USTC-ICTS/PCFT-22-14

Modular Anomaly Equation

for Schur Index of $\mathcal{N}=4$ Super-Yang-Mills

Min-xin<EMAIL_ADDRESS>

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

We propose a novel modular anomaly equation for the unflavored Schur index in
the $\mathcal{N}=4$ $SU(N)$ super-Yang-Mills theory. The vanishing conditions
overdetermine the modular ambiguity ansatz from the equation, thus together
they are sufficient to recursively compute the exact Schur indices for all
$SU(N)$ gauge groups. Using the representations as MacMahon’s generalized sum-
of-divisors functions and Jacobi forms, we then prove our proposal as well as
elucidate a general formula conjectured by Pan and Peelaers.

###### Contents

  1. 1 Introduction
  2. 2 Modular anomaly equation
  3. 3 The $SU(2N)$ case
  4. 4 Discussions
  5. A Eisenstein series and Jacobi theta functions

##  1 Introduction

As a type of Witten index, the superconformal indices [1] encode the BPS
spectrum of the theory, and have been studied extensively in the literature.
The case of $\mathcal{N}=4$ super-Yang-Mills theory with $SU(N)$ gauge group
is particularly interesting due to the holographic duality with type IIB
string theory on $AdS_{5}\times S^{5}$ background [2]. The superconformal
indices have many important applications. Most notably, they are essential for
the understandings of the microscopic entropy of supersymmetric $AdS_{5}$
black holes [3, 4, 5]. Their various expansions can be interpreted as the
contributions of D-branes, studied e.g. recently in [6, 7, 8, 9, 10].

For theories with a Lagrangian description, the $d$-dimensional superconformal
index can be computed by path integral formalism as the supersymmetric
partition function on $S^{1}\times S^{d-1}$, which localizes to a matrix
integral, see e.g. an early paper on the case of $\mathcal{N}=4$ super-Yang-
Mills [11]. A particular specialization of the 4d superconformal index, known
as the Schur index [12], has some further nice mathematical properties. For
example, in some cases it can be computed from the q-deformed 2d Yang-Mills
[13], or the vacuum character of a corresponding chiral algebra [14]. 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 $SU(2)_{F}\subset SU(4)_{R}$. In this paper we will simply consider
the unflavored index without the extra fugacity. Some remarkable
(quasi)-modular properties of the index are studied recently in [15, 16] in
the context of a larger class of theories, based on some earlier works in e.g.
[17, 18].

On the other hand, topological string theory on Calabi-Yau three-folds has
been an active research area for decades, with many sophisticated available
techniques. The goal of the present paper is to apply one of these techniques
to the calculations of Schur index. The relation between superconformal index
and topological string amplitude has appeared before, in e.g. [19, 20]. In
those cases, one has a 5d supersymmetric field theory from compactifying
M-theory on a Calabi-Yau three-fold, and the 5d Nekrasov partition function on
the Omega background $S^{1}\times\mathbb{R}^{4}_{\epsilon_{1},\epsilon_{2}}$
is simply equivalent to the refined topological string amplitude on the
Calabi-Yau space. The 5d superconformal index at the fixed point of
renormalization group flow can be computed by localization method as the
partition function of the 5d field theory on $S^{1}\times S^{4}$, and is
written as an integral of a product of two complex conjugate refined
topological string amplitudes. This is similar to Pestun’s calculation [21] of
$\mathcal{N}=2$ supersymmetric partition function on $S^{4}$, which localizes
to a matrix integral in terms of 4d Nekrasov partition function. Similar
relations appear also for 5d supersymmetric partition function on $S^{5}$ and
6d superconformal index, which are computed by an integral of a triple product
of refined topological string amplitudes [22].

Our setting is somewhat different from those of [19, 20, 22], as the 4d
superconformal index considered here seems much simpler than the 5d or 6d
cases. We will directly apply topological string method of modular anomaly
equation to the calculations of 4d Schur index, instead of writing it as an
integral of topological string amplitudes. We will encounter the Eisenstein
series and Jacobi Theta functions, where some of the basic properties are
listed in Appendix A. It is well known that the Eisenstein series
$E_{4},E_{6}$ freely generate the modular forms of $SL(2,\mathbb{Z})$. The
second Eisenstein series $E_{2}$ is not exactly modular but transforms with a
shift. The ring of polynomials of $E_{2},E_{4},E_{6}$, known as quasi-modular
forms, is closed under the derivative action $q\frac{d}{dq}$. For a general
introduction see [23]. The quasi-modular forms appear in many studies in
topological string theory, especially in geometries containing elliptic
curves, e.g. in early papers [24, 25, 26, 27]. In some cases there is a
modular anomaly equation containing derivative with respect to the quasi-
modular $E_{2}$, which is related to the holomorphic anomaly equation for
general Calabi-Yau geometries without necessarily elliptic curves [28]. See
e.g. the recent papers [29, 30] for more discussions.

We will propose an analogous modular anomaly equation for Schur index in our
context. During our study we will utilize the interesting connection to the
seemingly remote topic of number theory through the MacMahon’s sum-of-divisors
functions, whose mathematical properties [31, 32] provide a proof of our
proposal as well as elucidate the connections with available results in the
literature.

##  2 Modular anomaly equation

According to the localization method, the unflavored Schur index of the
$\mathcal{N}=4$ $SU(N)$ super-Yang-Mills theory can be written in terms of a
unitary matrix integral. As in the literature [15, 16], it is convenient to
treat the even and odd ranks of the gauge groups separately. We consider first
the simpler $SU(2N+1)$ case. The formula for Schur index is

$\mathcal{I}_{2N+1}(q)=\frac{q^{\frac{N(N+1)}{2}}}{(2N+1)!}\prod_{n=1}^{\infty}(\frac{1-q^{n-\frac{1}{2}}}{1-q^{n}})^{2}\oint\prod_{i=1}^{2N+1}\frac{dz_{i}}{2\pi
iz_{i}}\prod_{i\neq
j}(1-\frac{z_{i}}{z_{j}})\textrm{PE}[i_{V}(q^{\frac{1}{2}})(\sum_{i,j=1}^{2N+1}\frac{z_{i}}{z_{j}})],$
(2.1)

where $i_{V}(q)=\frac{2q}{1+q}$ is the $1/8$ BPS letter index, and PE denotes
the well known plethystic exponential applied to all variables $q,z_{i}$. Here
the factor $\prod_{n=1}^{\infty}(\frac{1-q^{n-\frac{1}{2}}}{1-q^{n}})^{2}$
accounts for the difference between special unitary group and unitary group.
We have also chosen the prefactor $q^{\frac{N(N+1)}{2}}$ in the convention so
that the results would have nice modular properties. For a finite $N$, it is
not difficult to perform the contour integrals which are residues around
$z_{i}\sim 0$ to obtain the $q$-expansion series to a finite order. For
special unitary group, the integration variables would satisfy the product
constraint $\prod_{i=1}^{2N+1}z_{i}=1$, so we only need to do the first $2N$
contour integrals and the last variable $z_{2N+1}$ will automatically drop
out. Although the formula appears to have half integer powers in the
$q$-expansion, the result actually has only integer powers. From the formula
(2.1) it is obvious that the $q$-expansion starts at a high power as

$\mathcal{I}_{2N+1}(q)=\mathcal{O}(q^{\frac{N(N+1)}{2}}).$ (2.2)

The exact calculations of (2.1) were first performed in [17] in terms of
elliptic integrals and there is also an all order $q$-series formula

$\mathcal{I}_{2N+1}(q)=\prod_{m=1}^{\infty}(1-q^{m})^{-3}\sum_{n=0}^{\infty}(-1)^{n}[\binom{2N+1+n}{2N+1}+\binom{2N+n}{2N+1}]q^{\frac{(n+N)(n+N+1)}{2}}.$
(2.3)

The results were organized into nice formulas in terms of quasi-modular forms
in [15, 16]. We can list the formulas in term of Eisenstein series for the
first few orders

$\displaystyle\mathcal{I}_{1}(q)=1,~{}~{}~{}~{}~{}~{}\mathcal{I}_{3}(q)=\frac{E_{2}}{2}+\frac{1}{24},$
(2.4)
$\displaystyle\mathcal{I}_{5}(q)=\frac{E_{2}^{2}}{8}-\frac{E_{4}}{4}+\frac{E_{2}}{16}+\frac{3}{640},$
$\displaystyle\mathcal{I}_{7}(q)=\frac{E_{2}^{3}}{48}-\frac{E_{2}E_{4}}{8}+\frac{E_{6}}{6}+\frac{5E_{2}^{2}}{192}-\frac{5E_{4}}{96}+\frac{37E_{2}}{3840}+\frac{5}{7168}.$

A general formula for all $N$’s is also conjectured by Pan and Peelaers [15]
as

$\mathcal{I}_{2N+1}=\sum_{k=0}^{N}\lambda^{(N)}_{k}\tilde{\mathbb{E}}_{2k},$
(2.5)

where $\lambda^{(N)}_{k}$’s are constants determined by some rather
complicated relations, and we will instead give a simpler recursion relation
as well as an elementary generating function for computing them below.
$\tilde{\mathbb{E}}_{2k}$ is a quasi-modular form of homogeneous weight $2k$
defined by

$\tilde{\mathbb{E}}_{0}=1,~{}~{}~{}~{}\tilde{\mathbb{E}}_{2k}=\sum_{\sum_{j\geq
1}jn_{j}=k}~{}\prod_{p\geq 1}\frac{1}{n_{p}!}(-\frac{E_{2p}}{2p})^{n_{p}}~{}.$
(2.6)

So the weight $2k$ component in the Schur index $\mathcal{I}_{2N+1}$ is
universal, i.e. independent of $N$ up to a constant factor.

Inspired particularly by the studies of the BPS partition functions of
E-strings in [26], we propose the following modular anomaly equation for the
Schur index

$\partial_{E_{2}}\mathcal{I}_{2N+1}=\sum_{k=1}^{N}c_{k}\mathcal{I}_{2N+1-2k},$
(2.7)

where $c_{k}$ are some constants to be determined in a moment. We note that by
string duality, the partition function in [26] is equivalent to genus zero
sector of topological string theory on a local half K3 Calabi-Yau space, and
the modular anomaly equation has been subsequently generalized to higher genus
[33] and to refined theory [34]. The modular anomaly equation in [26] is
recursive in the number of E-strings, which is identified with the rank of
gauge group in another equivalent description in terms of $\mathcal{N}=4$
topological Yang-Mills theories on a half K3 surface [27]. Therefore it is
reasonable that we can also have an equation (2.7) recursive in the rank of
the gauge group. There are certainly some notable differences with the usual
form of modular anomaly equation familiar in topological string theory. First,
the right hand side of our equation (2.7) is purely linear in the lower rank
indices, without the usual quadratic terms. Secondly, as seen from
(LABEL:Schurodd), the Schur index is inhomogeneous, i.e. a combination of
quasi-modular forms of different weights, unlike the usual homogenous forms.

The modular anomaly equation (2.7) determines the Schur index up to an $E_{2}$
independent term, a modular ambiguity which is polynomial of $E_{4},E_{6}$.
Since the index $\mathcal{I}_{2N+1}$ has a maximal weight of $2N$, the number
of unknown coefficients in the ansatz for modular ambiguity can be easily
counted. In general, the dimension of the space of modular forms of weight
$2N$ is no more than $[\frac{N}{6}]+1$. So in our case we can estimate the
number of unknown coefficients
$\sum_{k=0}^{N}([\frac{k}{6}]+1)\sim\frac{N^{2}}{12}$ for large $N$. On the
other hand, for a generic modular ambiguity, the $q$-expansion of the Schur
index starts from the lowest constant $q^{0}$ term. Similar to the case in
[27], the vanishing condition (2.2) imposes very strong constrains,
generically fixing $\frac{N(N+1)}{2}$ unknown coefficients, always
overdetermining the ansatz. Staring from a very simple initial condition
$\mathcal{I}_{1}(q)=1,c_{1}=\frac{1}{2}$, we can recursively efficiently
compute all Schur indices $\mathcal{I}_{2N+1}$ and also determine the
constants $c_{k}$’s in (2.7), which are
$\frac{1}{2},\frac{1}{24},\frac{1}{180},\frac{1}{1120},\frac{1}{6300},\cdots$.
We then observe a general formula for the constants

$c_{k}=\frac{(k-1)!^{2}}{(2k)!}.$ (2.8)

Our anomaly equation (2.7) is compatible with the general formula (2.5). It is
easy to see that
$\partial_{E_{2}}\tilde{\mathbb{E}}_{2k+2}=-\frac{1}{2}\tilde{\mathbb{E}}_{2k}$,
so the weight $2k$ components of each term in (2.7) are always proportional to
$\tilde{\mathbb{E}}_{2k}$. More precisely, comparing the coefficients in (2.7)
and (2.5) we find the relation

$\lambda^{(N)}_{k+1}=-2\sum_{l=1}^{N}c_{l}\lambda^{(N-l)}_{k},~{}~{}~{}k\geq
0.$ (2.9)

There is also another interesting method to compute the Schur index. It is
pointed out in [18] that in this case, the Schur index is simply a MacMahon’s
generalized sum-of-divisors function

$\mathcal{I}_{2N+1}(q)=\sum_{0<m_{1}<\cdots<m_{N}}\frac{q^{m_{1}+\cdots+m_{N}}}{(1-q^{m_{1}})^{2}\cdots(1-q^{m_{N}})^{2}}.$
(2.10)

In [31], a recursion relation for the MacMahon’s function is derived

$\mathcal{I}_{2N+1}(q)=\frac{1}{2N(2N+1)}[(6\mathcal{I}_{3}(q)+N(N-1))\mathcal{I}_{2N-1}(q)-2q\frac{d}{dq}\mathcal{I}_{2N-1}(q)].$
(2.11)

Using the derivative relations of quasi-modular forms (LABEL:Ramanujan), it is
the clear that $\mathcal{I}_{2N+1}$ is a inhomogeneous quasi-modular form of
weight $2N$, and it can be also easily computed recursively. The $q$-series
formula (2.3) was also proved in [31], therefore the equivalence of Schur
index and MacMahon’s function in this case is clear. The structure of formula
(2.5) of Schur index is preserved by the recursion (2.11) due to the following
derivative formula

$q\frac{d}{dq}\tilde{\mathbb{E}}_{2k-2}=k(2k+1)\tilde{\mathbb{E}}_{2k}-3\tilde{\mathbb{E}}_{2}\tilde{\mathbb{E}}_{2k-2},$
(2.12)

which is a generalization of Ramanujan formulas (LABEL:Ramanujan) and can be
certainly checked for any finite $k$. It should be derivable from the
differential equations of the twisted Eisenstein series used in [15]. The
relation (2.11) is then equivalent to a recursion for the coefficients

$\lambda^{(N)}_{k}=\frac{1}{8N(2N+1)}[(2N-1)^{2}\lambda^{(N-1)}_{k}-8k(2k+1)\lambda^{(N-1)}_{k-1}],$
(2.13)

where in the derivation we only need to look at the $E_{2}$ monomial term in
$\tilde{\mathbb{E}}_{2k}$ in (2.6). For $k<0$ or $k>N$ the coefficients are
defined as $\lambda^{(N)}_{k}=0$. From a simple initial condition
$\lambda^{(0)}_{0}=1$ we can then use the recursion (2.13) to compute all
coefficients. For the special cases $k=0$ or $k=N$, simple formulas
$\lambda^{(N)}_{0}=\frac{(2N)!}{2^{4N}(2N+1)N!^{2}}$ and
$\lambda^{(N)}_{N}=(-1)^{N}$ can be easily derived from the recursion. The
recursion (2.13) looks much simpler than those given in [15] but they should
certainly give the same result.

In the paper [32], Rose further considered more general MacMahon’s sum-of-
divisors functions, and provide formulas for the generating functions in terms
of Jacobi forms. A key ingredient in the proofs of the formulas in [31, 32] is
the well known Jacobi triple product identity. For an introduction of Jacobi
forms, see [35, 36]. This turns out to provide a proof of the anomaly equation
(2.7). In our case, the generating function for Schur index can be written in
terms of the Jacobi theta function as

$F(q,x):=\sum_{N=0}^{\infty}(-1)^{N}\mathcal{I}_{2N+1}(q)x^{2N+1}=\frac{i\theta_{1}(q,z)}{\eta(q)^{3}},$
(2.14)

with identification of parameters $x=e^{\pi iz}-e^{-\pi iz}$. It is known that
a Jacobi forms $\phi_{m}$ of index $m$ satisfies a modular anomaly equation

$(\partial_{E_{2}}-m(2\pi z)^{2})\phi_{m}=0.$ (2.15)

This has been applied successfully in topological string theory for making
ansatz, see e.g. [29]. In our context, the generating function is not exactly
a Jacobo form of $SL(2,\mathbb{Z})$, but of a subgroup with index
$\frac{1}{2}$ [32]. The modular anomaly equation can be still applied
similarly

$(\partial_{E_{2}}-\frac{1}{2}(2\pi z)^{2})F(q,x)=0.$ (2.16)

Using the relation $x=e^{\pi iz}-e^{-\pi iz}$ as mentioned below (2.14), we
can solve for the inverse relation

$\pi iz=\textrm{arcsinh}(\frac{x}{2})=\log[\frac{1}{2}(x+\sqrt{4+x^{2}})].$
(2.17)

Denoting $f(x):=(\pi z)^{2}$, it is easy to check that $f(x)$ satisfies a
differential equation $(x^{2}+4)f^{\prime\prime}(x)+xf^{\prime}(x)+2=0$. So we
can straightforwardly prove by induction that it has the following series
expansion

$f(x)=-\log^{2}[\frac{1}{2}(x+\sqrt{4+x^{2}})]=\frac{1}{2}\sum_{n=1}^{\infty}(-1)^{n}\frac{(n-1)!^{2}}{(2n)!}x^{2n}.$
(2.18)

Thus we have derived the modular anomaly equation (2.7) with the formulas
(2.8) for the coefficients.

We can define a generating function
$G(x,y):=\sum_{N=0}^{\infty}\sum_{k=0}^{N}\lambda^{(N)}_{k}x^{2N+1}y^{2k+1}$.
Using the relation (2.9), we have

$G(x,y)+2y^{2}G(x,y)f(ix)=\sum_{N=0}^{\infty}\frac{(2N)!}{2^{4N}(2N+1)N!^{2}}x^{2N+1}y=2yf(ix)^{\frac{1}{2}}.$
(2.19)

So we can also get a solution in terms of elementary functions

$G(x,y)=\frac{2yf(ix)^{\frac{1}{2}}}{1+4y^{2}f(ix)}.$ (2.20)

One can check the recursion (2.13) is satisfied due to the differential
equation

$[4\partial_{x}^{2}-(x\partial_{x})^{2}+4y^{2}\partial_{y}^{2}y^{2}]G(x,y)=0.$
(2.21)

##  3 The $SU(2N)$ case

Next we consider the $SU(2N)$ case, which is a little more complicated but
similar. The Schur index formula in our convention is

$\mathcal{I}_{2N}(q)=\frac{q^{\frac{N^{2}}{2}}}{(2N)!}\oint\prod_{i=1}^{2N}\frac{dz_{i}}{2\pi
iz_{i}}\prod_{i\neq
j}(1-\frac{z_{i}}{z_{j}})\textrm{PE}[i_{V}(q^{\frac{1}{2}})(\sum_{i,j=1}^{2N}\frac{z_{i}}{z_{j}})],$
(3.1)

similar to (2.1) but with a different prefactor. We have omitted the factor
$\prod_{n=1}^{\infty}(\frac{1-q^{n-\frac{1}{2}}}{1-q^{n}})^{2}$ so that the
expression would have better modular property in this case. So strictly
speaking this is a “rescaled” Schur index, but for convenience of notation we
simply work with this better definition in our context. The vanishing
constrains for the index is

$\mathcal{I}_{2N}(q)=\mathcal{O}(q^{\frac{N^{2}}{2}}).$ (3.2)

In this case, the $q$-expansion has half integer powers, so this generically
will impose $N^{2}$ constrains on the ansatz. In this case the $q$-series
expansion in [17] is

$\mathcal{I}_{2N}(q)=\prod_{m=1}^{\infty}\frac{1+q^{\frac{m}{2}}}{1-q^{\frac{m}{2}}}\sum_{n=0}^{\infty}(-1)^{n}[\binom{2N+n}{2N}+\binom{2N+n-1}{2N}]q^{\frac{(n+N)^{2}}{2}}.$
(3.3)

The modular group is now $\Gamma^{0}(2)$, whose modular forms are generated by

$\Theta_{r,s}(q)=\theta_{2}(q)^{4r}\theta_{3}(q)^{4s}+\theta_{2}(q)^{4s}\theta_{3}(q)^{4r},$
(3.4)

which has weight $2(r+s)$. Some low order formulas for the Schur indices are
also available in [15, 16]

$\displaystyle\mathcal{I}_{2}(q)=\frac{E_{2}}{2}+\frac{\Theta_{0,1}}{24},$
(3.5)
$\displaystyle\mathcal{I}_{4}(q)=\frac{E_{2}^{2}}{8}+\frac{E_{2}\Theta_{0,1}}{48}+\frac{\Theta_{0,2}}{1152}-\frac{\Theta_{1,1}}{576}+\frac{E_{2}}{24}+\frac{\Theta_{0,1}}{288}.$

Similarly, in this case we propose the modular anomaly equation

$\partial_{E_{2}}\mathcal{I}_{2N}=\sum_{k=1}^{N}c_{k}\mathcal{I}_{2N-2k},$
(3.6)

with a convention for initial index $\mathcal{I}_{0}=1$. The number of unknown
coefficients in the modular ambiguity in $\mathcal{I}_{2N}$ is counted by
$\Theta_{r,s}(q)$’s with $r+s\leq N,r\leq s$, and goes like $\frac{N^{2}}{4}$
for large $N$, much smaller than the number of constrains $N^{2}$. It also
turns out that there is no weight zero constant term in the modular ambiguity,
as can be seen from the examples in (LABEL:Schureven). So starting also from
the simple initial condition $\mathcal{I}_{0}=1,c_{1}=\frac{1}{2}$, we can
compute all Schur indices and fix the constants $c_{k}$’s which turn out to be
the same as in the $SU(2N+1)$ case (2.8). Of course we can also include the
constant term in the ansatz for modular ambiguity, then we simply require the
extra initial conditions for $\mathcal{I}_{2},c_{2}$ to start the recursive
algorithm.

The Schur index can be represented by another MacMahon’s generalized sum-of-
divisors function appeared in [31] as

$\mathcal{I}_{2N}(q)=\sum_{0<m_{1}<\cdots<m_{N}}\frac{q^{m_{1}+\cdots+m_{N}-\frac{N}{2}}}{(1-q^{m_{1}-\frac{1}{2}})^{2}\cdots(1-q^{m_{N}-\frac{1}{2}})^{2}},$
(3.7)

where the same series expansion (3.3) was also derived. Furthermore a
recursion relation is proved in [31]

$\mathcal{I}_{2N}=\frac{1}{2N(2N-1)}[(2\mathcal{I}_{2}+(N-1)^{2})\mathcal{I}_{2N-2}-2q\frac{d}{dq}\mathcal{I}_{2N-2}].$
(3.8)

From the recursion and the initial formula $\mathcal{I}_{2}$, it is clear that
the Schur index $\mathcal{I}_{2N}$ is an inhomogeneous $\Gamma^{0}(2)$ quasi-
modular form of weight $2N$. A general formula is also conjectured in [15] in
this case. In our convention it is

$\displaystyle\mathcal{I}_{2N}=\sum_{k=0}^{N}\frac{\tilde{\lambda}^{(N)}_{k}}{k!(2k-1)!!}\theta_{4}^{-1}(q\frac{d}{dq})^{k}\theta_{4}=\sum_{k=0}^{N}\frac{\tilde{\lambda}^{(N)}_{k}}{k!}(-\frac{E_{2}}{2})^{k}+\cdots,$
(3.9)

where analogous to the $SU(2N+1)$ case in the previous section, we denote the
coefficients of $E_{2}$ monomial terms with the same prefactor. It is easy to
check that the recursion (3.8) preserves the structure the general formula
(3.9) using derivative formula $q\frac{d}{dq}\log\theta_{4}=-\mathcal{I}_{2}$.
Furthermore it provides a recursion for the coefficients

$\tilde{\lambda}^{(N)}_{k}=\frac{1}{2N(2N-1)}[(N-1)^{2}\tilde{\lambda}^{(N-1)}_{k}-2k(2k-1)\tilde{\lambda}^{(N-1)}_{k-1}].$
(3.10)

From the initial condition $\tilde{\lambda}^{(0)}_{0}=1$ we can compute all
coefficients (again $\tilde{\lambda}^{(N)}_{k}=0$ for $k<0$ or $k>N$). In this
case it is easy to see $\tilde{\lambda}^{(N)}_{0}=0$ for all $N\geq 1$, which
explain the absence of constant term in the Schur index observed earlier. The
other special formula $\tilde{\lambda}^{(N)}_{N}=(-1)^{N}$ is the same as in
the previous section. Finally from the anomaly equation (3.6) we also have the
relation

$\tilde{\lambda}^{(N)}_{k+1}=-2\sum_{l=1}^{N}c_{l}\tilde{\lambda}^{(N-l)}_{k}.$
(3.11)

Similarly the Jacobi form formula in this case [32] provides a proof of the
anomaly equation (3.6), and we skip the details here.

Similarly we can define the generating function
$\tilde{G}(x,y):=\sum_{N=0}^{\infty}\sum_{k=0}^{N}\tilde{\lambda}^{(N)}_{k}x^{2N}y^{2k}$.
The anomaly equation (3.11) then provides a solution

$\tilde{G}(x,y)=\frac{1}{1+4y^{2}f(ix)},$ (3.12)

where $f(x)$ is the same function (2.18) appeared before. The recursion (3.10)
is satisfied since $\tilde{G}(x,y)$ is annihilated by the same differential
operator as in (2.21).

##  4 Discussions

Although the results for Schur index in the current study have been available
in the literature, we find our method of using the anomaly equation (2.7) and
the vanishing conditions (2.2) provides so far the simplest approach with
minimal assumptions. The vanishing conditions are in fact highly redundant,
providing consistency checks by themselves and automatically giving the
coefficients (2.8) in the anomaly equation. Furthermore, using the anomaly
equation we are able to solve the generating functions (2.20, 3.12) for the
coefficients in the general formulas (2.5, 3.9) conjectured in [15].

The non-trivial existence of such over-constrained systems likely suggests a
natural underlying geometric explanation, as mentioned in [31, 32]. It is
would also be interesting to check whether the more general MacMahon’s sum-of-
divisors functions studied in [32] have connections with Schur indices of some
other superconformal field theories.

Our anomaly equation (2.7) seems universally simple that it should have a
wider applicability. It would be interesting to apply our proposal to more
general superconformal indices, including more flavor fugacities. A better
understanding of the modular property would help the analysis of the
asymptotic behavior of the index, which is essential for accounting for the
black hole entropy in holographic duality.

Acknowledgments

We thank Bao-ning Du, Masazumi Honda, Jun-Hao Li, Yiwen Pan, Gao-fu Ren, Pei-
xuan Zeng for helpful discussions and correspondences, Sheldon Katz, Albrecht
Klemm, Kimyeong Lee, Yuji Sugimoto, Xin Wang for collaborations on related
topics. This work was supported in parts by the national Natural Science
Foundation of China (Grants No.11947301 and No.12047502).

##  Appendix A Eisenstein series and Jacobi theta functions

We use the following convention for the weight $2k$ Eisenstein series

$E_{2k}=-\frac{B_{2k}}{(2k)!}+\frac{2}{(2k-1)!}\sum_{n=1}^{\infty}\frac{n^{2k-1}q^{n}}{1-q^{n}}.$
(A.1)

The well known derivative formulas are due to Ramanujan

$\displaystyle
q\frac{d}{dq}E_{2}=-E_{2}^{2}+5E_{4},~{}~{}~{}~{}q\frac{d}{dq}E_{4}=-4E_{2}E_{4}+14E_{6},$
(A.2) $\displaystyle q\frac{d}{dq}E_{6}=-6E_{2}E_{6}+\frac{60E_{4}^{2}}{7}.$

In the modular anomaly equation we need to take derivative with respect to
$E_{2}$. Sometimes a commutation relation between the derivative actions is
potentially helpful, see e.g. [30]. For a homogeneous quasi-modular form
$G_{k}$ of weight $k$, in the current convention for $E_{2}$ we have

$\partial_{E_{2}}q\frac{d}{dq}G_{k}=(q\frac{d}{dq}\partial_{E_{2}}-k)G_{k}.$
(A.3)

The Jacobi theta function is defined by

$\theta\left[a\atop b\right](\tau,z)=\sum_{n\in\mathbb{Z}}e^{\pi
i(n+a)^{2}\tau+2\pi iz(n+a)+2\pi inb}\ ,$ (A.4)

with $q=e^{2\pi i\tau}$ and the usual auxiliary theta functions are
$\theta_{1}=-i\theta\left[\frac{1}{2}\atop\frac{1}{2}\right]$,
$\theta_{2}=\theta\left[\frac{1}{2}\atop 0\right]$,
$\theta_{3}=\theta\left[0\atop 0\right]$ and
$\theta_{4}=\theta\left[0\atop\frac{1}{2}\right]$. Often we set the elliptic
parameter $z=0$ and denote

$\theta_{2}(q)=\sum_{n=-\infty}^{\infty}q^{\frac{1}{2}(n+\frac{1}{2})^{2}},~{}~{}~{}\theta_{3}(q)=\sum_{n=-\infty}^{\infty}q^{\frac{n^{2}}{2}},~{}~{}~{}\theta_{4}(q)=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{\frac{n^{2}}{2}}.$
(A.5)

Then there is a relation $\theta_{3}^{4}=\theta_{2}^{4}+\theta_{4}^{4}$, and
the derivative formulas

$\displaystyle
q\frac{d}{dq}\log\theta_{2}=-\frac{1}{2}E_{2}+\frac{1}{24}(\theta_{3}^{4}+\theta_{4}^{4}),~{}~{}~{}q\frac{d}{dq}\log\theta_{3}=-\frac{1}{2}E_{2}+\frac{1}{24}(\theta_{2}^{4}-\theta_{4}^{4}),$
(A.6) $\displaystyle
q\frac{d}{dq}\log\theta_{4}=-\frac{1}{2}E_{2}-\frac{1}{24}(\theta_{2}^{4}+\theta_{3}^{4}).$

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