$
$\displaystyle+24867648\theta_{2}^{12}-3981312\theta_{3}^{12}+10734768\theta_{2}^{8}\theta_{3}^{4}-+5971968\theta_{2}^{4}\theta_{3}^{8}),$
$\displaystyle+\frac{E_{2}}{7705271992320}(35\Theta_{0,7}+245\Theta_{1,6}-4935\Theta_{2,5}+11431\Theta_{3,4}$
$\displaystyle+7560\Theta_{0,6}+26460\Theta_{1,5}-234360\Theta_{2,4}+24192\Theta_{3,3}$
$\displaystyle+1537074\theta_{2}^{20}-475776\theta_{3}^{20}-117306\theta_{2}^{16}\theta_{3}^{4}$
$\displaystyle+1600704\theta_{2}^{4}\theta_{3}^{16}-2311722\theta_{2}^{12}\theta_{3}^{8}-1030302\theta_{2}^{8}\theta_{3}^{12}$
$\displaystyle+24867648\theta_{2}^{16}-3981312\theta_{3}^{16}+138024\theta_{2}^{12}\theta_{3}^{4}+7962624\theta_{2}^{4}\theta_{3}^{12}+2150064\theta_{2}^{8}\theta_{3}^{8})$
$\displaystyle\frac{1}{739706111262720}(35\Theta_{0,8}+280\Theta_{1,7}-10360\Theta_{2,6}+65464\Theta_{3,5}-136546\Theta_{4,4}$
$\displaystyle+10080\Theta_{0,7}+40320\Theta_{1,6}-846720\Theta_{2,5}+1574496\Theta_{3,4}$
$\displaystyle+3074148\theta_{2}^{24}-951552\theta_{3}^{24}-710640\theta_{2}^{20}\theta_{3}^{4}+3870720\theta_{2}^{4}\theta_{3}^{20}-16414272\theta_{2}^{16}\theta_{3}^{8}$
$\displaystyle-12932892\theta_{2}^{8}\theta_{3}^{16}+2805264\theta_{2}^{12}\theta_{3}^{12}+99470592\theta_{2}^{20}-15925248\theta_{3}^{20}-41834880\theta_{2}^{16}\theta_{3}^{4}$
$\displaystyle+39813120\theta_{2}^{4}\theta_{3}^{16}-69999552\theta_{2}^{12}\theta_{3}^{8}-29719872\theta_{2}^{8}\theta_{3}^{12}+322407540\theta_{2}^{16}),$

$\displaystyle\widetilde{\mathcal{I}}_{SO(17)}$
$\displaystyle=\frac{191E_{2}^{8}}{2580480}+\frac{191E_{2}^{7}}{3870720}\Theta_{0,1}+\frac{E_{2}^{6}}{13271040}(191\Theta_{0,2}+211\theta_{2}^{4}\theta_{3}^{4}+4191\theta_{2}^{4}-1128\theta_{3}^{4})$
(D.6)
$\displaystyle+\frac{E_{2}^{5}}{79626240}(191\Theta_{0,3}+60\Theta_{1,2}12573\theta_{2}^{8}-3384\theta_{3}^{8}+5553\theta_{2}^{4}\theta_{3}^{4})$
$\displaystyle+\frac{E_{2}^{4}}{38220595200}(9550\Theta_{0,4}-13100\Theta_{1,3}-14925\theta_{2}^{8}\theta_{3}^{8}+1257300\theta_{2}^{12}$
$\displaystyle-338400\theta_{3}^{12}+653850\theta_{2}^{8}\theta_{3}^{4}-103500\theta_{2}^{4}\theta_{3}^{8}+6783327\theta_{2}^{8}-2043648\theta_{3}^{8}+2553408\theta_{2}^{4}\theta_{3}^{4})$
$\displaystyle+\frac{E_{2}^{3}}{114661785600}(1910\Theta_{0,5}-7550\Theta_{1,4}-1825\Theta_{2,3}+419100\theta_{2}^{16}-112800\theta_{3}^{16}$
$\displaystyle-14250\theta_{2}^{12}\theta_{3}^{4}-239700\theta_{2}^{4}\theta_{3}^{12}-139050\theta_{2}^{8}\theta_{3}^{8}+6783327\theta_{2}^{12}-2043648\theta_{3}^{12}$
$\displaystyle+3698703\theta_{2}^{8}\theta_{3}^{4}-171072\theta_{2}^{4}\theta_{3}^{8})$
$\displaystyle+\frac{E_{2}^{2}}{134842259865600}(+93590\Theta_{0,6}-695310\Theta_{1,5}+841575\Theta_{2,4}+581875\theta_{2}^{12}\theta_{3}^{12}$
$\displaystyle+30803850\theta_{2}^{20}-8290800\theta_{3}^{20}-37595250\theta_{2}^{16}\theta_{3}^{4}-31641750\theta_{2}^{4}\theta_{3}^{16}-11499075\theta_{2}^{12}\theta_{3}^{8}$
$\displaystyle-6637050\theta_{2}^{8}\theta_{3}^{12}+997149069\theta_{2}^{16}-300416256\theta_{3}^{16}+136431729\theta_{2}^{12}\theta_{3}^{4}$
$\displaystyle-393132096\theta_{2}^{4}\theta_{3}^{12}-154834659\theta_{2}^{8}\theta_{3}^{8}+3739608567\theta_{2}^{12}$
$\displaystyle-602505216\theta_{3}^{12}+1539809352\theta_{2}^{8}\theta_{3}^{4}+722608128\theta_{2}^{4}\theta_{3}^{8})$
$\displaystyle+\frac{E_{2}}{809053559193600}(13370\Theta_{0,7}-157780\Theta_{1,6}+512295\Theta_{2,5}$
$\displaystyle-259700\Theta_{3,4}+6160770\theta_{2}^{24}-1658160\theta_{3}^{24}-18724860\theta_{2}^{20}\theta_{3}^{4}$
$\displaystyle-8921430\theta_{2}^{4}\theta_{3}^{20}+12160575\theta_{2}^{16}\theta_{3}^{8}+15038100\theta_{2}^{8}\theta_{3}^{16}+5766075\theta_{2}^{12}\theta_{3}^{12}$
$\displaystyle+332383023\theta_{2}^{20}-100138752\theta_{3}^{20}-282160620\theta_{2}^{16}\theta_{3}^{4}-242867520\theta_{2}^{4}\theta_{3}^{16}-74943540\theta_{2}^{12}\theta_{3}^{8}$
$\displaystyle-53213265\theta_{2}^{8}\theta_{3}^{12}+3739608567\theta_{2}^{16}-602505216\theta_{3}^{16}+382624803\theta_{2}^{12}\theta_{3}^{4}$
$\displaystyle-696066048\theta_{2}^{4}\theta_{3}^{12}-247053240\theta_{2}^{8}\theta_{3}^{8}),$
$\displaystyle+\frac{1}{155338283365171200}(26740\Theta_{0,8}-456400\Theta_{1,7}+2680300\Theta_{2,6}$
$\displaystyle-6175960\Theta_{3,5}+5432875\theta_{2}^{16}\theta_{3}^{16}$
$\displaystyle+16428720\theta_{2}^{28}-4421760\theta_{3}^{28}-90205080\theta_{2}^{24}\theta_{3}^{4}-30140880\theta_{2}^{4}\theta_{3}^{24}+171302040\theta_{2}^{20}\theta_{3}^{8}$
$\displaystyle+155108520\theta_{2}^{8}\theta_{3}^{20}-72691500\theta_{2}^{16}\theta_{3}^{12}-73353000\theta_{2}^{12}\theta_{3}^{16}$
$\displaystyle+1329532092\theta_{2}^{24}-400555008\theta_{3}^{24}-3206708568\theta_{2}^{20}\theta_{3}^{4}$
$\displaystyle-1375411968\theta_{2}^{4}\theta_{3}^{20}+1608045642\theta_{2}^{16}\theta_{3}^{8}+1957516092\theta_{2}^{8}\theta_{3}^{16}+458968104\theta_{2}^{12}\theta_{3}^{12}$
$\displaystyle+29916868536\theta_{2}^{20}-4820041728\theta_{3}^{20}-20952823068\theta_{2}^{16}\theta_{3}^{4}-16917921792\theta_{2}^{4}\theta_{3}^{16}$
$\displaystyle-5785406856\theta_{2}^{12}\theta_{3}^{8}-3716922816\theta_{2}^{8}\theta_{3}^{12}+92227156875\theta_{2}^{16}).$

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