††institutetext: Enrico Fermi Institute and Department of Physics University
of Chicago
933 E 56th St., Chicago IL 60637

# Hecke Relations in Rational Conformal Field Theory

Jeffrey A. Harvey  and Yuxiao Wu<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We define Hecke operators on vector-valued modular forms of the type that
appear as characters of rational conformal field theories (RCFTs). These
operators extend the previously studied Galois symmetry of the modular
representation and fusion algebra of RCFTs to a relation between RCFT
characters. We apply our results to derive a number of relations between
characters of known RCFTs with different central charges and also explore the
relation between Hecke operators and RCFT characters as solutions to modular
linear differential equations. We show that Hecke operators can be used to
construct an infinite set of possible characters for RCFTs with two
independent characters and increasing central charge. These characters have
multiplicity one for the vacuum representation, positive integer coefficients
in their $q$ expansions, and are associated to a two-dimensional
representation of the modular group which leads to non-negative integer fusion
coefficients as determined by the Verlinde formula.

##  1 Introduction

Two-dimensional rational conformal field theory (RCFT) plays a central role in
many different areas of physics and mathematics ranging from critical
phenomenon, boundary modes in topological insulators and the world-sheet
description of classical string theory backgrounds in physics to the
modularity theorem and the study of sphere packing in mathematics, to name two
relatively recent examples. The characters of RCFTs are modular forms of a
type that will be described later. Hecke operators and their eigenfunctions
play a crucial role in the mathematical theory of modular forms, thus it is
natural to expect that they should play a role in RCFT as well. Hecke
operators have in fact appeared in physics, an important example is the work
Dijkgraaf:1996xw  where they were used in the computation of the elliptic
genus of the symmetric product of Kähler manifolds. Here we find a new
application: they can be used to relate characters of RCFTs with different
central charges and to construct infinite classes of modular functions which
have all the properties required of characters of consistent RCFTs.

There is a fundamental difference between the modular forms appearing in RCFT
and those appearing in many areas of mathematics. To explain this first recall
that $SL(2,{\mathbb{Z}})$ acts on $\tau$ in the upper half plane
${\mathbb{H}}$ by

$\tau\rightarrow\gamma\tau=\frac{a\tau+b}{c\tau+d}$ (1)

with

$\gamma=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in SL(2,{\mathbb{Z}})\,.$ (2)

A modular form of weight $k$ for the modular group $\Gamma=SL(2,{\mathbb{Z}})$
is a function $f:{\mathbb{H}}\rightarrow{\mathbb{C}}$ which is holomorphic
(except for a possible pole as $\tau\rightarrow i\infty$) in ${\mathbb{H}}$
and which obeys the transformation law

$f(\gamma\tau)=(c\tau+d)^{k}f(\tau)\,.$ (3)

Invariance under $\tau\rightarrow\tau+1$ implies that modular forms have an
expansion in $q=e^{2\pi i\tau}$ of the form

$f(\tau)=\sum_{n}a(n)q^{n}\,.$ (4)

Modular forms can be distinguished by their behavior as $q\rightarrow 0$ or
$\tau\rightarrow i\infty$ (the cusp at infinity). If $f$ is holomorphic away
from the cusp at infinity but diverges at the cusp with a finite number of
$a(n)\neq 0$ for $n<0$ then $f$ is said to be weakly holomorphic. If $f$ is
holomorphic everywhere including at the cusp, implying that $a(n)=0$ for $n<0$
then $f$ is usually just called a modular form. If $f$ vanishes at the cusp at
infinity so that $a(n)=0$ for $n\leq 0$ then $f$ is called a cusp form. We
will follow standard usage by referring to functions obeying eqn.(3) with
weight $k=0$ as modular functions rather than modular forms.

A famous example of a weight $12$ cusp form appears as the inverse of the
oscillator partition function of bosonic string theory,

$\Delta(\tau)=(\eta(\tau))^{24}=q\prod_{n=1}^{\infty}(1-q^{n})^{24}\,.$ (5)

Hecke operators will be described in more detail later, for the moment we
define the operator $T_{m}$ for $m\in{\mathbb{Z}}$ through its action on the
coefficients of the $q$ expansion of a modular form. If $f=\sum_{n}a(n)q^{n}$
is a weight $k$ modular form then we define 111Later we will find it
convenient to choose a different normalization for $T_{m}$ than that given
here.

$(T_{m}f)(\tau)=\sum_{n}a^{(m)}(n)q^{n}$ (6)

with

$a^{(m)}(n)=\sum_{s>0,s|\,\text{gcd}(m,n)}s^{k-1}a(mn/s^{2})\,.$ (7)

It can be shown that Hecke operators map cusp forms to cusp forms of the same
weight, and since the weight $12$ cusp form is the unique form of this weight,
it must be an eigenfunction of all the $T_{m}$. In fact one has

$T_{m}\Delta=c(m)\Delta$ (8)

where $c(m)$ is the coefficient of $q^{m}$ in the $q$ expansion of $\Delta$.
It is also clear that weakly holomorphic modular forms and functions cannot be
eigenfunctions of Hecke operators since if $n<0$ is the power of $q$ in the
most singular term in a weakly holomorphic modular form or function $f$ then
$T_{m}f$ will have most singular term $q^{mn}$. This applies in particular to
characters of conformal field theories. In this case the individual characters
are modular functions for the principal congruence subgroup $\Gamma(N)$ for a
finite $N$ defined below, and the components transform under a finite-
dimensional representation $\rho$ of the full modular group
$SL(2,{\mathbb{Z}})$ with a kernel containing $\Gamma(N)$. At least one of the
characters is always weakly holomorphic with leading term $q^{-c_{\rm
eff}/24}$ with $c_{\rm eff}>0$ and although the precise form of the Hecke
operators is different, the behavior is similar in that the resulting image
under the action of the Hecke operator has more singular behavior as
$q\rightarrow 0$. Therefore characters of rational conformal field theories
cannot be eigenfunctions of Hecke operators. However this does not rule out
the possibility that Hecke operators might relate the characters of RCFTs with
different values of $c_{\rm eff}$ and in fact we will find that suitably
defined Hecke operators do in fact relate characters of many RCFTs. This work
can be seen as a small step towards extending some of the mathematical
machinery used to study cusp forms to the world of weakly holomorphic weight
zero modular functions and the physics of two-dimensional conformal field
theory.

Our first main result consists of a purely mathematical statement about a
particular class of Hecke operators that act on vector-valued weakly
holomorphic modular functions of a particular type. We define a level $N$
vector-valued modular function $f$ with respect to a representation $\rho$
222This definition can obviously be extended to non-zero weight, but for
simplicity we restrict to weight zero since this is the case arising in RCFT.
to be a set $f_{i}(\tau)$, $i=1,\cdots n$ of weakly holomorphic modular
functions for $\Gamma(N)$ which transform under $SL(2,{\mathbb{Z}})$ as

$f_{i}(\gamma\tau)=\sum_{j}\rho(\gamma)_{ij}f_{j}(\tau)$ (9)

where $\rho$ is a $n$-dimensional unitary representation

$\rho:SL(2,{\mathbb{Z}})\rightarrow GL(V(\rho))$ (10)

with $V$ a complex vector space of finite dimension $n$ and where $N$ is the
finite order of $\rho(T)$ and $\Gamma(N)$ is in the kernel of $\rho$. Here

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

are the generators of $SL(2,{\mathbb{Z}})$. All characters of RCFTs are level
$N$ modular functions for some finite-dimensional representation $\rho$ by a
theorem of Bantay Bantay:2001ni  but the converse is not true.

If $f$ is a level $N$ modular function with respect to the representation
$\rho$, then for each $p$ such that $\text{gcd}(p,N)=1$ there are Hecke
operators ${\mathsf{T}}_{p}$ such that $({\mathsf{T}}_{p}f)_{i}$ are the
components of a level $N$ modular function with respect to the representation
$\rho^{(p)}$ defined by its values on the generators of $SL(2,{\mathbb{Z}})$,

$\rho^{(p)}(T)=\rho(T^{\bar{p}}),\qquad\rho^{(p)}(S)=\rho(\sigma_{p}S)\,,$
(12)

where $\bar{p}$ is the multiplicative inverse of $p$ mod $N$ and $\sigma_{p}$
is the preimage of

$\begin{pmatrix}\bar{p}&0\\\ 0&p\end{pmatrix}$ (13)

under the mod $N$ map from $SL(2,{\mathbb{Z}})$ to
$SL(2,{\mathbb{Z}}/N{\mathbb{Z}})$. The mod $N$ map takes an element of
$SL(2,{\mathbb{Z}})$ and reduces every element of the matrix mod $N$ to
produce an element of $SL(2,{\mathbb{Z}}/N{\mathbb{Z}})$. Note that
$\Gamma(N)$ is the kernel of the mod $N$ map and hence a normal subgroup of
$SL(2,{\mathbb{Z}})$. Note that $\sigma_{p}$ is not unique, but since two
choices differ by the action of $\Gamma(N)$ which is in the kernel of $\rho$,
the representation $\rho^{(p)}$ is uniquely determined by any choice of
$\sigma_{p}$. The Hecke operators ${\mathsf{T}}_{p}$ are defined in section 4
where a number of relations they obey are also discussed. Note that the order
of $\rho(T^{\bar{p}})$ is the same as the order of $\rho(T)$. Thus neither the
value of $N$ nor the dimension of the representation changes under the action
of ${\mathsf{T}}_{p}$.

Our second main result is that there are many sets of RCFT characters that are
related by the action of the Hecke operators ${\mathsf{T}}_{p}$. A number of
examples are discussed in Section 5. Our third result is that the Hecke
operators $\mathsf{T}_{p}$ for a certain infinite set of values of $p$ can be
used to construct infinite sets of level $N$ modular functions associated to a
finite set of representations $\rho^{(p)}$ that satisfy all the known
constraints for characters of a RCFT. In particular they have degeneracy one
for the vacuum representation, have all non-negative integer coefficients in
their $q$ expansions, and the fusion coefficients determined from the Verlinde
formula are all non-negative. The simplest example is an infinite set with two
independent characters which arise as Hecke images of the Yang-Lee model.

The outline of this paper is as follow. In section 2 we review the general
structure of RCFT and discuss two results of particular relevance to us,
namely the appearance of a Galois symmetry which relates representations of
$SL(2,{\mathbb{Z}})$ appearing in RCFTs and the fact that characters of a RCFT
are solutions to Modular Linear Differential Equations (MLDE) and we discuss
the role played by the number of zeros of the Wronskian in the classification
of MLDE. In Section 3 we review the most basic description of Hecke operators
for (scalar) modular forms for the full modular group
$\Gamma=SL(2,{\mathbb{Z}})$. We then discuss the extension to Hecke operators
for the principal congruence subgroup $\Gamma(N)$ and discuss the difficulties
in extending these results to Hecke operators for vector-valued modular forms.
Section 4 contains our first main result, showing that there is a specific
extension of Hecke operators to level $N$ modular forms with respect to a
representation $\rho$. We explain the relation of these Hecke operators to the
previously observed Galois symmetry and also discuss the interplay between
these Hecke operators and MLDE. We then turn in Section 5 to a number of
examples of RCFT that have characters related by Hecke operators, focusing for
simplicity on RCFT with two or three independent characters. We find a number
of relations between minimal models, WZW models and other RCFT that appear in
the literature. Finally in Section 6 we conclude and offer some thoughts on
future avenues of research that are suggested by our results.

##  2 RCFT

We first provide a brief summary of RCFT and set up our notation. For more in
depth reviews see Moore:1989vd ; Fuchs:2009iz ; Huang:2013jza .

In a two-dimensional rational conformal field theory there are chiral algebras
${\cal A}$, $\overline{\cal A}$ such that the Hilbert space decomposes into a
finite number of irreducible representations $V_{i},\overline{V_{\bar{i}}}$ of
the chiral algebras and has the form

${\cal H}=\bigoplus_{i\in{\cal I},\bar{i}\in\bar{\cal I}}{\cal
N}_{i,\bar{i}}V_{i}\otimes\overline{V_{\bar{i}}}$ (14)

with ${\cal N}_{i\bar{i}}\in{\mathbb{Z}}_{\geq 0}$. The partition function is
expressed in terms of characters

$\chi_{i}(\tau)={\rm Tr}_{V_{i}}q^{L_{0}-c/24},\qquad q=e^{2\pi i\tau}\,$ (15)

and their anti-holomorphic counterparts as

$Z=\sum_{i\in{\cal I},\bar{i}\in\bar{\cal I}}{\cal
N}_{i,\bar{i}}\chi_{i}(\tau)\overline{\chi_{\bar{i}}(\tau)}\,.$ (16)

Here we have introduced finite index sets ${\cal I}$, $\bar{\cal I}$ that we
use to label the irreducible representations of ${\cal A}$ and $\overline{\cal
A}$ as well as the associated characters. The central charge $c$ is positive
in a unitary RCFT and for unitary RCFT the character with most singular
behavior as $q\rightarrow 0$ is the vacuum character with leading behavior
$\chi_{0}\sim q^{-c/24}$. In non-unitary RCFT $c$ can be negative as can the
conformal weights $h_{i}$ of primary operators. In this case we define $c_{\rm
eff}$ so that the leading singular behavior of the most singular character is
$q^{-c_{\rm eff}/24}$.

We will often use Virasoro minimal models as examples of RCFTs. These are
determined by a pair of coprime integers $(p,q)$ with $p>1,q>1$ and have
central charge

$c(p,q)=1-6\frac{(p-q)^{2}}{pq}\,.$ (17)

These models are unitary provided that $|p-q|=1$. Both unitary and non-unitary
minimal models will enter into our considerations. In the following we denote
the minimal model labelled by $(p,q)$ by $\mathsf{M}(p,q)$.

###  2.1 Modular Properties of Characters

The modular properties of the characters $\chi_{i}(\tau)$ will play a central
role in what follows. Under modular transformations by $\gamma\in
SL(2,{\mathbb{Z}})$ they transform as

$\chi_{i}(\gamma\tau)=\sum_{j}\rho(\gamma)_{ij}\chi_{j}(\tau)\,.$ (18)

Here

$\rho:SL(2,{\mathbb{Z}})\rightarrow GL(V)$ (19)

is a unitary, finite-dimensional representation of the modular group with $V$
a complex vector space and $\rho(\gamma)_{ab}$ are the matrix elements of
$\rho$ with respect to a basis of $V$. The partition function is required to
be modular invariant, thus ${\cal N}$ is an intertwiner between $\rho$ and
$\bar{\rho}$. The representation $\rho$ is determined by its value on the
generators $S:\tau\rightarrow-1/\tau$ and $T:\tau\rightarrow\tau+1$ of
$SL(2,{\mathbb{Z}})$ and must satisfy

$(\rho(S))^{2}=(\rho(S)\rho(T))^{3}=C$ (20)

where $C$ is the conjugation matrix and is an involution, $C^{2}=1$.

In a RCFT the central charge $c$ and conformal weights $h_{i}$ of primary
operators are rational Anderson:1987ge  and as a result $\rho(T)$ will have
finite order which we denote by $N$. One can choose a basis in which $\rho(T)$
is diagonal and we will do so here. The diagonal elements of $\rho(T)$ are
then $N$th roots of unity.

A theorem of Bantay Bantay:2001ni  states that the individual characters
$\chi_{i}(\tau)$ are weight zero modular functions for the principal
congruence subgroup $\Gamma(N)$ defined as

$\Gamma(N)=\left\\{\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in
SL_{2}({\mathbb{Z}})|~{}a\equiv d\equiv 1~{}{\rm mod}~{}N,~{}b\equiv c\equiv
0~{}{\rm mod}~{}N\right\\}\,.$ (21)

Thus we have $\chi_{i}(\gamma\tau)=\chi_{i}(\tau)$ for $\gamma\in\Gamma(N)$.
It will be useful later to note that upon reduction mod $N$ of each element of
a matrix $\gamma\in SL(2,{\mathbb{Z}})$ we obtain an element of
$SL(2,{\mathbb{Z}}/N{\mathbb{Z}})$. The kernel of this mod $N$ map is
$\Gamma(N)$ which is thus a normal subgroup of $SL(2,{\mathbb{Z}})$.

While all characters of RCFT $\chi_{i}$ are individually modular forms for
$\Gamma(N)$ and transform according to a representation $\rho$ of
$SL(2,{\mathbb{Z}})$, the converse is not true. One constraint that any set of
modular forms of this type must obey in order to be characters of a RCFT
arises from the Verlinde formula Verlinde:1988sn . The fusion coefficients
${N_{ij}}^{k}$ which govern the fusion of primary operators $\phi_{i}$

$\phi_{i}\times\phi_{j}={N_{ij}}^{k}\phi_{k}\,$ (22)

are determined in terms of the representation $\rho(S)$ via

${N_{ij}}^{k}=\sum_{m}\frac{\rho(S)_{im}\rho(S)_{jm}\rho(S^{-1})_{km}}{\rho(S)_{0m}}$
(23)

where the index $0$ labels the vacuum representation. In an RCFT the fusion
coefficients must be non-negative integers.

###  2.2 Galois Symmetry of RCFT Characters

The idea of a Galois action on the modular representation $\rho$ of a RCFT
originated with work of De Boer and Goeree DeBoer:1990em . This Galois action
has implications not just for RCFT but also in the study of topological phases
of matter, see for example Freedman . The basic idea is to construct a field
$K$ which is an extension of the rationals ${\mathbb{Q}}$ obtained by
adjoining all of the matrix elements of $\rho(\gamma)$ for $\gamma\in
SL(2,{\mathbb{Z}})$ to ${\mathbb{Q}}$. One then uses Verlinde’s formula to
show that $K$ is a finite Abelian extension of ${\mathbb{Q}}$ DeBoer:1990em  .
A theorem of Kronecker and Weber states that any such $K$ is a subfield of
some cyclotomic field, that is an extension of ${\mathbb{Q}}$ by a primitive
$m$th root of unity. Writing $\xi_{m}=e^{2\pi i/m}$ we have
$K\subseteq{\mathbb{Q}}[\xi_{m}]$. In algebraic number theory the smallest $m$
such that a finite Abelian extension $K\subseteq{\mathbb{Q}}[\xi_{m}]$ is
known as the conductor of $K$ and following Bantay:2001ni  we will also refer
to it is as the conductor of a RCFT that gives rise to $K$ by the procedure
just described. One of the results of Bantay:2001ni  states that the conductor
of a RCFT is equal to the order of $\rho(T)$. This had been conjectured
earlier in Coste:1999yc ; Bauer . In what follows we use $N$ to denote the
order of $\rho(T)$ and hence the conductor of the underlying RCFT.

For a given RCFT the Galois group ${\rm Gal}(K/{\mathbb{Q}})$ is homomorphic
to ${\rm Gal}({\mathbb{Q}}[\xi_{N}])$ and the latter is isomorphic to
$({\mathbb{Z}}/N{\mathbb{Z}})^{\times}$, the group of multiplicative units in
${\mathbb{Z}}/N{\mathbb{Z}}$, or equivalently, the set of residue classes
$\ell$ in ${\mathbb{Z}}/N{\mathbb{Z}}$ with $\text{gcd}(\ell,N)=1$. Each
element of $({\mathbb{Z}}/N{\mathbb{Z}})^{\times}$ gives rise to a Frobenius
element $f_{N,\ell}$ that leaves ${\mathbb{Q}}$ fixed and maps $\xi_{N}$ to
$\xi_{N}^{\ell}$. In Coste:1993af  it was shown that these Frobenius elements
act in a simple way on the representation matrices $\rho(S)$, $\rho(T)$. Since
$\rho(T)$ is diagonal the action of $f_{N,\ell}$ is simply

$f_{N,\ell}:\rho(T)\rightarrow\rho(T)^{\ell}\,.$ (24)

The action on $\rho(S)$ has the general form

$f_{N,\ell}:\rho(S)_{i,j}\rightarrow\varepsilon_{\ell}(i)\rho(S)_{\pi_{\ell}(i),j}=\varepsilon_{\ell}(j)\rho(S)_{i,\pi_{\ell}(j)}$
(25)

where $\pi_{\ell}$ is some permutation of the elements of ${\cal I}$ and
$\varepsilon_{\ell}$ is a map from ${\cal I}$ to $\\{+1,-1\\}$. The equality
above follows from the fact that $\rho(S)$ is a symmetric matrix. We can write
this transformation in matrix form by defining the monomial matrix $G_{\ell}$:

$\displaystyle(G_{\ell})_{i,j}=\varepsilon_{\ell}(i)\delta_{\pi_{\ell}(i),j}\,,$
(26)

and writing the Galois action on $\rho(S)$ as 333Our convention for the matrix
$G_{\ell}$ is the same as that in Coste:1999yc .

$\displaystyle
f_{N,\ell}\big{(}\rho(S)\big{)}=\rho(S)\,G_{\ell}^{-1}=G_{\ell}\,\rho(S).$
(27)

In what follows we will show that this Galois action on the representation
$\rho$ is the same as an action resulting from Hecke operators on the
characters of the RCFT, at least for several classes of RCFT.

###  2.3 RCFT Characters and MLDE

It is known that the characters of a RCFT satisfy modular linear differential
equations (MLDE). This was used in Anderson:1987ge  to show that the central
charge $c$ and conformal weights $h_{i}$ of primary fields in a RCFT are
rational numbers. There are two points of view one can take. Given a specific
RCFT, the existence of null states in representations of the chiral algebra
can be used to derive MLDE obeyed by the characters of the theory. This point
of view was taken in Eguchi  where it was used to derive MLDE for the Virasoro
characters of minimal models. The other point of view is to use MLDE as a
method for exploring the space of RCFT characters, the idea being that
solutions of an $n$th order MLDE with positive integers coefficients of their
$q$ expansion will necessarily form a $n$-dimensional representation of the
modular group and are thus candidate characters of a RCFT. This point of view
was taken in Mathur:1988na ; Mathur:1988gt ; naculich  where it was used to
classify RCFT with two independent characters and in Hampapura:2016mmz  where
it was used to classify possible RCFT characters without dimension one
operators. Related work in the mathematical literature includes kanekozagier ;
Milas:2004 ; Mason ; kaneko ; franc ; mason2d ; francmason ; arike ; arikeII .

Acting on a modular form of weight $k$ the Serre derivative ${\cal
D}_{k}=d/d\tau-\frac{1}{6}i\pi kE_{2}$ (with $E_{2}$ the quasimodular
Eisenstein series of weight $2$) produces a modular form of weight $k+2$.
Therefore acting on weight zero modular functions the iterated derivative

${\cal D}^{n}={\cal D}_{2n-2}{\cal D}_{2n-4}\cdots{\cal D}_{2}{\cal D}_{0}$
(28)

produces a modular form of weight $2n$. A general $n$th order MLDE then takes
the form

${\cal D}^{n}f+\sum_{k=0}^{n-1}\phi_{k}(\tau){\cal D}^{k}f=0$ (29)

with the $\phi_{k}(\tau)$ modular forms of weight $2(n-k)$. The coefficients
$\phi_{k}(\tau)$ can be expressed in terms of the Wronskian determinants
constructed out of the $n$ linearly independent solutions $f_{1},\cdots,f_{n}$
of the MLDE as

$\displaystyle\phi_{k}=(-1)^{n-k}W_{k}/W\,,$ (30)

where

$W_{k}=\begin{vmatrix}f_{1}&f_{2}&\cdots&f_{n}\\\ {\cal D}f_{1}&{\cal
D}f_{2}&\cdots&{\cal D}f_{n}\\\ \vdots&\vdots&&\vdots\\\ {\cal
D}^{k-1}f_{1}&{\cal D}^{k-1}f_{2}&\cdots&{\cal D}^{k-1}f_{n}\\\ {\cal
D}^{k+1}f_{1}&{\cal D}^{k+1}f_{2}&\cdots&{\cal D}^{k+1}f_{n}\\\
\vdots&\vdots&&\vdots\\\ {\cal D}^{n}f_{1}&{\cal D}^{n}f_{2}&\cdots&{\cal
D}^{n}f_{n}\end{vmatrix}\,,$ (31)

and $W=W_{n}$ which we will call the modular Wronskian.

In RCFT the characters are known to be holomorphic in the upper half plane
except possibly at the cusp at infinity. Thus the Wronskians $W_{k}$ cannot
have poles except at infinity. However the coefficient functions $\phi_{k}$
may have poles coming from the zeros of $W$. It is useful to classify
solutions by the “number of zeros” of $W$ as has been done in earlier
literature Mathur:1988na ; naculich ; Hampapura:2015cea .

If $f$ is a meromorphic function in the upper half plane $\mathbb{H}$ and if
$p\in\mathbb{H}$ is a point at which $f$ has a pole or zero then the integer
$n$ such that $f/(\tau-p)^{n}$ is holomorphic and non-zero at $p$ is called
the order of $f$ at $p$ and denoted by ${\rm ord}_{p}(f)$. Now the modular
Wronskian $W$ is holomorphic except perhaps at infinity. Define ${\rm
ord}_{\infty}(W)$ be the leading coefficient $\alpha$ in the $q$ expansion
$W=q^{\alpha}\sum_{n=0}^{\infty}a(n)q^{n}$ 444$W$ transforms under the modular
group with a multiplier system given by $\det\rho(\gamma)$ for $\gamma\in
SL(2,{\mathbb{Z}})$.. Then following the notation of Mathur:1988gt  we define

$\ell(W)=6\left(\frac{1}{2}{\rm ord}_{i}(W)+\frac{1}{3}{\rm
ord}_{\omega}(W)+\sum^{\prime}_{p\in{\cal F}}{\rm ord}_{p}(W)\right)$ (32)

where $i=\sqrt{-1}$, $\omega=e^{2\pi i/3}$ are the orbifold points of orders
$2$ and $3$ (fixed by $S$ and $ST$) and $\sum^{\prime}$ indicates that the sum
runs over all points not equal to $i,\omega$ or $i\infty$. A standard theorem
in the theory of modular forms resulting from integrating $dW/W$ around the
boundary of the fundamental domain of the modular group as in serre  then
states that

${\rm ord}_{\infty}(W)+\frac{\ell(W)}{6}=\frac{k}{12}$ (33)

where $k$ is the weight of $W$.

For an $n$th order MLDE the modular Wronskian $W$ is a modular form of weight
$n(n-1)$ and if the leading behaviors of the the solutions $f_{i}$ are
$q^{\alpha_{i}}$ then this condition states that

$\sum_{i}\alpha_{i}+\frac{\ell(W)}{6}=\frac{n(n-1)}{12}.$ (34)

We first consider the $n=2$ MLDE with $\ell=0$. Then $W$ has no zeros and thus
$\phi_{0}$, $\phi_{1}$ have no poles. Therefore $\phi_{0}$ must be a weight
two modular form holomorphic in all of $\mathbb{H}$ and similarly for
$\phi_{1}$ of weight four. Since there are no modular forms of weight two and
the Eisenstein series $E_{4}$ is the unique modular form of weight four
holomorphic in $\mathbb{H}$, the general equation with $n=2$ and $\ell=0$
involves one free parameter $\mu$ and takes the form

$D^{2}f-\frac{1}{6}E_{2}Df-\frac{\mu}{4}E_{4}f=0$ (35)

in terms of

$D=\frac{1}{2\pi i}\frac{d}{d\tau}=q\frac{d}{dq}\,.$ (36)

Two solutions can be found in Frobenius form

$f^{\pm}=q^{\alpha^{\pm}}\sum_{n=0}^{\infty}f^{\pm}_{n}q^{n}$ (37)

with $\alpha_{\pm}=(1\pm x)/12$ where $x$ is the positive square root of
$1+36\mu$. Choices of $\mu$ that leads to solutions with integer $q$
expansions and which have a candidate vacuum character with leading
coefficient one were classified in Mathur:1988na . See also Mathur:1988gt ;
naculich . Excluding the degenerate case of affine $E_{8}$ at level one which
has a single character, nine solutions were found in Mathur:1988na  which we
label in order of increasing $c_{\rm eff}$ as
$X=\\{YL,A_{1},A_{2},G_{2},D_{4},F_{4},E_{6},E_{7},E_{7\frac{1}{2}}\\}$. The
first solution consists of the two characters of a non-unitary RCFT which is
the minimal model $\mathsf{M}(2,5)$ and is also known as the Yang-Lee model
with $c=-22/5$ and $c_{\rm eff}=2/5$. The last solution consists of the
characters of a model sometimes denoted by $E_{7\frac{1}{2}}$. This latter
model is not an RCFT because the fusion rules computed from the Verlinde
formula are not positive integers. It was recently given an interpretation as
an Intermediate Vertex Subalgebra in kawasetsu  following earlier work on
$E_{7\frac{1}{2}}$ as an Intermediate Lie Algebra sextonion . The remaining
seven solutions are characters of affine Lie algebras at level $k=1$ with two
independent characters. These seven affine Lie algebras contain as subalgebras
the Lie algebras of the Lie groups $A_{1},A_{2},G_{2},D_{4},F_{4},E_{6}$ and
$E_{7}$ 555Note that the $A_{2},D_{4},E_{6}$ cases have more than two
inequivalent simple modules, but only two independent characters.. These along
with $E_{8}$ form what is known as the Deligne exceptional series of Lie
groups and share a number of remarkable properties deligne . Further
information on these models can be found in the tables in Mathur:1988na ;
naculich .

We note two interesting connections of these two character theories to number
theory. First, it is well known that the characters of the Yang-Lee model are
given by

$\displaystyle\chi^{YL}_{0}(q)$ $\displaystyle=q^{-1/60}G(q)$ (38)
$\displaystyle\chi^{YL}_{1/5}(q)$ $\displaystyle=q^{11/60}H(q)$

where the functions $G(q)$, $H(q)$ obey the Rogers-Ramanujan identities

$\displaystyle G(q)$
$\displaystyle=\sum_{n=0}^{\infty}\frac{q^{n^{2}}}{(1-q)(1-q^{2})\cdots(1-q^{n})}=\prod_{n=0}^{\infty}\frac{1}{(1-q^{5n+1})(1-q^{5n+4})}$
(39) $\displaystyle H(q)$
$\displaystyle=\sum_{n=0}^{\infty}\frac{q^{n^{2}+n}}{(1-q)(1-q^{2})\cdots(1-q^{n})}=\prod_{n=0}^{\infty}\frac{1}{(1-q^{5n+2})(1-q^{5n+3})}$

and the ratio of characters $R(q)=\chi^{YL}_{1/5}(q)/\chi^{YL}_{0}(q)$ is
given by the continued fraction

$R(q)=\frac{q^{1/5}}{1+\dfrac{q}{1+\dfrac{q^{2}}{1+\dfrac{q^{3}}{1+\ddots}}}}\,.$
(40)

This is the starting point for many striking identities developed by Ramanujan
and others. For a brief overview see berndt . Second, if we define the
character with subleading behavior as $q\rightarrow 0$ to have leading
behavior $q^{-c_{\rm eff}/24+h_{\rm eff}}$ then the values of $h_{\rm eff}$
are

$\displaystyle h_{\rm eff}(X)=\\{1/5,1/4,1/3,2/5,1/2,3/5,2/3,3/4,4/5\\}$ (41)

and these numbers are exactly the entries not equal to $0$ or $1$ in the Farey
sequence of order $5$ 666Perhaps $\frac{0}{1}$ and $\frac{1}{1}$ in this
sequence should be identified with the trivial CFT and affine $E_{8}$ at level
$1$..

$F_{5}=\left\\{\frac{0}{1},\frac{1}{5},\frac{1}{4},\frac{1}{3},\frac{2}{5},\frac{1}{2},\frac{3}{5},\frac{2}{3},\frac{3}{4},\frac{4}{5},\frac{1}{1}\right\\}$
(42)

We do not know an explanation for this observation.

For $n=2$ and $\ell=2$ the modular Wronskian must have ${\rm
ord}_{\omega}(W)=1$ which implies it is proportional to $E_{4}$. We then have
$\phi_{0}\sim E_{6}/E_{4}$ and $\phi_{1}\sim E_{4}$. The resulting equation
appears to have two independent parameters but in fact this can be reduced to
a single parameter using the indicial equation as in Hampapura:2015cea . The
solutions that might correspond to RCFT characters were analyzed in naculich ;
Hampapura:2015cea , see for example Table 1 in Hampapura:2015cea . We will
show in section 4.3 that all of these characters can be simply obtained as
Hecke images of the $\ell=0$ characters described above. In Hampapura:2015cea
no solutions were found with $\ell=3,4,5$. We will comment later on the
significance of this result.

We now turn to third order MLDE. For $n=3$ and $\ell=0$ the MLDE involves two
free parameters $\mu_{1}$ and $\mu_{2}$, and takes the form

$\displaystyle\left(D^{3}-\frac{1}{2}E_{2}D^{2}+\Big{(}\frac{1}{2}(DE_{2})+\big{(}\frac{1}{18}-\frac{\mu_{1}}{4}\big{)}E_{4}D-\frac{\mu_{2}}{8}E_{6}\Big{)}\right)f=0.$
(43)

It can be solved by the same method as in the second order case. There will be
three solutions of Frobenius form:

$\displaystyle\chi_{i}=q^{\alpha_{i}}\sum_{n=0}^{\infty}c_{i}(n)\,q^{n},\qquad
i=0,1,2,$ (44)

with $c_{0}(0)=1$ for the vacuum character. The indicial equation for the
coefficients in the $q$ expansion gives

$\displaystyle\mu_{1}=\frac{2}{9}-4(\alpha_{0}\alpha_{1}+\alpha_{0}\alpha_{2}+\alpha_{1}\alpha_{2}),\qquad\mu_{2}=8\alpha_{0}\alpha_{1}\alpha_{2}.$
(45)

There is to our knowledge no complete list of solutions of this third order
equation of RCFT character type, that is with positive integer coefficients
$c_{i}(n)$ in the $q$ expansion (44). A number of solutions with $\ell=0$ were
studied and conjectures made in kaneko . In Hampapura:2016mmz  solutions with
$\ell=0$ that could correspond to RCFTs with no dimension one operators and
hence no Kac-Moody symmetry were classified. We will see that many of these
solutions can be obtained and some of their properties understood by viewing
them as Hecke images of solutions corresponding to known RCFTs. We are not
aware of a study of three character theories with $\ell>0$ but we will show
that many such characters can be constructed using Hecke operators.

In order to give examples of Hecke relations, in section 5 we will relate a
number of RCFT with three independent characters to the characters of three
simpler RCFT which we use to illustrate the Hecke relations. These simpler
models are the Ising model, that is the minimal model $\mathsf{M}(3,4)$, the
minimal model $\mathsf{M}(2,7)$ and the tensor product of the Yang-Lee model
of minimal model $\mathsf{M}(2,5)$ with itself. In the following we give the
characters of these three models with a subscript added to each character
indicating the (effective) weight of the corresponding primary field.

The characters of the Ising model are given by

$\displaystyle\begin{split}\chi^{I}_{0}&=\frac{1}{2}\left(\sqrt{\frac{\theta_{3}(\tau)}{\eta(\tau)}}+\sqrt{\frac{\theta_{4}(\tau)}{\eta(\tau)}}\right)=q^{-1/48}(1+q^{2}+q^{3}+2q^{4}+2q^{5}+\cdots),\\\
\chi^{I}_{1/2}&=\frac{1}{2}\left(\sqrt{\frac{\theta_{3}(\tau)}{\eta(\tau)}}-\sqrt{\frac{\theta_{4}(\tau)}{\eta(\tau)}}\right)=q^{23/48}(1+q+q^{2}+q^{3}+2q^{4}+2q^{5}+\cdots),\\\
\chi^{I}_{1/16}&=\frac{1}{\sqrt{2}}\sqrt{\frac{\theta_{2}(\tau)}{\eta(\tau)}}=q^{1/24}(1+q+q^{2}+2q^{3}+2q^{4}+3q^{5}+\cdots)\,.\end{split}$
(46)

In the minimal model $M(2,7)$, the three characters are
Hikami:Spherical_Seifert

$\displaystyle\begin{split}\chi_{0}(\tau)&=\frac{q^{1/56}}{\eta(\tau)}\prod_{n=1}^{\infty}(1-q^{7n})(1-q^{7n-4})(1-q^{7n-3})\\\
&=q^{-1/42}(1+q+2q^{2}+2q^{3}+3q^{4}+\cdots),\\\
\chi_{1/7}(\tau)&=\frac{q^{9/56}}{\eta(\tau)}\prod_{n=1}^{\infty}(1-q^{7n})(1-q^{7n-5})(1-q^{7n-2})\\\
&=q^{5/42}(1+q+q^{2}+2q^{3}+3q^{4}+\cdots),\\\
\chi_{3/7}(\tau)&=\frac{q^{25/56}}{\eta(\tau)}\prod_{n=1}^{\infty}(1-q^{7n})(1-q^{7n-6})(1-q^{7n-1})\\\
&=q^{17/42}(1+q^{2}+q^{3}+2q^{4}+\cdots)\,.\end{split}$ (47)

Finally, the characters of the tensor product of the Yang-Lee model with
itself are readily obtained as products of Yang-Lee characters (note the
representation with $h=1/5$ appears with multiplicity two in the tensor
product theory, so the tensor product theory has four characters, but only
three independent characters).

$\displaystyle\begin{split}\chi_{0}=&\chi^{YL}_{0}(\tau)^{2},\\\
\chi_{1/5}=&\chi^{YL}_{0}(\tau)\chi^{YL}_{1/5}(\tau),\\\
\chi_{2/5}=&\chi^{YL}_{1/5}(\tau)^{2}.\end{split}$ (48)

In Hampapura:2016mmz  the following solutions are presented, which although
not all linked to specific RCFTs, are solutions with positive integer
coefficients and thus possible characters of RCFTs or intermediate vertex
subalgebras. They are distinguished by the fact that the characters correspond
to a theory with no dimension one operators, that is no Kac-Moody symmetry.
The first solution consists of the characters of the Baby Monster VOA
constructed in hoehn  with $q$ expansions

$\displaystyle\begin{split}\chi_{VB^{\natural}_{(0)}}&=q^{-47/48}\left(1+96256q^{2}+9646891q^{3}+\cdots\right),\\\
\chi_{VB^{\natural}_{(1)}}&=q^{25/48}\left(4371+1143745q+64680601q^{2}+\cdots\right),\\\
\chi_{VB^{\natural}_{(2)}}&=q^{23/24}\left(96256+10602496q+420831232q^{2}+\cdots\right).\end{split}$
(49)

These are characters of a RCFT with central charge $c=47/2$ and conformal
weights $h_{1}=3/2$ and $h_{2}=31/16$ and arise as solutions of eqn.(43) with
$\mu_{1}=2315/176$ and $\mu_{2}=-27025/6912$. The second solution would
correspond to a RCFT with central charge $c=236/7=59*4/7$ and conformal
weights $h_{1}=16/7$ and $h_{2}=17/7$ and is a solution of eqn.(43) with
$\mu_{1}=461/63$ and $\mu_{2}=-93869/9261$. The characters are given by

$\displaystyle\begin{split}\chi^{236/7}_{0}&=q^{-59/42}\left(1+63366q^{2}+46421200q^{3}+5765081101q^{4}+\cdots\right)\,,\\\
\chi^{236/7}_{16/7}&=q^{37/42}\left(715139+257698784q+24078730130q^{2}+1120165372784q^{3}+\cdots\right)\,,\\\
\chi^{236/7}_{17/7}&=q^{43/42}\left(848656+232637826q+19201964416q^{2}+828747166732q^{3}+\cdots\right)\,.\end{split}$
(50)

The third solution would correspond to a RCFT with central charge
$c=164/5=41*4/5$ and conformal weights $h_{1}=11/5$ and $h_{2}=12/5$ and is a
solution to eqn.(43) with $\mu_{1}=1571/225$ and $\mu_{2}=-1271/135$. The
characters are given by

$\displaystyle\begin{split}\chi^{164/5}_{0}&=q^{-41/30}\left(1+90118q^{2}+53459408q^{3}+5940658961q^{4}+\cdots\right),\\\
\chi^{164/5}_{11/5}&=q^{5/6}\left(254200+92624125q+8457234000q^{2}+381896416750q^{3}+\cdots\right),\\\
\chi^{164/5}_{12/5}&=q^{31/30}\left(615164+152560672q+11717226984q^{2}+476717427120q^{3}+\cdots\right).\end{split}$
(51)

We will show in section 5 that the characters of these three examples are
images under Hecke operators $\mathsf{T}_{47},\mathsf{T}_{59}$ and
$\mathsf{T}_{41}$ of the characters of minimal models $\mathsf{M}(3,4)$,
$\mathsf{M}(2,7)$ and $\mathsf{M}(2,5)^{\otimes 2}$ respectively.

##  3 Scalar Hecke Operators

In this section we give a brief review of standard Hecke operators for
$\Gamma=SL(2,{\mathbb{Z}})$ and the principal congruence subgroup $\Gamma(N)$.
For $SL(2,{\mathbb{Z}})$ this is standard material and may be familiar from
discussions in serre ; zagier . Hecke operators for congruence subgroups are
probably less familiar to physicists. References we found useful include
diashur ; Rankin ; shimura .

Modular forms for $SL(2,{\mathbb{Z}})$ of weight $k$ can be viewed either as
holomorphic functions of $\tau\in{\mathbb{H}}$ which transform as

$f\left(\frac{a\tau+b}{c\tau+d}\right)=(c\tau+d)^{k}f(\tau)$ (52)

under modular transformations, or as functions of rank two lattices $L$ which
obey

$F(\lambda L)=\lambda^{-k}F(L)$ (53)

under rescaling of the lattice $L$. The two are related via
$F({\mathbb{Z}}\omega_{1}+{\mathbb{Z}}\omega_{2})=\omega_{2}^{-k}f(\omega_{1}/\omega_{2})$.

###  3.1 Hecke Operators for $SL(2,{\mathbb{Z}})$

The lattice viewpoint gives a simple definition of the Hecke operator $T_{n}$
acting on modular forms of weight $k$ for $SL(2,{\mathbb{Z}})$. For
$n\in{\mathbb{Z}}^{+}$, $T_{n}$ simply sums a function $F(L)$ over sublattices
$L^{\prime}$ of index $n$ in $L$:

$(T_{n}F)(L)=\sum_{\begin{subarray}{c}L^{\prime}\subset L\\\
|L/L^{\prime}|=n\end{subarray}}F(L^{\prime})$ (54)

If $\omega_{1},\omega_{2}$ are a basis of $L$ then we can generate any
sublattice $L^{\prime}$ of index $n$ as the lattice with basis vectors

$\displaystyle\begin{split}\omega^{\prime}_{1}&=a\omega_{1}+b\omega_{2}\\\
\omega^{\prime}_{2}&=c\omega_{1}+d\omega_{2}\end{split}$ (55)

with $a,b,c,d\in{\mathbb{Z}}$ and $ad-bc=n$. Write ${\cal
M}_{n}({\mathbb{Z}})$ for the set of two by two matrices with entries in
${\mathbb{Z}}$ and determinant $n$.

We can now translate the above definition into Hecke operators acting on
weight $k$ modular forms as functions of $\tau$. We first define an action of
$\mu\in{\cal M}_{n}({\mathbb{Z}})$ on such forms by

$(f|_{k}\mu)(\tau)=\frac{({\rm det}(\mu))^{k/2}}{(c\tau+d)^{k}}f(\mu\tau)$
(56)

with

$\mu=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\,,\qquad\mu\tau=\frac{a\tau+b}{c\tau+d}\,.$ (57)

The modular group $\Gamma$ acts on ${\cal M}_{n}({\mathbb{Z}})$ by left
multiplication and a set of representatives for $\Gamma\backslash{\cal
M}_{n}({\mathbb{Z}})$ is given by

$\Delta^{(n)}=\left\\{\begin{pmatrix}a&b\\\
0&d\end{pmatrix}\Big{|}ad=n,d>0,0\leq b<d\right\\}\,.$ (58)

The action of the $n$th Hecke operator is then

$\begin{split}(T_{n}f)(\tau)&=n^{k/2-1}\sum_{\delta\in\Delta^{(n)}}(f|_{k}\delta)(\tau)\\\
&=n^{k-1}\sum_{\begin{subarray}{c}ad=n\\\
a,d>0\end{subarray}}\frac{1}{d^{k}}\sum_{b~{}~{}(\text{mod
}d)}f\Big{(}\frac{a\tau+b}{d}\Big{)}.\end{split}$ (59)

Hecke operators $T_{m}$, $T_{n}$ for $m,n$ relatively prime obey

$T_{m}T_{n}f=T_{mn}f,$ (60)

and for $n=p^{m+1}$ with $p$ prime and $m\geq 1$

$T_{p^{m+1}}f=T_{p}T_{p^{m}}f-p^{k-1}T_{p^{m-1}}f\,.$ (61)

so it suffices to know the action of $T_{p}$ on weight $k$ forms $f$ for $p$
prime. For prime $p$ we have

$\Delta^{(p)}=\left\\{\begin{pmatrix}p&0\\\
0&1\end{pmatrix},\begin{pmatrix}1&b\\\ 0&p\end{pmatrix}\Big{|}0\leq
b<p\right\\}\,.$ (62)

To prove that $T_{p}f$ is again a modular form we note that the
transformations $T\tau$ and $S\tau$ acting on $(T_{p}f)(\tau)$ are equivalent
to replacing the sum over elements $\delta$ of $\Delta^{(p)}$ with a sum over
$\delta T$ or $\delta S$ respectively. Then we note that for each
$\delta\in\Delta^{(p)}$, $\delta T$ and $\delta S$ are equivalent to
$\gamma\delta^{\prime}$ for some $\gamma\in\Gamma$ and
$\delta^{\prime}\in\Delta^{(p)}$ and that each
$\delta^{\prime}\in\Delta^{(p)}$ arises this way. Thus acting on
$\Gamma\backslash{\cal M}_{p}({\mathbb{Z}})$ the right action of $T$, $S$
simply permute the elements of the representatives in $\Delta^{(p)}$ and hence
leave the sum defining $T_{p}f$ invariant.

The above is easily translated into an action on the coefficients of the $q$
expansion of a modular form. If $f=\sum_{n}a(n)q^{n}$ is a modular form of
weight $k$, then $(T_{p}f)(\tau)=\sum_{n}a^{(p)}(n)q^{n}$ with the Fourier
coefficients

$\displaystyle a^{(p)}(n)=\begin{cases}p^{k}a(pn)&\text{if }p\nmid n,\\\
p^{k-1}\big{(}p\,a(pn)+a(n/p)\big{)}&\text{if }p|n.\end{cases}$ (63)

Note that if $a(n)\in{\mathbb{Z}}$ then $a^{(p)}(n)\in{\mathbb{Z}}$ for $k\geq
1$. Since most mathematical applications of Hecke operators use the action on
weight $k$ modular forms for $k\geq 1$, this normalization is convenient.
However we wish to define Hecke operators on vector-valued weight zero modular
functions and for this case this standard normalization does not preserve
integrality of the coefficients. As a result the definition of a new class of
Hecke operators we give later replaces the factor of $n^{k-1}$ in eqn.(59) by
$1$ when acting on weight zero modular functions.

###  3.2 Hecke Operators for $\Gamma(N)$

To extend the treatment of Hecke operators for $SL(2,{\mathbb{Z}})$ to Hecke
operators for the principal congruence subgroup $\Gamma(N)$ it is useful to
recast the treatment of Hecke operators in terms of a double coset
construction. The general setup, as discussed say in diashur  involves two
congruence subgroups $\Gamma_{1},\Gamma_{2}$ of $\Gamma=SL(2,{\mathbb{Z}})$
and an element $\alpha\in GL^{+}(2,{\mathbb{Q}})$. The set

$\Gamma_{1}\alpha\Gamma_{2}=\\{\gamma_{1}\alpha\gamma_{2}|\gamma_{1}\in\Gamma_{1},\gamma_{2}\in\Gamma_{2}\\}$
(64)

is a double coset in $GL^{+}(2,{\mathbb{Q}})$ and can be used to define a map
from weight $k$ modular forms for $\Gamma_{1}$ to weight $k$ modular forms for
$\Gamma_{2}$. This map acting on a weight $k$ modular form $f$ for
$\Gamma_{1}$ is

$f[\Gamma_{1}\alpha\Gamma_{2}]_{k}=\sum_{j}f|_{k}\delta_{j}\,.$ (65)

Here we have extended the slash operator from ${\cal M}_{n}({\mathbb{Z}})$ to
$GL^{+}(2,{\mathbb{Q}})$ in the obvious way and the double coset
$\Gamma_{1}\alpha\Gamma_{2}$ is written as a disjoint union
$\bigcup_{j}\Gamma_{1}\delta_{j}$ over orbit representatives $\delta_{j}$.

The double coset machinery can be used to define Hecke operators $T_{p}$ for
modular forms for the principal congruence subgroup $\Gamma(N)$. Through out
this subsection, we restrict to $p$ prime. The extension to Hecke operators
$T_{n}$ for $n$ relatively prime to $N$ but not necessarily prime is presented
in Appendix A.

Following the discussion and notation of Rankin  we take
$\Gamma_{1}=\Gamma_{2}=\Gamma(N)$ and

$\alpha=J_{p}=\begin{pmatrix}1&0\\\ 0&p\end{pmatrix}$ (66)

and consider the double coset $(J_{p})=\Gamma(N)J_{p}\Gamma(N)$. We also
introduce

$[\\![J_{p}]\\!]=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\big{|}a,b,c,d\in{\mathbb{Z}},ad-bc=p,\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\equiv J_{p}~{}{\rm mod}~{}N\right\\}$ (67)

which is a union of double cosets. In Rankin  it is shown that

$[\\![J_{p}]\\!]=\Gamma(N)\cdot{\cal T}_{p}^{+}={}^{+}{\cal
T}_{p}\cdot\Gamma(N)$ (68)

and that

${\cal M}_{p}({\mathbb{Z}})=\Gamma(1)\cdot{\cal T}_{p}^{+}={\cal
T}_{p}^{+}\cdot\Gamma(1)$ (69)

where $\Gamma(1)=SL(2,{\mathbb{Z}})$, ${}^{+}{\cal T}_{p}$ is a left
transversal whose form will not be needed and ${\cal T}_{p}^{+}$ is a right
transversal given by

${\cal T}_{p}^{+}=\left\\{\sigma_{p}\beta_{p};U_{b},0\leq b\leq p-1\right\\},$
(70)

where

$\beta_{p}=\begin{pmatrix}p&0\\\ 0&1\end{pmatrix}\,,\qquad
U_{\nu}=\begin{pmatrix}1&\nu N\\\ 0&p\end{pmatrix}\,,$ (71)

we write $\bar{p}$ for the multiplicative inverse of $p$ in
$({\mathbb{Z}}/N{\mathbb{Z}})^{\times}$ and we denote by $\sigma_{p}$ the
preimage of

$\begin{pmatrix}\bar{p}&0\\\ 0&p\end{pmatrix}$ (72)

under the mod $N$ map from $SL(2,{\mathbb{Z}})$ to
$SL(2,{\mathbb{Z}}/N{\mathbb{Z}})$. The set of matrices ${\cal T}_{p}^{+}$ can
be regarded as a generalization to $\Gamma(N)$ of the $\Delta^{(p)}$ that
appeared in the description of Hecke operators for $SL(2,{\mathbb{Z}})$ and in
the following we will also denote this set of matrices by $\Delta^{(p)}_{N}$
in order to emphasize the dependence on $N$.

The Hecke operator $T_{p}$ acting on a weight $k$ modular form for $\Gamma(N)$
is then defined to be

$(T_{p}f)(\tau)=\sum_{\delta\in\Delta_{N}^{(p)}}(f|_{k}\delta)(\tau)\,.$ (73)

To show that $(T_{p}f)$ is again a modular form of weight $k$ for $\Gamma(N)$
we note that taking $\tau\rightarrow\gamma\tau$ with $\gamma\in\Gamma(N)$
replaces the sum over $\delta$ in (73) by a sum over $\delta\gamma$ but since
$[\\![J_{p}]\\!]\gamma=[\\![J_{p}]\\!]$ from eqn. (68), the sum over coset
representatives and hence $T_{p}f$ is unchanged, that is $T_{p}f$ is invariant
under the weight $k$ slash operator for $\Gamma(N)$ if $f$ is.

We note that $\Gamma(N)$ does not contain $T$, but does contain $T^{N}$ and
that therefore modular forms for $\Gamma(N)$ are invariant under
$\tau\rightarrow\tau+N$ which implies that the $q$ expansion of a modular form
for $\Gamma(N)$ can be written in the form

$f(\tau)=\sum_{n}b(n)q^{n/N}\,.$ (74)

However, because $(T_{p}f)(\tau)$ includes the term

$(f|_{k}\sigma_{p}\beta_{p})(\tau)=p^{k-1}(f|_{k}\sigma_{p})(p\tau)$ (75)

there is no simple way to write the action of Hecke operators on modular forms
for $\Gamma(N)$ in terms of an action on their Fourier coefficients $b(n)$.
This is dealt with in the mathematical literature by introducing modular forms
transforming according to a character $\chi$ such that there is a good action
of the Hecke operators on their Fourier coefficients. We will see in the
following section that this can also be remedied when we have level $N$
vector-valued modular functions $f$ with respect to a representation $\rho$ by
defining a variant of $T_{p}$ that leads to mixing of the vector components of
$f$ and does have a simple action on the Fourier coefficients of the
components $f_{i}$ of $f$. The relation between the two formulations is that
the representation by which the components transform can be diagonalized over
${\mathbb{C}}$ to give modular forms transforming according to specific
combinations of Dirichlet characters.

##  4 Hecke Operators for RCFT Characters

It is not obvious how one should extend the definition of Hecke operators to
vector-valued modular forms. The basic problem is that the definition of Hecke
operators involves a sum over matrices in ${\cal M}_{n}({\mathbb{Z}})$ while
vector-valued modular forms transform under modular transformations according
to a representation $\rho$ of $SL(2,{\mathbb{Z}})$ and it is not clear how to
extend this representation to a large enough group to include the elements of
${\cal M}_{n}({\mathbb{Z}})$ required to define Hecke operators. In the
mathematics literature some ways to deal with this have been discussed in
brunier ; mwr . The paper brunier  constructs Hecke operators for vector-
valued forms transforming according to Weil representations of either the
modular group (for integer weight modular forms) or the metaplectic group (for
half integer weight forms). They do this by extending a representation of
$SL(2,{\mathbb{Z}}/N{\mathbb{Z}})$ to a representation of a group $Q(N)$
isomorphic to
$SL(2,{\mathbb{Z}}/N{\mathbb{Z}})\times({\mathbb{Z}}/N{\mathbb{Z}})^{\times}$
and defining Hecke operators for elements $(M,r)\in
SL(2,{\mathbb{Z}}/N{\mathbb{Z}})\times({\mathbb{Z}}/N{\mathbb{Z}})^{\times}$
with $\det M=r^{2}~{}{\rm mod}~{}N$. The Hecke operators defined in mwr  take
a representation $\rho$ of $SL(2,{\mathbb{Z}})$ to a representation on a
complex vector space with basis elements labelled by the set of orbit
representatives of a double coset. Our Hecke operators in contrast take
representations of dimension $n$ of $SL(2,{\mathbb{Z}})$ to representations of
the same dimension and are defined without much of the mathematical machinery
employed in brunier ; mwr . Nonetheless, since the representations of
$SL(2,{\mathbb{Z}})$ that appear in RCFT can be recast in the language of Weil
representations (see e.g. eholzerI ; eholzerII ) it is quite likely that the
Hecke operators we define can be recast in the language of brunier ; mwr .
Hecke operators on vector-valued modular forms also appear in diacon .

###  4.1 Definition and Properties

In spite of the above difficulties, it is rather trivial to define Hecke
operators acting on level $N$ modular functions with respect to a
representation $\rho$, taking a modular function relative to the
representation $\rho$ to one relative to a representation $\rho^{(p)}$, also
with $\Gamma(N)\subset{\rm ker}(\rho^{(p)})$. We simply note that the first
term in the set of representatives of eqn.(70) acting on the components
$f_{i}$ gives
$f_{i}(\sigma_{p}p\tau)=\sum_{j}\rho(\sigma_{p})_{ij}f_{j}(p\tau)$ which mixes
the components of $f$ and does not require extending the definition of $\rho$
to a larger group. To be slightly more precise we now use the action of the
orbit representatives in eqn.(70) on weight zero vector-valued modular
functions relative to a representation $\rho$ whose kernel contains
$\Gamma(N)$ to define the Hecke operator for $p$ prime

$(\mathsf{T}_{p}f)_{i}(\tau)=\sum_{\delta\in\Delta_{N}^{(p)}}f_{i}(\delta\tau)=\sum_{j}\rho_{ij}(\sigma_{p})f_{j}(p\tau)+\sum_{b=0}^{p-1}f_{i}\left(\frac{\tau+bN}{p}\right).$
(76)

since by the definition of a vector-valued weight zero modular function $f$
relative to the representation $\rho$ we have

$f_{i}(\gamma\tau)=\sum_{j}\rho(\gamma)_{ij}f_{j}(\tau)$ (77)

for $\gamma\in SL(2,{\mathbb{Z}})$. The normalization of $\mathsf{T}_{p}$
differs from $T_{p}$ for scalar modular forms encountered before in order to
preserve the integrality of coefficients. One essential feature of these Hecke
operators is that they do lead to a nice action on the Fourier coefficients.
If

$f_{i}(\tau)=\sum_{n}b_{i}(n)q^{n/N}$ (78)

then using

$\sum_{b=0}^{p-1}e^{2\pi ibn/p}=\begin{cases}p&p|n\\\
0&\text{otherwise}\end{cases}$ (79)

in eqn.(76) gives

$(\mathsf{T}_{p}f)_{i}(\tau)=\sum_{n}b^{(p)}_{i}(n)q^{n/N}$ (80)

with

$b^{(p)}_{i}(n)=\begin{cases}pb_{i}(pn)&p\nmid n\\\
pb_{i}(pn)+\sum_{j}\rho_{ij}(\sigma_{p})b_{j}(n/p)&p|n\end{cases}$ (81)

We now prove that if $f$ is a level $N$ vector-valued modular function with
representation $\rho$ then $(\mathsf{T}_{p}f)$ is a level $N$ modular function
with representation $\rho^{(p)}$ of the form claimed earlier. First consider
the action of $\gamma\in\Gamma(N)$. The fact that each component of
$(\mathsf{T}_{p}f)$ is a modular form for $\Gamma(N)$ follows from our
discussion of Hecke operators for $\Gamma(N)$ and the fact that
$\Gamma(N)\subset{\rm ker}(\rho)$. To determine the representation
$\rho^{(p)}$ it suffices to determine it on the generators $T,S$ of
$SL(2,{\mathbb{Z}})$. Because of eqn.(69), for each $\gamma\in\Gamma$ and
$\delta\in\Delta_{N}^{(p)}$ we have
$\delta\gamma=\gamma^{\prime}\delta^{\prime}$ for some
$\gamma^{\prime}\in\Gamma$ and $\delta^{\prime}\in\Delta_{N}^{(p)}$. We want
to determine $\gamma^{\prime}$ for $\gamma=T\text{ and }S$ modulo the action
of $\Gamma(N)$.

In particular we show that

$\displaystyle\begin{split}\Delta_{N}^{(p)}\circ
T&=T^{\bar{p}}\circ\Delta_{N}^{(p)},\\\ \Delta_{N}^{(p)}\circ
S&=\sigma_{p}S\circ\Delta_{N}^{(p)}\,.\end{split}$ (82)

which shows that the Hecke image $\mathsf{T}_{p}f$ transforms under
$SL(2,{\mathbb{Z}})$ according to the representation $\rho^{(p)}$ with
$\rho^{(p)}(S)=\rho(\sigma_{p}S)$ and $\rho^{(p)}(T)=\rho(T^{\bar{p}})$.

We first consider the modular transformation $T:\tau\rightarrow\tau+1$ and its
right action on the second set of representatives in eqn. (70)

$\displaystyle U_{\nu}=\begin{pmatrix}1&\nu N\\\ 0&p\end{pmatrix},$ (83)

where $\nu$ runs over a complete set of residues mod $p$. We have

$\displaystyle U_{\nu}T=T^{\bar{p}}U_{\nu^{\prime}(\nu)}~{},$ (84)

with

$\displaystyle\nu^{\prime}(\nu)=\nu-\frac{p\bar{p}-1}{N}.$ (85)

Since $p\bar{p}=1~{}{\rm mod}~{}N$, when $\nu$ runs over a complete set of
residues ${\rm mod}~{}p$ so does $\nu^{\prime}$.

To deal with the first representative in eqn. (70) we note that

$\sigma_{p}\beta_{p}T=\gamma^{\prime}\sigma_{p}\beta_{p}$ (86)

with

$\gamma^{\prime}=\sigma_{p}\begin{pmatrix}1&p\\\
0&1\end{pmatrix}\sigma_{p}^{-1}$ (87)

and that $\gamma^{\prime}\in SL(2,{\mathbb{Z}})$ since $\sigma_{p}\in
SL(2,{\mathbb{Z}})$. Furthermore working mod $N$ we have

$\gamma^{\prime}=\begin{pmatrix}\bar{p}&0\\\
0&p\end{pmatrix}\begin{pmatrix}1&p\\\ 0&1\end{pmatrix}\begin{pmatrix}p&0\\\
0&\bar{p}\end{pmatrix}=\begin{pmatrix}1&\bar{p}\\\ 0&1\end{pmatrix}$ (88)

so $\gamma^{\prime}=T^{\bar{p}}$ up to the action of $\Gamma(N)$. This
establishes that

$\displaystyle\Delta_{N}^{(p)}\circ T=T^{\bar{p}}\circ\Delta_{N}^{(p)},$ (89)

and verifies the modular representation of $T$ for the Hecke image
$\mathsf{T}_{p}f$,

$\displaystyle\rho^{(p)}(T)=\rho(T^{\bar{p}}).$ (90)

Next we consider the transformation $S:\tau\rightarrow-1/\tau$ and first deal
with the action on the representatives $U_{\nu}$ for $\nu\neq 0~{}{\rm
mod}~{}p$. For each $\nu\neq 0\text{ mod }p$, we have

$U_{\nu}S=\gamma^{\prime}_{\nu}U_{\nu^{\prime}}$ (91)

with

$\displaystyle\gamma^{\prime}_{\nu}\equiv
U_{\nu}SU^{-1}_{\nu^{\prime}}=\begin{pmatrix}N\nu&-(1+N^{2}\nu\nu^{\prime})/p\\\
p&-N\nu^{\prime}\end{pmatrix}.$ (92)

To find a $\nu^{\prime}$ such that $\gamma^{\prime}_{\nu}\in
SL(2,{\mathbb{Z}})$ we note we must have

$\displaystyle 1+N^{2}\nu\nu^{\prime}=0\quad\text{ mod }p$ (93)

which has the solution $\nu^{\prime}=-(N^{2}\nu)^{p-2}$ by Fermat’s little
theorem. With this choice of $\nu^{\prime}$, $\gamma^{\prime}_{\nu}$ has
integral entries and determinant one and so is in $SL(2,{\mathbb{Z}})$. One
then checks that $\gamma^{\prime}_{\nu}S^{-1}$ is congruent to $\sigma_{p}$
mod $N$ so we can take $\gamma^{\prime}_{\nu}=\sigma_{p}S$. Since $p$ is a
prime number, we have that $\\{\nu^{\prime}(\nu)|1\leq\nu\leq p-1\\}$ is a
permutation of the elements of $({\mathbb{Z}}/p{\mathbb{Z}})^{\times}$.

There are two representatives in $\Delta_{N}^{(p)}$ left to deal with, i.e.

$\displaystyle U_{0}\quad\text{and}\quad\sigma_{p}\begin{pmatrix}p&0\\\
0&1\end{pmatrix}.$ (94)

One easily checks that

$\displaystyle\begin{split}U_{0}S&=\gamma^{\prime}_{0}\sigma_{p}\begin{pmatrix}p&0\\\
0&1\end{pmatrix},\\\ \sigma_{p}\begin{pmatrix}p&0\\\
0&1\end{pmatrix}S&=\gamma^{\prime}_{\sigma}U_{0},\end{split}$ (95)

with $\gamma^{\prime}_{0}$ and $\gamma^{\prime}_{\sigma}$ in
$SL(2,{\mathbb{Z}})$ and equal to $\sigma_{p}S$ up to the action of
$\Gamma(N)$.

Having verified

$\displaystyle\Delta_{N}^{(p)}\circ S=\sigma_{p}S\circ\Delta_{N}^{(p)},$ (96)

we find the modular representation of $S$ for the Hecke image
$\mathsf{T}_{p}f$,

$\displaystyle\rho^{(p)}(S)=\rho(\sigma_{p}S).$ (97)

and thus verify eqn. (12).

For simplicity we have restricted our discussion to $p$ prime, but Hecke
operators $\mathsf{T}_{n}$ for $n$ relatively prime to $N$ can also be
constructed, see Appendix A.

###  4.2 Relation to Galois Symmetry

In section 2.2 we discussed the Frobenius elements $f_{N,\ell}$ for
$\ell\in({\mathbb{Z}}/N{\mathbb{Z}})^{\times}$ which act on the representation
matrices $\rho(T),\rho(S)$ as

$f_{N,\ell}\big{(}\rho(T)\big{)}=\rho(T)^{\ell},\qquad
f_{N,\ell}\big{(}\rho(S)\big{)}=G_{\ell}\,\rho(S)\,.$ (98)

On the other hand, we have seen that the Hecke operators $\mathsf{T}_{p}$ for
$p$ relatively prime to $N$ also act to produce a new representation
$\rho^{(p)}$ from $\rho$. Here we establish that these two actions are related
by

$f_{N,p}\big{(}\rho(S)\big{)}=\rho^{(\bar{p})}(S),\qquad
f_{N,p}\big{(}\rho(T)\big{)}=\rho^{(\bar{p})}(T)\,.$ (99)

Thus the Hecke operators $\mathsf{T}_{\bar{p}}$ agree with the Galois action
$f_{N,p}$ on the modular representation $\rho$ but extend it to an action on
the characters of the RCFT rather than just an action on the modular
representation.

The main result we require to show this is Theorem 1 of Bantay:2001ni :

$\displaystyle
G_{\ell}\,\rho(T)\,G_{\ell}^{-1}=\rho(T)^{\ell^{2}},\qquad\ell\in(\mathbb{Z}/N\mathbb{Z})^{\times}\,.$
(100)

Writing the above as $\rho(T)=G_{\ell}^{-1}\rho(T)^{\ell^{2}}G_{\ell}$, taking
the $\bar{\ell}$ power of both sides and using the fact that $\rho(T)^{N}=1$
leads to the equivalent relation

$\displaystyle G_{p}^{-1}\,\rho(T)^{p}\,G_{p}=\rho(T)^{\bar{p}},\qquad
p\in(\mathbb{Z}/N\mathbb{Z})^{\times}\,.$ (101)

To establish eqn.(99) we first note that $C=\rho(S)^{2}$ is invariant under
the Frobenius map $f_{N,p}$ since

$\displaystyle\rho(S)\rho(S)\rightarrow\rho(S)\,G_{p}^{-1}G_{p}\,\rho(S)=\rho(S)^{2}.$
(102)

We now apply the Frobenius map $f_{N,p}$ to both sides of the equation:

$\displaystyle\big{(}\rho(S)\rho(T)\big{)}^{3}=C,$ (103)

to find

$\displaystyle\rho(S)\,G_{p}^{-1}\,\rho(T)^{p}\cdot
G_{p}\,\rho(S)\,\rho(T)^{p}\cdot\rho(S)\,G_{p}^{-1}\,\rho(T)^{p}=C=\rho(S)^{2},$
(104)

which using the identity eqn.(101), becomes

$\displaystyle
G_{p}=\rho(T)^{p}\,\rho(S)^{-1}\,\rho(T)^{\bar{p}}\,\rho(S)\,\rho(T)^{p}\,\rho(S)=\rho\big{(}T^{p}\,S^{-1}\,T^{\bar{p}}\,S\,T^{p}\,S\big{)}\,.$
(105)

An explicit computation of the argument gives

$\displaystyle
T^{p}\,S^{-1}\,T^{\bar{p}}\,S\,T^{p}\,S=\left(\begin{array}[]{cc}\left(1-p\bar{p}\right)p+p&p\bar{p}-1\\\
1-p\bar{p}&\bar{p}\\\ \end{array}\right)\equiv\begin{pmatrix}p&0\\\
0&\bar{p}\end{pmatrix}\text{ mod }N,$ (108)

which establishes that $T^{p}\,S^{-1}\,T^{\bar{p}}\,S\,T^{p}\,S\in
SL(2,{\mathbb{Z}})$ is the preimage of $\sigma_{\bar{p}}$ under the mod $N$
map from $SL(2,{\mathbb{Z}})$ to $SL(2,{\mathbb{Z}}/N{\mathbb{Z}})$ and thus
that $G_{p}=\rho(\sigma_{\bar{p}})$. This establishes the first relation in
eqn.(99) and the second relation follows trivially from the definitions since
$\rho(T)$ is diagonal.

It will be useful later to state a few results concerning the possible values
of the matrix $\rho(\sigma_{p})$ that appears in the Hecke transformation of
coefficients, eqn.(81), and in the relation between $\rho^{(p)}(S)$ and
$\rho(S)$. We note that $\rho(\sigma_{p})$ is periodic in $p$ with period $N$,
the conductor of the RCFT. In addition we have
$\rho(\sigma_{p_{1}})\rho(\sigma_{p_{1}})=\rho(\sigma_{p_{1}p_{2}})$. Thus
$\rho(\sigma_{p})$ provides a representation of the multiplicative group of
units, $({\mathbb{Z}}/N{\mathbb{Z}})^{\times}$. Since this group is Abelian,
all representations are one-dimensional over $\mathbb{C}$. Therefore, working
over $\mathbb{C}$ we can always decompose the representation provided by
$\rho(\sigma_{p})$ into a finite sum of Dirichlet characters mod $N$. As an
example consider Hecke images $\mathsf{T}_{p}\chi^{YL}$ of the Yang-Lee model
characters with $p\in({\mathbb{Z}}/60{\mathbb{Z}})^{\times}$. One easily finds
that

$\displaystyle\begin{split}\rho^{YL}(\sigma_{p})&=\begin{pmatrix}1&0\\\
0&1\end{pmatrix}~{}{\rm for}~{}p=1,19,41,59~{}{\rm mod~{}}60,\\\
&=\begin{pmatrix}-1&0\\\ 0&-1\end{pmatrix}~{}{\rm for}~{}p=11,29,31,49~{}{\rm
mod~{}}60,\\\ &=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}~{}{\rm
for}~{}p=7,13,47,53~{}{\rm mod~{}}60,\\\ &=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}~{}{\rm for}~{}p=17,23,37,43~{}{\rm mod~{}}60\,.\end{split}$
(109)

and finds that $\rho^{YL}(\sigma_{p})$ can be diagonalized over $\mathbb{C}$
into the direct sum $X_{10}(p)+X_{12}(p)$ where $X_{10}$ and $X_{12}$ are
Dirichlet characters mod $60$ which may be defined through their values on the
generators $7,19,11$ of $({\mathbb{Z}}/60{\mathbb{Z}})^{\times}\simeq
C_{4}\times C_{2}\times C_{2}$,

$\displaystyle\begin{split}&X_{10}(7)=i,\quad X_{10}(19)=1,\quad
X_{10}(11)=-1,\\\ &X_{12}(7)=-i,\quad X_{12}(19)=1,\quad
X_{12}(11)=-1\,.\end{split}$ (110)

###  4.3 Relation to Solutions of MLDE

Suppose we have a set of RCFT characters

$\chi_{i}=q^{\alpha_{i}}\sum_{n=0}^{\infty}c_{i}(n)q^{n},~{}~{}~{}~{}i=1,2,\cdots
n\,,$ (111)

that arise as the solution of an $n$th order MLDE such that the modular
Wronskian has $\ell=0$ as was the case for the minimal and affine models
discussed earlier. If the conductor of the RCFT is $N$ then we can write
$\alpha_{i}=m_{i}/N$ for some $m_{i}\in{\mathbb{Z}}$. The Hecke images
$\mathsf{T}_{p}$ are characters with the same conductor $N$ but with new
leading exponents $\alpha^{(p)}_{i}$ determined by the minimum integers
$n_{i}$ such that $b_{i}^{(p)}(n_{i})\neq 0$ in eqn.(81). As a result of
eqn.(34), the Hecke images will thus have to satisfy an MLDE of the same
order, but with different parameters and with in general a different value of
the parameter $\ell$ which counts the zeroes of the modular Wronskian. In
other words, the action of the Hecke operators on solutions induces an action
on the modular Wronskians which in general changes the structure of the
divisor of $W$.

As an example consider the Yang-Lee model. We then have for the leading
exponents of the Hecke image

$\alpha^{(p)}_{0}=-\frac{p}{60},\quad\alpha^{(p)}_{1}=\frac{(11p)~{}{\rm
mod~{}}60}{60}\,.$ (112)

We denote the corresponding $\ell$ values computed from eqn.(34) by
$\ell^{YL}(p)$ and provide a list of values for small $p$ in Table 1. This
suggests that the Hecke images of Yang-Lee characters with $p=7,13,19$ might
already be included in the list of solutions to the $n=2$ MLDE with $\ell=0$
and indeed we will find this to be the case. Similarly we will identify the
Hecke images with $p=41,47,53,59$ with some of the entries in Table 1 of
Hampapura:2015cea . The Hecke images with $p=11,17,23,29$ do not appear in
Hampapura:2015cea  because the characters contain negative integer
coefficients for reasons that will be explained in the following section. Note
also that no Hecke images of the YL model appear with $2<\ell<6$ which is
consistent with the result found in Hampapura:2015cea  on the absence of
solutions with these $\ell$ values.

Another important example is the Ising model. In this case the leading
exponents of the Hecke images are

$\displaystyle\alpha^{(p)}_{0}=-\frac{p}{48},\quad\alpha^{(p)}_{1}=\frac{(23p)~{}{\rm
mod~{}}48}{48},\quad\alpha^{(p)}_{2}=\frac{(2p)~{}{\rm mod~{}}48}{48}\,.$
(113)

Denote the corresponding $\ell$ values by $\ell^{I}(p)$, which are listed in
Table 2. Unlike the Hecke images of Yang-Lee characters, all $\ell^{I}(p)$
equal $0$ when $0<p<48$. This ensures that there are no zeros in the modular
Wronskian when the Hecke images with $p<48$ are viewed as solutions to a third
order MLDE. Increasing $p$ by a multiple of $48$ increases $\ell^{I}(p)$ by
the same multiple of $6$. Note that the Hecke image with $p=47$ coincides with
the Baby Monster characters obtained from MLDE in Hampapura:2016mmz .

$p$ | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 49 | 53 | 59 | 61 | 67 | 71
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\ell^{YL}(p)$ | 0 | 2 | 0 | 2 | 0 | 2 | 2 | 0 | 0 | 2 | 0 | 2 | 0 | 2 | 2 | 6 | 6 | 8
Table 1: Number of zeros in the modular Wronskian for Hecke images under $\mathsf{T}_{p}$ of Yang-Lee characters for small values of $p$. $p$ | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 25 | 29 | 31 | 35 | 37 | 41 | 43 | 47 | 49 | 53 | 55
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\ell^{I}(p)$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 6 | 6 | 6
Table 2: Number of zeros in the modular Wronskian for Hecke images under
$\mathsf{T}_{p}$ of Ising characters for small values of $p$.

##  5 Applications and Examples

###  5.1 One Character RCFT

RCFT with a single character can only exist for $c=0~{}{\rm mod}~{}8$ with the
simplest example being affine $E_{8}$ at level one with character

$\chi^{E_{8}}=\frac{E_{4}}{\eta^{8}}=\frac{\Theta_{E_{8}}}{\eta^{8}}$ (114)

with $E_{4}$ the weight four Eisenstein series (and the theta function of the
$E_{8}$ root lattice) and $\eta$ the Dedekind eta function. This is a somewhat
degenerate case of our formalism, but nonetheless Hecke operators still act in
an interesting way. $\chi^{E_{8}}$ transforms according to a one-dimensional
representation of the modular group with $\rho(S)=1$ and $\rho(T)=e^{2\pi
i/3}$. It thus has conductor $3$. By comparing $q$ expansions one deduces that

$\displaystyle\mathsf{T}_{2}\chi^{E_{8}}$ $\displaystyle=\chi^{E_{8}\times
E_{8}}=\chi^{Spin(32)/{\mathbb{Z}}_{2}}=\frac{E_{4}^{2}}{\eta^{16}}$ (115)
$\displaystyle\mathsf{T}_{4}\chi^{E_{8}}$ $\displaystyle=\chi^{{\rm
ext},32}=q^{-4/3}\left(1+139504q^{2}+69332992q^{3}+\cdots\right)$ (116)
$\displaystyle\mathsf{T}_{5}\chi^{E_{8}}$ $\displaystyle=\chi^{{\rm
ext},40}=q^{-5/3}\left(1+20620q^{2}+86666240q^{3}+\cdots\right)$ (117)

where $\chi^{E_{8}\times E_{8}}$ and $\chi^{Spin(32)/{\mathbb{Z}}_{2}}$ are
the characters of lattice VOAs associated to the two rank $16$ even, self-dual
lattices and $\chi^{{\rm ext},32}$ and $\chi^{{\rm ext},40}$ are the
characters of extremal VOAs of rank $32$ and $40$ appearing in Table 5.1 of
hoehn . Closely related results were obtained in Witten:2007kt  by looking at
the Hecke images of CFT with a single character at $c=24$ where the usual
Hecke operators for $SL(2,{\mathbb{Z}})$ can be used and again the images
correspond to certain extremal CFT.

Hecke operators $\mathsf{T}_{p}$ for $p\geq 7$ acting on the affine $E_{8}$
character lead to modular forms with a gap of more than $2$ between the powers
of $q$ of the leading and subleading terms with nonzero coefficients and thus
cannot be characters of a RCFT since the state $L_{-2}|0\rangle$ is always
present in an RCFT. This can be remedied using the same idea as in
Witten:2007kt  by adding together Hecke images
$\mathsf{T}_{p^{\prime}}\chi^{E_{8}}$ for $p^{\prime}<p$, $p^{\prime}=0~{}{\rm
mod~{}}3$ in a way that leads to a character consistent with the existence of
a decomposition into Virasoro characters.

###  5.2 Two Character RCFT

We now apply the Hecke operators of the previous section to derive relations
between the characters of RCFTs with more than one character. These examples
in general involve mixing between the different components of the characters.
We will start with the Yang-Lee model. The characters were given in eqns. (38,
39). The representations of $S$ and $T$ are

$\rho^{YL}(S)=\frac{2}{\sqrt{5}}\begin{pmatrix}~{}\sin(\frac{2\pi}{5})&~{}\sin(\frac{\pi}{5})\\\
\sin(\frac{\pi}{5})&-\sin(\frac{2\pi}{5})\end{pmatrix},\qquad\rho^{YL}(T)={\rm
diag}(\xi_{60}^{-1},\xi_{60}^{11}),$ (118)

where we use the notation $\xi_{N}$ for $e^{2\pi i/N}$. We thus see that the
conductor is $N=60$.

We now consider which $\mathsf{T}_{p}$ acting on the characters
$\chi^{YL}_{i}$ can lead to consistent RCFT characters by which we mean that
the vacuum has degeneracy one, the coefficients in the $q$ expansion are non-
negative integers and the Verlinde formula for the representation
$\rho(\sigma_{p})$ leads to non-negative fusion coefficients. Using eqns.
(109) and (81) we can rule out $p=11,29,31,49{\rm~{}mod~{}}60$ since these
lead to a negative coefficient
$b_{0}^{(p)}(-p)=\rho_{00}(\sigma_{p})b_{0}^{YL}(-1)=-1$. Similarly we can
rule out $p=17,23,37,43{\rm~{}mod~{}}60$ since then
$b_{1}^{(p)}(-p)=\rho_{10}(\sigma_{p})b_{0}(-1)=-1$. This leaves
$p=19,41,59{\rm~{}mod~{}}60$ with $\rho^{YL}(\sigma_{p})$ the identity matrix
so that the Hecke image has no mixing between different Yang-Lee characters
and $p=7,13,47,53{\rm~{}mod~{}}60$ with off-diagonal $\rho^{YL}(\sigma_{p})$
and hence mixing between the two characters in the Hecke image. Of these
values of $p$ only $p=7,13,19$ have $\ell=0$ according to Table 1.

For $p=7,13$ we have

$\rho^{YL}(\sigma_{7})=\rho^{YL}(\sigma_{13})=\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}\,.$ (119)

We can then use eqn.(81) to derive the coefficients $b^{(p)}_{i}(n)$ of the
Hecke image. Although the Hecke relations between coefficients are most easily
seen when characters are written as a series in $q^{1/N}$, it is more
conventional in conformal field theory to write characters in terms of a
leading fractional power of $q$ and a series with integer powers of $q$. Thus
for the Yang-Lee model we define coefficients of characters as

$\displaystyle\chi^{YL}_{0}=q^{-1/60}\sum_{n=0}^{\infty}c^{YL}_{0}(n)q^{n}\,,$
(120)
$\displaystyle\chi^{YL}_{1/5}=q^{11/60}\sum_{n=0}^{\infty}c^{YL}_{1/5}(n)q^{n}\,,$

and we similarly define coefficients $c^{G_{2}}_{0}(n)$ and
$c^{G_{2}}_{2/5}(n)$ of the vacuum character and character corresponding to
the primary of conformal weight $2/5$ for level one affine $G_{2}$ and
coefficients $c^{F_{4}}_{0}(n)$ and $c^{F_{4}}_{3/5}(n)$ for level one affine
$F_{4}$ characters.

We now show that the modular representation and characters of affine level one
$G_{2}$ and $F_{4}$ are the Hecke images of the Yang-Lee characters under the
Hecke operators $\mathsf{T}_{7}$ and $\mathsf{T}_{13}$ respectively. The
relation between modular representations is well known. We have

$\displaystyle\begin{split}&\rho^{G_{2}}(S)=\rho^{F_{4}}(S)=\rho^{YL}(\sigma_{p}S)=\rho^{YL}(\sigma_{p})\rho^{YL}(S)\\\
=&\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}\cdot\frac{2}{\sqrt{5}}\begin{pmatrix}\sin\left(\frac{2\pi}{5}\right)&\sin\left(\frac{\pi}{5}\right)\\\
\sin\left(\frac{\pi}{5}\right)&-\sin\left(\frac{2\pi}{5}\right)\end{pmatrix}=\frac{2}{\sqrt{5}}\begin{pmatrix}-\sin\left(\frac{\pi}{5}\right)&\sin\left(\frac{2\pi}{5}\right)\\\
\sin\left(\frac{2\pi}{5}\right)&\sin\left(\frac{\pi}{5}\right)\end{pmatrix}.\end{split}$
(121)

and

$\displaystyle\rho^{G_{2}}(T)$
$\displaystyle=\rho^{YL}(T^{\bar{7}})=\rho^{YL}(T^{43})={\rm
diag}\left(\xi_{60}^{17},~{}\xi_{60}^{-7}\right),$ (122)
$\displaystyle\rho^{F_{4}}(T)$
$\displaystyle=\rho^{YL}(T^{\bar{13}})=\rho^{YL}(T^{37})={\rm
diag}\left(\xi_{60}^{23},~{}\xi_{60}^{-13}\right).$

Note that the ordering of the $G_{2}$ and $F_{4}$ characters comes out with
the vacuum character in the second entry due to our choice of ordering of the
Yang-Lee characters.

Translating the Hecke relation into a relation between coefficients in the $q$
expansion above we find that the Hecke relations
$\chi^{G_{2}}=\mathsf{T}_{7}\chi^{YL}$ and
$\chi^{F_{4}}=\mathsf{T}_{13}\chi^{YL}$ are equivalent to the relations

$\displaystyle\begin{split}c_{0}^{G_{2}}(n)=&\begin{cases}7c_{1/5}^{YL}(7n-1)&\text{if
}7\nmid n,\\\ 7c_{1/5}^{YL}(7n-1)+c_{0}^{YL}(\frac{n}{7})&\text{if
}7|n;\end{cases}\\\
c_{2/5}^{G_{2}}(n)=&\begin{cases}7c_{0}^{YL}(7n+2)&\text{if }7\nmid(n-1),\\\
7c_{0}^{YL}(7n+2)-c_{1/5}^{YL}(\frac{n-1}{7})&\text{if
}7|(n-1).\end{cases}\end{split}$ (123)
$\displaystyle\begin{split}c_{0}^{F_{4}}(n)=&\begin{cases}13c_{1/5}^{YL}(13n-3)&\text{if
}13\nmid n,\\\ 13c_{1/5}^{YL}(13n-3)+c_{0}^{YL}(\frac{n}{13})&\text{if
}13|n;\end{cases}\\\
c_{3/5}^{F_{4}}(n)=&\begin{cases}13c_{0}^{YL}(13n+5)&\text{if
}13\nmid(n-2),\\\ 13c_{0}^{YL}(13n+5)-c_{1/5}^{YL}(\frac{n-2}{13})&\text{if
}13|(n-2).\end{cases}\end{split}$ (124)

which we claim are true.

We should clarify what we mean when we write down relations like these and
others to be presented later. From a practical point of view, sufficient for
most physicists, we have verified these relations to high order in their $q$
expansions, typically to order $q^{2000}$. This can be turned into a precise
mathematical statement although we have not done this for all cases. We have
done it for the example above as follows.

The Sturm bound sturm  states that two modular forms of weight $k$ for a
congruence subgroup $\Gamma$ of $SL(2,{\mathbb{Z}})$ are equal if the first
$k[SL(2,{\mathbb{Z}}):\Gamma]/12$ coefficients in their $q$ expansion agree
where $[SL(2,{\mathbb{Z}}):\Gamma]$ is the index of $\Gamma$ in
$SL(2,{\mathbb{Z}})$. We can apply this by noting that $\eta(\tau)^{2}$ is a
weight one modular form for $\Gamma(12)$ so we can multiply each term of the
putative identity which involves modular forms on $\Gamma(60)$ by $\eta^{2}$
to obtain weight one forms for $\Gamma(60)$. The index of $\Gamma(60)$ in
$SL(2,{\mathbb{Z}})$ is $138240$ so the Sturm bound implies that the two
modular forms are equal if their $q$ expansions match to order $q^{11520}$. We
have verified the above identities to this order thus proving that the Hecke
images of YL characters under $\mathsf{T}_{7}$ and $\mathsf{T}_{13}$ are the
affine level one $G_{2}$ and $F_{4}$ characters as claimed.

The same analysis with the obvious changes identifies the characters of the
$E_{7\frac{1}{2}}$ model with the Hecke image of the Yang-Lee characters under
$\mathsf{T}_{19}$:

$\displaystyle\begin{split}c_{0}^{E_{7+1/2}}(n)=&\begin{cases}19c_{0}^{YL}(19n-6)&\text{if
}19\nmid n,\\\ 19c_{0}^{YL}(19n-6)+c_{0}^{YL}(\frac{n}{19})&\text{if
}19|n;\end{cases}\\\
c_{4/5}^{E_{7+1/2}}(n)=&\begin{cases}19c_{1/5}^{YL}(19n+9)&\text{if
}19\nmid(n-3),\\\ 19c_{1/5}^{YL}(19n+9)+c_{1/5}^{YL}(\frac{n-3}{19})&\text{if
}19|(n-3).\end{cases}\end{split}$ (125)

There is no character mixing here, since

$\displaystyle\rho^{YL}(\sigma_{p})=\mathbb{I}_{2}~{},\qquad p=19.$ (126)

For $p=41,47,53,59$ which are the remaining $p<60$ which lead to Hecke images
with positive coefficients the Hecke images have $\ell=2$ and can be
identified with entries Nos. 1,4,6,9 in Table 1 of Hampapura:2015cea . Since
the representation matrix $\rho^{YL}(\sigma_{p}S)$ for the theories with
$p=41,59$ are the same as those of the $E_{7\frac{1}{2}}$ theory the resulting
theory will have some negative integer fusion coefficients and thus the
characters cannot be characters of a consistent RCFT but perhaps can be
identified with characters of Intermediate Vertex subalgebras as was done for
the $E_{7\frac{1}{2}}$ theory in kawasetsu .

While $\rho^{YL}(\sigma_{p})$ is periodic with period $N=60$, the Hecke
operators $\mathsf{T}_{p}$ are not. The Hecke images $\mathsf{T}_{p}\chi^{YL}$
for $p=7,13,47,53~{}{\rm mod~{}}60$ by a small extension of the above
discussion can be shown to have all positive integer coefficients in their $q$
expansion, have degeneracy one for the vacuum representation and have modular
S matrix $\rho^{YL}(\sigma_{p}S)$ which is the same as that for the affine
$G_{2}$ and $F_{4}$ theories which are known to lead to non-negative integer
fusion coefficients. However, as in the one character case these characters
are not consistent with a decomposition into Virasoro characters for
sufficiently large $p$ but again this can be remedied by combining together
different Hecke images. If we take $P=60M+p$ for $p=7,13,47,53~{}{\rm
mod~{}}60$ then the component of the Hecke image of the Yang-Lee characters we
would like to identify with the vacuum characters has a $q$ expansion of the
form

$q^{-p/60}(q^{-M}+O(q))\,.$ (127)

In a RCFT this $q$ expansion must contain at least the Virasoro vacuum
character at $c=2*P/5$ which is

${\rm
ch}_{c=2P/5,h=0}(q)=\frac{q^{-P/60}}{\prod_{n=2}^{\infty}(1-q^{n})}=q^{-P/60}\sum_{n=0}^{\infty}c(n)q^{n}$
(128)

The combination of Hecke images

$\mathsf{T}_{60M+p}\chi^{YL}+\sum_{k=0}^{M-2}d(k)\mathsf{T}_{p+60*k}\chi^{YL}$
(129)

will thus have a decomposition into a Virasoro vacuum character plus Virasoro
characters at larger conformal dimension $h$ provided that $d(k)\geq c(k)$.
The case $d(k)=c(k)$ might be regarded as a two-character generalization of
the one character extremal CFTs explored in hoehn ; Witten:2007kt  and later
references. These Hecke images thus provide a completely consistent set of
possible RCFT characters. It would clearly be interesting to see if these can
be identified with characters of specific RCFTs.

There are other characters related by Hecke relations in the Deligne-
Cvitanović exceptional series. For example, the $A_{1}$ and $E_{7}$ affine
characters at level $k=1$ are characters of RCFTs with conductor $N=24$ and
the Fourier coefficients of the $E_{7}$ characters are Hecke images under
$\mathsf{T_{7}}$ of those of $A_{1}$:

$\displaystyle\begin{split}c_{0}^{E_{7}}(n)=&\begin{cases}7c_{0}^{A_{1}}(7n-2)&\text{if
}7\nmid n,\\\ 7c_{0}^{A_{1}}(7n-2)+c_{0}^{A_{1}}(\frac{n}{7})&\text{if
}7|n;\end{cases}\\\
c_{3/4}^{E_{7}}(n)=&\begin{cases}7c_{1/4}^{A_{1}}(7n+3)&\text{if
}7\nmid(n-1),\\\
7c_{1/4}^{A_{1}}(7n+3)+c_{1/4}^{A_{1}}(\frac{n-1}{7})&\text{if
}7|(n-1).\end{cases}\end{split}$ (130)

where we have used

$\displaystyle\rho^{A_{1}}(\sigma_{7})=\mathbb{I}_{2}~{}.$ (131)

It also turns out that four more of the entries in Table 1 of
Hampapura:2015cea  can also be constructed as Hecke images of $n=2,\ell=0$
models. In addition to the examples mentioned earlier, the $c=17,23$ examples
are Hecke images under $\mathsf{T}_{17}$ and $\mathsf{T}_{23}$ respectively of
the affine $A_{1}$ characters and the $c=20,22$ examples are Hecke images
under $\mathsf{T}_{5}$ of the $D_{4}$ characters and under $\mathsf{T}_{11}$
of the $A_{2}$ characters respectively.

The only entries in the Deligne exceptional series we have not related by
Hecke operators are affine $E_{6}$ and affine $A_{2}$. Their characters are
related, but the relationship is more complicated, presumably because they
involve a Hecke operator for the prime $3$ which is not relatively prime to
the conductor $12$. We presume these characters can also be related to the
remaining $c=18$ example with $n=2,\ell=2$ but the details remain to be worked
out.

###  5.3 Three Character RCFT

Hecke relations between characters of RCFTs appear to be quite common. Here we
explore a few examples of relations between RCFTs with three independent
characters. Certainly the most well known RCFT with three independent
characters is the Ising model with $c=1/2$, equivalent to a free Majorana
fermion.

The well known characters and representation matrices of the Ising model show
that the conductor is $N=48$. Denoting the Ising model modular representation
matrices by $\rho^{I}$, there are only two different values of
$\rho^{I}(\sigma_{p})$ that occur depending on whether
$p^{2}=1{\rm~{}mod~{}}48$ or $p^{2}=25{\rm~{}mod~{}}48$:

$\displaystyle\rho^{I}(\sigma_{p})=\mathbb{I}_{3}~{},\qquad p^{2}=1\text{ mod
}48~{}.$ (132)
$\displaystyle\rho^{I}(\sigma_{p})=\left(\begin{array}[]{ccc}0&1&0\\\ 1&0&0\\\
0&0&-1\\\ \end{array}\right)~{},\qquad p^{2}=25\text{ mod }48~{}.$ (136)

The Hecke images $\mathsf{T}_{p}\chi^{I}$ for all prime $p$ with $(p,48)=1$
lead to characters obeying all of the consistency conditions for RCFT
characters. We mention here some examples where we have been able to match the
Hecke images to the characters of known RCFT. For all $p<24$ the Hecke images
are the characters of affine $so(p)$ algebras at level 1. While for all
$24<p<48$, $\mathsf{T}_{p}\chi^{I}$ is a shift from the $\hat{so}(p)_{1}$
characters :

$\displaystyle\begin{pmatrix}\chi_{0}\\\ \chi_{v}\\\
\chi_{s}\end{pmatrix}^{so(p)}-\begin{pmatrix}\chi_{0}\\\ \chi_{v}\\\
\chi_{s}\end{pmatrix}^{Y=(p)}=p\,{\cal P}\cdot\begin{pmatrix}\chi_{0}\\\
\chi_{v}\\\ \chi_{s}\end{pmatrix}^{so(p-24)},$ (137)

where

$\displaystyle{\cal P}=\left(\begin{array}[]{ccc}0&1&0\\\ 1&0&0\\\ 0&0&-1\\\
\end{array}\right)~{}.$ (141)

The super index $Y=(p)$ represents that it is the Hecke image under
$\mathsf{T}_{p}$. The subindices $0,v,s$ of $\chi$ stand for the vacuum,
vector and spinor representation respectively.

For $p=31$ the Hecke image of the Ising model characters are the characters of
affine $E_{8}$ at level $2$:

$\displaystyle\begin{split}\chi^{E_{8}^{(2)}}_{0}=&q^{-31/48}(1+248q+31124q^{2}+871627q^{3}+13496501q^{4}+\cdots),\\\
\chi^{E_{8}^{(2)}}_{3/2}=&q^{41/48}(3875+181753q+3623869q^{2}+46070247q^{3}+\cdots),\\\
\chi^{E_{8}^{(2)}}_{15/16}=&q^{7/24}(248+34504q+1022752q^{2}+16275496q^{3}+\cdots).\end{split}$
(142)

with the Hecke relations to the Ising model characters given explicitly by

$\displaystyle\begin{split}c_{0}^{E_{8}^{(2)}}(n)=&\begin{cases}31c_{0}^{I}(31n-20)&\text{if
}31\nmid n,\\\ 31c_{0}^{I}(31n-20)+c_{0}^{I}(\frac{n}{31})&\text{if
}31|n;\end{cases}\\\
c_{3/2}^{E_{8}^{(2)}}(n)=&\begin{cases}31c_{1/2}^{I}(31n+26)&\text{if
}31\nmid(n-14),\\\ 31c_{1/2}^{I}(31n+26)+c_{1/2}^{I}(\frac{n-14}{31})&\text{if
}31|(n-14);\end{cases}\\\
c_{15/16}^{E_{8}^{(2)}}(n)=&\begin{cases}31c_{1/16}^{I}(31n+9)&\text{if
}31\nmid(n-1),\\\ 31c_{1/16}^{I}(31n+9)+c_{1/16}^{I}(\frac{n-1}{31})&\text{if
}31|(n-1).\end{cases}\end{split}$ (143)

For $p=47$ case the Hecke images are the characters of the Baby Monster vertex
algebra constructed in hoehn . The Baby Monster characters are

$\displaystyle\begin{split}\chi^{BM}_{0}=&q^{-47/48}(1+96256q^{2}+9646891q^{3}+366845011q^{4}+\cdots),\\\
\chi^{BM}_{3/2}=&q^{25/48}(4371+1143745q+64680601q^{2}+1829005611q^{3}+\cdots),\\\
\chi^{BM}_{31/16}=&q^{23/24}(96256+10602496q+420831232q^{2}+9685952512q^{3}+\cdots).\end{split}$
(144)

and are given in terms of Ising model characters by

$\displaystyle\begin{split}c_{0}^{BM}(n)=&\begin{cases}47c_{0}^{I}(47n-46)&\text{if
}47\nmid n,\\\ 47c_{0}^{I}(47n-46)+c_{0}^{I}(\frac{n}{47})&\text{if
}47|n;\end{cases}\\\
c_{3/2}^{BM}(n)=&\begin{cases}47c_{1/2}^{I}(47n+24)&\text{if
}47\nmid(n-22),\\\ 41c_{1/2}^{I}(47n+24)+c_{1/2}^{I}(\frac{n-22}{47})&\text{if
}47|(n-22);\end{cases}\\\
c_{31/16}^{BM}(n)=&\begin{cases}47c_{1/16}^{I}(47n+45)&\text{if
}47\nmid(n-1),\\\ 47c_{1/16}^{I}(47n+45)+c_{1/16}^{I}(\frac{n-1}{47})&\text{if
}47|(n-1);\end{cases}\end{split}$ (145)

We now give a few example of Hecke images of the minimal model
$\mathsf{M}(2,7)$ characters. The representations of $S$ and $T$ are

$\displaystyle\rho^{\mathsf{M}(2,7)}(S)$
$\displaystyle=\frac{2}{\sqrt{7}}\begin{pmatrix}\sin(\frac{4\pi}{7})&\sin(\frac{2\pi}{7})&\sin(\frac{\pi}{7})\\\
\sin(\frac{2\pi}{7})&-\sin(\frac{\pi}{7})&-\sin(\frac{4\pi}{7})\\\
\sin(\frac{\pi}{7})&-\sin(\frac{4\pi}{7})&\sin(\frac{2\pi}{7})\end{pmatrix},$
$\displaystyle\qquad\rho^{\mathsf{M}(2,7)}(T)$ $\displaystyle={\rm
diag}(\xi_{42}^{-1},\xi_{42}^{5},\xi_{42}^{17}).$ (146)

For integers $p$ with $\text{gcd}(p,42)=1$ we find

$\displaystyle\rho^{\mathsf{M}(2,7)}(\sigma_{p})=\mathbb{I}_{3}~{},\qquad
p^{2}=1\text{ mod }42~{}.$ (147)
$\displaystyle\rho^{\mathsf{M}(2,7)}(\sigma_{p})=\left(\begin{array}[]{ccc}0&1&0\\\
0&0&-1\\\ -1&0&0\\\ \end{array}\right)~{},\qquad p^{2}=25\text{ mod }42~{}.$
(151)
$\displaystyle\rho^{\mathsf{M}(2,7)}(\sigma_{p})=\left(\begin{array}[]{ccc}0&0&-1\\\
1&0&0\\\ 0&-1&0\\\ \end{array}\right)~{},\qquad p^{2}=37\text{ mod }42~{}.$
(155)

The Hecke image under $p=5$ gives rise to a vector-valued modular form that is
relevant to the WRT invariant of the Brieskorn homology sphere $\Sigma(2,3,7)$
Hikami:Spherical_Seifert . The Hecke image under $p=59$ recovers the three
character model without Kac-Moody symmetry

$\displaystyle\begin{split}\chi_{0}^{(59)}&=q^{-59/42}(1+63366q^{2}+46421200q^{3}+5765081101q^{4}+\cdots),\\\
\chi_{16/7}^{(59)}&=q^{37/42}(715139+257698784q+24078730130q^{2}+\cdots),\\\
\chi_{17/7}^{(59)}&=q^{43/42}(848656+232637826q+19201964416q^{2}+\cdots),\end{split}$
(156)

obtained in Hampapura:2016mmz  by analyzing the third order MLDE.

###  5.4 Bilinear Relations between Hecke Images and Modular Functions

In Hampapura:2016mmz  some bilinear relations between RCFT characters and the
modular $J$ function were exhibited. These turn out to follow simply from
properties of characters and their Hecke images. Let $f_{1}(\tau)$ and
$f_{2}(\tau)$ be the Hecke images of some RCFT character $\chi(\tau)$:

$\displaystyle f_{i}(\tau)=(\mathsf{T}_{p_{i}}\chi)(\tau),\qquad i=1,2.$ (157)

The induced modular representations are

$\displaystyle\rho^{(p_{i})}(T)=\rho(T^{\bar{p}_{i}}),\qquad\rho^{(p_{i})}(S)=G_{p_{i}}^{-1}\rho(S)=\rho(S)G_{p_{i}}\,.$
(158)

Under the action of $SL(2,{\mathbb{Z}})$ generators $T$ and $S$, the bilinear
form $f_{2}(\tau)^{T}G_{\ell}f_{1}(\tau)$ transforms as

$\displaystyle\begin{split}f_{2}(\tau+1)^{T}G_{\ell}f_{1}(\tau+1)&=f_{2}(\tau)^{T}\big{[}\rho(T^{\bar{p}_{2}})G_{\ell}\rho(T^{\bar{p}_{1}})\big{]}f_{1}(\tau),\\\
f_{2}(-1/\tau)^{T}G_{\ell}f_{1}(-1/\tau)&=f_{2}(\tau)^{T}\big{[}G_{p_{2}}^{-1}\rho(S)G_{\ell}\rho(S)G_{p_{1}}\big{]}f_{1}(\tau)\,.\end{split}$
(159)

and will be modular invariant provided that

$\displaystyle\begin{split}\rho(T^{\bar{p}_{2}})G_{\ell}\rho(T^{\bar{p}_{1}})=&G_{\ell},\\\
G_{p_{2}}^{-1}\rho(S)G_{\ell}\rho(S)G_{p_{1}}=&G_{\ell}\,,\end{split}$ (160)

or equivalently

$\displaystyle\rho\big{(}T^{\bar{p}_{2}+\bar{p}_{1}\ell^{2}}\big{)}=$
$\displaystyle\mathbb{I}_{n},$ (161) $\displaystyle
G_{p_{2}}^{-1}G_{-\bar{\ell}}G_{p_{1}}G_{\bar{\ell}}=$
$\displaystyle\mathbb{I}_{n}.$ (162)

Since the set of $G_{\ell}$ is homomorphic to
$({\mathbb{Z}}/N{\mathbb{Z}})^{\times}$ and hence abelian, eqn.(162) amounts
to

$\displaystyle G_{-p_{1}\bar{p}_{2}\bar{\ell}^{2}}=\mathbb{I}_{n}.$ (163)

While eqn.(161) states the fact

$\displaystyle\bar{p}_{2}+\bar{p}_{1}\ell^{2}\equiv 0\text{ mod }N,$ (164)

which necessarily implies $-p_{1}\bar{p}_{2}\bar{\ell}^{2}\equiv 1\text{ mod
}N$ therefore solves eqn.(162).

As an application consider the bilinear combination presented in
Hampapura:2016mmz  of characters of $\mathsf{M}(2,7)$ and its dual
$\widetilde{\mathsf{M}}(2,7)$. The $\widetilde{\mathsf{M}}(2,7)$ characters
are the Hecke images of the $\mathsf{M}(2,7)$ characters under
$\mathsf{T}_{59}$. A modular invariant bilinear form can be constructed by
taking

$\displaystyle p_{1}=1,\quad p_{2}=59,\quad\ell^{2}=-17\text{ mod }42,$ (165)

which satisfy the criteria eqn.(164). The leading coefficient is easily seen
to be $q^{-1}$ so this combination must be the modular $J$ function

$\displaystyle J(q)=\frac{1}{q}+196884q+21493760q^{2}+864299970q^{3}+\cdots$
(166)

up to an additive constant.

For the Ising model with conductor $N$ equals $48$ one can also construct
modular invariant combinations. For example, when $\ell$ is taken to be 1,
eqn.(164) boils down to

$\displaystyle p_{1}+p_{2}=0\text{ mod }48.$ (167)

As a result, $(\mathsf{T}_{p}\chi)^{T}(\mathsf{T}_{48-p}\chi)$ is modular
invariant for $p\in({\mathbb{Z}}/48{\mathbb{Z}})^{\times}$. Since it has the
most singular term $q^{-1}$, this bilinear form also gives $J(q)$ up to an
additive constant, namely

$\displaystyle(\mathsf{T}_{p}\chi)^{T}(\mathsf{T}_{48-p}\chi)=J(q)+c_{0}^{(p)}(1)+c_{0}^{(48-p)}(1).$
(168)

##  6 Conclusions and Outlook

We have established that the known Galois symmetry connecting modular
representations of certain RCFTs has an extension to a relation via Hecke
operators between their characters. We explored these connection for theories
with two or three independent characters and showed that many of the
characters are Hecke images of the characters of minimal models, their tensor
products or characters of affine Lie algebras. We also showed that Hecke
operators connect solutions to MLDE which have different numbers of zeroes in
the modular Wronskian and used this to show that all the $\ell=2$ characters
found in Table 1 of Hampapura:2015cea  are Hecke images of $\ell=0$ solutions.
We also showed that some earlier observations regarding the pairing of
characters of different RCFTs into the modular $J$ function have a simple
interpretation in terms of Hecke relations between characters. In general it
appears that given the characters $\chi_{i}$ of a known RCFT with $n$
independent characters then there will be an infinite sequence of characters
$(\mathsf{T}_{p}\chi)$ that will, for particular values of $p$ depending on
$\rho(\sigma_{p})$, provide characters that obey all the conditions required
for them to be characters of a RCFT, that is vacuum degeneracy one, positive
integer coefficients in the $q$ expansion and non-negative integer fusion
coefficients. For the Yang-Lee model we gave an explicit description of the
allowed $p$ values. This can clearly be extended to other RCFT in a
straightforward way. It will be interesting to see if these infinite sets of
Hecke images can be connected with explicit RCFTs. We would also like to
achieve a more global picture of how Hecke operators relate the characters of
different RCFTs and also whether the Hecke relations extend more broadly to
the full structure of the RCFT. In particular the results presented here are
compatible with the idea that the characters of all RCFT with $n$ independent
characters are Hecke images of the characters of a finite set of RCFTs. In the
cases examined here this finite set consists of minimal models and models with
affine Lie algebras.

From a physical point of view perhaps the most interesting question is whether
there is a physical origin of the Hecke operators $\mathsf{T}_{p}$. Hecke
operators have appeared previously in the computation of the elliptic genus of
symmetric products of Kähler manifolds where they do have a clear physical
interpretation in terms of instanton sectors but that interpretation does not
apply in any obvious way here. Characters of vertex operator algebras have
recently appeared in the context of the Schur index of $N=2$ superconformal
theories in four dimensions Beem  and this seems like an interesting context
in which to look for applications of the Hecke relations presented here. If
the Hecke images of characters $\mathsf{T}_{p}\chi$ that satisfy all the
consistency conditions for characters of a RCFT actually are characters of
RCFT then at large $p$ the central charge becomes large and it is then natural
to ask about a dual $AdS_{3}$ description of the Hecke images.

## Acknowledgements

We would like to thank A. Blass, P. Bantay, F. Brunault, J. Chi, T. Creutzig,
M. Emerton, M. Kaneko, M. Kim, G. Moore, K. Ono, M. Raum and J. Rouse for
helpful discussions and correspondence. J.H. and Y.W. acknowledge support from
the NSF 777Any opinions, findings, and conclusions or recommendations
expressed in this material are those of the author(s) and do not necessarily
reflect the views of the National Science Foundation. under grant PHY 1520748.

##  Appendix A Properties of the Hecke operator $\mathsf{T}_{p}$

The properties eqn.(60) and (61) are special cases of the general identity

$\displaystyle T_{m}T_{n}=\sum_{d|\text{gcd}(m,n)}d^{k-1}T_{mn/d^{2}}~{},$
(169)

for the standard Hecke operators acting on modular forms of weight $k$.

In this section we provide the analog of eqn.(169), based on which the
counterparts of eqn.(60) and (61) can be found as particular cases. Namely the
Hecke operators $\mathsf{T}_{m}$, $\mathsf{T}_{n}$ for $m,n$ relatively prime
obey

$\displaystyle\mathsf{T}_{m}\mathsf{T}_{n}f=\mathsf{T}_{mn}f$ (170)

and for $n=p^{m}$ with $p$ prime and $m\geq 1$

$\displaystyle\mathsf{T}_{p^{m+1}}f=\mathsf{T}_{p}\mathsf{T}_{p^{m}}f-p^{1-k}\sigma_{p}\circ\mathsf{T}_{p^{m-1}}f\,.$
(171)

The form of the equation above slightly differs from eqn.(61) due to the
normalization of the Hecke operator $\mathsf{T}_{p}$ for RCFT characters.

For an positive integer $n$ with $\text{gcd}(n,N)=1$, we denote by $J_{n}$ the
diagonal matrix

$\displaystyle\begin{pmatrix}1&0\\\ 0&n\end{pmatrix}$ (172)

and define

$\displaystyle[\\![J_{n}]\\!]:=$ $\displaystyle\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\Bigg{|}~{}\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\equiv
J_{n}\text{ mod }N,~{}\det\begin{pmatrix}a&b\\\ c&d\end{pmatrix}=\det
J_{n}\right\\},$ (173) $\displaystyle(J_{n}):=$
$\displaystyle\Gamma(N)J_{n}\Gamma(N).$ (174)

It is proved in Rankin  that $[\\![J_{n}]\\!]$ and $(J_{n})$ obey

$\displaystyle[\\![J_{n}]\\!]=$ $\displaystyle\Gamma(N)\cdot\Delta^{(n)}_{N},$
(175) $\displaystyle[\\![J_{n}]\\!]=$
$\displaystyle\sum_{q^{2}|n}(q\,\sigma_{q})(J_{n/q^{2}}).$ (176)

The Hecke operator $\mathsf{T}_{n}$ on $\Gamma(N)$ modular forms $f$ is
defined as

$\displaystyle(\mathsf{T}_{n}\,f)(\tau)=n^{-k/2}\sum_{\delta\in\Delta_{N}^{(n)}}(f|_{k}\delta)(\tau).$
(177)

For a composite number $n$ relatively prime to $N$, the right transversal
${\cal T}_{n}^{+}\equiv\Delta_{N}^{(n)}$ reads

$\displaystyle\Delta_{N}^{(n)}=\left\\{\sigma_{a}\begin{pmatrix}a&b\\\
0&d\end{pmatrix}\Bigg{|}~{}ad=n,a>0,b=\nu N~{}~{}(0\leq\nu<d)\right\\}.$ (178)

As a divisor of $n$, the integer $a$ can take values other than $1$ and $n$,
adding to the complexity of the representatives as well as the Fourier
coefficients of the Hecke images. Following eqn.(170) and (171), we can write
down the prime factorization of $n$ and express $\mathsf{T}_{n}$ in terms of
polynomials of $\mathsf{T}_{p}$ with $p|n$. It can also be shown that the
Hecke image $\mathsf{T}_{n}f$ has the modular representation in the same way
as eqn(12)

$\displaystyle\rho^{(n)}(T)=\rho(T^{\bar{n}}),\qquad\rho^{(n)}(S)=\rho(\sigma_{n}S)\,.$
(179)

By algebraic manipulation of $[\\![J_{n}]\\!]$ we can deduce identities obeyed
by the Hecke operator $\mathsf{T}_{n}$. We move on to prove eqn(170) and
(171). For coprime $m$ and $n$, the computation of double cosets proceeds as

$\displaystyle\begin{split}[\\![J_{n}]\\!][\\![J_{m}]\\!]=&\sum_{q^{2}|n,~{}r^{2}|m}(q\,\sigma_{q})(J_{n/q^{2}})(r\,\sigma_{r})(J_{m/r^{2}})\\\
=&\sum_{q^{2}|n,~{}r^{2}|m}(q\,\sigma_{q})(r\,\sigma_{r})(J_{n/q^{2}})(J_{m/r^{2}})\\\
=&\sum_{q^{2}|n,~{}r^{2}|m}(qr\,\sigma_{qr})(J_{(nm)/(qr)^{2}})\\\
=&[\\![J_{nm}]\\!].\end{split}$ (180)

The second equality comes from the commutativity Rankin

$\displaystyle(J_{n})(\sigma_{p})=(\sigma_{p})(J_{n}),\qquad\text{gcd}(p,N)=1.$
(181)

Because $m$ and $n$ are coprime, $(qr)^{2}$ exhausts all the square divisors
of $mn$ when $q^{2}$ and $r^{2}$ run over square divisors of $m$ and $n$
respectively, verifying the last equality in eqn.(180). Since the Hecke image
is a sum over representatives in the right coset
$\Delta^{(n)}_{N}=\Gamma(N)\backslash[\\![J_{n}]\\!]$, the multiplication rule
eqn.(170) follows as a result of eqn.(180).

To prove eqn.(171), we only need to verify the relation among the sets of
representatives. Recall the representative associated with the prime power
$p^{n}$

$\displaystyle\Delta_{N}^{(p^{n})}=\bigcup_{j=0}^{n}\left\\{\sigma_{p^{n-j}}\begin{pmatrix}p^{n-j}&\lambda_{j}N\\\
0&p^{j}\end{pmatrix}\Bigg{|}0\leq\lambda_{j}<p^{j}\right\\}.$ (182)

One can show that

$\displaystyle\begin{split}&\Delta_{N}^{(p)}\circ\left\\{\sigma_{p^{n-j}}\begin{pmatrix}p^{n-j}&\lambda_{j}N\\\
0&p^{j}\end{pmatrix}\Bigg{|}0\leq\lambda_{j}<p^{j}\right\\}\\\
=&\sigma_{p^{n-j}}\circ\Delta_{N}^{(p)}\circ\left\\{\begin{pmatrix}p^{n-j}&\lambda_{j}N\\\
0&p^{j}\end{pmatrix}\Bigg{|}0\leq\lambda_{j}<p^{j}\right\\}\\\
=&\left\\{\sigma_{p^{n-j}}\begin{pmatrix}1&(\lambda_{j}+p^{j}\nu)N\\\
0&p^{j+1}\end{pmatrix}\Bigg{|}0\leq\lambda_{j}<p^{j},~{}0\leq\nu<p\right\\}\\\
&+\begin{pmatrix}p&0\\\
0&p\end{pmatrix}\sigma_{p}\circ\left\\{\sigma_{p^{n-j}}\begin{pmatrix}p^{n-j}&\lambda_{j}N\\\
0&p^{j-1}\end{pmatrix}\Bigg{|}0\leq\lambda_{j}<p^{j}\right\\}\end{split}$
(183)

Summing over $j$ from $0$ to $n$, one establishes the relation among double
cosets

$\displaystyle[\\![J_{p}]\\!][\\![J_{p^{n}}]\\!]=[\\![J_{p^{n+1}}]\\!]+p(p\,\sigma_{p})[\\![J_{p^{n-1}}]\\!],$
(184)

where the factor $p$ in front of $(p\,\sigma_{p})$ means $p$ copies. The
$p\,\sigma_{p}$ in the parentheses is understood as
$p\mathbb{I}_{2}\cdot\sigma_{p}$, a composition of the homothety operator
$p\mathbb{I}_{2}$ and the action $\sigma_{p}$. The homothety operator
$p\mathbb{I}_{2}$ is a rescaling of the lattice and acts trivially on $\tau$.
However, it induces a factor $p^{-k}$ via the factor $(c\tau+d)^{-k}$ upon a
$GL(2,{\mathbb{Z}})$ action, completing the proof of eqn.(171).

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