#  Cone Vertex Algebras, Mock theta functions,
and Umbral Moonshine modules

Miranda C. N. Cheng Corresponding author. Institute for Mathematics, Academica
Sinica, Taipei, Taiwan  Gabriele Sgroi First author.

###### Abstract

We describe a family of indefinite theta functions of signature $(1,1)$ that
can be expressed in terms of trace functions of vertex algebras built from
cones in lattices. The family of indefinite theta functions considered has
interesting connections with mock theta functions and Appell-Lerch sums. We
use these relations to write the McKay-Thompson series of umbral moonshine at
lambency $\ell=8,12,16$ in terms of trace functions of vertex algebras
modules, and thereby provide the modules for these instances of umbral
moonshine.

###### Contents

  1. 1 Introduction
  2. 2 Background
    1. 2.1 Lattice Vertex Algebras
    2. 2.2 Indefinite Theta Functions
  3. 3 Indefinite Theta Functions and Cone Vertex Algebras
  4. 4 Umbral McKay-Thompson Series, Mock Theta Functions and Indefinite Thetas
    1. 4.1 Lambency Eight
    2. 4.2 Lambency Twelve
    3. 4.3 Lambency Sixteen
  5. 5 Moonshine Modules
    1. 5.1 Heisenberg, Clifford and Weyl Characters
    2. 5.2 Lambency Eight
    3. 5.3 Lambency Twelve
    4. 5.4 Lambency Sixteen
  6. 6 Conclusion and Outlook
  7. A Indefinite Theta Representations of Mock Theta Functions

##  1 Introduction

Mock theta functions [1] were introduced by Ramanujan in 1920 in his deathbed
letter to Hardy [2], in which he constructed 17 examples and provided a series
of identities satisfied by them. The mysterious nature of these functions,
together with the lack of explanation on how he discovered those examples (see
[3] for an interesting account), turned the subject into one that has
fascinated mathematicians for over a century. The theory underlying mock theta
functions remained elusive until it was elucidated by Zwegers in his PhD
thesis [4] and they are now understood in the context of harmonic Maass forms
[5], [6]. The holomorphic parts of harmonic Maass forms are often referred to
as mock modular forms and in this language mock theta functions are
essentially mock modular forms with shadows given by theta functions. In
recent years, mock theta functions have made their appearance in a variety of
fields with numerous applications, see e.g. [6], [7], [8], [9]. One of the
most intriguing appearances of mock theta functions is in the context of
Umbral Moonshine [10], [11], [12]. Umbral Moonshine consists of a family of 23
moonshine instances associated to appropriate quotients of the automorphism
groups of Niemeier lattices, the 23 even unimodular positive-definite lattices
of rank 24 with non-trivial root systems. Given the root system $X$ of a
Niemeier lattice $L^{X}$, the umbral group $G^{X}$ associated to it is given
by the quotient of the automorphisms group of $L^{X}$ by the Weyl group
$W^{X}$ associated to the root system

$G^{X}:=\text{Aut}(L^{X})/W^{X}.$ (1.1)

Following [10, 11], we will often refer to twenty-three instances as the
different lambencies of umbral moonshine. To each conjugacy class $[g]$ of
$G^{X}$ is associated a vector-valued mock modular form, the umbral McKay-
Thompson series $H^{X}_{g}$. The umbral moonshine conjecture predicts, for
each Neiemeier lattice, the existence of a naturally defined bi-graded
$G^{X}$-module

$\check{K}^{X}:=\bigoplus\limits_{r\in
I^{X}}\bigoplus_{\begin{subarray}{c}D\in\mathbb{Z},D\leq 0\\\ D=r^{2}\mod
4m\end{subarray}}\check{K}^{X}_{r,-D/4m}$ (1.2)

such that the corresponding McKay-Thompson series $H^{X}_{g}=(H^{X}_{g,r})$ is
related to the graded trace of $g$ over $\check{K}$ by

$H^{X}_{g,r}(\tau)=-2q^{-\frac{1}{4m}}\delta_{r,1}+\sum_{\begin{subarray}{c}D\in\mathbb{Z},D\leq
0\\\ D=r^{2}\mod
4m\end{subarray}}\text{tr}_{\check{K}^{X}_{r,-D/4m}}(g)q^{-\frac{D}{4m}}$
(1.3)

where $m$ is the Coxeter number of any simple component of the Niemeier root
system $X$, and $I^{X}\subset\mathbb{Z}/2m\mathbb{Z}$ is specified by

$I^{X}:=\begin{cases}\\{1,2,3,\ldots,m-1\\}&\text{if $X$ has an A-type
component,}\\\ \\{1,3,5,\ldots,\frac{m}{2}\\}&\text{if $X$ has a D-type
component and no A-type components,}\\\ \\{1,4,5\\}&\text{if
$X=E^{4}_{6}$,}\\\ \\{1,7\\}&\text{if $X=E^{3}_{8}$.}\end{cases}$ (1.4)

The existence of the modules (1.2) has been proven in [13] for the case of
Mathieu Moonshine, and then in [14] for the remaining cases. These proofs,
however, do not prescribe how such modules can be built nor offer much insight
on possible further algebraic structure.
Since Monstrous Moonshine [15], the very first example of a moonshine
phenomenon, vertex operator algebras have proven to be an invaluable tool to
understand the underlying structure behind the moonshine properties [16],
[17], [18], [19]. It is thus natural to ask if a similar approach could
provide interesting insights in the case of umbral moonshine. For some
instances of umbral moonshine it has already been shown that suitable (super)
vertex operator algebras can be used to explicitly construct the modules
$\check{K}^{X}$ [20], [21] or to solve the so called “meromorphic module
problem”, i.e. building modules such that specific trace functions give the
meromorphic Jacobi forms associated to the $H_{g}$ of Umbral Moonshine [22],
[23], [24]. In particular, in [20] the authors built the module
$\check{K}^{E^{3}_{8}}$ through the means of particular vertex operator
algebras obtained from lattice cones. Their construction makes use of the
relations between the umbral McKay-Thompson series for $E^{3}_{8}$, the fifth
order Ramanujan’s mock theta functions $\phi_{0}$, $\phi_{1}$ and their
expressions in terms of indefinite theta functions. It is natural to ask if
the techinques of [20] can be extended to build modules for other instances of
umbral moonshine. In this work we employ a particular class of cone vertex
algebras and construct modules for instances of umbral moonshine corresponding
to root systems $A^{2}_{7}D^{2}_{5}$, $A_{11}D_{7}E_{6}$, $A_{15}D_{9}$. In
order to achieve this, we will establish intermediate results relating cone
vertex algebras to indefinite theta functions that are mock theta functions.
In particular, we first describe a specific family of indefinite theta
functions can be expressed in terms of trace functions of cone vertex
algebras. Then, expressing the umbral McKay-Thompson series $H^{X}_{g}$ in
terms of indefinite theta functions, we relate $H^{X}_{g}$ to suitable linear
combinations of the traces of cone vertex algebra and other known (super)
vertex operator algebras. In the cases considered, we find that the respective
umbral groups act trivially on the underlying cone vertex algebra modules.
Thus the modules realizing the McKay-Thompson series appearing in these
examples have the structure of a tensor product $R\otimes M$ of a finite group
representation $R$ and a (super) vertex algebra module $M$. In particular, the
umbral finite group $G$ acts on $R\otimes M$ as $G\otimes{\bf 1}_{\cal V}$,
while the vertex algebra ${\cal V}$ acts as ${\bf 1}_{G}\otimes{\cal V}$. This
makes the analysis particularly simple as the representation of the umbral
group can be determined independently from the relevant cone vertex algebra
structure.
As an intermediate result, we also show that the following Appell-Lerch sums

$\displaystyle\mu(z_{1},z_{2};\tau):=\frac{y_{1}^{\frac{1}{2}}}{\theta(z_{2};\tau)}\sum\limits_{n\in\mathbb{Z}}\frac{(-1)^{n}y^{n}_{2}q^{\frac{n(n+1)}{2}}}{1-y_{1}q^{n}},$
(1.5)
$\displaystyle\mu_{m,0}(z,\tau):=\sum\limits_{k\in\mathbb{Z}}y^{2km}q^{mk^{2}}\frac{yq^{k}+1}{yq^{k}-1},$
(1.6)

admit an expression in terms of indefinite theta functions and cone vertex
algebra characters. These are distinguished examples connecting cone vertex
algebras to mock theta functions and umbral moonshine. In fact, all Ramanujan
mock thetas can be expressed in terms of (1.5) [4], [25], [26], [6], while
(1.6) appears in the construction of the optimal meromorphic Jacobi forms
associated to the umbral McKay-Thompson series [11], [14]. The latter fact
allows us to draw a connection between the construction of modules for the
McKay-Thompson series (as considered in this paper) and the meromorphic module
problem considered in [22], [24]. Furthermore, the specialized Appell-Lerch
sum (1.6) is also interesting because it captures the non-modular part of the
elliptic genus of non-compact supersymmetric coset models, as featured in
[27], [28], [29]. The techniques used in this paper can be easily used to
build an alternative module for the elliptic genus of such theories in terms
of cone vertex algebras trace functions.
The paper is organized as follows: in section 2 we recall basic notions and
notations of cone vertex algebras and indefinite theta functions that will be
used in the rest of the paper; in section 3 we present a core result of the
paper (Theorem 3.2) relating trace functions of cone vertex algebras to
indefinite theta functions; in section 4 we give expression specifying the
umbral McKay-Thompson series for lambencies $\ell=8,12,16$ in terms of
indefinite theta functions and modular forms; finally, in section 5 we specify
the umbral McKay-Thompson series considered in section 4 in terms of trace
functions of vertex algebra modules (Theorems 5.3, 5.4, 5.5).

##  2 Background

###  2.1 Lattice Vertex Algebras

In this subsection we will briefly summarize the construction of vertex
algebras associated to lattices, closely following the exposition in [20]. The
main goal is to introduce the notation and conventions that will be used in
the rest of the paper. More details can be found in, e.g, [18], [30], and
[31].
Consider a lattice $L$. Let’s define
$\mathfrak{h}:=L\otimes_{\mathbb{Z}}\mathbb{C}$ with the symmetric
$\mathbb{C}$-bilinear form $\braket{\cdot,\cdot}$ naturally inherited from the
bilinear form on $L$. Given a formal variable $t$, define
$\hat{\mathfrak{h}}:=\mathfrak{h}[t,t^{-1}]\oplus\mathbb{C}{\bf c}$ with the
Lie algebra structure given by $[u\otimes t^{m},v\otimes
t^{n}]=m\braket{u,v}\delta_{m+n,0}{\bf c}$ with ${\bf c}$ a central element.
The algebra $\hat{\mathfrak{h}}$ has a natural decomposition as
$\hat{\mathfrak{h}}=\hat{\mathfrak{h}}^{-}\oplus\hat{\mathfrak{h}}^{0}\oplus\hat{\mathfrak{h}}^{+}$
with $\hat{\mathfrak{h}}^{\pm}:=\mathfrak{h}[t^{\pm 1}]t^{\pm}$ and
$\mathfrak{h}^{0}:=\mathfrak{h}\oplus\mathbb{C}{\bf c}$. Given an ordered
integral basis $\\{\epsilon_{j}\\}$ for the lattice $L$, define

$b(\epsilon_{i},\epsilon_{j})=\begin{cases}0&\text{ if }i\leq j\\\ 1&\text{ if
}i>j\end{cases}$ (2.1)

extended linearly over $L$, and set
$\beta(\lambda,\mu):=(-1)^{b(\lambda,\mu)}$. We then consider the ring
$\mathbb{C}_{\beta}[L]$ generated by ${\bf v}_{\lambda}$, $\lambda\in L$,
satisfying ${\bf v}_{\lambda}{\bf v}_{\mu}=\beta(\lambda,\mu){\bf
v}_{\lambda+\mu}$. Give $\mathbb{C}_{\beta}[L]$ a
$\hat{\mathfrak{h}}^{0}\oplus\hat{\mathfrak{h}}^{+}$-module structure by
setting, for $h\in\mathfrak{h}$ and $\lambda\in L$,
$\mathbb{c}\mathbb{v}_{\lambda}=\mathbf{v}_{\lambda}$ and
$u(m)\mathbb{v}_{\lambda}=\delta_{m,0}\braket{u,\lambda}\mathbb{v}_{\lambda}$,
where we have used the standard notation $u(m)=u\otimes t^{m}$. Finally, we
consider the module

$V_{L}:=U(\hat{\mathfrak{h}})\otimes_{U(\hat{\mathfrak{h}}^{0}\otimes{\hat{\mathfrak{h}}^{+})}}\mathbb{C}_{\beta}[L].$
(2.2)

We can equip this module with a (unique) vertex algebra structure with vacuum
vector $1\otimes\mathbb{v}_{0}$, vertex operator map $Y:V_{L}\to({\rm
End}V_{L})[[z,z^{-1}]]$ given by, for $u\in\mathfrak{h}$ and $\lambda\in L$,

$\begin{split}&Y(u(-1)\otimes\mathbb{v}_{0},z)=\sum\limits_{n\in\mathbb{Z}}u(n)z^{-n-1}\\\
&Y(\mathbb{1}\otimes\mathbb{v}_{\lambda},z)=\text{exp}\left(-\sum\limits_{n<0}\frac{\lambda(n)}{n}z^{-n}\right)\text{exp}\left(-\sum\limits_{n>0}\frac{\lambda(n)}{n}z^{-n}\right)\mathbb{v}_{\lambda}z^{\lambda(0)}\end{split}$
(2.3)

where $\mathbb{v}_{\lambda}$ in the right hand side denotes the operator
$p\otimes\mathbf{v}_{\mu}\to\beta(\lambda,\mu)p\otimes\mathbf{v}_{\lambda+\mu}$,
and
$z^{\lambda(0)}(p\otimes\mathbf{v}_{\mu}):=(p\otimes\mathbf{v}_{\mu})z^{\braket{\lambda,\mu}}$.
Furthermore, given the basis $\\{\epsilon_{j}\\}$ for $L$ and the dual basis
$\\{\epsilon^{\prime}_{j}\\}$, $\epsilon^{\prime}_{j}\in
L\otimes_{\mathbb{Z}}\mathbb{Q}$ satisfying
$\braket{\epsilon^{\prime}_{i},\epsilon_{j}}=\delta_{ij}$, we can define the
conformal element

$\omega:=\frac{1}{2}\sum\limits_{i}\epsilon^{\prime}_{i}(-1)\epsilon_{i}(-1)\otimes\mathbb{v}_{0}.$
(2.4)

Writing $Y(\omega,z)=\sum\limits_{n\in\mathbb{Z}}L(n)z^{-n-2}$, we have
$[L(0),v(n)]=-nv(n)$ and
$L(0)1\otimes\mathbf{v}_{\lambda}=\frac{\braket{\lambda,\lambda}}{2}1\otimes\mathbf{v}_{\lambda}$.
In particular, when the bilinear form on $L$ is positive definite, this give
$V_{L}$ the structure of a vertex operator algebra. In the more general case,
vector of zero length give infinite dimensional eigenspaces for $L(0)$. We can
define a finite order automorphism of $V_{L}$ by choosing $h\in
L\otimes_{\mathbb{Z}}{\mathbb{Q}}$ acting as
$h(0)p\otimes\mathbb{v}_{\lambda}=\braket{h,\lambda}p\otimes\mathbb{v}_{\lambda}$
with $p\in S(\hat{\mathfrak{h}}^{-})$ and defining

$g_{h}:=e^{2\pi ih(0)}.$ (2.5)

In order to build twisted modules for the lattice vertex algebra , let’s
consider $\mathbb{C}_{\beta}[L+h]$ generated by $\mathbb{v}_{\mu+h}$, with
$\mu\in L$ and $h\in L\otimes_{\mathbb{Z}}\mathbb{Q}$, equipped with the
$\mathbb{C}_{\beta}[L]$-module structure given by
$\mathbb{v}_{\lambda}\mathbb{v}_{\mu+h}=\beta(\lambda,\mu)\mathbb{v}_{\lambda+\mu+h}$
and the $U(\hat{\mathfrak{h}}^{0}\otimes{\hat{\mathfrak{h}}^{+})}$-module
structure $\mathbb{c}\mathbb{v}_{\mu+h}=\mathbb{v}_{\mu+h}$,
$u(m)\mathbb{v}_{\mu+h}=\delta_{m,0}\braket{u,\mu+h}\mathbb{v}_{\mu+h}$ for
$u\in\mathfrak{h}$, $\mu,\lambda\in L$. We can then define $g_{h}$-twisted
modules for the lattice vertex algebra $V_{L}$ by setting
$V_{L+h}:=U(\hat{\mathfrak{h}})\otimes_{U(\hat{\mathfrak{h}}^{0}\otimes{\hat{\mathfrak{h}}^{+})}}\mathbb{C}_{\beta}[L+h]$
and defining $Y_{h}:=V_{L}\to(\text{End}V_{L+h})[[z,z^{-1}]]$ similarly as
before but with $\mathbb{v}_{\lambda}$ acting as
$\mathbb{v}_{\lambda}(p\otimes\mathbf{v}_{\mu+h})=\beta(\lambda,\mu)p\otimes\mathbb{v}_{\lambda+\mu+h}$.
When $h$ belongs to the dual lattice $L^{*}=\\{\lambda\in
L\otimes_{\mathbb{Z}}{Q}|\braket{\lambda,L}\in\mathbb{Z}\\}$, the modules are
untwisted. Furthermore, all the $g_{h}$-twisted modules of $V_{L}$ are given
by $V_{L+h^{\prime}}$ for some $h^{\prime}\in L\otimes_{\mathbb{Z}}\mathbb{Q}$
congruent to $h$ modulo $L^{*}$. The action of $L\otimes_{Z}\mathbb{Q}$ on
$V_{L}$ specified by $g_{h}$ extends to $g_{h^{{}^{\prime}}}$-twisted modules
through

$g_{h}(p\otimes\mathbb{v}_{\lambda+h^{{}^{\prime}}}):=e^{2\pi
i\braket{h,\lambda}}p\otimes\mathbb{v}_{\lambda+h^{{}^{\prime}}}.$ (2.6)

In order to include vertex algebras associated to cones, as opposed to the
full lattice, we will describe a family of sub-vertex algebras of $V_{L}$. For
a $K\subset L$ that is closed under addition that contains 0, the submodule
$V_{K}$ of $V_{L}$ generated by $\mathbb{v}_{\lambda}$ for $\lambda\in K$ has
the structure of a sub-vertex algebra of $V_{L}$ with the same conformal
element. Furthermore, given $\gamma\in L\otimes_{\mathbb{Z}}{\mathbb{Q}}$, for
any $K^{\prime}\subset L+\gamma$ such that $K+K^{\prime}\subset K^{\prime}$,
the corresponding $V_{K^{{}^{\prime}}}$ with the restriction of the vertex
operators $a\otimes b\mapsto Y(a,z)b$ to $V_{K}\otimes V_{K^{{}^{\prime}}}$
has the structure of a twisted module over $V_{K}$.

###  2.2 Indefinite Theta Functions

In this subsection we will review some basic properties of indefinite theta
functions of lattices with signature $(r-1,1)$. We will mostly follow the
exposition in chapter 8 of [6].
We will start by introducing some notation. For the rest of the paper we will
set $q:=e^{2\pi i\tau}$ and $y:=e^{2\pi iz}$. Given a symmetric matrix $A$
with integer coefficients, we define the bilinear form
$B(\mathbb{v},\mathbb{w}):=\mathbb{v}^{T}A\mathbb{w}$ and, correspondingly,
the quadratic form $Q(\mathbf{v}):=\frac{1}{2}B(\mathbb{v},\mathbb{v})$. It is
a well known fact that, when $Q$ is positive definite, given
$\mathbb{x}_{0}\in\mathbb{Z}^{r}$ we have that

$\Theta_{Q,\mathbb{x}_{0}}(\tau;z):=\sum\limits_{\mathbb{n}\in\mathbb{Z}^{r}}q^{Q(\mathbb{n})}y^{B(\mathbb{n},\mathbb{x}_{0})}$
(2.7)

is a Jacobi form of weight $\frac{r}{2}$ and index $Q(\mathbb{x}_{0})$ [32].
This result does not hold when $Q$ is not positive definite. When the
quadratic form is of signature $(r-1,1)$, i.e. when the largest linear
subspace on which $Q$ is negative definite has dimension $1$, generalisations
of (2.7) were studied by Zwegers [4]. For such quadratic forms, the set
$\\{\mathbb{c}\in\mathbb{R}^{r}:Q(\mathbb{c})<0\\}$ splits into two connected
components, we fix one of these and denote it $C_{Q}$. Explicitly we choose a
$\mathbb{c}_{0}$ such that $Q(\mathbb{c}_{0})<0$ and define

$C_{Q}:=\\{\mathbb{c}\in\mathbb{R}^{r}:Q(\mathbb{c})<0,B(\mathbb{c},\mathbb{c}_{0})<0\\}.$
(2.8)

We also define

$S_{Q}:=\\{\mathbb{c}=(c_{1},\ldots,c_{r})\in\mathbb{Z}^{r}:\text{gcd}(c_{1},\ldots,c_{r})=1,Q(\mathbb{c})=0,B(\mathbb{c},\mathbb{c}_{0})<0\\}$
(2.9)

and consider the compactification of $C_{Q}$, $\overline{C}_{Q}:=C_{Q}\cup
S_{Q}$. We furthermore define, $\forall\mathbb{c}\in\overline{C}_{Q}$,

$R(\mathbb{c}):=\begin{cases}\mathbb{R}^{r}&\text{if }\mathbb{c}\in C_{Q},\\\
\\{\mathbb{a}\in\mathbb{R}^{r}:B(\mathbb{a},\mathbb{c})\not\in\mathbb{Z}\\}&\text{if
}\mathbb{c}\in S_{Q}.\end{cases}$ (2.10)

With the above notation, given a symmetric matrix $A$,
$\mathbb{c}_{1},\mathbb{c}_{2}\in\overline{C}_{Q}$, $\mathbb{a}\in
R(\mathbb{c}_{1})\cap R(\mathbb{c}_{2})$, and $\mathbb{b}\in\mathbb{R}^{r}$,
we can define the indefinite theta functions

$\Theta^{(\rho)}_{\mathbb{a},\mathbb{b}}(\tau):=\sum\limits_{\mathbb{n}\in\mathbb{a}+\mathbb{Z}^{r}}\rho(\mathbb{n};\tau)e^{2\pi
iB(\mathbb{b},\mathbb{n})}q^{Q(\mathbb{n})}$ (2.11)

where we have written

$\rho(\mathbb{n};\tau)=\rho^{\mathbb{c}_{1}}(\mathbb{n};\tau)-\rho^{\mathbb{c}_{2}}(\mathbb{n};\tau)$
(2.12)

with

$\rho^{\mathbb{c}}(\mathbb{n};\tau):=\begin{cases}E\left(\frac{B(\mathbb{c},\mathbb{n})v^{\frac{1}{2}}}{\sqrt{-Q(\mathbb{c})}}\right)&\text{if
}\mathbb{c}\in C_{Q}\\\ \text{sgn}(B(\mathbb{c},\mathbb{n}))&\text{if
}\mathbb{c}\in S_{Q}\end{cases}$ (2.13)

in which $v=\text{Im}(\tau)$ and $E$ is the error function

$E(z):=2\int_{0}^{z}e^{-\pi t^{2}}dt.$ (2.14)

It has been shown [4] that, with the assumptions above, the series in (2.11)
converges absolutely. Furthermore, it has been shown [4], [6] that for
$\mathbb{c}_{1},\mathbb{c}_{2}\in\mathbb{Z}^{r}\cap\overline{C}_{Q}$ with
relatively prime coordinates, $\mathbb{a},\mathbb{b}\in R(\mathbb{c}_{1})\cap
R(\mathbb{c}_{2})$, the indefinite theta in (2.11) is a component of a vector-
valued mixed harmonic Maass form of weight $\frac{r}{2}$ for
SL${}_{2}({\mathbb{Z}})$, with holomorphic part given by

$\Theta^{+}_{\mathbb{a},\mathbb{b}}(\tau)=\sum\limits_{\mathbb{n}\in\mathbb{Z}^{r}+\mathbb{a}}\left[\text{sgn}(B(\mathbb{c}_{1},\mathbb{n}))-\text{sgn}(B(\mathbb{c}_{2},\mathbb{n}))\right]e^{2\pi
iB(\mathbb{b},\mathbb{n})}q^{Q(\mathbb{n})}.$ (2.15)

It is also shown [4], [25], [26], [6] that all Ramanujan’s mock theta
functions (and a further number of mock theta functions discovered later) can
be written in terms of a linear combination of modular forms and indefinite
theta functions (2.15) with $r=2$. This can be viewed as a generalization of
the following relation between the indefinite theta functions and the Appell-
Lerch sum

$\mu(z_{1},z_{2};\tau):=\frac{y_{1}^{\frac{1}{2}}}{\theta(z_{2};\tau)}\sum\limits_{n\in\mathbb{Z}}\frac{(-1)^{n}y^{n}_{2}q^{\frac{n(n+1)}{2}}}{1-y_{1}q^{n}},$
(2.16)

where $z_{1},z_{2}\in\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})$, $y_{j}=e^{2\pi
iz_{j}}$ for $j=1,2$, and $\theta(z;\tau)$ is the Jacobi theta function

$\theta(z;\tau):=\sum\limits_{n\in\mathbb{Z}+\frac{1}{2}}e^{\pi
in^{2}\tau+2\pi in\left(z+\frac{1}{2}\right)}.$ (2.17)

Namely, defining

$\Theta^{+}_{A,\mathbb{c}_{1},\mathbb{c}_{2}}(\mathbb{z};\tau):=\sum\limits_{n\in\mathbb{Z}^{r}}\left[\text{sgn}\left(B\left(\mathbb{c}_{1},\mathbb{n}+\frac{\text{Im}(\mathbb{z})}{\text{Im}(\tau)}\right)\right)-\text{sgn}\left(B\left(\mathbb{c}_{2},\mathbb{n}+\frac{\text{Im}(\mathbb{z})}{\text{Im}(\tau)}\right)\right)\right]e^{2\pi
iB(\mathbb{z},\mathbb{n})}q^{Q(\mathbb{n})}$ (2.18)

it can be shown [6] that, for
$0<\frac{\text{Im}(z_{1})}{\text{Im}(\tau)},\frac{\text{Im}(z_{2})-\text{Im}(z_{1})}{\text{Im}(\tau)}+\frac{1}{2}<1$,
we have the following relation

$\mu(z_{1},z_{2};\tau)=\frac{y_{1}^{\frac{1}{2}}}{2\theta(z_{2};\tau)}\Theta^{+}_{A,\mathbb{c}_{1},\mathbb{c}_{2}}\left(z_{1},z_{2}-z_{1}+\frac{\tau+1}{2};\tau\right)$
(2.19)

for $A=\left(\begin{smallmatrix}1&1\\\ 1&0\end{smallmatrix}\right)$,
$\mathbb{c}_{1}=(0,1)$, $\mathbb{c}_{2}=(-1,1)$.
Notice that, writing $\mathbb{z}=\mathbb{a}\tau+\mathbb{b}$ with
$\mathbb{a}\in R(\mathbb{c}_{1})\cap
R(\mathbb{c}_{2}),\mathbb{b}\in\mathbb{R}^{r}$, equation (2.18) can be related
to expression (2.15) through

$\Theta^{+}_{A,{\bf c}_{1},{\bf c}_{2}}({\bf a}\tau+{\bf b},\tau)=e^{-2\pi
iB({\bf a},{\bf b})}q^{-Q({\bf a})}\Theta^{+}_{{\bf a},{\bf b}}(\tau).$ (2.20)

The relation to Ramanujan’s mock theta functions follows via their expression
in terms of the universal mock theta functions $g_{2}$, $g_{3}$ [33] which in
turn can be related to the Appell-Lerch sum (2.16) [34].

##  3 Indefinite Theta Functions and Cone Vertex Algebras

In this section we will describe a family of trace functions of vertex
algebras modules that can be expressed in terms of indefinite theta functions.
Consider a symmetric $2\times 2$ matrix $A$ with integer coefficients, the
associated bilinear and quadratic forms $B$ and $Q$ as in section 2.2, and the
vectors $\mathbb{c}_{1},\mathbb{c}_{2}\in\bar{C}_{Q}$ satisfying

$\begin{array}[]{ll}\mathbb{c}_{1}^{\rm T}A=k(1,0),&\mathbb{c}_{2}^{\rm
T}A=k^{\prime}(0,-1),\end{array}$ (3.1)

with $k,k^{\prime}\in\mathbb{R}^{*}$ and
$\text{sgn}(k)=\text{sgn}(k^{\prime})$. With the above constraints, we will
consider the family of rank 2 indefinite theta functions 111From now on we
will omit the $+$ apex from the symbol $\Theta^{+}_{\mathbf{a},\mathbf{b}}$

$\Theta_{{\bf a},{\bf b}}(N\tau)=\sum\limits_{{\bf
n}\in\mathbb{Z}^{2}+\mathbb{a}}\left[\text{sgn}(B(\mathbb{c}_{1},\mathbb{n}))-\text{sgn}(B(\mathbb{c}_{2},\mathbb{n}))\right]e^{2\pi
iB({\bf n},{\bf b})}q^{NQ({\bf n})}$ (3.2)

with $N$ a positive integer, $\mathbb{b}\in\mathbb{R}^{2}$, and
$\mathbb{a}=(a_{1},a_{2})\in\mathbb{Q}^{2}\cap R(\mathbb{c}_{1})\cap
R(\mathbb{c_{2}})$. As we will show in later sections, in the cases considered
the components of umbral McKay-Thompson series can be rewritten in terms of
indefinite theta functions with such quadratic form $A$ and vectors
$\mathbb{c}_{1},\mathbb{c}_{2}$.
Let’s now define the relevant cone vertex algebras. Following the construction
in section 2.1, we start by defining the underlying lattice. We consider the
rank 2 lattice $L^{(N)}$ generated by $\epsilon_{1},\epsilon_{2}$ with the
bilinear form $\braket{\cdot,\cdot}$ specified by the matrix $A$ and a
positive integer $N$ as

$\braket{\epsilon_{i},\epsilon_{j}}=NA_{ij}.$ (3.3)

Consider the sublattice of $L^{(N)}$ given by the cone

$P^{(N)}=\\{\sum\limits_{i=1}^{2}\alpha_{i}\epsilon_{i}\in
L^{(N)}\otimes\mathbb{Q}:\alpha_{i}\geq 0,\forall i=1,2\\}$ (3.4)

and its shifted version $P^{(N)}+\gamma:=\\{\mu+\gamma|\mu\in P^{(N)}\\}$. As
described in section 2.1, $V_{P^{(N)}}$, generated by $\mathbb{v}_{\lambda}$
for $\lambda\in P^{(N)}$, is a sub-vertex operator algebra of $V_{L^{(N)}}$.
For $\mathbf{a}:=(a_{1},a_{2})\in\mathbb{Q}^{2}$ let’s define
$\rho^{+}_{\mathbf{a}}:=a_{1}\epsilon_{1}+a_{2}\epsilon_{2}$ and
$\rho^{-}_{\mathbf{a}}:=(1-a_{1})\epsilon_{1}+(1-a_{2})\epsilon_{2}$. To any
lattice $L^{(N)}$ we thus associate a module $V_{\mathbf{a}}^{(N)}$ given by
the following direct sum

$V_{\mathbf{a}}^{(N)}:=V_{P^{(N)}+\rho^{+}_{\mathbf{a}}}\oplus
V_{P^{(N)}+\rho^{-}_{\mathbf{a}}}$ (3.5)

where $V_{P^{(N)}+\rho^{+}_{\mathbf{a}}}$ and
$V_{P^{(N)}+\rho^{-}_{\mathbf{a}}}$ are the modules of the vertex algebra
$V_{P^{(N)}}$ built from $P^{(N)}+\rho^{+}_{\mathbf{a}}$ and
$P^{(N)}+\rho^{-}_{\mathbf{a}}$ respectively. Notice that, when
$\rho^{\pm}_{\mathbb{a}}\in{L^{(N)}}^{*}$, as is the case when
$a_{1},a_{2}\in\left\\{0,\frac{1}{N},\frac{2}{N},\ldots,\frac{N-1}{N}\right\\}$,
the modules $V_{P^{(N)}+\rho^{\pm}_{\mathbf{a}}}$ are untwisted. For
$\lambda=n_{1}\epsilon_{1}+n_{2}\epsilon_{2}+\rho^{\pm}_{\mathbf{a}}\in
P^{(N)}+\rho^{\pm}_{\mathbf{a}}$, $\mathbf{b}=(b_{1},b_{2})\in\mathbb{Q}^{2}$
and write $\mathbf{n}=(n_{1},n_{2})$, let’s define the operator
$g_{\mathbf{b}}:V_{\mathbf{a}}^{(N)}\to V_{\mathbf{a}}^{(N)}$ acting as

$g_{\bf{b}}(p\otimes\mathbf{v}_{\lambda}):=\begin{cases}e^{2\pi iB\left({\bf
n},{\bf b}\right)}p\otimes\mathbf{v}_{\lambda}&\text{if
}\mathbf{v}_{\lambda}\in V_{P^{(N)}+\rho^{+}_{\mathbf{a}}}\\\ -e^{-2\pi
iB\left({\bf n}+\mathbf{1},{\bf
b}\right)}p\otimes\mathbf{v}_{\lambda}&\text{if }\mathbf{v}_{\lambda}\in
V_{P^{(N)}+\rho^{-}_{\mathbf{a}}}\end{cases}$ (3.6)

The main object we will be interested in is the trace function

$\displaystyle T^{(N)}_{\mathbb{a},{\bf b}}(\tau)$
$\displaystyle:=\text{Tr}_{V_{\mathbf{a}}^{(N)}}\left(g_{\bf{b}}q^{L(0)-c/24}\right).$
(3.7)

Notice that, in general, when there are non-trivial vectors with non-positive
norm in $P+\rho^{\pm}_{\mathbb{a}}$ the trace will not converge. We will thus
restrict to matrices $A$ that are positive definite on
$P+\rho^{\pm}_{\mathbb{a}}$, i.e. $v^{T}Av>0$ $\forall v\in
P+\rho^{\pm}_{\mathbb{a}}$. Under such assumptions, we will now show how the
trace functions $T^{(N)}_{{\bf a},{\bf b}}(\tau)$ are related to $\Theta_{{\bf
a},{\bf b}}(N\tau)$.

######  Lemma 3.1.

Let $\mathbb{a}=(a_{1},a_{2})\in\mathbb{Q}^{2}$ with $0<a_{1},a_{2}<1$, $A$ a
symmetric matrix positive definite on $P^{(N)}+\rho^{\pm}_{\mathbb{a}}$, and
$\mathbf{c}_{1},\mathbf{c}_{2}\in\overline{C}_{Q}$ such that

$\begin{array}[]{ll}\mathbb{c}_{1}^{\rm T}A=k(1,0),&\mathbb{c}_{2}^{\rm
T}A=k^{\prime}(0,-1),\end{array}$ (3.8)

for some $k,k^{\prime}\in\mathbb{R}^{*}$ with ${\rm sgn}(k)={\rm
sgn}(k^{\prime})$. We have

$T^{(N)}_{\mathbb{a},{\bf b}}(\tau)={\rm sgn}(k)\frac{e^{-2\pi
iB(\mathbb{a},{\bf b})}}{2\eta(\tau)^{2}}\Theta_{\mathbb{a},{\bf b}}(N\tau),$
(3.9)

###### Proof.

Explicitly, (3.7) equals

$\begin{split}T^{(N)}_{\mathbb{a},{\bf
b}}(\tau)&=\frac{1}{\eta(\tau)^{2}}\left(\sum\limits_{\mu\in
P^{(N)}+\rho^{+}_{{\bf a}}}e^{2\pi iB({\bf n},{\bf
b})}q^{\frac{\braket{\mu,\mu}}{2}}-\sum\limits_{\mu\in P^{(N)}+\rho^{-}_{{\bf
a}}}e^{-2\pi iB({\bf n}+\mathbf{1},{\bf
b})}q^{\frac{\braket{\mu,\mu}}{2}}\right)\\\
&=\frac{1}{\eta(\tau)^{2}}\left(\sum_{\begin{subarray}{c}\mathbf{n}\in\mathbb{Z}^{2}\\\
n_{1},n_{2}\geq 0\end{subarray}}e^{2\pi iB({\bf n},{\bf
b})}q^{Q(\mathbf{n}+\mathbf{a})}-\sum_{\begin{subarray}{c}\mathbf{n}\in\mathbb{Z}^{2}\\\
n_{1},n_{2}\geq 0\end{subarray}}e^{-2\pi iB({\bf n}+\mathbf{1},{\bf
b})}q^{Q(\mathbf{n}+\mathbf{1}-\mathbf{a})}\right)\\\
&=\frac{1}{\eta(\tau)^{2}}\left(\sum_{\begin{subarray}{c}\mathbf{n}\in\mathbb{Z}^{2}\\\
n_{1},n_{2}\geq 0\end{subarray}}e^{2\pi iB({\bf n},{\bf
b})}q^{Q(\mathbf{n}+\mathbf{a})}-\sum_{\begin{subarray}{c}\mathbf{n}\in\mathbb{Z}^{2}\\\
n_{1},n_{2}\leq 0\end{subarray}}e^{2\pi iB({\bf n}-\mathbf{1},{\bf
b})}q^{Q(\mathbf{n}-\mathbf{1}+\mathbf{a})}\right)\\\
&=\frac{1}{\eta(\tau)^{2}}\left(\sum_{\begin{subarray}{c}\mathbf{n}\in\mathbb{Z}^{2}\\\
n_{1},n_{2}\geq
0\end{subarray}}-\sum_{\begin{subarray}{c}\mathbf{n}\in\mathbb{Z}^{2}\\\
n_{1},n_{2}<0\end{subarray}}\right)e^{2\pi iB({\bf n},{\bf
b})}q^{Q(\mathbf{n}+\mathbf{a})}\end{split}$ (3.10)

where we have written
$\mu=(n_{1}+a_{1})\epsilon_{1}+(n_{2}+a_{2})\epsilon_{2}$ and
$\mathbb{n}=(n_{1},n_{2})$.
On the other hand, since $0<a_{1},a_{2}<1$, the factor $\rho^{{\bf c},{\bf
c^{\prime}}}({\bf n})$ in (3.2) equals, using (3.1),

$\begin{split}\text{sgn}(B({\bf c},{\bf n}))-\text{sgn}(B({\bf
c^{\prime}},{\bf
n}))&=\text{sgn}(k)\text{sgn}(n_{1}+a_{1})+\text{sgn}(k^{\prime})\text{sgn}(n_{2}+a_{2})\\\
&=\begin{cases}2\text{sgn}(k)&\text{if }n_{1},n_{2}\geq 0\\\
-2\text{sgn}(k)&\text{if }n_{1},n_{2}<0\\\
0&\text{otherwise}\end{cases}\end{split}$ (3.11)

where we have also used $\text{sgn}(k)=\text{sgn}(k^{\prime})$. By comparison
it is immediate to see that the difference between the two sums in (3.10)
equals the indefinite theta function defined in (3.2) up to the overall
$\frac{e^{-2\pi iB(\bf{a},{\bf b})}}{2}\text{sgn}(k)$ factor. ∎

The result can be easily generalized to the cases where $a_{1}$ or $a_{2}$ is
equal to 0. In that case we will have an extra one-dimensional theta series
appearing in the right hand side of (3.9). In fact, let’s consider for example
the case $a_{1}=0,a_{2}\neq 0$. We have

$\begin{split}\text{sgn}(B({\bf c},{\bf n}))-\text{sgn}(B({\bf
c^{\prime}},{\bf
n}))&=\text{sgn}(k)\text{sgn}(n_{1})+\text{sgn}(k^{\prime})\text{sgn}(n_{2}+a_{2})\\\
&=\begin{cases}2\text{sgn}(k)&\text{if }n_{1}\geq 0,n_{2}>0\\\
-2\text{sgn}(k)&\text{if }n_{1},n_{2}<0\\\ \text{sgn}(k)&\text{if }n_{1}\geq
0,n_{2}=0\\\ -\text{sgn}(k)&\text{if }n_{1}<0,n_{2}=0\\\
0&\text{otherwise}\end{cases}\end{split}$ (3.12)

so we have to add some series to account the case $n_{2}=0$ correctly. An easy
calculation for the general case shows that we have the following

######  Theorem 3.2.

Let $\mathbb{a}=(a_{1},a_{2})\in\mathbb{Q}^{2}$ with $0\leq a_{1},a_{2}<1$,
$A$ a symmetric matrix positive definite on $P^{(N)}+\rho^{\pm}_{\mathbf{a}}$,
and $\mathbf{c}_{1},\mathbf{c}_{2}\in\overline{C}_{Q}$ such that

$\begin{array}[]{ll}\mathbb{c}_{1}^{T}A=k(1,0),&\mathbb{c}_{2}^{T}A=k^{\prime}(0,-1),\end{array}$
(3.13)

for some $k,k^{\prime}\in\mathbb{R}^{*}$ with
$\text{sgn}(k)=\text{sgn}(k^{\prime})$. We have

$\begin{split}T^{(N)}_{\mathbb{a},{\bf b}}(\tau)=&\text{sgn}(k)\frac{e^{-2\pi
iB(\mathbb{a},{\bf b})}}{2\eta(\tau)^{2}}\Biggl{[}\Theta_{\mathbb{a},{\bf
b}}(N\tau)+\delta_{a_{1}}\sum_{\begin{subarray}{c}n_{1}=0\\\
n_{2}\in\mathbb{Z}\end{subarray}}e^{2\pi
iB(\mathbb{n}+\mathbb{a},\mathbb{b})}q^{NQ(\mathbb{n}+\mathbb{a})}\\\
&+\delta_{a_{2}}\sum_{\begin{subarray}{c}n_{1}\in\mathbb{Z}\\\
n_{2}=0\end{subarray}}e^{2\pi
iB(\mathbb{n}+\mathbb{a},\mathbb{b})}q^{NQ(\mathbb{n}+\mathbb{a})}-\delta_{a_{1}}\delta_{a_{2}}e^{2\pi
iB(\mathbb{a},\mathbb{b})}q^{Q(\mathbb{a})}\Biggr{]},\end{split}$ (3.14)

where $\delta_{i}$ is the Kronecker delta $\delta_{i,0}$.

We will now show that the Appell-Lerch sums (1.5) and (1.6) can be written in
terms of the trace functions (3.7). These functions will also be important for
later sections. Let’s consider the Appell-Lerch sum Let’s first consider the
Appell-Lerch sum (1.5). We have the following

$\mu(z_{1},z_{2};\tau)=\frac{y_{1}^{\frac{1}{2}}}{\theta(z_{2};\tau)}\sum\limits_{n\in\mathbb{Z}}\frac{(-1)^{n}y^{n}_{2}q^{\frac{n(n+1)}{2}}}{1-y_{1}q^{n}}.$
(3.15)

We have the following

######  Corollary 3.3.

Let $\mathbb{\tilde{a}}=(\tilde{a}_{1},\tilde{a}_{2})\in\mathbb{Q}^{2}$ such
that $0<\tilde{a}_{1}<1$, $0\leq\tilde{a}_{2}-\tilde{a}_{1}+\frac{1}{2}<1$,
$\mathbb{\tilde{b}}=(\tilde{b}_{1},\tilde{b}_{2})\in\mathbb{R}^{2}$,
$N\in\mathbb{N}^{*}$ . Let $T^{(N)}_{\mathbb{a},\mathbb{b}}$ be the trace
function (3.7) associated to the lattice with quadratic form
$N\left(\begin{smallmatrix}1&1\\\ 1&0\end{smallmatrix}\right)$. We have

$\mu(\tilde{\mathbb{a}}N\tau+\tilde{\mathbb{b}};N\tau)=\frac{2q^{\frac{Na_{1}}{2}}\eta(\tau)^{2}}{\theta((a_{2}+a_{1}-1/2)\tau+b_{2}+b_{1}-1/2;N\tau)}q^{-NQ(\mathbb{a})}T^{(N)}_{\mathbb{a},\mathbb{b}}(\tau).$
(3.16)

where
$\mathbb{a}:=(a_{1},a_{2})=\left(\tilde{a}_{1},\tilde{a}_{2}-\tilde{a}_{1}+\frac{1}{2}\right)$
and
$\mathbb{b}:=(b_{1},b_{2})=\left(\tilde{b}_{1},\tilde{b}_{2}-\tilde{b}_{1}+\frac{1}{2}\right)$.

###### Proof.

The result follows by the rewriting of $\mu$ in terms of indefinite theta
functions. In fact, using equations (2.19) and (2.20), we have

$\mu(\tilde{\mathbb{a}}\tau+\tilde{\mathbb{b}};\tau)=\frac{q^{\frac{a_{1}}{2}}e^{\pi
ib_{1}}}{\theta((a_{2}+a_{1}-1/2)\tau+b_{2}+b_{1}-1/2;\tau)}e^{-2\pi
iB(\mathbb{a},\mathbb{b})}q^{-Q(\mathbb{a})}\Theta_{\mathbb{a},\mathbb{b}}(\tau).$
(3.17)

The choice $A=\left(\begin{smallmatrix}1&1\\\ 1&0\end{smallmatrix}\right)$,
$\mathbb{c}_{1}=(0,1)$, $\mathbb{c}_{2}=(-1,1)$ satisfies (3.1). Furthermore,
while $P^{(N)}$ has infinitely many vectors of the form $n_{2}\epsilon_{2}$
$\forall n_{2}\in\mathbb{Z}$, that have null norm, the scalar product
$\braket{\lambda,\lambda}=N(n_{1}+a_{1})^{2}+2N(n_{1}+a_{1})(n_{2}+a_{2})$ is
strictly positive $\forall\lambda\in P^{(N)}+\rho^{\pm}_{\mathbb{a}}$ with for
$0\leq a_{2}<1$, $0<a_{1}<1$. Thus, using Theorem 3.2, the conclusion follows.
∎

As already mentioned, all Ramanujan’s mock theta functions can be written in
terms of the Appell-Lerch sum (2.16) (up to modular functions) with the choice
of $\mathbb{z}=\mathbb{\tilde{a}}\tau+\mathbb{\tilde{b}}$ discussed above [6],
[4], [25], [26], thus they can be expressed in terms of cone vertex algebras
trace functions using the previous Corollary.

Let’s now consider the specialized Appell-Lerch sum (1.6). This function
appears in the definition of the meromorphic Jacobi forms associated to the
umbral McKay-Thompson series [11], [14], [35]. We will see that it also admits
an expression in terms of the trace function (3.7). Specifically, we have the
following

######  Corollary 3.4.

Let $a\in\mathbb{Q}^{*}$, $|a|<1$, $b\in\mathbb{R}$, $N\in\mathbb{N}^{*}$.
Consider the lattice with quadratic form $A=N\left(\begin{smallmatrix}2m&1\\\
1&0\end{smallmatrix}\right)$ with $m\in\mathbb{N}$, and the trace function
(3.7) $T^{(N)}_{\mathbb{a},\mathbb{b}}$ associated to it. We have

$\mu_{m,0}(aN\tau+b,N\tau)=-2f(b)q^{-2mNa^{2}}\eta(\tau)^{2}T^{(N)}_{\mathbb{a},\mathbb{b}}(\tau)-\sum\limits_{n\in\mathbb{Z}+a}e^{2\pi
inb}q^{2mNn^{2}}$ (3.18)

where $\mathbb{b}=(b,0)$ and $\mathbf{a}=(a,0)$, $f(b)=1$ when $a>0$ while
$\mathbf{a}=(1+a,0)$, $f(b)=e^{-4\pi ib}$ when $a<0$.

###### Proof.

Let’s show first that, for
$\left|\frac{\text{Im}(z)}{\text{Im}(\tau)}\right|<1$, $\text{Im}(z)\neq 0$,
we can write $\mu_{m,0}(z,\tau)$ in terms of indefinite theta functions
satisfying the conditions of Theorem 3.2. We write
$\mu_{m,0}(z,\tau)=f_{1}(z,\tau)+f_{2}(z,\tau)$, with

$\begin{array}[]{ll}f_{1}(z,\tau):=-\sum\limits_{k\in\mathbb{Z}}\frac{y^{2km}q^{mk^{2}}}{1-yq^{k}},&f_{2}(z,\tau):=-\sum\limits_{k\in\mathbb{Z}}\frac{y^{2km+1}q^{mk^{2}+k}}{1-yq^{k}}\end{array}$
(3.19)

Let us also set

$\begin{array}[]{lll}A=\begin{pmatrix}2m&1\\\
1&0\end{pmatrix},&\mathbb{c}_{1}=\begin{pmatrix}0\\\
1\end{pmatrix},&\mathbb{c}_{2}=\begin{pmatrix}-1\\\
2m\end{pmatrix}.\end{array}$ (3.20)

Let us first focus on the domain
$0<\frac{{\text{Im}}(z)}{{\text{Im}}(\tau)}<1$. Using the geometric series
expansion for the denominator in the range
$0<\frac{\text{Im}(z)}{\text{Im}(\tau)}<1$, we can rewrite $f_{1}$ as

$\begin{split}f_{1}(z,\tau)&=-\left(\sum\limits_{k,l\geq
0}-\sum\limits_{k,l<0}\right)y^{2km+l}q^{mk^{2}+kl}\\\
&=-\frac{1}{2}\sum\limits_{(k,l)\in\mathbb{Z}^{2}}\left[\text{sgn}\left(k+\frac{\text{Im}(z)}{\text{Im}(\tau)}\right)+\text{sgn}\left(l\right)\right]e^{2\pi
iB[(k,l),(z,0)]}q^{Q((k,l))}-\frac{1}{2}\sum\limits_{k\in\mathbb{Z}}y^{2mk}q^{mk^{2}}\end{split}$
(3.21)

where the second sum has to be introduced to fix the contributions for $l=0$.
It is then immediate to see that we can write

$f_{1}(z,\tau)=-\frac{1}{2}\Theta_{A,\mathbb{c}_{1},\mathbb{c}_{2}}(z,0;\tau)-\frac{1}{2}\sum\limits_{k\in\mathbb{Z}}y^{2mk}q^{mk^{2}}.$
(3.22)

Analogously

$\begin{split}f_{2}(z,\tau)&=-\left(\sum\limits_{k,l\geq
0}-\sum\limits_{k,l<0}\right)y^{2km+l+1}q^{mk^{2}+k(l+1)}\\\
&=-\left(\sum\limits_{k\geq 0,l\geq
1}-\sum\limits_{k<0,l<1}\right)y^{2km+l}q^{mk^{2}+kl}\\\
&=-\frac{1}{2}\sum\limits_{(k,l)\in\mathbb{Z}^{2}}\left[\text{sgn}\left(k+\frac{\text{Im}(z)}{\text{Im}(\tau)}\right)+\text{sgn}\left(l\right)\right]e^{2\pi
iB[(k,l),(z,0)]}q^{Q((k,l))}+\frac{1}{2}\sum\limits_{k\in\mathbb{Z}}y^{2mk}q^{mk^{2}}\\\
&=-\frac{1}{2}\Theta_{A,\mathbb{c}_{1},\mathbb{c}_{2}}(z,0;\tau)+\frac{1}{2}\sum\limits_{k\in\mathbb{Z}}y^{2mk}q^{mk^{2}}\end{split}$
(3.23)

where in the second line we have sent $l+1\to l$ and the second sum is again
due to the $l=0$ terms. Interestingly, when summing $f_{1}$ and $f_{2}$, only
the contribution of the indefinite theta survives, and we have

$\mu_{m,0}(z,\tau)=-\Theta_{A,\mathbb{c}_{1},\mathbb{c}_{2}}(z,0;\tau).$
(3.24)

In particular, notice that $\mathbb{c}_{1}$ and $\mathbb{c_{2}}$ satisfy
(3.1). When $z=a\tau+b$ with $a\in\mathbb{Q}$, $0<a<1$, $b\in\mathbb{R}$, for
any $N\in\mathbb{N}$, using Theorem 3.2 we have

$\begin{split}\mu_{m,0}(aN\tau+b,N\tau)&=-2e^{-4\pi
imab}q^{-2mNa^{2}}\Theta_{\mathbf{a},{\bf b}}(N\tau)\\\
&=-2q^{-2mNa^{2}}\eta(\tau)^{2}T^{(N)}_{\mathbb{a},\mathbb{b}}(\tau)-\sum\limits_{n\in\mathbb{Z}+a}e^{2\pi
inb}q^{2mNn^{2}}\end{split}$ (3.25)

with $\mathbb{a}=(a,0)$ and $\mathbb{b}=(b,0)$.
The same result still holds in the domain
$0<-\frac{\text{Im}(z)}{\text{Im}(\tau)}<1$. In this case we have $|yq^{k}|<1$
for $k>0$ and $|yq^{k}|>1$ for $k\leq 0$. Thus we get

$f_{1}(z,\tau)=-\left(\sum_{\begin{subarray}{c}k>0\\\ l\geq
0\end{subarray}}-\sum_{\begin{subarray}{c}k\leq 0\\\
l<0\end{subarray}}\right)y^{2km+l}q^{mk^{2}+kl}.$ (3.26)

On the other side, in this domain

$\text{sign}\left(k+\frac{\text{Im}(z)}{\text{Im}(\tau)}\right)+\text{sign}(l)=\begin{cases}2&\text{if
}k>0,l>0,\\\ 1&\text{if }k>0,l=0,\\\ -1&\text{if }k\leq 0,l=0,\\\ -2&\text{if
}k\leq 0,l<0,\\\ 0&\text{otherwise}.\end{cases}$ (3.27)

So we have again

$f_{1}(z,\tau)=-\frac{1}{2}\sum\limits_{(k,l)\in\mathbb{Z}^{2}}\left[\text{sgn}\left(k+\frac{\text{Im}(z)}{\text{Im}(\tau)}\right)+\text{sgn}\left(l\right)\right]e^{2\pi
iB[(k,l),(z,0)]}q^{Q((k,l))}-\frac{1}{2}\sum\limits_{k\in\mathbb{Z}}y^{2mk}q^{mk^{2}}.$
(3.28)

Proceeding in the same way for $f_{2}(z,\tau)$, it is possible to show that
equation (3.24) still holds in the domain
$0<-\frac{\text{Im}(z)}{\text{Im}(\tau)}<1$. In particular, we have

$\mu_{m,0}(a\tau+b,\tau)=-2e^{-4\pi imab}q^{-2ma^{2}}\Theta_{\mathbf{a},{\bf
b}}(\tau)$ (3.29)

where $\mathbf{a}=(1+a,0)$ and $\mathbf{b}=(b,0)$ and we have used the
property
$\Theta_{\mathbf{a},\mathbf{b}}=\Theta_{\mathbf{a}+\mathbf{s},\mathbf{b}}$ for
all $\mathbf{s}\in\mathbb{Z}^{2}$. Notice that $1+a>0$ and thus we can use
Theorem 3.2. We then get

$\mu_{m,0}(aN\tau+b,N\tau)=-2e^{-4\pi
imb}q^{-2mNa^{2}}\eta(\tau)^{2}T^{(N)}_{\mathbb{a},\mathbb{b}}(\tau)-\sum\limits_{n\in\mathbb{Z}+a}e^{2\pi
inb}q^{2mNn^{2}}.$ (3.30)

∎

##  4 Umbral McKay-Thompson Series, Mock Theta Functions and Indefinite Thetas

In this section we will write the umbral McKay-Thompson series appearing for
lambency $\ell=8,12,16$ in terms of mock theta functions, eta quotients and
Jacobi theta functions. In particular, all the mock theta functions
encountered in these cases can be rewritten in terms of the indefinite theta
functions [4], [6], [25], [26], with data satisfying the properties of Theorem
3.2. All the indefinite theta function have bilinear form
$A=\left(\begin{smallmatrix}1&1\\\ 1&0\end{smallmatrix}\right)$ and vectors
$\mathbb{c}_{1}=\left(0,1\right),\mathbb{c}_{2}=(-1,1)$. The relations between
mock theta functions and indefinite theta functions relevant for the cases
considered are collected in appendix A.
In some cases it is not possible to directly specify the individual Umbral
McKay-Thompson series in terms of mock theta functions. When this happens, we
will specify suitable linear combinations of the umbral McKay-Thompson series
with disjoint sets of $q$-powers. In this way, the individual series can be
retrieved by projecting onto the desired set of $q$-powers. In fact, given an
instance of umbral moonshine with Coxeter number $m$, the $r$-th component of
the corresponding mock modular form will have a series expansion in which the
appearing $q$-powers will have the general form $q^{-\frac{r^{2}}{4m}+N}$ with
$N\in\mathbb{N}$. Thus, the $q$-series of components with different values of
$r^{2}\mod 4m$ will have no common $q$-powers and therefore a linear
combination of such components contains the same information as the set of the
individual components.
The expressions provided are obtained by making use of the explicit
specification of some umbral McKay-Thompson series in terms of mock theta
functions combined with the multiplicative relations among different
lambencies, as provided in [11].

###  4.1 Lambency Eight

Lambency $\ell=8$ corresponds to the Niemeier root system $A^{2}_{7}D^{2}_{5}$
with umbral group $Dih_{4}$. The McKay-Thompson series appearing for $\ell=8$
can be expressed in terms of mock theta functions and eta quotients by making
use of the multiplicative relations with $\ell=4$ and the explicit
specifications in [11]. In particular, we encounter the order 2 mock theta
functions

$\displaystyle
A(q):=\sum\limits_{n=0}^{\infty}\frac{q^{n+1}(-q^{2};q^{2})_{n}}{(q;q^{2})_{n+1}},$
(4.1) $\displaystyle
B(q):=\sum\limits_{n=0}^{\infty}\frac{q^{n}(-q;q^{2})_{n}}{(q;q^{2})_{n+1}},$
(4.2)

and the order 8 mock theta functions

$\displaystyle
S_{0}(q):=\sum\limits_{n=0}^{\infty}\frac{q^{n^{2}}(-q;q^{2})_{n}}{(-q^{2};q^{2})_{n}},$
(4.3) $\displaystyle
S_{1}(q):=\sum\limits_{n=0}^{\infty}\frac{q^{n(n+2)}(-q;q^{2})_{n}}{(-q^{2};q^{2})},$
(4.4) $\displaystyle
T_{0}(q):=\sum\limits_{n=0}^{\infty}\frac{q^{(n+1)(n+2)}(-q^{2};q^{2})_{n}}{(-q;q^{2})_{n+1}},$
(4.5) $\displaystyle
T_{1}(q):=\sum\limits_{n=0}^{\infty}q^{n(n+1)}\frac{(-q^{2};q^{2})_{n}}{(-q;q^{2})_{n+1}}.$
(4.6)

The expressions specifying all the components for all conjugacy classes of
$Dih_{4}$ in terms of the previous functions are

$\begin{split}&(H^{(8)}_{1A,1}-H^{(8)}_{1A,7})(2\tau)=H^{(4)}_{2C,1}(\tau)=q^{-\frac{1}{16}}(-2S_{0}(q)+4T_{0}(q)),\\\
&H^{(8)}_{1A,2}(\tau)=H^{(8)}_{1A,6}(\tau)=4q^{-\frac{1}{4}}A(q),\\\
&H^{(8)}_{1A,4}(\tau)=4q^{\frac{1}{2}}B(q),\\\
&(H^{(8)}_{1A,3}-H^{(8)}_{1A,5})(2\tau)=H^{(4)}_{2C,3}(\tau)=q^{\frac{7}{16}}(2S_{1}(q)-4T_{1}(q)),\\\
&(H^{(8)}_{2BC,1}-H^{(8)}_{2BC,7})(2\tau)=H^{(4)}_{4C,1}(\tau)=-2q^{-\frac{1}{16}}S_{0}(q)\\\
&(H^{(8)}_{2BC,3}-H^{(8)}_{2BC,5})(2\tau)=H^{(4)}_{4C,3}(\tau)=2q^{\frac{7}{16}}S_{1}(q)\\\
&(H^{(8)}_{4A,1}-H^{(8)}_{4A,7}-H^{(8)}_{4A,3}+H^{(8)}_{4A,5})(2\tau)=H^{(4)}_{4B,1}(\tau)-H^{(4)}_{4B,3}(\tau),\\\
\end{split}$ (4.7)

together with the identities $H^{8}_{2BC,r}=H^{8}_{4A,r}=0$ for $r$ even, and
the pairing relation

$H^{(8)}_{2A,r}+(-1)^{r}H^{(8)}_{1A,r}=0.$ (4.8)

We can furthermore express the difference between components $r=1$ and $r=3$
for $\ell=4$ appearing in the relation for class 4A in terms of an eta
quotient 222This formula has a typo in the original paper. [11]

$H^{(4)}_{4B,1}(\tau)-H^{(4)}_{4B,3}(\tau)=-2\frac{\eta(\frac{\tau}{2})\eta(2\tau)^{4}}{\eta(\tau)^{2}\eta(4\tau)^{2}}.$
(4.9)

We can further simplify the previous expressions by making use of the
following lemma

######  Lemma 4.1.

The order 8 mock theta functions $S_{0},S_{1},T_{0},T_{1}$ satisfy

$\begin{split}&S_{0}(q)+2T_{0}(q)=\frac{q^{\frac{1}{16}}}{2}\left(\frac{\eta(\frac{\tau}{2})^{3}}{\eta(\tau)\eta(2\tau)}+\frac{\eta(\tau)^{8}}{\eta(\frac{\tau}{2})^{3}\eta(2\tau)^{4}}\right),\\\
&S_{1}(q)+2T_{1}(q)=\frac{q^{-\frac{7}{16}}}{2}\left(-\frac{\eta(\frac{\tau}{2})^{3}}{\eta(\tau)\eta(2\tau)}+\frac{\eta(\tau)^{8}}{\eta(\frac{\tau}{2})^{3}\eta(2\tau)^{4}}\right).\end{split}$
(4.10)

###### Proof.

Using the expression in appendix A of [6]

$\begin{split}&S_{0}(q)=-2iq^{\frac{1}{2}}g_{2}(iq^{\frac{1}{2}};q^{4})+\frac{(-iq^{\frac{1}{2}};-q)_{\infty}^{2}(-q;-q)_{\infty}(-q^{3};q^{8})_{\infty}(-q^{5};q^{8})_{\infty}}{(-q;q^{4})_{\infty}(-q^{3};q^{4})_{\infty}(q^{4};q^{4})_{\infty}},\\\
&S_{1}(q)=-2iq^{\frac{1}{2}}g_{2}(-iq^{\frac{3}{2}};q^{4})+\frac{(-iq^{\frac{1}{2}};-q)_{\infty}^{2}(-q;-q)_{\infty}(-q;q^{8})_{\infty}(-q^{7};q^{8})_{\infty}}{(-q;q^{4})_{\infty}(-q^{3};q^{4})_{\infty}(q^{4};q^{4})_{\infty}},\\\
&\begin{split}T_{0}(q)=&iq^{\frac{1}{2}}g_{2}(iq^{\frac{1}{2}};q^{4})-\frac{(-iq^{\frac{1}{2}};-q)_{\infty}^{2}(-q;-q)_{\infty}(-q^{3};q^{8})_{\infty}(-q^{5};q^{8})_{\infty}}{2(-q;q^{4})_{\infty}(-q^{3};q^{4})_{\infty}(q^{4};q^{4})_{\infty}}\\\
&+\frac{1}{4}\frac{(q^{\frac{1}{2}};q^{\frac{1}{2}})^{3}_{\infty}}{(q;q)_{\infty}(q^{2};q^{2})_{\infty}}+\frac{1}{4}\frac{(q;q)^{8}_{\infty}}{(q^{\frac{1}{2}};q^{\frac{1}{2}})^{3}_{\infty}(q^{2};q^{2})^{4}_{\infty}},\end{split}\\\
&\begin{split}T_{1}(q)=&iq^{\frac{1}{2}}g_{2}(-iq^{\frac{3}{2}};q^{4})-\frac{(-iq^{\frac{1}{2}};-q)_{\infty}^{2}(-q;-q)_{\infty}(-q;q^{8})_{\infty}(-q^{7};q^{8})_{\infty}}{2(-q;q^{4})_{\infty}(-q^{3};q^{4})_{\infty}(q^{4};q^{4})_{\infty}}\\\
&-\frac{q^{-\frac{1}{2}}}{4}\frac{(q^{\frac{1}{2}};q^{\frac{1}{2}})^{3}_{\infty}}{(q;q)_{\infty}(q^{2};q^{2})_{\infty}}+\frac{q^{-\frac{1}{2}}}{4}\frac{(q;q)^{8}_{\infty}}{(q^{\frac{1}{2}};q^{\frac{1}{2}})^{3}_{\infty}(q^{2};q^{2})^{4}_{\infty}},\\\
\end{split}\end{split}$ (4.11)

where $g_{2}$ is the universal mock theta function

$g_{2}(\zeta;q)=\sum\limits_{n=0}^{\infty}\frac{(-q)_{n}q^{\frac{n(n+1)}{2}}}{(\zeta)_{n+1}(\zeta^{-1}q)_{n+1}},$
(4.12)

we can express $S_{0}$ ($S_{1}$ respectively) in terms of $T_{0}$ ($T_{1}$)
and eta quotients. In fact, we can express the linear combinations
$S_{0}+2T_{0}$, $S_{1}+2T_{1}$ as

$\begin{split}&S_{0}(q)+2T_{0}(q)=\frac{1}{2}\frac{(q^{\frac{1}{2}};q^{\frac{1}{2}})^{3}_{\infty}}{(q;q)_{\infty}(q^{2};q^{2})_{\infty}}+\frac{1}{2}\frac{(q;q)^{8}_{\infty}}{(q^{\frac{1}{2}};q^{\frac{1}{2}})^{3}_{\infty}(q^{2};q^{2})^{4}_{\infty}},\\\
&S_{1}(q)+2T_{1}(q)=-\frac{q^{-\frac{1}{2}}}{2}\frac{(q^{\frac{1}{2}};q^{\frac{1}{2}})^{3}_{\infty}}{(q;q)_{\infty}(q^{2};q^{2})_{\infty}}+\frac{q^{-\frac{1}{2}}}{2}\frac{(q;q)^{8}_{\infty}}{(q^{\frac{1}{2}};q^{\frac{1}{2}})^{3}_{\infty}(q^{2};q^{2})^{4}_{\infty}},\end{split}$
(4.13)

from which the conclusion since $\eta(\tau)=q^{\frac{1}{24}}(q;q)_{\infty}$. ∎

Using the previous relations we can rewrite the expressions for the components
specifying the Umbral McKay-Thompson series for all conjugacy class of the
Umbral group $Dih_{4}$ as

$\begin{split}&(H^{(8)}_{1A,1}-H^{(8)}_{1A,7}-H^{(8)}_{1A,3}+H^{(8)}_{1A,5})(2\tau)=q^{-\frac{1}{16}}8T_{0}(q)+q^{\frac{7}{16}}8T_{1}(q)-2\frac{\eta(\tau)^{8}}{\eta(\frac{\tau}{2})^{3}\eta(2\tau)^{4}},\\\
&(H^{(8)}_{2A,1}-H^{(8)}_{2A,7}-H^{(8)}_{2A,3}+H^{(8)}_{2A,5})(2\tau)=q^{-\frac{1}{16}}8T_{0}(q)+q^{\frac{7}{16}}8T_{1}(q)-2\frac{\eta(\tau)^{8}}{\eta(\frac{\tau}{2})^{3}\eta(2\tau)^{4}},\\\
&(H^{(8)}_{2BC,1}-H^{(8)}_{2BC,7}-H^{(8)}_{2BC,3}+H^{(8)}_{2BC,5})(2\tau)=4q^{-\frac{1}{16}}T_{0}(q)+4q^{\frac{7}{16}}T_{1}(q)-2\frac{\eta(\tau)^{8}}{\eta(\frac{\tau}{2})^{3}\eta(2\tau)^{4}},\\\
&(H^{(8)}_{4A,1}-H^{(8)}_{4A,7}-H^{(8)}_{4A,3}+H^{(8)}_{4A,5})(2\tau)=-2\frac{\eta(\frac{\tau}{2})\eta(\tau)^{4}}{\eta(\tau)^{2}\eta(4\tau)^{2}},\\\
&H^{(8)}_{1A,2}(\tau)=H^{(8)}_{1A,6}(\tau)=4q^{-\frac{1}{8}}A(q),\\\
&H^{(8)}_{1A,4}(\tau)=4q^{\frac{1}{2}}B(q).\\\ \end{split}$ (4.14)

We can finally use the relations collected in appendix A to write all the
appearing mock theta functions in terms of indefinite theta functions.

######  Proposition 4.2.

The expression specifying all the Mc-Kay Thompson series for $\ell=8$ at all
conjugacy classes of the umbral group $Dih_{4}$ are

$\begin{split}&(H^{(8)}_{1A,1}-H^{(8)}_{1A,7}-H^{(8)}_{1A,3}+H^{(8)}_{1A,5})(2\tau)=\\\
&8e^{-\frac{3\pi
i}{4}}\frac{\eta(4\tau)}{2\eta(2\tau)\eta(8\tau)}\left[\Theta_{\left(\frac{5}{8},\frac{1}{8}\right),\left(\frac{1}{2},0\right)}(8\tau)-i\Theta_{\left(\frac{7}{8},\frac{3}{8}\right),\left(\frac{1}{2},0\right)}(8\tau)\right]-2\frac{\eta(\tau)^{8}}{\eta(\frac{\tau}{2})^{3}\eta(2\tau)^{4}},\\\
&(H^{(8)}_{2A,1}-H^{(8)}_{2A,7}-H^{(8)}_{2A,3}+H^{(8)}_{2A,5})(2\tau)=\\\
&8e^{-\frac{3\pi
i}{4}}\frac{\eta(4\tau)}{2\eta(2\tau)\eta(8\tau)}\left[\Theta_{\left(\frac{5}{8},\frac{1}{8}\right),\left(\frac{1}{2},0\right)}(8\tau)-i\Theta_{\left(\frac{7}{8},\frac{3}{8}\right),\left(\frac{1}{2},0\right)}(8\tau)\right]-2\frac{\eta(\tau)^{8}}{\eta(\frac{\tau}{2})^{3}\eta(2\tau)^{4}},\\\
&(H^{(8)}_{2BC,1}-H^{(8)}_{2BC,7}-H^{(8)}_{2BC,3}+H^{(8)}_{2BC,5})(2\tau)=\\\
&4e^{-\frac{3\pi
i}{4}}\frac{\eta(4\tau)}{2\eta(2\tau)\eta(8\tau)}\left[\Theta_{\left(\frac{5}{8},\frac{1}{8}\right),\left(\frac{1}{2},0\right)}(8\tau)-i\Theta_{\left(\frac{7}{8},\frac{3}{8}\right),\left(\frac{1}{2},0\right)}(8\tau)\right]-2\frac{\eta(\tau)^{8}}{\eta(\frac{\tau}{2})^{3}\eta(2\tau)^{4}},\\\
&(H^{(8)}_{4A,1}-H^{(8)}_{4A,7}-H^{(8)}_{4A,3}+H^{(8)}_{4A,5})(2\tau)=-2\frac{\eta(\frac{\tau}{2})\eta(2\tau)^{4}}{\eta(\tau)^{2}\eta(4\tau)^{2}},\\\
&H^{(8)}_{1A,2}(\tau)=H^{(8)}_{1A,6}(\tau)=2e^{-\frac{3\pi
i}{4}}\frac{\eta(4\tau)}{\eta(2\tau)^{2}}\Theta_{\left(\frac{3}{4},\frac{1}{4}\right),\left(0,\frac{1}{2}\right)}(4\tau),\\\
&H^{(8)}_{1A,4}(\tau)=2e^{-\frac{3\pi
i}{4}}\frac{\eta(2\tau)}{\eta(\tau)\eta(4\tau)}\Theta_{\left(\frac{3}{4},\frac{1}{2}\right),\left(0,\frac{1}{2}\right)}(4\tau).\\\
\end{split}$ (4.15)

We observe, in particular, that the indefinite theta functions appearing for
the same components at different conjugacy classes are the same, thus the
indefinite theta structure is invariant under the action of the umbral group.

###  4.2 Lambency Twelve

At lambency $\ell=12$, we have Niemeier root system $A_{11}D_{7}E_{6}$ and
umbral group $\mathbb{Z}_{2}$. The mock theta functions relevant in this case
are the order 3

$\displaystyle
f(q):=\sum\limits_{n=0}^{\infty}\frac{q^{n^{2}}}{(-q;q)_{n}^{2}},$ (4.16)
$\displaystyle\omega(q):=\sum\limits_{n=0}^{\infty}\frac{q^{2n(n+1)}}{(q;q^{2})^{2}_{n+1}},$
(4.17)

and the order 6

$\displaystyle\sigma(q):=\sum\limits_{n=0}^{\infty}\frac{q^{\frac{(n+1)(n+2)}{2}}(-q;q)_{n}}{(q;q^{2})_{n+1}},$
(4.18)
$\displaystyle\psi_{6}(q):=\sum\limits_{n=0}^{\infty}\frac{(-1)^{n}q^{(n+1)^{2}}(q;q^{2})_{n}}{(-q;q)_{2n+1}}.$
(4.19)

All the McKay-Thompson series for conjugacy class $2A$ are specified in terms
of the ones for conjugacy class $1A$ by the pairing relation
$H^{(12)}_{2A,r}+(-1)^{r}H^{(12)}_{1A,r}=0$. In [11] we find the following
identities in terms of mock theta functions

$\begin{split}&H^{(12)}_{1A,2}(\tau)=H^{(12)}_{1A,10}(\tau)=-2q^{-\frac{4}{48}}\sigma(q),\\\
&H^{(12)}_{1A,4}(\tau)=H^{(12)}_{1A,8}(\tau)=2q^{\frac{2}{3}}\omega(q).\\\
\end{split}$ (4.20)

The multiplicative relations between $\ell=12$ and $\ell=6$

$\begin{split}&(H^{(12)}_{1A,1}-H^{(12)}_{1A,11})(2\tau)=H^{(6)}_{2B,1}(\tau),\\\
&(H^{(12)}_{1A,5}-H^{(12)}_{1A,7})(2\tau)=H^{(6)}_{2B,5}(\tau),\\\
&(H^{(12)}_{1A,3}-H^{(12)}_{1A,9})(2\tau)=H^{(6)}_{2B,3}(\tau),\end{split}$
(4.21)

together with the multiplicative relations for $\ell=6$

$\begin{split}&H^{(6)}_{2B,1}(3\tau)-H^{(6)}_{2B,3}(3\tau)+H^{(6)}_{2B,5}(3\tau)=H^{(2)}_{6A,1}(\tau),\\\
&H^{(6)}_{2B,1}(2\tau)-H^{(6)}_{2B,5}(2\tau)=H^{(3)}_{4C}(\tau),\end{split}$
(4.22)

and the following further explicit expressions in terms of mock theta
functions

$\begin{split}&H^{(3)}_{4C,1}(\tau)=-2q^{-\frac{1}{12}}f(q^{2}),\\\
&H^{(6)}_{2B,3}(\tau)=-2q^{-\frac{3}{8}}\psi_{6}(q),\end{split}$ (4.23)

allow to specify all the components with $r$ odd in terms of mock theta
functions and the function $H^{(2)}_{6A,1}$ as

$\begin{split}&\left(H^{(12)}_{1A,1}-H^{(12)}_{1A,11}\right)(2\tau)=\frac{1}{2}H^{(2)}_{6A,1}\left(\frac{\tau}{3}\right)-q^{-\frac{3}{8}}\psi_{6}(q)-q^{-\frac{1}{24}}f(q),\\\
&\left(H^{(12)}_{1A,5}-H^{(12)}_{1A,7}\right)(2\tau)=\frac{1}{2}H^{(2)}_{6A,1}\left(\frac{\tau}{3}\right)-q^{-\frac{3}{8}}\psi_{6}(q)+q^{-\frac{1}{24}}f(q),\\\
&\left(H^{(12)}_{1A,3}-H^{(12)}_{1A,9}\right)(2\tau)=H^{(6)}_{2B,3}(\tau)=-2q^{-\frac{3}{8}}\psi_{6}(q).\end{split}$
(4.24)

Finally, the multiplicative relations with $\ell=4$ give the component $r=6$

$H^{(12)}_{1A,6}(3\tau)=H^{(12)}_{1A,2}(3\tau)+H^{(12)}_{1A,10}(3\tau)-H^{(4)}_{3A,2}(\tau)=-4q^{\frac{1}{4}}\sigma(q^{3})-H^{(4)}_{3A,2}(\tau).$
(4.25)

For $H^{(4)}_{3A,2}$ and $H^{(2)}_{6A,1}$ a simple expression in terms of mock
theta functions and/or eta quotients is not known, so we need to deal with
them separately. It is convenient to write everything in terms of $\ell=4$
functions by using the multiplicative relation

$\left(H^{(4)}_{3A,1}-H^{(4)}_{3A,3}\right)(2\tau)=H^{(2)}_{6A,1}(\tau).$
(4.26)

Components of the McKay-Thompson series at $\ell=4$ for conjugacy class $3A$
are specified by different powers of $y=e^{2\pi iz}$ in [14]

$2i\theta_{1}(3\tau,6z)\theta_{1}(z,\tau)^{-1}\theta_{1}(3\tau,3z)^{-1}\eta(\tau)^{3}=-2\mu^{0}_{4,0}(z,\tau)-2\mu^{1}_{4,0}(z,\tau)+\sum\limits_{r\mod
8}H^{(4)}_{3A,r}\theta_{4,r}(z,\tau)$ (4.27)

Where we have made use of the following functions

$\begin{split}&\theta_{1}(z,\tau):=-iq^{\frac{1}{8}}y^{\frac{1}{2}}\prod\limits_{n>0}(1-y^{-1}q^{n-1})(1-yq^{n})(1-q^{n}),\\\
&\theta_{2}(z,\tau):=q^{\frac{1}{8}}y^{\frac{1}{2}}\prod\limits_{n>0}(1+y^{-1}q^{n-1})(1+yq^{n})(1-q^{n}),\\\
&\theta_{m,r}(z,\tau):=\sum\limits_{k\in\mathbb{Z}}y^{2mk+r}q^{\frac{(2mk+r)^{2}}{4m}},\\\
&\mu^{k}_{m,0}(z,\tau):=\frac{1}{2}\left(\mu_{m,0}(z,\tau)+(-1)^{k}\mu_{m,0}\left(z,\tau+\frac{1}{2}\right)\right).\end{split}$
(4.28)

We recall that the function $\mu_{m,0}(z,\tau)$, defined in (1.6), has an
expression in terms of indefinite theta functions. In fact, for
$\left|\frac{\text{Im}(z)}{\text{Im}(\tau)}\right|<1$, $\text{Im}(z)\neq 0$,
setting $z=a\tau+b$ with $a\in\mathbb{Q}^{*}$, $|a|<1$, $b\in\mathbb{R}$ we
can use the result in equation (3.24) to write

$\begin{split}\sum\limits_{r\mod
8}H^{(4)}_{3A,r}(\tau)\theta_{4,r}(a\tau+b,\tau)=&-2\Theta^{+}_{A^{(4)},\mathbb{c}^{(4)}_{1},\mathbb{c}^{(4)}_{2}}(a\tau+b,0;\tau)\\\
&+2i\theta_{1}(6a\tau+6b,3\tau)\theta_{1}(a\tau+b,\tau)^{-1}\theta_{1}(3a\tau+3b,3\tau)^{-1}\eta(\tau)^{3}.\end{split}$
(4.29)

with $A^{(m)}=\left(\begin{smallmatrix}2m&1\\\ 1&0\end{smallmatrix}\right)$,
$\mathbb{c}_{1}^{(m)}=\left(0,1\right)$,
$\mathbb{c}_{2}^{(m)}=\left(-1,2m\right)$. Notice also that equation (4.27)
implies that $H^{(4)}_{3A,r}$ have even coefficients. We can thus rewrite the
umbral McKay Thompson series in terms of indefinite theta functions using the
relations in appendix A as follows.

######  Proposition 4.3.

The expression specifying all the McKay-Thompson series for $\ell=12$ at all
conjugacy classes of the umbral group $\mathbb{Z}_{2}$ are given by

$\begin{split}&\begin{split}(H^{(12)}_{1A,1}-H^{(12)}_{1A,11})(2\tau)=&-e^{-\frac{7\pi
i}{6}}\frac{\eta(\tau)\eta(6\tau)}{2\eta(2\tau)\eta(3\tau)^{2}}\Theta_{\left(\frac{1}{3},\frac{1}{2}\right),\left(\frac{1}{2},\frac{1}{2}\right)}(3\tau)+\frac{2e^{-\frac{5\pi
i}{6}}}{\eta(\tau)}\Theta_{\left(\frac{2}{3},\frac{1}{6}\right),\left(\frac{1}{2},0\right)}(3\tau)\\\
&-\frac{\eta(3\tau)^{4}}{\eta(\tau)\eta(6\tau)^{2}}+\frac{\left(H^{(4)}_{3A,1}-H^{(4)}_{3A,3}\right)}{2}\left(\frac{2}{3}\tau\right)\end{split}\\\
&\begin{split}(H^{(12)}_{1A,5}-H^{(12)}_{1A,7})(2\tau)=&-e^{-\frac{7\pi
i}{6}}\frac{\eta(\tau)\eta(6\tau)}{2\eta(2\tau)\eta(3\tau)^{2}}\Theta_{\left(\frac{1}{3},\frac{1}{2}\right),\left(\frac{1}{2},\frac{1}{2}\right)}(3\tau)\\\
&-\frac{2e^{-\frac{5\pi
i}{6}}}{\eta(\tau)}\Theta_{\left(\frac{2}{3},\frac{1}{6}\right),\left(\frac{1}{2},0\right)}(3\tau)+\frac{\eta(3\tau)^{4}}{\eta(\tau)\eta(6\tau)^{2}}+\frac{\left(H^{(4)}_{3A,1}-H^{(4)}_{3A,3}\right)}{2}\left(\frac{2}{3}\tau\right),\end{split}\\\
&(H^{(12)}_{1A,3}-H^{(12)}_{1A,9})(2\tau)=-e^{-\frac{7\pi
i}{6}}\frac{\eta(\tau)\eta(6\tau)}{\eta(2\tau)\eta(3\tau)^{2}}\Theta_{\left(\frac{1}{3},\frac{1}{2}\right),\left(\frac{1}{2},\frac{1}{2}\right)}\left(3\tau\right),\\\
&H^{(12)}_{1A,2}(\tau)=H^{(12)}_{1A,10}(\tau)=-2e^{-i\frac{\pi}{2}}\frac{\eta(2\tau)\eta(3\tau)}{2\eta(\tau)\eta(6\tau)^{2}}\Theta_{\left(\frac{1}{2},\frac{1}{6}\right),\left(0,\frac{1}{2}\right)}(6\tau),\\\
&H^{(12)}_{1A,4}(\tau)=H^{(12)}_{1A,8}(\tau)=\frac{2e^{-\frac{\pi
i}{2}}}{\eta(\tau)}\Theta_{\left(\frac{1}{2},\frac{1}{3}\right),\left(0,\frac{1}{2}\right)}(6\tau)+2\frac{\eta(6\tau)^{4}}{\eta(2\tau)\eta(3\tau)^{2}},\\\
&H^{(12)}_{1A,6}(3\tau)=-2e^{\frac{-\pi
i}{2}}\frac{\eta(6\tau)\eta(9\tau)}{\eta(3\tau)\eta(18\tau)^{2}}\Theta_{\left(\frac{1}{2},\frac{1}{6}\right),\left(0,\frac{1}{2}\right)}\left(18\tau\right)-H^{(4)}_{3A,2}(\tau).\\\
\end{split}$ (4.30)

together with the pairing relation
$H^{(12)}_{2A,r}+(-1)^{r}H^{(12)}_{1A,r}=0$.

Again, we observe that in each component $H^{(12)}_{g,r}$ for given $r$, the
indefinite theta part is invariant under the action of the umbral group, and
the only difference between conjugacy class $1A$ and conjugacy class $2A$ is
at most an overall minus sign.

###  4.3 Lambency Sixteen

At $\ell=16$ we have Niemeier root system $A_{15}D_{9}$ and umbral group
$\mathbb{Z}_{2}$. Again, all the McKay-Thompson series for conjugacy class
$2A$ are related to the one for class $1A$ by the pairing relation
$H^{(16)}_{2A,r}+(-1)^{r}H^{(16)}_{1A,r}=0$. As a result, we only need to
specify $H^{(16)}_{1A,r}$ explicitly. Using the expressions in [11] we can
specify all the components of the Umbral McKay-Thompson series for class $1A$
in terms of order 8 mock thetas: $T_{0}(q)$ and $T_{1}(q)$ already defined in
the previous section and

$\displaystyle
U_{0}(q):=\sum\limits_{n=0}^{\infty}\frac{q^{n^{2}}(-q;q^{2})_{n}}{(-q^{4},q^{4})_{n}},$
(4.31) $\displaystyle
V_{0}(q):=-1+2\sum\limits_{n=0}^{\infty}\frac{q^{n^{2}}(-q;q^{2})_{n}}{(q;q^{2})_{n+1}},$
(4.32) $\displaystyle
V_{1}(q):=\sum\limits_{n=0}^{\infty}\frac{q^{(n+1)^{2}}(-q;q^{2})_{n}}{(q;q^{2})_{n+1}},$
(4.33)

as

$\begin{split}&H^{(16)}_{1A,2}(\tau)=H^{(16)}_{1A,14}(\tau)=2q^{-\frac{1}{16}}T_{0}(-q),\\\
&H^{(16)}_{1A,4}(\tau)=H^{(16)}_{1A,12}(\tau)=2q^{-\frac{1}{4}}V_{1}(q),\\\
&H^{(16)}_{1A,6}(\tau)=H^{(16)}_{1A,10}(\tau)=2q^{\frac{7}{16}}T_{1}(-q),\\\
&H^{(16)}_{1A,8}(\tau)=V_{0}(q),\\\
&\sum\limits_{n=0,7}(-1)^{n}H^{(16)}_{1A,2n+1}(8\tau)=H^{(2)}_{8A,1}(\tau)=-2q^{-\frac{1}{8}}U_{0}(q).\end{split}$
(4.34)

Using the relations in appendix A we easily obtain

######  Proposition 4.4.

The expression specifying all the Mc-Kay Thompson series for $\ell=16$ at all
conjugacy classes of the umbral group $\mathbb{Z}_{2}$ are

$\begin{split}&H^{(16)}_{1A,2}\left(\tau-\frac{1}{2}\right)=H^{(16)}_{1A,14}\left(\tau-\frac{1}{2}\right)=2e^{-\frac{3\pi
i}{4}}\frac{\eta(4\tau)}{2\eta(2\tau)\eta(8\tau)}\Theta_{\left(\frac{5}{8},\frac{1}{8}\right),\left(\frac{1}{2},0\right)}(8\tau),\\\
&H^{(16)}_{1A,4}(\tau)=H^{(16)}_{1A,12}(\tau)=2ie^{-\frac{3\pi
i}{8}}\frac{q^{-\frac{1}{16}}}{2\theta_{1}(-\tau,8\tau)}\Theta_{\left(\frac{3}{8},\frac{1}{4}\right),\left(0,\frac{1}{2}\right)}(8\tau),\\\
&H^{(16)}_{1A,6}\left(\tau-\frac{1}{2}\right)=H^{(16)}_{1A,10}\left(\tau-\frac{1}{2}\right)=2e^{-\frac{5\pi
i}{4}}\frac{\eta(4\tau)}{2\eta(2\tau)\eta(8\tau)}\Theta_{\left(\frac{7}{8},\frac{3}{8}\right),\left(\frac{1}{2},0\right)}(8\tau),\\\
&H^{(16)}_{1A,8}(\tau)=-ie^{-\frac{\pi
i}{8}}\frac{q^{-\frac{1}{16}}}{\theta_{1}(-\tau,8\tau)}\Theta_{\left(\frac{1}{8},\frac{1}{2}\right),\left(0,\frac{1}{2}\right)}(8\tau)-\frac{\eta(2\tau)^{3}\eta(4\tau)}{\eta(\tau)^{2}\eta(8\tau)},\\\
&\sum\limits_{n=0,7}(-1)^{n}H^{(16)}_{1A,2n+1}(8\tau)=H^{(2)}_{8A,1}(\tau)=-2\frac{\eta(4\tau)}{2\eta(8\tau)^{2}}\Theta_{\left(\frac{1}{4},\frac{1}{4}\right),(0,0)}(4\tau)\end{split}$
(4.35)

together with the pairing relation
$H^{(16)}_{2A,r}+(-1)^{r}H^{(16)}_{1A,r}=0$.

We observe that also in this case the indefinite thetas appearing in all
components are invariant under the action of the umbral group.

######  Remark 4.5.

The quantity $q^{\frac{1}{16}}\theta_{1}(-\tau,8\tau)$ is modular under the
congruence subgroup

$\Gamma_{1}(8):=\left\\{\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\text{SL}(2,\mathbb{Z}):a,d=1\bmod 8,c=0\bmod
8\right\\}$ (4.36)

generated by the transformations $T:\tau\to\tau+1$,
$\tilde{S}:\tau\to\frac{\tau}{8\tau+1}$, as is easy to see that from the
transformation properties

$\begin{array}[]{ll}\theta_{1}(z,\tau+1)=e^{\frac{\pi
i}{4}}\theta_{1}(z,\tau),&\theta_{1}\left(\frac{z}{\tau},-\frac{1}{\tau}\right)=-i\sqrt{-i\tau}e^{\frac{i\pi
z^{2}}{\tau}}\theta_{1}(z,\tau).\end{array}$ (4.37)

##  5 Moonshine Modules

In this section we will build modules whose trace functions reproduce the
specifying expressions for the McKay-Thompson series provided in the previous
section for lambency $\ell=8,12,16$. As mentioned in the previous section, in
these cases we found that the the umbral groups act trivially on all the
indefinite theta functions appearing in the McKay-Thompson series. Thus, we
can construct modules that have the structure of a tensor product between an
appropriate linear representation of the umbral group and a direct sum of
vertex algebras modules on which the umbral group acts trivially. In the
following, all the trace functions defined as in (3.7) are trace functions of
modules of subalgebras of the vertex algebra associated to the two-dimensional
lattice with the indefinite quadratic form $A=\left(\begin{smallmatrix}1&1\\\
1&0\end{smallmatrix}\right)$.
We first start by introducing some vertex algebra modules that will appear in
our construction, and then provide explicit expressions for the relevant
umbral moonshine modules.

###  5.1 Heisenberg, Clifford and Weyl Characters

In this section we collect formulas for characters of (super) vertex algebras
that will recover some of the functions appearing in the McKay-Thompson series
specified in the previous section. Here we will follow the notation and
definitions in [20], [22], [23] for the super vertex operator algebras and
their modules.
The simplest character we will need is the character of the Heisenberg vertex
operator algebra $\mathcal{H}$

$\chi^{H}(\tau):={\rm
tr}_{\mathcal{H}}\left(q^{L(0)-\frac{c}{24}}\right)=\frac{1}{q^{\frac{1}{24}}\prod_{n>0}(1-q^{n})}=\frac{1}{\eta(\tau)}.$
(5.1)

Next, we will consider the graded characters of the irreducible canonically-
twisted modules of the Clifford vertex operator algebra $A^{\pm}_{\rm tw}$
[20]

$\chi^{A^{\pm}}(\tau):={\rm tr}_{A_{\rm
tw}^{\pm}}\left(p(0)q^{L(0)-\frac{c}{24}}\right)=\pm
q^{\frac{1}{24}}\prod\limits_{n>0}(1-q^{n})=\pm\eta(\tau)$ (5.2)

as well as the character of the (d-dimensional) Clifford super vertex operator
algebra canonically-twisted module $A_{\rm tw}$ [22]

$\chi^{A_{\rm tw}}(z,\tau):={\rm tr}_{A_{\rm
tw}}\left(y^{J(0)}q^{L(0)-\frac{d}{24}}\right)=y^{\frac{d}{4}}q^{\frac{d}{24}}\prod\limits_{n>0}(1+y^{-1}q^{n-1})^{\frac{d}{2}}(1+yq^{n})^{\frac{d}{2}}.$
(5.3)

Finally, we will also make use of the canonically twisted d-dimensional Weyl
modules ${\text{\rotatebox[origin={c}]{180.0}{$A$}}}_{\rm tw}$ [22]

$\chi^{\text{\rotatebox[origin={c}]{180.0}{$A$}}_{\rm tw}}(z,\tau):={\rm
tr_{\text{\rotatebox[origin={c}]{180.0}{$A$}}_{\rm
tw}}}(y^{J(0)}q^{L(0)-\frac{d}{24}})=y^{-\frac{d}{4}}q^{-\frac{d}{24}}\prod_{n>0}(1-y^{-1}q^{n-1})^{-\frac{d}{2}}(1-yq^{n})^{-\frac{d}{2}}.$
(5.4)

######  Remark 5.1.

The previous formula holds when each factor $(1-X)^{-1}$ is interpreted as
$\sum\limits_{n\geq 0}X^{n}$, which is possible in the domain $0<-{\rm
Im}(z)<{\rm Im}(\tau)$.

From now on, let’s fix $d=2$ since this is the case that will be needed the
following subsections. In particular, for $d=2$, we get the following
relations with the Jacobi theta functions defined in (4.28)

$\begin{split}&\chi^{\text{\rotatebox[origin={c}]{180.0}{$A$}}_{\rm
tw}}(z,\tau)=-i\frac{\eta(\tau)}{\theta_{1}(z,\tau)},\\\ &\chi^{A_{\rm
tw}}\left(z+\frac{1}{2},\tau\right)=-\frac{\theta_{1}\left(z,\tau\right)}{\eta(\tau)}.\end{split}$
(5.5)

We will also need characters of 1-dimensional lattice vertex algebras. Let’s
consider the general 1-dimensional (even) lattice
$L^{1}:=\\{\alpha\epsilon:\alpha\in\mathbb{Z}\\}$ with scalar product
$\braket{\epsilon,\epsilon}=2m$. Let’s recall the operator $g_{h}$ for
$h:=\epsilon\otimes h\in L^{1}\otimes_{\mathbb{Z}}\mathbb{Q}$ defined in
(2.6). We have

$\chi^{L^{1}}_{h}(\tau):=\text{Tr}_{V_{L^{1}}}(g_{h}q^{L_{0}-\frac{c}{24}})=\frac{1}{\eta(\tau)}\sum\limits_{n\in\mathbb{Z}}e^{4\pi
imhn}q^{mn^{2}}.$ (5.6)

Furthermore, the characters of the modules $V_{L^{1}+\frac{r}{2m}}$, for
$0<r<2m$

$\chi^{L^{1}+\frac{r}{2m}}_{h}(\tau):=\text{Tr}_{V_{L^{1}+\frac{r}{2m}}}(g_{h}q^{L_{0}-\frac{c}{24}})=\frac{1}{\eta(\tau)}\sum\limits_{n\in\mathbb{Z}}e^{2\pi
ih(2mn+r)}q^{\frac{(2mn+r)^{2}}{4m}}.$ (5.7)

give the theta functions $\theta_{m,r}$ defined in (4.28). Since they will
appear frequently later, let’s give special names to the following characters
of the vertex algebra $V_{L^{1}}$ associated to the 1 dimensional lattice
$L^{1}:=\\{\alpha\epsilon:\alpha\in\mathbb{Z}\\}$ with scalar product
$\braket{\epsilon,\epsilon}=2$, and the vertex algebra $V_{K}$ associated to
the sublattice $K\subset L^{1}=\\{\alpha\epsilon:\alpha\in\mathbb{Z}_{\geq
0}\\}$. Introducing the operator

$g_{\frac{1}{4}}(p\otimes n\epsilon)=(-1)^{n}(p\otimes n\epsilon)$ (5.8)

which corresponds to (2.6) with the choice $h=\frac{1}{4}\epsilon$, we define

$\begin{split}&\chi^{L^{1}}(\tau):={\rm
Tr}_{V_{L^{1}}}\left(q^{L_{0}-\frac{c}{24}}\right)=\frac{1}{\eta(\tau)}\sum\limits_{n\in\mathbb{Z}}q^{n^{2}},\\\
&\chi^{K}(\tau):={\rm
Tr}_{V_{K}}\left(q^{L_{0}-\frac{c}{24}}\right)=\frac{1}{\eta(\tau)}\sum\limits_{n\geq
0}q^{n^{2}},\\\ &\tilde{\chi}^{K}(\tau):={\rm
Tr}_{V_{K}}\left(g_{\frac{1}{4}}q^{L_{0}-\frac{c}{24}}\right)=\frac{1}{\eta(\tau)}\sum\limits_{n\geq
0}(-1)^{n}q^{n^{2}}.\\\ \end{split}$ (5.9)

###  5.2 Lambency Eight

The umbral group for lambency $\ell=8$ is $G=Dih_{4}$. We will use the
conventions for the names of conjugacy classes and irreducible representations
that are specified in the character table 1.

Table 1: Character table of $Dih_{4}$ | 1A | 2A | 2B | 2C | 4A
---|---|---|---|---|---
$A_{1}$ | 1 | 1 | 1 | 1 | 1
$A_{2}$ | 1 | 1 | -1 | -1 | 1
$B_{1}$ | 1 | 1 | -1 | 1 | -1
$B_{2}$ | 1 | 1 | 1 | -1 | -1
$E$ | 2 | -2 | 0 | 0 | 0

Using the results of the previous sections, we can specify the McKay-Thompson
series for $\ell=8$ in terms of characters of the VOAs introduced before. The
even components can be directly rewritten as

$\begin{split}&H^{(8)}_{g,2}(\tau)=H^{(8)}_{g,6}(\tau)=2{\rm
tr}_{E_{2}}(g)\chi^{A^{+}}(4\tau)\chi^{A^{+}}(\tau)^{2}\chi^{H}(2\tau)^{2}T^{(4)}_{\left(\frac{3}{4},\frac{1}{4}\right),\left(0,\frac{1}{2}\right)}(\tau),\\\
&H^{(8)}_{g,4}(\tau)=2{\rm
tr}_{E_{2}}(g)\chi^{A^{+}}(2\tau)\chi^{H}(\tau)\chi^{H}(4\tau)\chi^{A^{+}}(\tau)^{2}T^{(4)}_{\left(\frac{3}{4},\frac{1}{2}\right),\left(0,\frac{1}{2}\right)}(\tau),\end{split}$
(5.10)

while the odd components are specified by

$\begin{split}&(H^{(8)}_{g,1}-H^{(8)}_{g,7}-H^{(8)}_{g,3}+H^{(8)}_{g,5})(2\tau)=\\\
&2{\rm tr}_{2A_{1}\oplus B_{1}\oplus
B_{2}}(g)\chi^{A^{+}}(4\tau)\chi^{H}(2\tau)\chi^{H}(8\tau)\chi^{A^{+}}(\tau)\left[\chi^{A^{+}}(\tau)T^{(8)}_{\left(\frac{5}{8},\frac{1}{8}\right),\left(\frac{1}{2},0\right)}(\tau)+\chi^{A^{-}}(\tau)T^{(8)}_{\left(\frac{7}{8},\frac{3}{8}\right),\left(\frac{1}{2},0\right)}(\tau)\right]\\\
&+\left[\chi^{A^{+}}\left(\frac{\tau}{2}\right)tr_{A_{1}\oplus
A_{2}}(g)+\chi^{A^{-}}\left(\frac{\tau}{2}\right)tr_{B_{1}\oplus
B_{2}}(g)\right]\Biggl{[}\tilde{\chi}^{K}(\tau)\chi^{L^{1}}(\tau)+\\\
&\chi^{A^{-}}\left(\frac{\tau}{2}\right)\chi^{A^{+}}\left(\frac{\tau}{2}\right)\chi^{\mathcal{H}}(\tau)^{2}\chi^{K}\left(\frac{\tau}{2}\right)\chi^{L^{1}}\left(\frac{\tau}{2}\right)+\chi^{\mathcal{H}}(\tau)\chi^{K}(\tau)+\chi^{A^{+}}\left(\frac{\tau}{2}\right)\chi^{\mathcal{H}}(\tau)^{2}\chi^{K}\left(\frac{\tau}{2}\right)\Biggr{]}\\\
&+2{\rm
tr}_{A_{1}}(g)\chi^{A^{-}}(\tau)\chi^{A^{+}}(\tau)^{7}\chi^{\mathcal{H}}\left(\frac{\tau}{2}\right)^{3}\chi^{H}(2\tau)^{4}.\end{split}$
(5.11)

In rewriting the second addend we have used the following lemma so that the
prefactor multiplying the characters is integer

######  Lemma 5.2.

$\begin{split}\frac{\eta(\frac{\tau}{2})\eta(2\tau)^{4}}{\eta(\tau)^{2}\eta(4\tau)^{2}}-\frac{\eta(\tau)^{8}}{\eta(\frac{\tau}{2})^{3}\eta(2\tau)^{4}}=2\chi^{A^{+}}\left(\frac{\tau}{2}\right)\Biggl{[}&\tilde{\chi}^{K}(\tau)\chi^{L^{1}}(\tau)-\chi^{A^{+}}\left(\frac{\tau}{2}\right)^{2}\chi^{\mathcal{H}}(\tau)^{2}\chi^{K}\left(\frac{\tau}{2}\right)\chi^{L^{1}}\left(\frac{\tau}{2}\right)\\\
&+\chi^{\mathcal{H}}(\tau)\chi^{K}(\tau)+\chi^{A^{+}}\left(\frac{\tau}{2}\right)\chi^{\mathcal{H}}(\tau)^{2}\chi^{K}\left(\frac{\tau}{2}\right)\Biggr{]}\end{split}$
(5.12)

###### Proof.

Using the identities [36]

$\begin{array}[]{ll}\frac{\eta(2\tau)^{5}}{\eta(\tau)^{2}\eta(4\tau)^{2}}=\sum\limits_{n\in\mathbb{Z}}q^{n^{2}}=:\theta^{1}(\tau),&\frac{\eta(\tau)^{2}}{\eta(2\tau)}=\sum\limits_{n\in\mathbb{Z}}(-1)^{n}q^{n^{2}},\end{array}$
(5.13)

we get

$\begin{split}&\frac{\eta(\frac{\tau}{2})\eta(2\tau)^{4}}{\eta(\tau)^{2}\eta(4\tau)^{2}}-\frac{\eta(\tau)^{8}}{\eta(\frac{\tau}{2})^{3}\eta(2\tau)^{4}}=\frac{\eta\left(\frac{\tau}{2}\right)}{\eta(2\tau)}\frac{\eta(2\tau)^{5}}{\eta(\tau)^{2}\eta(4\tau)^{2}}-\frac{\eta\left(\frac{\tau}{2}\right)}{\eta(\tau)^{2}}\left(\frac{\eta(\tau)^{5}}{\eta\left(\frac{\tau}{2}\right)^{2}\eta(2\tau)^{2}}\right)^{2}\\\
&=\frac{\eta\left(\frac{\tau}{2}\right)}{\eta(\tau)^{2}}\left[\frac{\eta(\tau)^{2}}{\eta(2\tau)}\theta_{1}(\tau)-\theta_{1}\left(\frac{\tau}{2}\right)^{2}\right]=\frac{\eta\left(\frac{\tau}{2}\right)}{\eta(\tau)^{2}}\left[\sum\limits_{m,n\in\mathbb{Z}}(-1)^{n}q^{m^{2}+n^{2}}-q^{\frac{m^{2}+n^{2}}{2}}\right]\\\
&=2\frac{\eta(\frac{\tau}{2})}{\eta(\tau)^{2}}\left[\sum_{\begin{subarray}{c}n,m\in\mathbb{Z}\\\
n\geq
0\end{subarray}}(-1)^{n}q^{m^{2}+n^{2}}-\sum_{\begin{subarray}{c}n,m\in\mathbb{Z}\\\
n\geq 0\end{subarray}}q^{\frac{m^{2}+n^{2}}{2}}-\sum\limits_{n\geq
0}q^{n^{2}}+\sum\limits_{n\geq 0}q^{\frac{n^{2}}{2}}\right]\end{split}$ (5.14)

and the conclusion follows easily using the expressions for the characters
provided in (5.9). ∎

In order to specify the trace functions that will give us the relevant umbral
McKay-Thompson series at $\ell=8$, let’s define the modules

$\displaystyle\mathfrak{M}_{1,1}^{(8)}$ $\displaystyle:={A^{+}_{tw}}^{\otimes
3}\otimes\mathcal{H}^{\otimes 2}\otimes
V^{(8)}_{\left(\frac{5}{8},\frac{1}{8}\right)},$
$\displaystyle\mathfrak{M}_{1,2}^{(8)}$ $\displaystyle:={A^{+}_{tw}}^{\otimes
2}\otimes A^{-}_{tw}\otimes\mathcal{H}^{\otimes 2}\otimes
V^{(8)}_{\left(\frac{7}{8},\frac{3}{8}\right)},$
$\displaystyle\mathfrak{M}_{1,3}^{(8)}$ $\displaystyle:={A^{+}_{tw}}\otimes
K\otimes L^{1},$ $\displaystyle\mathfrak{M}_{1,4}^{(8)}$
$\displaystyle:={A^{+}_{tw}}^{\otimes 2}\otimes
A_{tw}^{-}\otimes\mathcal{H}^{\otimes 2}\otimes K\otimes L^{1},$
$\displaystyle\mathfrak{M}^{(8)}_{1,5}$
$\displaystyle:=A^{+}_{tw}\otimes\mathcal{H}\otimes K,$
$\displaystyle\mathfrak{M}^{(8)}_{1,6}$ $\displaystyle:={A^{+}_{tw}}^{\otimes
2}\otimes\mathcal{H}^{\otimes 2}\otimes K,$
$\displaystyle\mathfrak{M}_{1,7}^{(8)}$ $\displaystyle:={A^{-}_{tw}}\otimes
K\otimes L^{1},$ $\displaystyle\mathfrak{M}_{1,8}^{(8)}$
$\displaystyle:={A^{-}_{tw}}^{\otimes 2}A^{+}_{tw}\otimes\mathcal{H}^{\otimes
2}\otimes K\otimes L^{1}$ $\displaystyle\mathfrak{M}^{(8)}_{1,9}$
$\displaystyle:=A^{-}_{tw}\otimes\mathcal{H}\otimes K,$
$\displaystyle\mathfrak{M}^{(8)}_{1,10}$ $\displaystyle:=A^{+}_{tw}\otimes
A^{-}_{tw}\otimes\mathcal{H}^{\otimes 2}\otimes K,$
$\displaystyle\mathfrak{M}^{(8)}_{1,11}$
$\displaystyle:=A^{-}_{tw}\otimes{A^{+}_{tw}}^{\otimes
7}\otimes\mathcal{H}^{\otimes 7},$ $\displaystyle\mathfrak{M}^{(8)}_{2}$
$\displaystyle:={A^{+}_{tw}}^{\otimes 3}\otimes\mathcal{H}^{\otimes 2}\otimes
V^{(4)}_{\left(\frac{3}{4},\frac{1}{4}\right)},$
$\displaystyle\mathfrak{M}^{(8)}_{4}$ $\displaystyle:={A^{+}_{tw}}^{\otimes
3}\otimes\mathcal{H}^{\otimes 2}\otimes
V^{(4)}_{\left(\frac{3}{4},\frac{1}{2}\right)},$

and for each of them let’s define the vectors

$\displaystyle\omega_{1,1}^{(8)}$
$\displaystyle:=2\hat{\omega}^{(1)}+\frac{1}{2}\hat{\omega}^{(2)}+\frac{1}{2}\hat{\omega}^{(3)}+\hat{\omega}^{(4)}+4\hat{\omega}^{(5)}+\frac{1}{2}\hat{\omega}^{(6)},$
$\displaystyle\omega_{1,2}^{(8)}$
$\displaystyle:=2\hat{\omega}^{(1)}+\frac{1}{2}\hat{\omega}^{(2)}+\frac{1}{2}\hat{\omega}^{(3)}+\hat{\omega}^{(4)}+4\hat{\omega}^{(5)}+\frac{1}{2}\hat{\omega}^{(6)},$
$\displaystyle\omega^{(8)}_{1,3}$
$\displaystyle:=\frac{1}{4}\hat{\omega}^{(1)}+\frac{1}{2}\hat{\omega}^{(2)}+\frac{1}{2}\hat{\omega}^{(3)},$
$\displaystyle\omega^{(8)}_{1,4}$
$\displaystyle:=\frac{1}{4}\hat{\omega}^{(1)}+\frac{1}{4}\hat{\omega}^{(2)}+\frac{1}{4}\hat{\omega}^{(3)}+\frac{1}{2}\hat{\omega}^{(4)}+\frac{1}{2}\hat{\omega}^{(5)}+\frac{1}{4}\hat{\omega}^{(6)}+\frac{1}{4}\hat{\omega}^{(7)},$
$\displaystyle\omega^{(8)}_{1,5}$
$\displaystyle:=\frac{1}{4}\hat{\omega}^{(1)}+\frac{1}{4}\hat{\omega}^{(2)}+\frac{1}{4}\hat{\omega}^{(3)},$
$\displaystyle\omega^{(8)}_{1,6}$
$\displaystyle:=\frac{1}{4}\hat{\omega}^{(1)}+\frac{1}{4}\hat{\omega}^{(2)}+\frac{1}{2}\hat{\omega}^{(3)}+\frac{1}{2}\hat{\omega}^{(4)}+\frac{1}{4}\hat{\omega}^{(5)},$
$\displaystyle\omega^{(8)}_{1,7}$
$\displaystyle:=\frac{1}{4}\hat{\omega}^{(1)}+\frac{1}{2}\hat{\omega}^{(2)}+\frac{1}{2}\hat{\omega}^{(3)},$
$\displaystyle\omega^{(8)}_{1,8}$
$\displaystyle:=\frac{1}{4}\hat{\omega}^{(1)}+\frac{1}{4}\hat{\omega}^{(2)}+\frac{1}{4}\hat{\omega}^{(3)}+\frac{1}{2}\hat{\omega}^{(4)}+\frac{1}{2}\hat{\omega}^{(5)}+\frac{1}{4}\hat{\omega}^{(6)}+\frac{1}{4}\hat{\omega}^{(7)},$
$\displaystyle\omega^{(8)}_{1,9}$
$\displaystyle:=\frac{1}{4}\hat{\omega}^{(1)}+\frac{1}{2}\hat{\omega}^{(2)}+\frac{1}{2}\hat{\omega}^{(3)},$
$\displaystyle\omega^{(8)}_{1,10}$
$\displaystyle:=\frac{1}{4}\hat{\omega}^{(1)}+\frac{1}{4}\hat{\omega}^{(2)}+\frac{1}{2}\hat{\omega}^{(3)}+\frac{1}{2}\hat{\omega}^{(4)}+\frac{1}{4}\hat{\omega}^{(5)},$
$\displaystyle\begin{split}\omega^{(8)}_{1,11}&:=\frac{1}{2}\hat{\omega}^{(1)}+\frac{1}{2}\hat{\omega}^{(2)}+\frac{1}{2}\hat{\omega}^{(3)}+\frac{1}{2}\hat{\omega}^{(4)}+\frac{1}{2}\hat{\omega}^{(5)}+\frac{1}{2}\hat{\omega}^{(6)}+\frac{1}{2}\hat{\omega}^{(7)}+\frac{1}{2}\hat{\omega}^{(8)}+\frac{1}{4}\hat{\omega}^{(9)}+\frac{1}{4}\hat{\omega}^{(10)}+\frac{1}{4}\hat{\omega}^{(11)}\\\
&+\hat{\omega}^{(12)}+\hat{\omega}^{(13)}+\hat{\omega}^{(14)}+\hat{\omega}^{(15)},\end{split}$
$\displaystyle\omega^{(8)}_{2}$
$\displaystyle:=4\hat{\omega}^{(1)}+\hat{\omega}^{(2)}+\hat{\omega}^{(3)}+2\hat{\omega}^{(4)}+2\hat{\omega}^{(5)}+\hat{\omega}^{(6)},$
$\displaystyle\omega^{(8)}_{4}$
$\displaystyle:=2\hat{\omega}^{(1)}+\hat{\omega}^{(2)}+\hat{\omega}^{(3)}+\hat{\omega}^{(4)}+4\hat{\omega}^{(5)}+\hat{\omega}^{(6)},$

where, for brevity, we have written
$\hat{\omega}^{(i)}=\mathbb{v}\otimes\cdots\otimes\left(\omega^{(i)}-\frac{c^{(i)}}{24}\mathbf{v}\right)\otimes\cdots\otimes\mathbb{v}$
to indicate the tensor product of vectors that at position $i$ has the factor
$\omega^{(i)}-\frac{c^{(i)}}{24}\mathbf{v}$, where $\omega$ and $c$ are
respectively the conformal vector and central charge of the module at the
$i$-th position, and the remaining factors are the vacuum vectors $\mathbb{v}$
of the other modules. Let’s consider the operators333To make the notation
lighter we will not write the indices in $\hat{L}$. It is understood that, for
each module, $\hat{L}$ corresponds to the vector associated to the module.
$\hat{L}(0)$ corresponding to the 0-modes of the vertex operators associated
to the previous vectors. With this notation we get

######  Theorem 5.3.

The umbral McKay-Thompson series at lambency $\ell=8$ are specified by

$\begin{split}&H^{(8)}_{g,2}(\tau)=H^{(8)}_{g,6}(\tau)=2{\rm
tr}_{E_{2}}(g){\rm
tr}_{\mathfrak{M}^{(8)}_{2}}\left(g_{\left(0,\frac{1}{2}\right)}q^{\hat{L}(0)}\right),\\\
&H^{(8)}_{g,4}(\tau)=2{\rm tr}_{E_{2}}(g){\rm
tr}_{\mathfrak{M}^{(8)}_{4}}\left(g_{\left(0,\frac{1}{2}\right)}q^{\hat{L}(0)}\right),\\\
&\begin{split}&(H^{(8)}_{g,1}-H^{(8)}_{g,7}-H^{(8)}_{g,3}+H^{(8)}_{g,5})(\tau)=2{\rm
tr}_{2A_{1}\oplus B_{1}\oplus B_{2}}(g){\rm
tr}_{\mathfrak{M}^{(8)}_{1,1}\oplus\mathfrak{M}^{(8)}_{1,2}}\left(g_{\left(\frac{1}{2},0\right)}q^{\hat{L}(0)}\right)\\\
&+{\rm tr}_{A_{1}\oplus A_{2}}(g){\rm
tr}_{\mathfrak{M}^{(8)}_{1,3}}\left(g_{\frac{1}{4}}q^{\hat{L}(0)}\right)+{\rm
tr}_{B_{1}\oplus B_{2}}(g){\rm
tr}_{\mathfrak{M}^{(8)}_{1,7}}\left(g_{\frac{1}{4}}q^{\hat{L}(0)}\right)\\\
&+{\rm tr}_{A_{1}\oplus A_{2}}(g){\rm
tr}_{\mathfrak{M}^{(8)}_{1,4}\oplus\mathfrak{M}^{(8)}_{1,5}\oplus\mathfrak{M}^{(8)}_{1,6}}\left(q^{\hat{L}(0)}\right)+{\rm
tr}_{B_{1}\oplus B_{2}}(g){\rm
tr}_{\mathfrak{M}^{(8)}_{1,8}\oplus\mathfrak{M}^{(8)}_{1,9}\oplus\mathfrak{M}^{(8)}_{1,10}}\left(q^{\hat{L}(0)}\right)\\\
&+2{\rm tr}_{A_{1}}(g){\rm
tr}_{\mathfrak{M}^{(8)}_{1,11}}\left(q^{\hat{L}(0)}\right),\end{split}\end{split}$
(5.15)

where $g_{\mathbb{b}}$ acts as specified in (3.6) on the cone vertex algebra
module in the tensor product and trivially on all the others. Analogously
$g_{\frac{1}{4}}$ is specified by (5.8) and only acts non-trivially on the
module $K$.

###  5.3 Lambency Twelve

The umbral group corresponding to $\ell=12$ is $\mathbb{Z}/2\mathbb{Z}$. There
are only 2 irreducible representations, we will call $A$ the trivial
representation and $B$ the sign representation.
We can specify the McKay-Thompson series in terms of characters of vertex
algebras and $H^{(4)}$ functions. Let’s write

$\displaystyle e_{4}(\tau)=-H^{(4)}_{3A,2}(\tau)$ (5.16) $\displaystyle
o_{4}(\tau)=\left(\frac{H^{(4)}_{3A,1}-H^{(4)}_{3A,3}}{2}\right)\left(\frac{2}{3}\tau\right)$
(5.17)

The odd components are specified by

$\begin{split}&\left(H^{(12)}_{g,1}-H^{(12)}_{g,11}\right)(2\tau)=\\\
&\begin{split}{\rm
tr}_{A}(g)\Biggl{[}&\chi^{A^{-}}(\tau)\chi^{A^{+}}\left(\frac{\tau}{2}\right)^{2}\chi^{A^{+}}(6\tau)\chi^{\mathcal{H}}(2\tau)\chi^{\mathcal{H}}(3\tau)T^{(6)}_{\left(\frac{1}{3},\frac{1}{2}\right),\left(\frac{1}{2},\frac{1}{2}\right)}\left(\frac{\tau}{2}\right)\\\
&+4\chi^{A^{+}}\left(\frac{\tau}{2}\right)^{2}\chi^{\mathcal{H}}(\tau)T^{(6)}_{\left(\frac{2}{3},\frac{1}{6}\right),\left(\frac{1}{2},0\right)}\left(\frac{\tau}{2}\right)\\\
&+\chi^{A^{+}}(3\tau)^{2}\chi^{A^{-}}(3\tau)\chi^{\mathcal{H}}(\tau)\chi^{\mathcal{H}}(6\tau)^{2}+o_{4}(\tau)\Biggr{]},\end{split}\\\
&\left(H^{(12)}_{g,5}-H^{(12)}_{g,7}\right)(2\tau)=\\\ &\begin{split}{\rm
tr}_{A}(g)\Biggl{[}&\chi^{A^{-}}(\tau)\chi^{A^{+}}\left(\frac{\tau}{2}\right)^{2}\chi^{A^{+}}(6\tau)\chi^{\mathcal{H}}(2\tau)\chi^{\mathcal{H}}(3\tau)T^{(6)}_{\left(\frac{1}{3},\frac{1}{2}\right),\left(\frac{1}{2},\frac{1}{2}\right)}\left(\frac{\tau}{2}\right)\\\
&+4\chi^{A^{-}}\left(\frac{\tau}{2}\right)\chi^{A^{+}}\left(\frac{\tau}{2}\right)\chi^{\mathcal{H}}(\tau)T^{(6)}_{\left(\frac{2}{3},\frac{1}{6}\right),\left(\frac{1}{2},0\right)}\left(\frac{\tau}{2}\right)+\chi^{A^{+}}(3\tau)^{2}\chi^{A^{+}}(3\tau)\chi^{\mathcal{H}}(\tau)\chi^{\mathcal{H}}(6\tau)^{2}\\\
&+o_{4}(\tau)\Biggr{]},\end{split}\\\
&\left(H^{(12)}_{g,3}-H^{(12)}_{g,9}\right)(2\tau)=2{\rm
tr}_{A}(g)\chi^{A^{-}}(\tau)\chi^{A^{+}}\left(\frac{\tau}{2}\right)\chi^{A^{+}}(6\tau)\chi^{\mathcal{H}}(2\tau)\chi^{\mathcal{H}}(3\tau)^{2}T^{(6)}_{\left(\frac{1}{3},\frac{1}{2}\right),\left(\frac{1}{2},\frac{1}{2}\right)}\left(\frac{\tau}{2}\right),\\\
\end{split}$ (5.18)

The even components are instead given by

$\begin{split}&H^{(12)}_{g,2}(\tau)=H^{(12)}_{g,10}(\tau)=2tr_{B}(g)\chi^{\mathcal{H}}(6\tau)^{2}\chi^{A^{-}}(\tau)\chi^{A^{+}}(2\tau)\chi^{A^{+}}(3\tau)T^{(6)}_{\left(\frac{1}{2},\frac{1}{6}\right),\left(0,\frac{1}{2}\right)}(\tau),\\\
&H^{(12)}_{g,4}(\tau)=H^{(12)}_{g,8}(\tau)=4{\rm
tr}_{B}(g)\left[\chi^{A^{+}}(\tau)T^{(6)}_{\left(\frac{1}{2},\frac{1}{3}\right),\left(0,\frac{1}{2}\right)}(\tau)+2\chi^{A^{+}}(6\tau)^{4}\chi^{\mathcal{H}}(2\tau)\chi^{\mathcal{H}}(3\tau)^{2}\right],\\\
&H^{(12)}_{g,6}(3\tau)={\rm
tr}_{B}(g)\left[4\chi^{A^{-}}(6\tau)\chi^{A^{+}}(9\tau)\chi^{A^{+}}(\tau)^{2}\chi^{\mathcal{H}}(3\tau)\chi^{\mathcal{H}}(18\tau)^{2}T^{(18)}_{\left(\frac{1}{2},\frac{1}{6}\right),\left(0,\frac{1}{2}\right)}(\tau)+e_{4}(\tau)\right].\end{split}$
(5.19)

We define the modules

$\displaystyle\mathfrak{M}^{(12)}_{1,1}:=A^{-}_{tw}\otimes{A^{+}_{tw}}^{\otimes
3}\otimes\mathcal{H}^{\otimes 2}\otimes
V^{(6)}_{\left(\frac{1}{3},\frac{1}{2}\right)},$
$\displaystyle\mathfrak{M}^{(12)}_{1,2}:={A^{+}_{tw}}^{\otimes
2}\otimes\mathcal{H}\otimes V^{(6)}_{\left(\frac{2}{3},\frac{1}{6}\right)},$
$\displaystyle\mathfrak{M}^{(12)}_{1,3}:={A^{+}_{tw}}^{\otimes 2}\otimes
A^{-}_{tw}\otimes\mathcal{H}^{\otimes 3},$
$\displaystyle\mathfrak{M}^{(12)}_{2}:=A^{-}_{tw}\otimes{A^{+}_{tw}}^{\otimes
2}\otimes\mathcal{H}^{\otimes 2}\otimes
V^{(6)}_{\left(\frac{1}{2},\frac{1}{6}\right)},$
$\displaystyle\mathfrak{M}^{(12)}_{3}:=A^{-}_{tw}\otimes{A^{+}_{tw}}^{\otimes
2}\otimes\mathcal{H}^{\otimes 3}\otimes
V^{(6)}_{\left(\frac{1}{3},\frac{1}{2}\right)},$
$\displaystyle\mathfrak{M}^{(12)}_{4,1}:=A^{+}_{tw}\otimes
V^{(6)}_{\left(\frac{1}{2},\frac{1}{3}\right)},$
$\displaystyle\mathfrak{M}^{(12)}_{4,2}:={A^{+}_{tw}}^{\otimes
4}\otimes\mathcal{H}^{\otimes 3}$
$\displaystyle\mathfrak{M}^{(12)}_{5,1}:=A^{-}_{tw}\otimes{A^{+}_{tw}}^{\otimes
3}\otimes\mathcal{H}^{\otimes 2}\otimes
V^{(6)}_{\left(\frac{1}{3},\frac{1}{2}\right)},$
$\displaystyle\mathfrak{M}^{(12)}_{5,2}:=A^{-}_{tw}\otimes
A^{+}_{tw}\otimes\mathcal{H}\otimes
V^{(6)}_{\left(\frac{2}{3},\frac{1}{6}\right)}$
$\displaystyle\mathfrak{M}^{(12)}_{5,3}:={A^{+}_{tw}}^{\otimes
3}\otimes\mathcal{H}^{\otimes 3}$
$\displaystyle\mathfrak{M}^{(12)}_{6}:=A^{-}_{tw}\otimes{A^{+}_{tw}}^{\otimes
3}\otimes\mathcal{H}^{\otimes 3}\otimes
V^{(18)}_{\left(\frac{1}{2},\frac{1}{6}\right)},$

and, to account for the different coefficients in front of $\tau$, the vectors

$\displaystyle\omega_{1,1}^{(12)}:=\frac{1}{2}\hat{\omega}^{(1)}+\frac{1}{4}\hat{\omega}^{(2)}+\frac{1}{2}\hat{\omega}^{(3)}+3\hat{\omega}^{(4)}+\hat{\omega}^{(5)}+\frac{3}{2}\hat{\omega}^{(6)}+\frac{1}{4}\hat{\omega}^{(7)},$
$\displaystyle\omega_{1,2}^{(12)}:=\frac{1}{4}\hat{\omega}^{(1)}+\frac{1}{4}\hat{\omega}^{(2)}+\frac{1}{2}\hat{\omega}^{(3)}+\frac{1}{4}\hat{\omega}^{(4)},$
$\displaystyle\omega_{1,3}^{(12)}:=\frac{3}{2}\hat{\omega}^{(1)}+\frac{3}{2}\hat{\omega}^{(2)}+\frac{3}{2}\hat{\omega}^{(3)}+\frac{1}{2}\hat{\omega}^{(4)}+3\hat{\omega}^{(5)}+3\hat{\omega}^{(6)},$
$\displaystyle\omega_{2}^{(12)}:=\hat{\omega}^{(1)}+2\hat{\omega}^{(2)}+3\hat{\omega}^{(3)}+6\hat{\omega}^{(4)}+6\hat{\omega}^{(5)}+\hat{\omega}^{(6)},$
$\displaystyle\omega_{3}^{(12)}:=\frac{1}{2}\hat{\omega}^{(1)}+\frac{1}{4}\hat{\omega}^{(2)}+3\hat{\omega}^{(3)}+\hat{\omega}^{(4)}+\frac{3}{2}\hat{\omega}^{(5)}+\frac{3}{2}\hat{\omega}^{(6)}+\frac{1}{4}\hat{\omega}^{(7)},$
$\displaystyle\omega_{4,1}^{(12)}:=\hat{\omega}^{(1)}+\hat{\omega}^{(2)},$
$\displaystyle\omega_{4,2}^{(12)}:=6\hat{\omega}^{(1)}+6\hat{\omega}^{(2)}+6\hat{\omega}^{(3)}+6\hat{\omega}^{(4)}+2\hat{\omega}^{(5)}+3\hat{\omega}^{(6)}+3\hat{\omega}^{(7)},$
$\displaystyle\omega_{5,1}^{(12)}:=\frac{1}{2}\hat{\omega}^{(1)}+\frac{1}{4}\hat{\omega}^{(2)}+\frac{1}{4}\hat{\omega}^{(3)}+3\hat{\omega}^{(4)}+\hat{\omega}^{(5)}+\frac{3}{2}\hat{\omega}^{(6)}+\frac{1}{4}\hat{\omega}^{(7)},$
$\displaystyle\omega_{5,2}^{(12)}:=\frac{1}{4}\hat{\omega}^{(1)}+\frac{1}{4}\hat{\omega}^{(2)}+\frac{1}{2}\hat{\omega}^{(3)}+\frac{1}{4}\hat{\omega}^{(4)}$
$\displaystyle\omega_{5,3}^{(12)}:=\frac{3}{2}\hat{\omega}^{(1)}+\frac{3}{2}\hat{\omega}^{(2)}+\frac{3}{2}\hat{\omega}^{(3)}+\frac{1}{2}\hat{\omega}^{(4)}+3\hat{\omega}^{(5)}+3\hat{\omega}^{(6)},$
$\displaystyle\omega_{6}^{(12)}:=2\hat{\omega}^{(1)}+3\hat{\omega}^{(2)}+\frac{1}{3}\hat{\omega}^{(3)}+\frac{1}{3}\hat{\omega}^{(4)}+1\hat{\omega}^{(5)}+6\hat{\omega}^{(6)}+6\hat{\omega}^{(7)}+\frac{1}{3}\hat{\omega}^{(8)},$

where again we have written
$\hat{\omega}^{(i)}=\mathbb{v}\otimes\cdots\otimes\left(\omega^{(i)}-\frac{c^{(i)}}{24}\mathbf{v}\right)\otimes\cdots\otimes\mathbb{v}$.
As before we write $\hat{L}(0)$ to indicate the 0-mode of the vertex operators
associated to the previous vectors. We also need modules for $e_{4}(\tau)$ and
$o_{4}(\tau)$. It is possible to specify these modules implicitly by making
use of equation (4.29). In fact, using Corollary 3.4 we can rewrite
$\mu_{m,0}(z,\tau)$ in terms of characters of cone vertex algebras and
1-dimensional lattice vertex algebras. Furthermore, the theta functions
$\theta_{m,r}$ also admits expressions in terms of trace functions of 1d
lattice vertex algebras as described in section 5.1. It remains to find a
module for the meromorphic Jacobi form

$\psi^{(4)}_{3A}(z,\tau):=2i\theta_{1}(6z,3\tau)\theta_{1}(z,\tau)^{-1}\theta_{1}(3z,3\tau)^{-1}\eta(\tau)^{3}$
(5.20)

featuring in equation (4.27). Notice that constructing modules for these
meromorphic functions is what is referred to as the “meromorphic module
problem” in [22]. It is easy to see that (5.20) also admits an expression in
terms of characters of the modules discussed in 5.1. In fact we have, for
$0<-{\rm Im}(z)<{\rm Im}(\tau)$,

$\psi^{(4)}_{3A}(z,\tau)=2i\chi^{A^{+}}(\tau)\chi^{A^{-}}(\tau)\chi^{A_{tw}}\left(6z+\frac{1}{2},3\tau\right)\chi^{\text{\rotatebox[origin={c}]{180.0}{$A$}}_{tw}}(z,\tau)\chi^{\text{\rotatebox[origin={c}]{180.0}{$A$}}_{tw}}(3z,3\tau).$
(5.21)

Using the relations $H^{(4)}_{3A,r}(\tau)=-H^{(4)}_{3A,-r}(\tau)$, and
$\theta_{m,r}(z,\tau)=\theta_{m,-r}(z,\tau)$, we can give a prescription for
the construction of modules444We can also express modules for $H^{(4)}$
implicitly in terms of vertex algebra modules by writing, for $z=a\tau+b$ with
$a\in\mathbb{Q}^{*}$, $|a|<1$, $b\in\mathbb{R}$
$\begin{split}&\sum\limits_{r=1}^{3}H^{(4)}_{3A,r}(\tau)\left[\theta_{4,r}(z,\tau)-\theta_{4,r}(-z,\tau)\right]=-4e^{-16\pi
ib}q^{-2ma^{2}}\chi^{\mathcal{H}}(\tau)^{2}\tilde{T}^{(1)}_{\mathbf{a},\mathbf{b}}(\tau)\\\
&+2i\chi^{A^{+}}(\tau)\chi^{A^{-}}(\tau)\chi^{A_{tw}}\left(6z+\frac{1}{2},3\tau\right)\chi^{\text{\rotatebox[origin={c}]{180.0}{$A$}}_{tw}}(z,\tau)\chi^{\text{\rotatebox[origin={c}]{180.0}{$A$}}_{tw}}(3z,3\tau)-2\chi_{\frac{b}{16}}^{L^{1}+a}(\tau)\chi^{H}(\tau)\end{split}$
(5.22) where $\mathbf{a}=(1+a,0)$, $\mathbf{b}=(b,0)$ and we have written
$\tilde{T}_{\mathbf{a},\mathbf{b}}$ to indicate the cone vertex algebra trace
function with quadratic form $\tilde{A}=\left(\begin{smallmatrix}8&1\\\
1&0\end{smallmatrix}\right)$ in order to distinguish it from the trace
functions with respect to $A=\left(\begin{smallmatrix}1&1\\\
1&0\end{smallmatrix}\right)$. Thus the McKay-Thompson series $H^{(4)}_{3A,r}$
are specified by the different $y$-powers in the right hand side of (5.22).
Notice that the $z$ dependence influences, through $a$, which cone vertex
algebra and one dimensional lattice modules will appear in the right hand side
of (5.22) for $H^{(4)}_{3A,r}$ starting from equation (4.29). In fact, we can
write $\Theta^{+}$ as

$\Theta^{+}_{A^{(4)},\mathbb{c}^{(4)}_{1},\mathbb{c}^{(4)}_{2}}(a\tau+b,0;\tau)=2\sum\limits_{(n_{1},n_{2})\in\mathcal{C}}(-1)^{s(n_{1},n_{2})}y^{8n_{1}+n_{2}}q^{4n_{1}^{2}+n_{1}n_{2}}-\sum\limits_{n\in\mathbb{Z}}y^{8n}q^{4n^{2}}$
(5.23)

where $\mathcal{C}$ is the cone
$\mathcal{C}:=\left\\{(n_{1},n_{2})\in\mathbb{Z}^{2}:n_{1}\geq 0,n_{2}\geq
0\right\\}\cup\left\\{(n_{1},n_{2})\in\mathbb{Z}^{2}:n_{1}<0,n_{2}<0\right\\}$
and $s$ corresponds to the sign automorphism

$s(n_{1},n_{2}):=\begin{cases}1&\text{if }n_{1}\geq 0,n_{2}\geq 0,\\\
-1&\text{if }n_{1}<0,n_{2}<0.\end{cases}$ (5.24)

The vector space interpretation of the indefinite theta function (5.23), the
vertex algebra interpretation of $\psi^{(4)}_{3A}$ (5.21), together with
(4.29) give a definition of a bi-graded vector space
$\mathcal{H}=\bigoplus\limits_{n,l}\mathcal{H}_{n,l}$ with an additional
${\mathbb{Z}}_{2}$-grading, that satisfies

$\sum\limits_{r=1}^{3}\frac{H^{(4)}_{3A,r}(\tau)}{2}\left[\theta_{4,r}(z,\tau)-\theta_{4,r}(-z,\tau)\right]=\sum_{n,l}\text{sdim}(\mathcal{H}_{n,l})q^{n}y^{l}$
(5.25)

where sdim stands for the super dimension that takes the ${\mathbb{Z}}_{2}$
grading into account by including additional sign factors. We now define the
operators $\tilde{L}_{0}$ and $\tilde{J}_{0}$ acting as
$\tilde{L}_{0}\mathbf{v}=n\mathbf{v}$, $\tilde{J}_{0}\mathbf{v}=l\mathbf{v}$
$\forall\mathbf{v}\in\mathcal{H}_{n,l}$. We can thus define a supertrace on
$\mathcal{H}$ through

$\text{sTr}_{\mathcal{H}}q^{\tilde{L}_{0}}y^{\tilde{J}_{0}}:=\sum_{n,l}\text{sdim}(\mathcal{H}_{n,l})q^{n}y^{l}.$
(5.26)

Noticing that

$\left[\theta_{4,r}(z,\tau)-\theta_{4,r}(-z,\tau)\right]=\sum\limits_{k\in\mathbb{Z}}\left(y^{4k+r}-y^{-(4k+r)}\right)q^{\frac{(4k+r)^{2}}{16}}$
(5.27)

we can specify $H^{(4)}_{3A,r}$ for $r=1,2,3$ with the previous notation
through

$\frac{H^{(4)}_{3A,r}(\tau)}{2}=\text{sTr}_{\tilde{\mathcal{H}}_{r}}q^{\tilde{L}_{0}-\left(\frac{\tilde{J}_{0}}{4}\right)^{2}}$
(5.28)

where

$\mathcal{H}_{r}=\bigoplus\limits_{n}\mathcal{H}_{n,l=r}.$ (5.29)

With this notation, we can rewrite the functions (5.16), and (5.17) as

$\displaystyle
e_{4}(\tau)=-2\text{sTr}_{\tilde{\mathcal{H}}_{2}}q^{\tilde{L}_{0}-\left(\frac{\tilde{J}_{0}}{4}\right)^{2}},$
(5.30) $\displaystyle
o_{4}\left(\frac{3}{2}\tau\right)=\text{sTr}_{\tilde{\mathcal{H}}_{1}}q^{\tilde{L}_{0}-\left(\frac{\tilde{J}_{0}}{4}\right)^{2}}-\text{sTr}_{\tilde{\mathcal{H}}_{3}}q^{\tilde{L}_{0}-\left(\frac{\tilde{J}_{0}}{4}\right)^{2}}.$
(5.31)

We thus have

######  Theorem 5.4.

The umbral McKay-Thompson series at lambency $\ell=12$ are specified by

$\begin{split}&\begin{split}\left(H^{(12)}_{g,1}-H^{(12)}_{g,11}\right)(\tau)={\rm
tr}_{A}(g)\Biggl{[}&{\rm
tr}_{\mathfrak{M}^{(12)}_{1,1}}\left(g_{\left(\frac{1}{2},\frac{1}{2}\right)}q^{\hat{L}(0)}\right)+4{\rm
tr}_{\mathfrak{M}^{(12)}_{1,2}}\left(g_{\left(\frac{1}{2},0\right)}q^{\hat{L}(0)}\right)\\\
&+{\rm
tr}_{\mathfrak{M}^{(12)}_{1,3}}\left(q^{\hat{L}(0)}\right)\Biggr{]}+{\rm
tr}_{A}(g)o_{4}\left(\frac{\tau}{2}\right),\end{split}\\\
&\left(H^{(12)}_{g,3}-H^{(12)}_{g,9}\right)(\tau)=2{\rm
tr}_{A}(g)tr_{\mathfrak{M}^{(12)}_{3}}\left(g_{\left(\frac{1}{2},\frac{1}{2}\right)}q^{\hat{L}(0)}\right),\\\
&\begin{split}\left(H^{(12)}_{g,5}-H^{(12)}_{g,7}\right)(\tau)={\rm
tr}_{A}(g)\Biggl{[}&{\rm
tr}_{\mathfrak{M}^{(12)}_{5,1}}\left(g_{\left(\frac{1}{2},\frac{1}{2}\right)}q^{\hat{L}(0)}\right)+4{\rm
tr}_{\mathfrak{M}^{(12)}_{5,2}}\left(g_{\left(\frac{1}{2},0\right)}q^{\hat{L}(0)}\right)\\\
&+{\rm
tr}_{\mathfrak{M}^{(12)}_{5,3}}\left(q^{\hat{L}(0)}\right)\Biggr{]}+{\rm
tr}_{A}(g)o_{4}\left(\frac{\tau}{2}\right),\end{split}\\\
&H^{(12)}_{g,2}(\tau)=H^{(12)}_{g,10}(\tau)=2{\rm tr}_{B}(g){\rm
tr}_{\mathfrak{M}^{(12)}_{2}}\left(g_{\left(0,\frac{1}{2}\right)}q^{\hat{L}(0)}\right),\\\
&H^{(12)}_{g,4}(\tau)=H^{(12)}_{g,8}(\tau)=4{\rm tr}_{B}(g)\left[{\rm
tr}_{\mathfrak{M}^{(12)}_{4,1}}\left(g_{\left(0,\frac{1}{2}\right)}q^{\hat{L}(0)}\right)+2{\rm
tr}_{\mathfrak{M}^{(12)}_{4,2}}\left(q^{\hat{L}(0)}\right)\right],\\\
&H^{(12)}_{g,6}(\tau)=4{\rm tr}_{B}(g){\rm
tr}_{\mathfrak{M}^{(12)}_{6}}\left(g_{\left(0,\frac{1}{2}\right)}q^{\hat{L}(0)}\right)+{\rm
tr}_{B}(g)e_{4}\left(\frac{\tau}{3}\right),\end{split}$ (5.32)

where $g_{\mathbb{b}}$ acts as specified in (3.6) on the cone vertex algebra
module in the tensor product and trivially on all the others.

###  5.4 Lambency Sixteen

The umbral group is $G=\mathbb{Z}/2\mathbb{Z}$. Using the same notation as
before for the irreducible representations, we can write all the McKay-
Thomposon series in terms of characters as

$\begin{split}&H^{(16)}_{g,2}\left(\tau-\frac{1}{2}\right)=H^{(16)}_{g,14}\left(\tau-\frac{1}{2}\right)=2{\rm
tr}_{B}(g)\chi^{A^{+}}(4\tau)\chi^{A^{+}}(\tau)^{2}\chi^{H}(2\tau)\chi^{H}(8\tau)T^{(8)}_{\left(\frac{5}{8},\frac{1}{8}\right),\left(\frac{1}{2},0\right)}(\tau),\\\
&H^{(16)}_{g,4}(\tau)=H^{(16)}_{g,12}(\tau)=2{\rm
tr}_{B}(g)q^{-\frac{1}{16}}\chi^{A^{+}}(\tau)\chi^{\text{\rotatebox[origin={c}]{180.0}{$A$}}}(8\tau,-\tau)T^{(8)}_{\left(\frac{3}{8},\frac{1}{4}\right),\left(0,\frac{1}{2}\right)}(\tau),\\\
&H^{(16)}_{g,6}\left(\tau-\frac{1}{2}\right)=H^{(16)}_{g,10}\left(\tau-\frac{1}{2}\right)=2{\rm
tr}_{B}(g)\chi^{A^{+}}(4\tau)\chi^{A^{+}}(\tau)^{2}\chi^{H}(2\tau)\chi^{H}(8\tau)T^{(8)}_{\left(\frac{7}{8},\frac{3}{8}\right),\left(\frac{1}{2},0\right)}(\tau),\\\
&H^{(16)}_{g,8}(\tau)={\rm
tr}_{B}(g)\left(2q^{-\frac{1}{16}}\chi^{A^{+}}(\tau)\chi^{\text{\rotatebox[origin={c}]{180.0}{$A$}}}(8\tau,-\tau)T^{(8)}_{\left(\frac{1}{8},\frac{1}{2}\right),\left(0,\frac{1}{2}\right)}(\tau)+\chi^{A^{+}}(2\tau)^{3}\chi^{A^{-}}(4\tau)\chi^{H}(\tau)^{2}\chi^{H}(8\tau)\right),\\\
&\sum\limits_{n=0,7}(-1)^{n}H^{(16)}_{g,2n+1}(8\tau)=2{\rm
tr}_{A}(g)\chi^{A^{+}}(4\tau)\chi^{A^{+}}(\tau)^{2}\chi^{H}(8\tau)T^{(8)}_{\left(\frac{1}{4},\frac{1}{4}\right),\left(0,0\right)}(\tau).\end{split}$
(5.33)

Let’s now consider the following tensor products of modules

$\displaystyle\mathfrak{M}^{(16)}_{1}:={A^{+}_{tw}}^{\otimes
3}\otimes\mathcal{H}\otimes V^{(8)}_{\left(\frac{1}{4},\frac{1}{4}\right)},$
$\displaystyle\mathfrak{M}^{(16)}_{2}:={A^{+}_{tw}}^{\otimes
3}\otimes\mathcal{H}^{\otimes 2}\otimes
V^{(8)}_{\left(\frac{5}{8},\frac{1}{8}\right)},$
$\displaystyle\mathfrak{M}^{(16)}_{4}:=A^{+}_{tw}\otimes\text{\rotatebox[origin={c}]{180.0}{$A$}}_{tw}\otimes
V^{(8)}_{\left(\frac{3}{8},\frac{1}{4}\right)}$
$\displaystyle\mathfrak{M}^{(16)}_{6}:={A^{+}_{tw}}^{\otimes
3}\otimes\mathcal{H}^{\otimes 2}\otimes
V^{(8)}_{\left(\frac{7}{8},\frac{3}{8}\right)},$
$\displaystyle\mathfrak{M}^{(16)}_{8,1}:=A^{+}_{tw}\otimes\text{\rotatebox[origin={c}]{180.0}{$A$}}_{tw}\otimes
V^{(8)}_{\left(\frac{1}{8},\frac{1}{2}\right)},$
$\displaystyle\mathfrak{M}^{(16)}_{8,2}:={A^{+}_{tw}}^{\otimes 3}\otimes
A^{-}_{tw}\otimes\mathcal{H}^{\otimes 3},$

and the respective vectors

$\displaystyle\omega_{1}^{(16)}:=\frac{1}{2}\hat{\omega}^{(1)}+\frac{1}{8}\hat{\omega}^{(2)}+\frac{1}{8}\hat{\omega}^{(3)}+\hat{\omega}^{(4)}+\frac{1}{8}\hat{\omega}^{(5)},$
$\displaystyle\omega_{2}^{(16)}:=4\hat{\omega}^{(1)}+\hat{\omega}^{(2)}+\hat{\omega}^{(3)}+2\hat{\omega}^{(4)}+8\hat{\omega}^{(5)}+\hat{\omega}^{(6)},$
$\displaystyle\omega_{4}^{(16)}:=\hat{\omega}^{(1)}+8\hat{\omega}^{(2)}+\hat{\omega}^{(3)},$
$\displaystyle\omega_{6}^{(16)}:=4\hat{\omega}^{(1)}+\hat{\omega}^{(2)}+\hat{\omega}^{(3)}+8\hat{\omega}^{(4)}+\hat{\omega}^{(5)},$
$\displaystyle\omega_{8,1}^{(16)}:=\hat{\omega}^{(1)}+8\hat{\omega}^{(2)}+\hat{\omega}^{(3)},$
$\displaystyle\omega_{8,2}^{(16)}:=2\hat{\omega}^{(1)}+2\hat{\omega}^{(2)}+2\hat{\omega}^{(3)}+4\hat{\omega}^{(4)}+\hat{\omega}^{(5)}+\hat{\omega}^{(6)}+8\hat{\omega}^{(7)},$

using the same notation as before. Defining $\hat{L}(0)$ as usual we get

######  Theorem 5.5.

The umbral McKay-Thompson series at lambency $\ell=16$ are specified by

$\begin{split}&H_{g,2}\left(\tau\right)=H_{g,14}\left(\tau\right)=2{\rm
tr}_{B}(g){\rm
tr}_{\mathfrak{M}^{(16)}_{2}}\left(g_{\left(\frac{1}{2},0\right)}e^{\pi
i\hat{L}(0)}q^{\hat{L}(0)}\right),\\\
&H_{g,4}(\tau)=H_{g,12}(\tau)=2q^{-\frac{1}{16}}{\rm tr}_{B}(g){\rm
tr}_{\mathfrak{M}^{(16)}_{4}}\left(g_{\left(0,\frac{1}{2}\right)}q^{-J(0)}q^{\hat{L}(0)}\right),\\\
&H_{g,6}\left(\tau\right)=H_{g,10}\left(\tau\right)=2{\rm tr}_{B}(g){\rm
tr}_{\mathfrak{M}^{(16)}_{6}}\left(g_{\left(\frac{1}{2},0\right)}e^{\pi
i\hat{L}(0)}q^{\hat{L}(0)}\right),\\\ &H_{g,8}(\tau)={\rm
tr}_{B}(g)\left[2q^{-\frac{1}{16}}{\rm
tr}_{\mathfrak{M}^{(16)}_{8,1}}\left(g_{\left(0,\frac{1}{2}\right)}q^{-J(0)}q^{\hat{L}(0)}\right)+{\rm
tr}_{\mathfrak{M}^{(16)}_{8,2}}\left(q^{\hat{L}(0)}\right)\right],\\\
&\sum\limits_{n=0,7}(-1)^{n}H_{g,2n+1}(\tau)=2{\rm tr}_{A}(g){\rm
tr}_{\mathfrak{M}^{(16)}_{1}}\left(g_{\left(0,0\right)}q^{\hat{L}(0)}\right),\end{split}$
(5.34)

where $g_{\mathbb{b}}$ acts as specified in (3.6) on the cone vertex algebra
module in the tensor product and trivially on all the others. Analogously,
$J(0)$ acts non-trivially only on the Weyl module
$\text{\rotatebox[origin={c}]{180.0}{$A$}}_{tw}$.

##  6 Conclusion and Outlook

In this paper we showed how certain trace functions of cone vertex algebras
are related to a certain family of indefinite theta functions of signature
$(1,1)$. This family possesses interesting number theoretic properties and it
is related to Appell-Lerch sums and Ramanujan’s mock theta functions. For
three instances of umbral moonshine, those with lambency $\ell=8,12,16$, this
allowed us to construct modules for the relevant finite groups in terms of
cone vertex algebras and other known super vertex operator algebras modules.
We end the paper with a collection of open questions and possible future
directions.

  * •

We expect that the family of indefinite theta functions expressible as trace
functions of cone vertex algebras can be extended by studying vertex algebras
associated to cones with a more general shape than what used in (3.4). The
condition in (3.1) on the choice of $\mathbf{c}$ is chosen to restrain the sum
over the lattice vectors on the first and third quadrant of the plane. More
general choices for the vectors $\mathbf{c}$ will lead to a sum on different
cones.

  * •

Another natural generalization is to investigate more general cone vertex
algebras that can reproduce, through trace functions, indefinite theta
functions of general signature $(r-n,n)$. In particular, it is worth
investigating whether cone vertex algebras could be useful to gain a better
understanding of the umbral moonshine phenomenon more generally, including the
potential moonshine phenomenon involving all the optimal Jacobi theta
functions classified in [37]. As remarked in previous sections, all mock theta
functions appearing in the McKay-Thompson series of umbral moonshine can be
written in terms of the traces of cone vertex algebras discussed in this
paper. The remaining challenge is thus to find expressions of the McKay-
Thompson series that are compatible with the umbral group actions. While here
we have limited our analysis to three instances of umbral moonshine with small
umbral groups that turn out to act trivially on the cone vertex algerba
structure, more involved group actions can certainly appear in other examples,
akin to what happens in [20]. Furthermore, we note that the trace functions of
the cone vertex algebras seem to connect the McKay-Thompson series to the
meromorphic Jacobi forms associated to various instances of umbral moonshine,
as a consequence of Corollary 3.4.

  * •

Finally, it would be interesting to investigate the physical significance of
the cone vertex algebras. Vertex operator algebras provide a mathematical
axiomatization of the chiral algebra of conformal field theories in two
dimension and it would be interesting to understand what kind of conformal
fields theories cone vertex algebras are related to. For instance, it is known
that the specialized Appell-Lerch sum (1.6) captures the non-modular part of
the elliptic genus of non-compact supersymmetric coset models [28], [29]. This
could shed light on the still mysterious relation between umbral moonshine and
string theories compactified on $K3$ surfaces [29] (see also [38] for more
complete references).

#### Acknowledgments:

We thank John Duncan for interesting comments on an earlier version of the
draft. The work of M.C. and G.S. is supported by the NWO vidi grant (number
016.Vidi.189.182). The work of M.C. has also received support from ERC
starting grant H2020 #640159.

##  Appendix A Indefinite Theta Representations of Mock Theta Functions

For completeness, we include expressions for the mock theta functions used in
this work in terms of indefinite theta functions. A more extensive list of
expressions including all Ramanujan’s mock theta functions can be found in
[6]. We have555Notice that in our notation
$\theta_{1}(z,\tau)=\theta(-z,\tau)$ with $\theta(z,\tau)$ defined as in [6].

#### Order 2

$\displaystyle A(q)$
$\displaystyle=\frac{q^{2}\eta(4\tau)}{2\eta(2\tau)^{2}}\Theta^{+}_{\left(\begin{smallmatrix}1&1\\\
1&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0\\\
1\end{smallmatrix}\right),\left(\begin{smallmatrix}-1\\\
1\end{smallmatrix}\right)}\left(3\tau,\tau+\frac{1}{2},4\tau\right),$
$\displaystyle B(q)$
$\displaystyle=\frac{q^{\frac{17}{8}}\eta(2\tau)}{2\eta(\tau)\eta(4\tau)}\Theta^{+}_{\left(\begin{smallmatrix}1&1\\\
1&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0\\\
1\end{smallmatrix}\right),\left(\begin{smallmatrix}-1\\\
1\end{smallmatrix}\right)}\left(3\tau,2\tau+\frac{1}{2},4\tau\right).$

#### Order 3

$\displaystyle f(q)$
$\displaystyle=-2\frac{q^{\frac{25}{24}}}{\eta(\tau)}\Theta^{+}_{\left(\begin{smallmatrix}1&1\\\
1&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0\\\
1\end{smallmatrix}\right),\left(\begin{smallmatrix}-1\\\
1\end{smallmatrix}\right)}\left(2\tau+\frac{1}{2},\frac{1}{2}\tau,3\tau\right)+q^{\frac{1}{24}}\frac{\eta(3\tau)^{4}}{\eta(\tau)\eta(6\tau)^{2}},$
$\displaystyle\omega(q)$
$\displaystyle=\frac{q^{\frac{13}{12}}}{\eta(\tau)}\Theta^{+}_{\left(\begin{smallmatrix}1&1\\\
1&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0\\\
1\end{smallmatrix}\right),\left(\begin{smallmatrix}-1\\\
1\end{smallmatrix}\right)}\left(3\tau,2\tau+\frac{1}{2},6\tau\right)+q^{-\frac{2}{3}}\frac{\eta(6\tau)^{4}}{\eta(2\tau)\eta(3\tau)^{2}}.$

#### Order 6

$\displaystyle\sigma(q)$
$\displaystyle=q^{\frac{4}{3}}\frac{\eta(2\tau)\eta(3\tau)}{2\eta(\tau)\eta(6\tau)^{2}}\Theta^{+}_{\left(\begin{smallmatrix}1&1\\\
1&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0\\\
1\end{smallmatrix}\right),\left(\begin{smallmatrix}-1\\\
1\end{smallmatrix}\right)}\left(3\tau,\tau+\frac{1}{2},6\tau\right),$
$\displaystyle\psi_{6}(q)$
$\displaystyle=q^{\frac{25}{24}}\frac{\eta(\tau)\eta(6\tau)}{2\eta(2\tau)\eta(3\tau)^{2}}\Theta^{+}_{\left(\begin{smallmatrix}1&1\\\
1&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0\\\
1\end{smallmatrix}\right),\left(\begin{smallmatrix}-1\\\
1\end{smallmatrix}\right)}\left(\tau+\frac{1}{2},\frac{3}{2}\tau+\frac{1}{2},3\tau\right).$

#### Order 8

$\displaystyle T_{0}(q)$
$\displaystyle=\frac{q^{\frac{9}{4}}\eta(4\tau)}{2\eta(2\tau)\eta(8\tau)}\Theta^{+}_{\left(\begin{smallmatrix}1&1\\\
1&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0\\\
1\end{smallmatrix}\right),\left(\begin{smallmatrix}-1\\\
1\end{smallmatrix}\right)}\left(5\tau+\frac{1}{2},\tau,8\tau\right),$
$\displaystyle T_{1}(q)$
$\displaystyle=-\frac{q^{\frac{21}{4}}\eta(4\tau)}{2\eta(2\tau)\eta(8\tau)}\Theta^{+}_{\left(\begin{smallmatrix}1&1\\\
1&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0\\\
1\end{smallmatrix}\right),\left(\begin{smallmatrix}-1\\\
1\end{smallmatrix}\right)}\left(7\tau+\frac{1}{2},3\tau,8\tau\right),$
$\displaystyle U_{0}(q)$
$\displaystyle=\frac{q^{\frac{1}{2}}\eta(4\tau)}{2\eta(8\tau)^{2}}\Theta^{+}_{\left(\begin{smallmatrix}1&1\\\
1&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0\\\
1\end{smallmatrix}\right),\left(\begin{smallmatrix}-1\\\
1\end{smallmatrix}\right)}\left(\tau,\tau,4\tau\right),$ $\displaystyle
V_{0}(q)$
$\displaystyle=-\frac{iq^{\frac{1}{2}}}{\theta_{1}(-\tau,8\tau)}\Theta^{+}_{\left(\begin{smallmatrix}1&1\\\
1&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0\\\
1\end{smallmatrix}\right),\left(\begin{smallmatrix}-1\\\
1\end{smallmatrix}\right)}\left(\tau,4\tau+\frac{1}{2},8\tau\right)-\frac{\eta(2\tau)^{3}\eta(4\tau)}{\eta(\tau)^{2}\eta(8\tau)},$
$\displaystyle V_{1}(q)$
$\displaystyle=-\frac{iq^{\frac{3}{2}}}{2\theta_{1}(-\tau,8\tau)}\Theta^{+}_{\left(\begin{smallmatrix}1&1\\\
1&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0\\\
1\end{smallmatrix}\right),\left(\begin{smallmatrix}-1\\\
1\end{smallmatrix}\right)}\left(3\tau,2\tau+\frac{1}{2},8\tau\right).$

To make contact with the notation used in section 4, we write the function
$\Theta^{+}_{A,{\bf c},{\bf c^{\prime}}}(\mathbb{z},\tau)$ in terms of
indefinite thetas functions (3.2) through relation (2.20). We can thus rewrite

#### Order 2

$\displaystyle A(q)$ $\displaystyle=e^{-\frac{3\pi
i}{4}}q^{\frac{1}{8}}\frac{\eta(4\tau)}{2\eta(2\tau)^{2}}\Theta_{\left(\frac{3}{4},\frac{1}{4}\right),\left(0,\frac{1}{2}\right)}(4\tau),$
$\displaystyle B(q)$ $\displaystyle=e^{-\frac{3\pi
i}{4}}q^{-\frac{1}{2}}\frac{\eta(2\tau)}{2\eta(\tau)\eta(4\tau)}\Theta_{\left(\frac{3}{4},\frac{1}{2}\right),\left(0,\frac{1}{2}\right)}(4\tau).$

#### Order 3

$\displaystyle f(q)$ $\displaystyle=-2e^{-\frac{5\pi
i}{6}}\frac{q^{\frac{1}{24}}}{\eta(\tau)}\Theta_{\left(\frac{2}{3},\frac{1}{6}\right),\left(\frac{1}{2},0\right)}(3\tau)+q^{\frac{1}{24}}\frac{\eta(3\tau)^{4}}{\eta(\tau)\eta(6\tau)^{2}},$
$\displaystyle\omega(q)$ $\displaystyle=e^{-\frac{\pi
i}{2}}\frac{q^{-\frac{2}{3}}}{\eta(2\tau)}\Theta_{\left(\frac{1}{2},\frac{1}{3}\right),\left(0,\frac{1}{2}\right)}(6\tau)+q^{-\frac{2}{3}}\frac{\eta(6\tau)^{4}}{\eta(2\tau)\eta(3\tau)^{2}}.$

#### Order 6

$\displaystyle\sigma(q)$ $\displaystyle=e^{-\frac{\pi
i}{2}}q^{\frac{1}{12}}\frac{\eta(2\tau)\eta(3\tau)}{2\eta(\tau)\eta(6\tau)^{2}}\Theta_{\left(\frac{1}{2},\frac{1}{6}\right),\left(0,\frac{1}{2}\right)}\left(6\tau\right),$
$\displaystyle\psi_{6}(q)$ $\displaystyle=e^{-\frac{7\pi
i}{6}}q^{\frac{3}{8}}\frac{\eta(\tau)\eta(6\tau)}{2\eta(2\tau)\eta(3\tau)^{2}}\Theta_{\left(\frac{1}{3},\frac{1}{2}\right),\left(\frac{1}{2},\frac{1}{2}\right)}\left(3\tau\right).$

#### Order 8

$\displaystyle T_{0}(q)$ $\displaystyle=e^{-\frac{3\pi
i}{4}}q^{\frac{1}{16}}\frac{\eta(4\tau)}{2\eta(2\tau)\eta(8\tau)}\Theta_{\left(\frac{5}{8},\frac{1}{8}\right),\left(\frac{1}{2},0\right)}(8\tau),$
$\displaystyle T_{1}(q)$ $\displaystyle=-e^{-\frac{5\pi
i}{4}}q^{-\frac{7}{16}}\frac{\eta(4\tau)}{2\eta(2\tau)\eta(8\tau)}\Theta_{\left(\frac{7}{8},\frac{3}{8}\right),\left(\frac{1}{2},0\right)}(8\tau),$
$\displaystyle U_{0}(q)$
$\displaystyle=q^{\frac{1}{8}}\frac{\eta(4\tau)}{2\eta(8\tau)^{2}}\Theta_{\left(\frac{1}{4},\frac{1}{4}\right),(0,0)}(4\tau),$
$\displaystyle V_{0}(q)$ $\displaystyle=-ie^{-\frac{\pi
i}{8}}\frac{q^{-\frac{1}{16}}}{\theta_{1}(-\tau,8\tau)}\Theta_{\left(\frac{1}{8},\frac{1}{2}\right),\left(0,\frac{1}{2}\right)}(8\tau)-\frac{\eta(2\tau)^{3}\eta(4\tau)}{\eta(\tau)^{2}\eta(8\tau)},$
$\displaystyle V_{1}(q)$ $\displaystyle=-ie^{-\frac{3\pi
i}{8}}\frac{q^{\frac{3}{16}}}{2\theta_{1}(-\tau,8\tau)}\Theta_{\left(\frac{3}{8},\frac{1}{4}\right),\left(0,\frac{1}{2}\right)}(8\tau).$

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