††footnotetext: _MSC2010:_ 11F37, 11F50, 17B69, 17B81, 20C35.

#  Meromorphic Jacobi Forms of Half-Integral Index and Umbral Moonshine
Modules

Miranda C. N. Cheng<EMAIL_ADDRESS>Institute of Physics and Korteweg-de Vries
Institute for Mathematics
University of Amsterdam, Amsterdam, the Netherlands111On leave from CNRS,
France..  John F. R. Duncan<EMAIL_ADDRESS>Department of Mathematics
and Computer Science
Emory University, Atlanta, GA 30322, U.S.A.

###### Abstract

In this work we consider an association of meromorphic Jacobi forms of half-
integral index to the pure D-type cases of umbral moonshine, and solve the
module problem for four of these cases by constructing vertex operator
superalgebras that realise the corresponding meromorphic Jacobi forms as
graded traces. We also present a general discussion of meromorphic Jacobi
forms with half-integral index and their relationship to mock modular forms.

###### Contents

  1. 1 Introduction
  2. 2 Meromorphic and Mock Jacobi Forms
    1. 2.1 Elliptic Forms
    2. 2.2 Meromorphic Jacobi Forms
    3. 2.3 Residues
    4. 2.4 Mock Modularity
    5. 2.5 Umbral Meromorphic Forms
  3. 3 Umbral Moonshine Modules
    1. 3.1 Lambency $10+5$
    2. 3.2 Lambency $14+7$
    3. 3.3 Lambency $22+11$
    4. 3.4 Lambency $46+23$
  4. 4 Discussion
  5. Acknowledgements

##  1 Introduction

Moonshine connects modular objects and finite groups in an interesting way.
Starting from 2010, the discovery of Mathieu moonshine for $M_{24}$ [1] has
initiated a new wave of activity in the study of moonshine. One development
arising is the realisation that this $M_{24}$ moonshine is but one of 23
instances of the so-called umbral moonshine [2, 3]. Umbral moonshine
associates a vector-valued mock modular form
$H^{(\ell)}_{g}=(H^{(\ell)}_{g,r})$, with a certain level and multiplier
system, to each conjugacy class $[g]$ of a finite group $G^{(\ell)}$, where
$\ell$ is a symbol (called lambency) which indexes the 23 Niemeier lattices
(and also indexes certain genus zero groups of isometries of the upper half-
plane, cf. [4]). The group $G^{(\ell)}$ can be defined explicitly as the group
of outer automorphisms of the Niemeier lattice corresponding to $\ell$ (cf.
§2.4 of [3]). One of the umbral moonshine conjectures, proven for Mathieu
moonshine in [5] and proven for umbral moonshine in general in [6], then
states that the $H^{(\ell)}_{g}$ can be identified with the graded characters
of a certain infinite-dimensional $G^{(\ell)}$-module.

This is reminiscent of monstrous moonshine [7], which attaches to each
conjugacy class $[g]$ of the monster group a certain Hauptmodul $T_{g}(\tau)$
for a certain subgroup $\Gamma_{g}$ of
$\operatorname{\textsl{SL}}_{2}({\mathbb{R}})$ that defines a genus zero
quotient of the upper half-plane. These functions $T_{g}(\tau)$ were
ultimately shown by Borcherds [8] to coincide with the graded characters of a
certain infinite-dimensional module $V^{\natural}$ for the monster group,
which was constructed earlier by Frenkel–Lepowsky–Meurman [9, 10] and which
famously possesses vertex operator algebra structure [11, 12].

We would like to have a similar understanding of the modules for umbral
moonshine. Arguably, this is the most important outstanding question in the
study of umbral moonshine. The first case of umbral moonshine for which the
module problem was solved is the case where $\ell=30+6,10,15$, the Niemeier
lattice is $E_{8}^{\oplus 3}$, and $G^{(\ell)}\cong S_{3}$ [13]. This is the
unique case for which the Niemeier lattice coincides with its root lattice. So
far the techniques of [13] have not been successfully extended to other cases.

In the more recent work [14], and in the present paper, the focus is shifted
from the vector-valued mock modular forms $H^{(\ell)}_{g}$ to certain
meromorphic Jacobi forms. The former may be obtained from the latter once a
canonically defined polar part that captures the poles of the latter is
removed. We explain this in detail in §2. The meromorphic Jacobi forms
involved in the cases of umbral moonshine discussed here and in [14] admit
expressions as sums of ratios of infinite products, and this makes it possible
to realise them as (twined) partition functions of certain free chiral
conformal field theories. Using this strategy, vertex operator superalgebras
are constructed that possess automorphisms containing the corresponding umbral
groups $G^{(\ell)}$, and are such that the twined partition functions arising
coincide with the meromorphic Jacobi forms specified by umbral moonshine. In
this way, the umbral moonshine modules for two of the pure A-type cases,
corresponding to $\ell=7$ and $\ell=13$, have been constructed in [14] (while
partial solutions to the module problem are given in loc. cit. for $\ell=4$
and $\ell=5$). The corresponding Niemeier lattices are those with the root
systems $A_{6}^{\oplus 4}$ and $A_{12}^{\oplus 2}$, respectively.

The main objective of this work is the construction of umbral moonshine
modules for four of the five pure D-type umbral moonshine cases, corresponding
to $\ell=10+5$, $\ell=14+7$, $\ell=22+11$ and $\ell=46+23$, for which the
corresponding Niemeier lattices are those whose roots systems are
$D_{6}^{\oplus 4}$, $D_{8}^{\oplus 3}$, $D_{12}^{\oplus 2}$ and $D_{24}$,
respectively. We achieve this in §3. See Theorems 3.1, 3.2, 3.3 and 3.4. Our
methods are similar to those of [14], but diverge from loc. cit. in an
apparently powerful aspect, by considering meromorphic Jacobi forms of half-
integral index, rather than those of integral index which are attached to the
pure D-type cases of umbral moonshine in [3].

In order to support our approach we present results on meromorphic Jacobi
forms of half-integral index and their relationship to mock modular forms in
§2. Theorem 2.4 represents an extension to half-integral index of the
canonical splitting of a meromorphic Jacobi form into polar and finite parts,
which was first given for integral index in [16]. Theorem 2.6 precisely
formulates the relationship between mock modular forms and meromorphic Jacobi
forms of weight $1$ and half-integral index that have simple poles only at
torsion points, and satisfy a certain growth condition.

The paper is structured in the following way. In §2 we discuss meromorphic
Jacobi forms of half-integral index and their relationship to mock modular
forms. We review elliptic forms, theta series and Eicher–Zagier operators of
half-integral index in §2.1. We review holomorphic, skew-holomorphic and
meromorphic Jacobi forms in §2.2, and also review there the construction of
the finite part of a meromorphic Jacobi form due to Dabholkar–Murthy–Zagier
[16]. In §2.3 we review the construction of the polar part of a meromorphic
Jacobi form from its poles, assuming that they are simple and constrained to
lie at torsion points. This extends the discussion of §8.2 of [16] to half-
integral indexes. In §2.4 we specialise to meromorphic Jacobi forms of weight
$1$ satisfying a growth condition, and prove our main result (see Theorem 2.6)
on the mock modularity of the finite parts and theta coefficients of such
forms. In §2.5 we explain the construction which converts the vector-valued
mock modular forms of pure D-type umbral moonshine into meromorphic Jacobi
forms of half-integral index.

Our main constructions appear in §3, where we obtain vertex algebraic
realisations of the half-integral index meromorphic Jacobi forms presented in
§2.5, for four of the pure D-type cases of umbral moonshine. The case of
$D_{6}^{\oplus 4}$ is treated in §3.1 (see Theorem 3.1), the case of
$D_{8}^{\oplus 3}$ is treated in §3.2 (see Theorem 3.2), the case of
$D_{12}^{\oplus 2}$ is treated in §3.3 (see Theorem 3.3), and $D_{24}$ is
treated in §3.4 (see Theorem 3.4). We conclude in §4 with discussions on the
relation between the known modules for pure A-type and pure D-type umbral
moonshine, as well as the possible physical relevance of our constructions.

##  2 Meromorphic and Mock Jacobi Forms

In this section we describe a relationship between meromorphic Jacobi forms
and mock modular forms. This relationship goes back to work of Zwegers [15],
which was subsequently developed by Dabholkar–Murthy–Zagier in [16]. Here we
follow [16] quite closely, but extend the discussion to include half-integral
index, and refine it for meromorphic Jacobi forms with weight $1$.

###  2.1 Elliptic Forms

We first discuss elliptic forms with half-integral index. For this set
$\operatorname{e}(x):=e^{2\pi ix}$ and let
${\mathbb{H}}:=\\{\tau\in{\mathbb{C}}\mid\Im(\tau)>0\\}$ denote the upper
half-plane. For $m\in{1\over 2}{\mathbb{Z}}$ define the index $m$ elliptic
action of ${\mathbb{Z}}^{2}$ on functions
$\phi:{\mathbb{H}}\times{\mathbb{C}}\to{\mathbb{C}}$ by setting

$(\phi\lvert_{m}(\lambda,\mu))(\tau,z):=e\left(m\lambda^{2}\tau+2m\lambda
z+{(\lambda+\mu)m}\right)\,\phi(\tau,z+\lambda\tau+\mu)$ (2.1)

for $(\lambda,\mu)\in{\mathbb{Z}}^{2}$. With this definition we call a smooth
function $\phi:{\mathbb{H}}\times{\mathbb{C}}\to{\mathbb{C}}$ an elliptic
form222Our nomenclature follows [16], except that elliptic forms are assumed
to be holomorphic in that work, and the focus there is on integral index. with
index $m$ if $(\phi\lvert_{m}(\lambda,\mu))=\phi$ for all
$(\lambda,\mu)\in{\mathbb{Z}}^{2}$. Write $\mathcal{E}_{m}$ for the space of
elliptic forms of index $m\in\frac{1}{2}{\mathbb{Z}}$.

For $m\in{1\over 2}{\mathbb{Z}}$ and $r\in{\mathbb{Z}}+m$ define the index $m$
theta function $\theta_{m,r}:{\mathbb{H}}\times{\mathbb{C}}\to{\mathbb{C}}$ by
setting

$\displaystyle\theta_{m,r}(\tau,z):=\sum_{\begin{subarray}{c}\ell\in{\mathbb{Z}}+m\\\
\ell=r{\rm\;mod\;}2m\end{subarray}}\operatorname{e}(m\ell)y^{\ell}q^{\frac{\ell^{2}}{4m}}$
(2.2)

where $q:=\operatorname{e}(\tau)$ and $y:=\operatorname{e}(z)$. Also define
$\theta^{1}_{m,r}(\tau):=\frac{1}{2\pi i}{\partial\over\partial
z}\theta_{m,r}(\tau,z)\lvert_{z=0}$. Then a smooth function
$\phi:{\mathbb{H}}\times{\mathbb{C}}\to{\mathbb{C}}$ belongs to
$\mathcal{E}_{m}$ if and only if it can be written in the form

$\displaystyle\phi(\tau,z)=\sum_{r{\rm\;mod\;}2m}h_{r}(\tau)\theta_{m,r}(\tau,z)$
(2.3)

for some $2m$ theta coefficient functions $h_{r}:{\mathbb{H}}\to{\mathbb{C}}$.
We emphasise here that $m\in\frac{1}{2}{\mathbb{Z}}$, and the summation in
(2.3) is over representatives for ${\mathbb{Z}}+m$ modulo translation by $2m$.

For an explicit example note that the $m=r=\frac{1}{2}$ case of (2.2) recovers
a classical Jacobi theta function

$\displaystyle\begin{split}\theta_{\frac{1}{2},\frac{1}{2}}(\tau,z)&=\sum_{n\in{\mathbb{Z}}}i(-1)^{n+1}y^{n+\frac{1}{2}}q^{\frac{1}{2}(n+\frac{1}{2})^{2}}\\\
&=-iy^{\frac{1}{2}}q^{\frac{1}{8}}\prod_{n>0}(1-y^{-1}q^{n-1})(1-yq^{n})(1-q^{n}).\end{split}$
(2.4)

We will denote $\theta_{\frac{1}{2},\frac{1}{2}}(\tau,z)$ by
$\theta_{1}(\tau,z)$ in §§3,4 (cf. (3.5)).

Elliptic forms come equipped with some naturally defined symmetry. To explain
this define a function $m\mapsto\tilde{m}$ on $\frac{1}{2}{\mathbb{Z}}$ by
setting

$\displaystyle\tilde{m}:=\begin{cases}m&\text{ if $m\in{\mathbb{Z}}$,}\\\
2m&\text{ if $m\in{\mathbb{Z}}+\frac{1}{2}$.}\end{cases}$ (2.5)

Then the group $O_{\tilde{m}}:=\\{a\in{\mathbb{Z}}/2\tilde{m}{\mathbb{Z}}\mid
a^{2}=1{\rm\;mod\;}4\tilde{m}\\}$ acts naturally on ${\mathcal{E}}_{m}$ via

$\displaystyle\phi\cdot a:=\sum_{r{\rm\;mod\;}2m}h_{r}\theta_{m,ra}$ (2.6)

when $\phi=\sum_{r{\rm\;mod\;}2m}h_{r}\theta_{m,r}$ is the theta decomposition
(2.3) of $\phi\in{\mathcal{E}}_{m}$.

It develops that $O_{\tilde{m}}$ is naturally isomorphic to the group
$\operatorname{Ex}(\tilde{m})$ of exact divisors of $\tilde{m}$. We obtain an
explicit isomorphism $O_{\tilde{m}}\to\operatorname{Ex}(\tilde{m})$, to be
denoted $n\mapsto a(n)$, by letting $a(n)$ be the unique element of
${\mathbb{Z}}/2\tilde{m}{\mathbb{Z}}$ such that
$a(n)=1{\rm\;mod\;}2\frac{\tilde{m}}{n}$ and $a(n)=-1{\rm\;mod\;}2n$. With
this definition we have

$\displaystyle\phi|\mathcal{W}_{m}(n)=\phi\cdot a(n)$ (2.7)

for $\phi\in\mathcal{E}_{m}$ and $n\in\operatorname{Ex}(\tilde{m})$, where

$\displaystyle(\phi|\mathcal{W}_{m}(n))(\tau,z):=\tfrac{1}{n}\sum_{a,b=0}^{n-1}\operatorname{e}(m(\tfrac{a^{2}}{n^{2}}\tau+2\tfrac{a}{n}z+\tfrac{ab}{n^{2}}+ab+a+b))\phi(\tau,z+\tfrac{a}{n}\tau+\tfrac{b}{n}).$
(2.8)

We call $\mathcal{W}_{m}(n)$ an Eichler–Zagier operator of index $m$. The
$\mathcal{W}_{m}(n)$ for integral index first appeared in [17]. Note that
(2.8) is well-defined and stabilises $\mathcal{E}_{m}$ for $n$ an arbitrary
divisor of $\tilde{m}$.

###  2.2 Meromorphic Jacobi Forms

We now explain precisely what we mean by a meromorphic Jacobi form. Recall
that a holomorphic Jacobi form of weight $k\in{\mathbb{Z}}$ and index
$m\in\frac{1}{2}{\mathbb{Z}}$ for a group
$\Gamma<\operatorname{\textsl{SL}}_{2}({\mathbb{Z}})$ is an elliptic form
$\phi=\sum_{r{\rm\;mod\;}2m}h_{r}\theta_{m,r}$ of index $m$ that is invariant
for the weight $k$ index $m$ modular action

$\displaystyle\left(\phi|_{k,m}\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\right)(\tau,z):=\phi\left(\tfrac{a\tau+b}{c\tau+d},\tfrac{z}{c\tau+d}\right)e\left(-m\tfrac{cz^{2}}{c\tau+d}\right)(c\tau+d)^{-k}$
(2.9)

of $\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right)\in\Gamma$, and
whose theta coefficients (2.3) are holomorphic and bounded near any cusp of
$\Gamma$. A skew-holomorphic Jacobi form is defined similarly, just replacing
(2.9) with the weight $k$ index $m$ skew-modular action

$\displaystyle\left(\phi|_{k,m}^{\rm sk}\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\right)(\tau,z):=\phi\left(\tfrac{a\tau+b}{c\tau+d},\tfrac{z}{c\tau+d}\right)e\left(-m\tfrac{cz^{2}}{c\tau+d}\right)(c\tau+d)^{-k+1}|c\tau+d|^{-1}$
(2.10)

of $\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right)\in\Gamma$, and
requiring the theta coefficients (2.3) to be anti-holomorphic and bounded near
any cusp of $\Gamma$. For us a meromorphic Jacobi form is a quotient
$\psi=\frac{\phi_{1}}{\phi_{2}}$ of holomorphic Jacobi forms such that for any
fixed $\tau\in{\mathbb{H}}$ the function $z\mapsto\phi_{2}(\tau,z)$ is not
identically zero. With this definition $z\mapsto\psi(\tau,z)$ is meromorphic
on ${\mathbb{C}}$ for all $\tau\in{\mathbb{H}}$. If $k_{i}$ and $m_{i}$ are
the weight and index (respectively) of $\phi_{i}$ then $k=k_{1}-k_{2}$ and
$m=m_{1}-m_{2}$ are the weight and index (respectively) of $\psi$.

A weak holomorphic Jacobi form of weight $k$ and index $m$ is a holomorphic
function $\phi\in\mathcal{E}_{m}$ that is invariant for the weight $k$ modular
action (2.9), and is such that $\tau\mapsto(\phi|_{k,m}\gamma)(\tau,z)$ is
bounded as $\Im(\tau)\to\infty$ for any fixed $z\in{\mathbb{C}}$ and
$\gamma\in\operatorname{\textsl{SL}}_{2}({\mathbb{Z}})$. We will say that a
meromorphic Jacobi form $\psi$ of weight $k$ and index $m$ is a weak
meromorphic Jacobi form if, for fixed $z\in{\mathbb{C}}$ and arbitrary
$\gamma\in\operatorname{\textsl{SL}}_{2}({\mathbb{Z}})$, the function
$\tau\mapsto(\psi|_{k,m}\gamma)(\tau,z)$ is bounded as $\Im(\tau)\to\infty$
along any path that avoids poles. In general we have that
$\tau\mapsto(\psi|_{k,m}\gamma)(\tau,z)$ is $O(e^{C\Im(\tau)})$ as
$\Im(\tau)\to\infty$ (along paths that avoid poles), for some $C>0$ that is
independent of $\gamma$ and $z$. In the rest of this section we will focus on
meromorphic Jacobi forms for $\operatorname{\textsl{SL}}_{2}({\mathbb{Z}})$,
but meromorphic Jacobi forms for proper subgroups
$\Gamma<\operatorname{\textsl{SL}}_{2}({\mathbb{Z}})$ will appear in §3.

If $\psi$ is a meromorphic Jacobi form then from invariance under the index
$m$ elliptic action (2.1) it follows that the function
$z\mapsto\psi(\tau,z)\operatorname{e}(-rz)$ is invariant under $z\mapsto z+1$
for $r\in{\mathbb{Z}}+m$. So for fixed $\tau\in{\mathbb{H}}$ the integral

$\displaystyle
h_{r}(\tau):=\operatorname{e}(mr)q^{\frac{r^{2}}{4m}}\int_{{\mathbb{R}}/{\mathbb{Z}}}\psi(\tau,z-\tfrac{r}{2m}\tau)\operatorname{e}(-rz){\rm
d}z$ (2.11)

is well-defined for $r\in{\mathbb{Z}}+m$, so long as $z\mapsto\psi(\tau,z)$
has no poles of the form $z=-\tfrac{r}{2m}\tau+\beta$ for
$\beta\in{\mathbb{R}}$. If $z=-\tfrac{r}{2m}\tau+\beta$ is a pole for some
$\beta\in{\mathbb{R}}$ then choose $\beta_{0}\in{\mathbb{R}}$ so that
$z=-\tfrac{r}{2m}\tau+\beta_{0}$ is not a pole, let $\gamma$ be a deformation
of the line segment $[\beta_{0},\beta_{0}+1]$ that passes just above the poles
of $z\mapsto\psi(\tau,z)$, let $\bar{\gamma}$ be the image of $\gamma$ under
complex conjugation, and replace $\int_{{\mathbb{R}}/{\mathbb{Z}}}$ with
$\frac{1}{2}\left(\int_{\gamma}+\int_{\bar{\gamma}}\right)$ in (2.11). For
$m\in{\mathbb{Z}}$ this recovers the definition (8.2) in [16]. For general
$m\in\frac{1}{2}{\mathbb{Z}}$ we check using the behavior of $\psi$ under
$z\mapsto z+\tau$ (cf. (2.1)) that $h_{r}(\tau)$ depends only on
$r{\rm\;mod\;}2m$.

If $\psi$ happens to be holomorphic then the $h_{r}$ of (2.11) are the theta
coefficients (2.3) of $\psi$. If $\psi$ has poles it cannot admit such a
decomposition, but following [16] we may define the finite part $\psi^{F}$ of
$\psi$ by setting

$\displaystyle\psi^{F}(\tau,z):=\sum_{r{\rm\;mod\;}2m}h_{r}(\tau)\theta_{m,r}(\tau,z),$
(2.12)

where the sum is over $r\in{\mathbb{Z}}+m$ modulo $2m$, and the $\theta_{m,r}$
are as in (2.2). The construction (2.11) does not preserve modular invariance
(2.9) in general. It will develop in §2.4 that $\psi^{F}$ is a mock Jacobi
form, and the $h_{r}$ are mock modular forms, when certain hypotheses are
satisfied by $\psi$ (cf. Theorem 2.6).

###  2.3 Residues

In order to relate meromorphic Jacobi forms to mock modular forms we follow
[16] by restricting to meromorphic Jacobi forms $\psi$ such that the poles of
$z\mapsto\psi(\tau,z)$, for fixed $\tau\in{\mathbb{H}}$, are constrained to
lie in the set of torsion points ${\mathbb{Q}}\tau+{\mathbb{Q}}$. For
simplicity we further restrict to meromorphic Jacobi forms whose poles are
simple. (Single poles and double poles are discussed in [16].) So for
$m\in\frac{1}{2}{\mathbb{Z}}$ let $J^{\rm mst}_{k,m}$ denote the space of
meromorphic Jacobi forms $\psi$ of weight $k$ and index $m$ for
$\Gamma=\operatorname{\textsl{SL}}_{2}({\mathbb{Z}})$ such that the poles of
the functions $z\mapsto\psi(\tau,z)$ are simple, and occur only at torsion
points $z\in{\mathbb{Q}}\tau+{\mathbb{Q}}$.

Our next objective is to present an explicit construction of the difference
$\psi^{P}=\psi-\psi^{F}$, which is called the polar part of $\psi$. To prepare
for this observe that
${\mathbb{Z}}^{2}\rtimes\operatorname{\textsl{SL}}_{2}({\mathbb{Z}})$ acts
naturally from the right on ${\mathbb{Q}}^{2}$ according to the rules

$\displaystyle(\alpha,\beta)\cdot(\lambda,\mu):=(\alpha+\lambda,\beta+\mu),\quad(\alpha,\beta)\cdot\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right):=(\alpha a+\beta c,\alpha b+\beta d).$ (2.13)

Here $(\alpha,\beta)\in{\mathbb{Q}}^{2}$, $(\lambda,\mu)\in{\mathbb{Z}}^{2}$
and $\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\operatorname{\textsl{SL}}_{2}({\mathbb{Z}})$.
It will be useful to have a description of the orbits of this action. With
this in mind we set

$\displaystyle
S_{n}:=\left\\{(\tfrac{k}{n},\tfrac{l}{n})\in{\mathbb{Q}}^{2}\mid
k,l\in{\mathbb{Z}},\,\gcd(k,l,n)=1\right\\}$ (2.14)

for $n\in{\mathbb{Z}}^{+}$.

######  Lemma 2.1.

The sets $S_{n}$ for $n\in{\mathbb{Z}}^{+}$ are the orbits for the action of
${\mathbb{Z}}^{2}\rtimes\operatorname{\textsl{SL}}_{2}({\mathbb{Z}})$ on
${\mathbb{Q}}^{2}$.

###### Proof.

It is straightforward to check that $S_{n}$ is stable under the action of
${\mathbb{Z}}^{2}\rtimes\operatorname{\textsl{SL}}_{2}({\mathbb{Z}})$. To see
that ${\mathbb{Z}}^{2}\rtimes\operatorname{\textsl{SL}}_{2}({\mathbb{Z}})$
acts transitively observe that if $s=(\tfrac{k}{n},\tfrac{l}{n})\in S_{n}$
where $k,l\in{\mathbb{Z}}$ and $\gcd(k,l,n)=1$ then there exists
$j\in{\mathbb{Z}}$ such that $\gcd(k,l+jn)=1$. So after replacing $s$ with
$s\cdot(0,j)$ we may assume that $\gcd(k,l)=1$. Then for $x,y\in{\mathbb{Z}}$
such that $kx+ly=1$ we have $s\cdot\left(\begin{smallmatrix}x&-l\\\
y&k\end{smallmatrix}\right)=(\frac{1}{n},0)$. So $S_{n}$ is just the orbit of
$(\frac{1}{n},0)$ under
${\mathbb{Z}}^{2}\rtimes\operatorname{\textsl{SL}}_{2}({\mathbb{Z}})$.

To see that the $S_{n}$ exhaust ${\mathbb{Q}}^{2}$ let
$S\subset{\mathbb{Q}}^{2}$ be an arbitrary orbit for
${\mathbb{Z}}^{2}\rtimes\operatorname{\textsl{SL}}_{2}({\mathbb{Z}})$. If
$(\frac{k}{n},\frac{l}{n})\in S$ where $k,l,n\in{\mathbb{Z}}$ (and the
components are not necessarily fractions in lowest form), then $n\alpha$ and
$n\beta$ are integers for every $(\alpha,\beta)\in S$. So there is a smallest
positive integer $n$ such that $n\alpha$ and $n\beta$ are integers for every
$(\alpha,\beta)\in S$. For this $n$ there must be
$(\frac{k}{n},\frac{l}{n})\in S$ such that $\gcd(k,l,n)=1$. Then $S=S_{n}$ by
the above. ∎

Given $\psi\in J^{\rm mst}_{k,m}$ define $S(\psi)\subset{\mathbb{Q}}^{2}$ to
be the set of pairs $s=(\alpha,\beta)$ such that $z\mapsto\psi(\tau,z)$ has a
pole at $z_{s}(\tau):=\alpha\tau+\beta$ for some $\tau\in{\mathbb{H}}$. When
there is no risk of confusion we write $z_{s}$ in place of $z_{s}(\tau)$. For
$s=(\alpha,\beta)\in S(\psi)$ we follow [16] in defining

$\displaystyle D_{s}(\tau):=2\pi i\operatorname{e}(m\alpha
z_{s})\operatorname{Res}_{z=z_{s}}\psi(\tau,z).$ (2.15)

From the invariance of $\psi$ under the elliptic (2.1) and modular (2.9)
actions we deduce that

$\displaystyle D_{s\cdot(\lambda,\mu)}(\tau)$
$\displaystyle=\operatorname{e}(m(\alpha\mu-\beta\lambda+\lambda\mu+\lambda+\mu))D_{s}(\tau),$
(2.16) $\displaystyle D_{s\cdot\gamma}(\tau)$
$\displaystyle=D_{s}(\gamma\tau)(c\tau+d)^{1-k},$ (2.17)

for $(\lambda,\mu)\in{\mathbb{Z}}^{2}$ and
$\gamma=\left(\begin{smallmatrix}*&*\\\
c&d\end{smallmatrix}\right)\in\operatorname{\textsl{SL}}_{2}({\mathbb{Z}})$.
(See Proposition 8.3 of [16] for details on (2.17).) So in particular,
$S(\psi)$ is stable for the action of
${\mathbb{Z}}^{2}\rtimes\operatorname{\textsl{SL}}_{2}({\mathbb{Z}})$, and the
$D_{s}(\tau)$ are modular forms of weight $k-1$. From Lemma 2.1 we deduce that
$S(\psi)$ is a union of finitely many of the $S_{n}$. We formulate the
modularity of the $D_{s}(\tau)$ more precisely as follows.

######  Proposition 2.2.

Let $\psi\in J_{k,m}^{\rm mst}$ for $k\in{\mathbb{Z}}$ and
$m\in\frac{1}{2}{\mathbb{Z}}$. If $s\in S(\psi)\cap S_{n}$ then $D_{s}(\tau)$
is a weakly holomorphic modular form of weight $k-1$ for $\Gamma(2n^{2})$. If
$\psi$ is weak then $D_{s}(\tau)$ is a holomorphic modular form of weight
$k-1$ for $\Gamma(2n^{2})$.

###### Proof.

The growth conditions on $D_{s}(\tau)$ near cusps follow from the definitions
of meromorphic and weak meromorphic Jacobi form (cf. §2.2). To check
modularity write $s=(\frac{k}{n},\frac{l}{n})$ where $\gcd(k,l,n)=1$ and let
$\gamma=\left(\begin{smallmatrix}1+2n^{2}a^{\prime}&2n^{2}b^{\prime}\\\
2n^{2}c^{\prime}&1+2n^{2}d^{\prime}\end{smallmatrix}\right)\in\Gamma(2n^{2})$.
Then

$\displaystyle
s\cdot\gamma=(\tfrac{k}{n}+2n\lambda,\tfrac{l}{n}+2n\mu)=s\cdot(2n\lambda,2n\mu)$
(2.18)

where $\lambda=ka^{\prime}+lc^{\prime}$ and $\mu=kb^{\prime}+ld^{\prime}$ are
integers. Now $D_{s\cdot(2n\lambda,2n\mu)}(\tau)=D_{s}(\tau)$ according to
(2.16), so from (2.17) we have

$\displaystyle
D_{s}(\gamma\tau)(c\tau+d)^{1-k}=D_{s\cdot\gamma}(\tau)=D_{s\cdot(2n\lambda,2n\mu)}(\tau)=D_{s}(\tau)$
(2.19)

where $c=2n^{2}c^{\prime}$ and $d=1+2n^{2}d^{\prime}$. This proves the claim.
∎

The next result will be applied in §2.4.

######  Lemma 2.3.

Let $\psi\in J_{k,m}^{\rm mst}$ for $m\in\frac{1}{2}{\mathbb{Z}}$ and suppose
that $S_{n}\subset S(\psi)$. Then for $s=(\frac{1}{n},0)$ the Fourier
expansion of $D_{s}(\tau)$ takes the form

$\displaystyle D_{s}(\tau)=\sum_{\begin{subarray}{c}N\in{\mathbb{Z}}\\\
N=2m(n+1){\rm\;mod\;}2n\end{subarray}}c_{s}(N)q^{\frac{N}{2n^{2}}}.$ (2.20)

In particular, the constant term of $D_{s}(\tau)$ vanishes unless $n$ divides
$\tilde{m}$.

###### Proof.

By Proposition 2.2 we know that $D_{s}(\tau)$ has a Fourier expansion of the
form $D_{s}(\tau)=\sum_{N\in{\mathbb{Z}}}c_{s}(N)q^{\frac{N}{2n^{2}}}$. We
also have $s\cdot(0,1)=s\cdot\gamma$ for
$\gamma=\left(\begin{smallmatrix}1&n\\\ 0&1\end{smallmatrix}\right)$. So
applying (2.16) and (2.17) we obtain

$\displaystyle
D_{s}(\tau+n)=D_{s\cdot\gamma}(\tau)=D_{s\cdot(0,1)}(\tau)=\operatorname{e}(m(\tfrac{1}{n}+1))D_{s}(\tau).$
(2.21)

This shows that $c_{s}(N)=0$ unless
$\frac{N}{2n}=m\frac{n+1}{n}{\rm\;mod\;}1$. This is the same as
$N=2m(n+1){\rm\;mod\;}2n$ so (2.20) holds. Of course
$2m(n+1)=2m{\rm\;mod\;}2n$ if $m\in{\mathbb{Z}}$ so $c_{s}(0)$ must vanish
unless $n$ divides $m=\tilde{m}$ when $m\in{\mathbb{Z}}$. If
$m\in{\mathbb{Z}}+\frac{1}{2}$ then $\tilde{m}=2m$ so $c_{s}(0)=0$ unless
$\tilde{m}(n+1)$ is divisible by $2n$. This forces $n$ to be a divisor of
$\tilde{m}$. The proof is complete. ∎

Roughly speaking, we will obtain $\psi^{P}$ by averaging the residues of
$z\mapsto\psi(\tau,z)$ that appear in some fundamental parallelogram for the
action of ${\mathbb{Z}}\tau+{\mathbb{Z}}$ on ${\mathbb{C}}$. In order to
realise this we define the index $m$ averaging operator

$\displaystyle\operatorname{Av}_{m}(F(y)):=\sum_{k\in{\mathbb{Z}}}(-1)^{2mk}q^{mk^{2}}y^{2mk}F(q^{k}y).$
(2.22)

For the purpose at hand, $F(y)$ will be a rational function in $y$ when
$m\in{\mathbb{Z}}$, and a rational function in $y^{\frac{1}{2}}$ when
$m\in{\mathbb{Z}}+\frac{1}{2}$. Indeed, for $m\in\frac{1}{2}{\mathbb{Z}}$ and
$c\in{\mathbb{Q}}$ we set

$\displaystyle{\cal
R}_{m,c}(y):=\begin{cases}y^{c}\dfrac{1}{2}\dfrac{y+1}{y-1}&\text{ if
$m-c\in{\mathbb{Z}}$,}\\\ y^{\lceil c\rceil}\dfrac{1}{y-1}&\text{ if
$m\in{\mathbb{Z}}$, $c\notin{\mathbb{Z}}$,}\\\ y^{\lceil
c-\frac{1}{2}\rceil}\dfrac{y^{\frac{1}{2}}}{y-1}&\text{ if
$m\in{\mathbb{Z}}+\frac{1}{2}$,
$c\notin{\mathbb{Z}}+\frac{1}{2}$.}\end{cases}$ (2.23)

We then define the universal Appell–Lerch sum ${\cal A}^{s}_{m}(\tau,z)$ for
$m\in\frac{1}{2}{\mathbb{Z}}$ and $s=(\alpha,\beta)\in{\mathbb{Q}}^{2}$ by
setting

$\displaystyle{\cal A}^{s}_{m}(\tau,z):=\operatorname{e}(-m\alpha
z_{s})\operatorname{Av}_{m}({\cal R}_{m,-2m\alpha}(yy_{s}^{-1}))$ (2.24)

where $y_{s}:=\operatorname{e}(z_{s})=\operatorname{e}(\beta)q^{\alpha}$. Note
that (2.24) reduces to (8.17) of [16] when $m\in{\mathbb{Z}}$. Also, the
function ${\cal R}_{m,c}$ admits a more uniform description as a series:

$\displaystyle{\cal R}_{m,c}(y)=\begin{cases}-\sum_{\ell\geq
c}y^{\ell}(1-\frac{1}{2}\delta_{\ell,c})&\text{ for $|y|<1$,}\\\
\quad\sum_{\ell\leq c}y^{\ell}(1-\frac{1}{2}\delta_{\ell,c})&\text{ for
$|y|>1$,}\end{cases}$ (2.25)

where the summations are over $\ell\in{\mathbb{Z}}+m$.

Using the identity ${\cal R}_{m,c+k}(y)={\cal R}_{m,c}(y)y^{k}$ for
$k\in{\mathbb{Z}}$ we obtain

$\displaystyle{\cal
A}_{m}^{s\cdot(\lambda,\mu)}(\tau,z)=\operatorname{e}(-m(\alpha\mu-\beta\lambda+\lambda\mu+\lambda+mu)){\cal
A}_{m}^{s}(\tau,z)$ (2.26)

for $s=(\alpha,\beta)\in{\mathbb{Q}}^{2}$ and
$(\lambda,\mu)\in{\mathbb{Z}}^{2}$. So comparing (2.16) with (2.26) we see
that it makes sense to define

$\displaystyle\psi^{P}(\tau,z):=\sum_{s\in
S(\psi)/{\mathbb{Z}}^{2}}D_{s}(\tau){\cal A}_{m}^{s}(\tau,z)$ (2.27)

where the sum is over any set of representatives for the action of
${\mathbb{Z}}^{2}$ on $S(\psi)$.

The next result recovers Theorem 8.1 of [16] when $m\in{\mathbb{Z}}$. Given
the definitions we have made, the proof for $m\in{\mathbb{Z}}+\frac{1}{2}$ is
essentially the same.

######  Theorem 2.4.

Let $\psi\in J^{\rm mst}_{k,m}$ for $k\in{\mathbb{Z}}$ and
$m\in\frac{1}{2}{\mathbb{Z}}$. Then we have

$\displaystyle\psi(\tau,z)=\psi^{F}(\tau,z)+\psi^{P}(\tau,z),$ (2.28)

where $\psi^{F}$ is defined by (2.12) and $\psi^{P}$ is defined by (2.27).

###  2.4 Mock Modularity

It is shown in §8.3 of [16] that if $m\in{\mathbb{Z}}$ then the theta
coefficients $h_{r}$ (cf. (2.11)) of the finite part $\psi^{F}$ (cf. (2.12))
of a meromorphic Jacobi form $\psi\in J_{k,m}^{\rm mst}$ are (mixed) mock
modular forms, whose shadows admit concrete expressions in terms of the
residue forms $D_{s}$ (cf. (2.15)) and the theta functions $\theta_{m,r}^{1}$
(cf. (2.2)). Our focus in this work is on weak meromorphic Jacobi forms of
weight $1$ (with half-integral index), so in this section we discuss mock
modularity in that specific setting. We begin by showing that $S(\psi)$ and
the $D_{s}$ are greatly constrained by these hypotheses.

######  Proposition 2.5.

If $\psi$ is a weak meromorphic Jacobi form in $J^{\rm mst}_{1,m}$ for
$m\in\frac{1}{2}{\mathbb{Z}}$ then $D_{s}(\tau)$ is a constant function for
all $s\in S(\psi)$, and if $S_{n}\subset S(\psi)$ then $n$ divides
$\tilde{m}$.

###### Proof.

Let $s\in S(\psi)$. Then $s\in S_{n}$ for some $n$, and Proposition 2.2 shows
that $D_{s}(\tau)$ is a bounded weakly holomorphic modular form of weight $0$
for $\Gamma(2n^{2})$. We conclude that $D_{s}(\tau)=D_{s}$ is constant. Note
that $D_{s}\neq 0$ for otherwise $s$ would not be in $S(\psi)$. From (2.16)
and (2.17) and the fact that
${\mathbb{Z}}^{2}\rtimes\operatorname{\textsl{SL}}_{2}({\mathbb{Z}})$ acts
transitively on $S_{n}$ (Lemma 2.1) we have that $D_{s}$ is a non-zero
constant times $D_{(\frac{1}{n},0)}$. Applying Lemma 2.3 we obtain that $n$
divides $\tilde{m}$ (recall (2.5)). The proof is complete. ∎

We now aim to show that if $m\in\frac{1}{2}{\mathbb{Z}}$ and $\psi\in J^{\rm
mst}_{1,m}$ is weak then the finite part $\psi^{F}$ is a mock Jacobi form of
weight $1$, and its theta coefficients $h_{r}$ constitute a vector-valued mock
modular form $h=(h_{r})$ of weight $\frac{1}{2}$. To identify their shadows
let us focus momentarily on the poles lying on lattice points
${\mathbb{Z}}\tau+{\mathbb{Z}}$. For half-integral $m$, the mock modular
property of the polar part (and hence also the finite part) can be analyzed
using properties of the higher level Appell functions introduced by Zwegers
(see Definition 1.3. of [18]). Define

$\mu_{{m},0}(\tau,z):={\cal A}_{m}^{(0,0)}(\tau,z),$ (2.29)

and define its completion by setting

$\hat{\mu}_{{m},0}(\tau,z):=\mu_{{m},0}(\tau,z)+{i\over
2\sqrt{2m}}\sum_{r{\rm\;mod\;}2m}\theta_{m,r}(\tau,z)\int_{-\bar{\tau}}^{i\infty}{\rm
d}\tau^{\prime}{\theta^{1}_{m,r}(\tau^{\prime})\over\sqrt{-i(\tau+\tau^{\prime})}}$
(2.30)

where the sum is over $r\in{\mathbb{Z}}+m$. Then $\hat{\mu}_{m,0}$ transforms
as a Jacobi form of weight $1$ and index $m$. That is, $\hat{\mu}_{m,0}$
belongs to $\mathcal{E}_{m}$ (cf. §2.1) and is invariant for the weight $1$
modular action (2.9) of $\operatorname{\textsl{SL}}_{2}({\mathbb{Z}})$ on
$\mathcal{E}_{m}$.

From (2.30) we see that if $\psi$ as in Proposition 2.5 has poles only at
lattice points (i.e. $S(\psi)=S_{1}={\mathbb{Z}}^{2}$), then the shadow of
$h_{r}$ is proportional to the theta series $\theta^{1}_{m,r}(\tau)$ of weight
$\frac{3}{2}$, and the shadow of $\psi^{F}$ is proportional to the skew-
holomorphic Jacobi form
$\sum_{r{\rm\;mod\;}2m}\overline{\theta_{m,r}^{1}(\tau)}\theta_{m,r}(\tau,z)$
of weight $2$ and index $m$.

To describe the shadows of forms with poles at general torsion points we first
note that for $m\in\frac{1}{2}{\mathbb{Z}}$ and $n$ a divisor of $\tilde{m}$
we have

$(\theta_{m,r}\lvert{\cal
W}_{m}{(n)})(\tau,z)=\sum_{r^{\prime}{\rm\;mod\;}2m}(\Omega_{m}(n)_{r,r^{\prime}})\theta_{m,r^{\prime}}(\tau,z)$
(2.31)

where $\mathcal{W}_{m}(n)$ is as in (2.8), and
$\Omega_{m}(n)=(\Omega_{m}(n)_{r,r^{\prime}})$ is the $2m\times 2m$ Omega
matrix defined by setting

$\displaystyle\Omega_{m}(n)_{r,r^{\prime}}:=\begin{cases}\delta^{[2n]}_{r+r^{\prime},0}\delta^{[{2m\over
n}]}_{r-r^{\prime},0}&\text{ if $m\in{\mathbb{Z}}$,}\\\
\delta^{[n]}_{r+r^{\prime},0}\delta^{[{2m\over n}]}_{r-r^{\prime},0}&\text{ if
$m\in{\mathbb{Z}}+{1\over 2}$.}\end{cases}$ (2.32)

In (2.32) the indices $r$ and $r^{\prime}$ range over ${\mathbb{Z}}+m$ modulo
translation by $2m$, and we use the notation

$\delta^{[N]}_{a,b}:=\begin{cases}1&\text{ if $a=b{\rm\;mod\;}{N}$,}\\\
0&\text{ otherwise,}\end{cases}$ (2.33)

for $a,b,N\in{\mathbb{Z}}$. For convenience let us also define the (non-
holomorphic) Eichler integral

$\displaystyle\theta_{m,r}^{1,*}(\tau):=\int_{-\bar{\tau}}^{i\infty}{\rm
d}\tau^{\prime}{\theta^{1}_{m,r}(\tau^{\prime})\over\sqrt{-i(\tau+\tau^{\prime})}}$
(2.34)

for $m\in\frac{1}{2}{\mathbb{Z}}$ and $r\in{\mathbb{Z}}+m$.

######  Theorem 2.6.

Let $m\in\frac{1}{2}{\mathbb{Z}}$ and let $\psi\in J^{\rm mst}_{1,m}$ be a
weak meromorphic Jacobi form. Then

$\psi^{P}=\sum_{n|\tilde{m}}c_{n}{\cal A}^{(0,0)}_{m}\lvert{\cal W}_{m}(n)$
(2.35)

for some $c_{n}\in{\mathbb{C}}$. Moreover, the completion

$\hat{\psi}^{P}(\tau,z):=\psi^{P}(\tau,z)+{i\over
2\sqrt{2m}}\sum_{r{\rm\;mod\;}2m}\theta_{m,r}(\tau,z)\sum_{n|\tilde{m}}c_{n}\sum_{r^{\prime}{\rm\;mod\;}2m}\Omega_{m}(n)_{r,r^{\prime}}\theta_{m,r^{\prime}}^{1,*}(\tau)$
(2.36)

transforms as a weight 1 index $m$ Jacobi form.

###### Proof.

First note that if $\phi\in\mathcal{E}_{m}$ is invariant for the weight $k$
index $m$ modular action (2.9) of
$\operatorname{\textsl{SL}}_{2}({\mathbb{Z}})$ then so is
$\phi|\mathcal{W}_{m}(n)$ for $n$ a divisor of $\tilde{m}$. So the second
statement follows from the first, together with the fact that
$\hat{\mu}_{m,0}$ (cf. (2.30)) transforms as a weight $1$ index $m$ Jacobi
form.

For the first statement we first use (2.16) and (2.17) to check that

$D_{(-{a\over n},-{b\over
n})}=\operatorname{e}\left(m(a+1)(b+1)\right)D_{({1\over n},{1\over n})}$
(2.37)

when $\gcd(a,b,n)=1$. Taking this together with Proposition 2.5 we obtain that

$\begin{split}\psi^{P}(\tau,z):=&\sum_{s\in
S(\psi)/{\mathbb{Z}}^{2}}D_{s}{\cal A}^{s}_{m}(\tau,z)\\\
=&\sum_{n|\tilde{m}}{c_{n}\over
n}\sum_{a,b=0}^{n-1}\operatorname{e}(m(ab+a+b)){\cal
A}^{(-\frac{a}{n},-\frac{b}{n})}_{m}(\tau,z)\end{split}$ (2.38)

for some $c_{n}\in{\mathbb{C}}$. Next we check that

$\displaystyle{\cal A}^{(-{a\over n},-{b\over n})}_{m}(\tau,z)$
$\displaystyle=\operatorname{e}\left(m\left(\tfrac{a^{2}}{n^{2}}\tau+2\tfrac{a}{n}z+\tfrac{ab}{n^{2}}\right)\right){\cal
A}^{(0,0)}\left(\tau,z+\tfrac{a}{n}\tau+\tfrac{b}{n}\right)$ (2.39)

for $n$ a divisor of $\tilde{m}$. Then the first claim (2.35) follows from the
definition (2.8) of $\mathcal{W}_{m}(n)$. ∎

From Theorem 2.6 we see that if $\psi$ is as in Proposition 2.5 then there are
$c_{n}\in{\mathbb{C}}$ for $n|\tilde{m}$ such that the shadow of $h_{r}$ is
proportional to

$\displaystyle\sum_{n|\tilde{m}}\sum_{r^{\prime}{\rm\;mod\;}2m}\overline{c_{n}}\Omega_{m}(n)_{r,r^{\prime}}\theta_{m,r}^{1}(\tau),$
(2.40)

and the shadow of $\psi^{F}$ is proportional to the skew-holomorphic Jacobi
form

$\displaystyle\sum_{r{\rm\;mod\;}2m}\sum_{n|\tilde{m}}\sum_{r^{\prime}{\rm\;mod\;}2m}c_{n}\Omega_{m}(n)_{r,r^{\prime}}\overline{\theta_{m,r^{\prime}}^{1}(\tau)}\theta_{m,r}(\tau,z).$
(2.41)

###  2.5 Umbral Meromorphic Forms

Here we specialise the preceding analysis to the specific meromorphic Jacobi
forms that will appear in §3.

To begin we note that (2.30) and the identity $\theta^{1}_{m,r}={1\over
2}\operatorname{e}({-\tfrac{r}{2}})(\theta^{1}_{4m,2r}+\theta^{1}_{4m,4m-2r})$
suggests a relation between half-integral meromorphic Jacobi forms and the
umbral moonshine functions of pure D-type. To make this concrete, define
$M^{\prime}:=\frac{1}{2}M+1$ for $M$ an even positive integer. Then for each
positive $M$ such that $M=2{\rm\;mod\;}4$ and $M^{\prime}$ is a divisor of
$24$ there is a pure D-type Niemeier root system $X=D_{M^{\prime}}^{\oplus
d^{\prime}}$ where $M^{\prime}d^{\prime}=24$. (In case $M=M^{\prime}=2$ we
should interpret $D_{2}$ as $A_{1}^{\oplus 2}$.) Given such an $M$ let $\ell$
be the symbol $M+\frac{M}{2}$. For instance, we have $\ell=6+3$ for $M=6$ and
$M^{\prime}=4$, corresponding to the Niemeier root system $D_{4}^{\oplus 6}$.
We refer to $\ell$ as a lambency following [2, 3]. Umbral moonshine [3]
attaches a $2M$-vector-valued mock modular form
$H^{(\ell)}_{g}(\tau)=(H^{(\ell)}_{g,r}(\tau))_{r{\rm\;mod\;}2M}$ to each
$g\in G^{(\ell)}$, where
$G^{(\ell)}:=\operatorname{Aut}(N^{X})/\operatorname{Inn}(N^{X})$, for $N^{X}$
the Niemeier lattice with root system $X=D_{M^{\prime}}^{\oplus d^{\prime}}$,
and $\operatorname{Inn}(N^{X})$ the subgroup of $\operatorname{Aut}(N^{X})$
generated by reflections in roots.

For the particular lambencies $\ell=M+\frac{M}{2}$ under consideration, the
mock modular property of $H^{(\ell)}_{g}$ can be formulated in the following
way. First note that $G^{(\ell)}\simeq S_{d}$ according to §2.4 of [3]. Denote
the permutation character of $G^{(\ell)}\simeq S_{d}$ by $\chi^{(\ell)}_{g}$.
(This is $\chi^{X_{D}}_{g}$ in the notation of §B.2 of [3].) Then for every
conjugacy class $[g]$ of $G^{(\ell)}$,

$\displaystyle\psi^{(\frac{M}{4})}_{g}(\tau,z):=-2\chi^{(\ell)}_{g}{\rm
Av}_{\frac{M}{4}}\left(1\over
y^{\frac{1}{2}}-y^{-\frac{1}{2}}\right)+\sum_{s=-\frac{M}{4}+1,-\frac{M}{4}+2,\dots,\frac{M}{4}}\operatorname{e}(-\tfrac{s}{2})H^{(\ell)}_{g,2s}(\tau)\theta_{\frac{M}{4},s}(\tau,z)$
(2.42)

is a meromorphic Jacobi form of weight $1$ and (half-integral) index
$\frac{M}{4}$, with some level, and the above decomposition corresponds
precisely to the splitting of this meromorphic Jacobi form into its polar and
finite parts, cf. Theorem 2.4.

Returning to the theta function identity mentioned above (take
$m=\frac{M}{4}$), note that the discussion in §§4,5 of [3] attaches a
meromorphic Jacobi form $\psi^{(M+\frac{M}{2})}_{g}$ of weight $1$ and index
$M$ to each $g\in G^{(M+\frac{M}{2})}$. While the polar parts of
$\psi^{(\frac{M}{4})}$ and $\psi^{(M+\frac{M}{2})}$ do not seem to be related
in a simple way, we do have
$\psi^{(\frac{M}{4}),F}_{g}(\tau,2z)=\psi^{(M+{M\over 2}),F}_{g}(\tau,z)$ for
the finite parts. The half-integral index forms $\psi^{(\frac{M}{4})}$ seem to
be better suited to explicit realisation in vertex algebraic terms.

Indeed, in §3 we will recover series expansions of the functions
$\psi^{(\frac{M}{4})}_{g}$ for
$\frac{M}{4}\in\left\\{\frac{5}{2},\frac{7}{2},\frac{11}{2},\frac{23}{2}\right\\}$
and $g\in G^{(M+\frac{M}{2})}$ as traces on twisted modules
$W^{(\frac{M}{4})}_{\mathrm{tw}}$ for explicitly constructed vertex operator
superalgebras $W^{(\frac{M}{4})}$.

##  3 Umbral Moonshine Modules

In this section we present our main constructions, which are similar to those
of [14]. We adopt all the notational conventions and terminology of §2 of loc.
cit. in what follows.

###  3.1 Lambency $10+5$

Let $\mathfrak{e}$ be a $2$-dimensional complex vector space equipped with a
non-degenerate symmetric bilinear form, let $\mathfrak{a}$ be a
$4$-dimensional complex vector space equipped with a non-degenerate symmetric
bilinear form, and let $\mathfrak{b}$ be a $6$-dimensional complex vector
space equipped with a non-degenerate anti-symmetric bilinear form. Choose
polarisations $\mathfrak{e}=\mathfrak{e}^{+}\oplus\mathfrak{e}^{-}$,
$\mathfrak{a}=\mathfrak{a}^{+}\oplus\mathfrak{a}^{-}$ and
$\mathfrak{b}=\mathfrak{b}^{+}\oplus\mathfrak{b}^{-}$, and let
$\\{e^{\pm}\\}$, $\\{a^{\pm}_{i}\\}$ and $\\{b_{i}^{\pm}\\}$ be bases for
$\mathfrak{e}^{\pm}$, $\mathfrak{a}^{\pm}$ and $\mathfrak{b}^{\pm}$,
respectively, such that ${\langle}e^{-},e^{+}{\rangle}=1$ and
${\langle}a_{i}^{-},a_{j}^{+}{\rangle}=\langle\\!\langle
b_{i}^{-},b_{j}^{+}{\rangle}=\delta_{i,j}$. Define a vertex operator
superalgebra and a canonically twisted module for it by setting

$\displaystyle\begin{split}W^{(\frac{5}{2})}&:=A(\mathfrak{e})\otimes
A(\mathfrak{a})\,\otimes\mathrel{\raisebox{0.0pt}{\rotatebox{180.0}{$\\!\\!A$}}}(\mathfrak{b}),\\\
W^{(\frac{5}{2})}_{\mathrm{tw}}&:=A(\mathfrak{e})_{\mathrm{tw}}\otimes
A(\mathfrak{a})_{\mathrm{tw}}\,\otimes\mathrel{\raisebox{0.0pt}{\rotatebox{180.0}{$\\!\\!A$}}}(\mathfrak{b})_{\mathrm{tw}},\end{split}$
(3.1)

and by equipping $W^{(\frac{5}{2})}$ with the Virasoro element
$\omega^{(\frac{5}{2})}:=\omega\otimes\mathbf{v}\otimes\mathbf{v}+\mathbf{v}\otimes\omega\otimes\mathbf{v}+\mathbf{v}\otimes\mathbf{v}\otimes\omega$.
Set $\jmath_{\mathfrak{e}}:=\jmath\otimes\mathbf{v}\otimes\mathbf{v}$ and
$\jmath^{(\frac{5}{2})}:=2\mathbf{v}\otimes\jmath\otimes\mathbf{v}+\mathbf{v}\otimes\mathbf{v}\otimes\jmath$.
Then the group
${\textsl{GL}}(\mathfrak{e}^{+})\otimes{\textsl{GL}}(\mathfrak{a}^{+})\otimes{\textsl{GL}}(\mathfrak{b}^{+})$
acts naturally on $W^{(\frac{5}{2})}$ and $W^{(\frac{5}{2})}_{\mathrm{tw}}$,
respecting the vertex operator superalgebra module structures and preserving
the bigradings defined by the zero modes of $\omega^{(\frac{5}{2})}$ and
$\jmath^{(\frac{5}{2})}$.

Table 1: Character table of ${G}^{(10+5)}\simeq S_{4}$ $[g]$ | 1A | 2A | 3A | 2B | 4A
---|---|---|---|---|---
$[g^{2}]$ | 1A | 1A | 3A | 1A | 2A
${\chi}_{1}$ | $1$ | $1$ | $1$ | $1$ | $1$
${\chi}_{2}$ | $1$ | $1$ | $1$ | $-1$ | $-1$
${\chi}_{3}$ | $2$ | $2$ | $-1$ | $0$ | $0$
${\chi}_{4}$ | $3$ | $-1$ | $0$ | $1$ | $-1$
${\chi}_{5}$ | $3$ | $-1$ | $0$ | $-1$ | $1$

The group $G^{(10+5)}$ is isomorphic to $S_{4}$ according to §2.4 of [3].
Choose homomorphisms $\varrho:G^{(10+5)}\to{\textsl{GL}}(\mathfrak{a}^{+})$
and
$\mathrel{\raisebox{0.0pt}{\rotatebox{180.0}{$\\!\\!\varrho$}}}\;\;:G^{(10+5)}\to{\textsl{GL}}(\mathfrak{b}^{+})$
such that the corresponding characters are $\chi_{3}$ and $\chi_{4}$,
respectively, in the character table, Table 1. Then the assignment $g\mapsto
I\otimes\varrho(g)\,\otimes\mathrel{\raisebox{0.0pt}{\rotatebox{180.0}{$\\!\\!\varrho$}}}(g)$
defines faithful and compatible actions of $G^{(10+5)}$ on $W^{(\frac{5}{2})}$
and $W^{(\frac{5}{2})}_{\mathrm{tw}}$. Set $(-1)^{F}:=(-I)\otimes(-I)\otimes
I$. Let $J_{\mathfrak{e}}(0)$ be the coefficient of $z^{-1}$ in
$Y_{\mathrm{tw}}(\jmath_{\mathfrak{e}},z)$, let $J(0)$ be the coefficient of
$z^{-1}$ in $Y_{\mathrm{tw}}(\jmath^{(\frac{5}{2})},z)$, and let $L(0)$ be the
coefficient of $z^{-2}$ in $Y_{\mathrm{tw}}(\omega^{(\frac{5}{2})},z)$. For
$g\in G^{(10+5)}$ we consider the formal series
$\widetilde{\psi}^{(\frac{5}{2})}_{g}\in{\mathbb{C}}[y][[y^{-1}]][[q]]$
defined by

$\displaystyle\widetilde{\psi}^{(\frac{5}{2})}_{g}:=-2\operatorname{tr}(gJ_{\mathfrak{e}}(0)(-1)^{F}y^{J(0)}q^{L(0)}|W^{(\frac{5}{2})}_{\mathrm{tw}}).$
(3.2)

######  Theorem 3.1.

For $g\in G^{(10+5)}$ the series $\widetilde{\psi}^{(\frac{5}{2})}_{g}$ is the
expansion of $\psi^{(\frac{5}{2})}_{g}$ in the domain $0<-\Im(z)<\Im(\tau)$.

###### Proof.

For $g\in G^{(10+5)}$ let $\\{\lambda_{1},\lambda_{2}\\}$ be the eigenvalues
for the action of $g$ on $\mathfrak{a}^{+}$, and let
$\\{\mathrel{\raisebox{0.0pt}{\rotatebox{180.0}{$\\!\lambda$}}}_{1},\mathrel{\raisebox{0.0pt}{\rotatebox{180.0}{$\\!\lambda$}}}_{2},\mathrel{\raisebox{0.0pt}{\rotatebox{180.0}{$\\!\lambda$}}}_{3}\\}$
be the eigenvalues for its action on $\mathfrak{b}^{+}$. Then

$\displaystyle\widetilde{\psi}^{(\frac{5}{2})}_{g}=2y^{-\frac{1}{2}}\prod_{n>0}\frac{(1-q^{n})^{2}\prod_{i=1}^{2}(1-\bar{\lambda}_{i}y^{-2}q^{n-1})(1-\lambda_{i}y^{2}q^{n})}{\prod_{j=1}^{3}(1-\bar{\mathrel{\raisebox{0.0pt}{\rotatebox{180.0}{$\\!\lambda$}}}}_{j}y^{-1}q^{n-1})(1-\mathrel{\raisebox{0.0pt}{\rotatebox{180.0}{$\\!\lambda$}}}_{j}\\!yq^{n})},$
(3.3)

where $(1-X)^{-1}$ is to be understood as a shorthand for $\sum_{k\geq
0}X^{k}$. This series converges in the domain $0<-\Im(z)<\Im(\tau)$ once we
substitute $\operatorname{e}(\tau)$ for $q$ and $\operatorname{e}(z)$ for $y$.
So it remains to check that the right-hand side of (3.3) agrees with the
meromorphic Jacobi form $\psi^{(\frac{5}{2})}_{g}$ when regarded as a function
of $\tau$ and $z$. According to §B.3.11 of [6], the $\psi^{(\frac{5}{2})}_{g}$
are given explicitly by

$\displaystyle\begin{split}\psi^{(\frac{5}{2})}_{1A}(\tau,z)&:=2i\eta(\tau)^{3}{\theta_{1}(\tau,2z)^{2}}{\theta_{1}(\tau,z)^{-3}},\\\
\psi^{(\frac{5}{2})}_{2A}(\tau,z)&:=-2i\eta(\tau)^{3}{\theta_{1}(\tau,2z)^{2}}{\theta_{1}(\tau,z)^{-1}\theta_{2}(\tau,z)^{-2}},\\\
\psi^{(\frac{5}{2})}_{3A}(\tau,z)&:=2i\eta(\tau)^{3}{\theta_{1}(3\tau,6z)}{\theta_{1}(\tau,2z)^{-1}\theta_{1}(3\tau,3z)^{-1}},\\\
\psi^{(\frac{5}{2})}_{2B}(\tau,z)&:=2i\eta(\tau)^{3}{\theta_{1}(\tau,2z)\theta_{2}(\tau,2z)}{\theta_{1}(\tau,z)^{-2}\theta_{2}(\tau,z)^{-1}},\\\
\psi^{(\frac{5}{2})}_{4A}(\tau,z)&:=-2i\eta(\tau)\eta(2\tau){\theta_{1}(\tau,2z)\theta_{2}(\tau,2z)}{\theta_{2}(2\tau,2z)^{-1}},\end{split}$
(3.4)

where

$\displaystyle\begin{split}\eta(\tau)&:=q^{\frac{1}{24}}\prod_{n>0}(1-q^{n}),\\\
\theta_{1}(\tau,z)&:=-iq^{\frac{1}{8}}y^{\frac{1}{2}}\prod_{n>0}(1-y^{-1}q^{n-1})(1-yq^{n})(1-q^{n}),\\\
\theta_{2}(\tau,z)&:=q^{\frac{1}{8}}y^{\frac{1}{2}}\prod_{n>0}(1+y^{-1}q^{n-1})(1+yq^{n})(1-q^{n}).\end{split}$
(3.5)

By applying these product formula definitions of $\eta$, $\theta_{1}$ and
$\theta_{2}$ to the formulas in (3.4) we obtain product formulas for the
$\psi^{(\frac{5}{2})}_{g}$. For each $g$ we find agreement with the product
formula (3.3) for $\widetilde{\psi}^{(\frac{5}{2})}_{g}$ obtained by
substituting the corresponding values for $\lambda_{i}$ and
$\mathrel{\raisebox{0.0pt}{\rotatebox{180.0}{$\\!\lambda$}}}_{j}$, as are
given in Table 2. This completes the proof. ∎

Table 2: Eigenvalues for $\ell=10+5$ $[g]$ | $\\{\lambda_{i}\\}$ | $\\{\mathrel{\raisebox{0.0pt}{\rotatebox{180.0}{$\\!\lambda$}}}_{j}\\}$
---|---|---
1A | $\\{1,1\\}$ | $\\{1,1,1\\}$
2A | $\\{1,1\\}$ | $\\{1,-1,-1\\}$
3A | $\\{\omega,\omega^{2}\\}$ | $\\{1,\omega,\omega^{2}\\}$
2B | $\\{1,-1\\}$ | $\\{1,1,-1\\}$
4A | $\\{1,-1\\}$ | $\\{-1,{\bf i},-{\bf i}\\}$

###  3.2 Lambency $14+7$

Let $\mathfrak{e}$ and $\mathfrak{a}$ be $2$-dimensional complex vector spaces
equipped with non-degenerate symmetric bilinear forms, and let $\mathfrak{b}$
be a $4$-dimensional complex vector space equipped with a non-degenerate anti-
symmetric bilinear form. Choose polarisations
$\mathfrak{e}=\mathfrak{e}^{+}\oplus\mathfrak{e}^{-}$,
$\mathfrak{a}=\mathfrak{a}^{+}\oplus\mathfrak{a}^{-}$ and
$\mathfrak{b}=\mathfrak{b}^{+}\oplus\mathfrak{b}^{-}$, and let
$\\{e^{\pm}\\}$, $\\{a^{\pm}\\}$ and $\\{b_{i}^{\pm}\\}$ be bases for
$\mathfrak{e}^{\pm}$, $\mathfrak{a}^{\pm}$ and $\mathfrak{b}^{\pm}$,
respectively, such that
${\langle}e^{-},e^{+}{\rangle}={\langle}a^{-},a^{+}{\rangle}=1$ and
$\langle\\!\langle b_{i}^{-},b_{j}^{+}{\rangle}=\delta_{i,j}$. Define a vertex
operator superalgebra and a canonically twisted module for it by setting

$\displaystyle\begin{split}W^{(\frac{7}{2})}&:=A(\mathfrak{e})\otimes
A(\mathfrak{a})\,\otimes\mathrel{\raisebox{0.0pt}{\rotatebox{180.0}{$\\!\\!A$}}}(\mathfrak{b}),\\\
W^{(\frac{7}{2})}_{\mathrm{tw}}&:=A(\mathfrak{e})_{\mathrm{tw}}\otimes
A(\mathfrak{a})_{\mathrm{tw}}\,\otimes\mathrel{\raisebox{0.0pt}{\rotatebox{180.0}{$\\!\\!A$}}}(\mathfrak{b})_{\mathrm{tw}},\end{split}$
(3.6)

and by equipping $W^{(\frac{7}{2})}$ with the usual tensor product Virasoro
element, which we denote $\omega^{(\frac{7}{2})}$. Set
$\jmath_{\mathfrak{e}}:=\jmath\otimes\mathbf{v}\otimes\mathbf{v}$ and
$\jmath^{(\frac{7}{2})}:=3\mathbf{v}\otimes\jmath\otimes\mathbf{v}+\mathbf{v}\otimes\mathbf{v}\otimes\jmath$.
The group
${\textsl{GL}}(\mathfrak{e}^{+})\otimes{\textsl{GL}}(\mathfrak{a}^{+})\otimes{\textsl{GL}}(\mathfrak{b}^{+})$
acts naturally on $W^{(\frac{7}{2})}$ and $W^{(\frac{7}{2})}_{\mathrm{tw}}$,
respecting the vertex operator superalgebra module structures and preserving
the bigradings defined by the Virasoro element $\omega^{(\frac{7}{2})}$ and
the zero mode of $\jmath^{(\frac{7}{2})}$.

Table 3: Character table of ${G}^{(14+7)}\simeq S_{3}$ $[g]$ | 1A | 2A | 3A
---|---|---|---
$\chi_{1}$ | $1$ | $1$ | $1$
$\chi_{2}$ | $1$ | $-1$ | $1$
$\chi_{3}$ | $2$ | $0$ | $-1$

The group $G^{(14+7)}$ is isomorphic to $S_{3}$ according to §2.4 of [3].
Choose homomorphisms $\varrho:G^{(14+7)}\to{\textsl{GL}}(\mathfrak{a}^{+})$
and
$\mathrel{\raisebox{0.0pt}{\rotatebox{180.0}{$\\!\\!\varrho$}}}\;\;:G^{(14+7)}\to{\textsl{GL}}(\mathfrak{b}^{+})$
such that the corresponding characters are the sign character and the unique
irreducible character of dimension $2$, respectively (i.e. $\chi_{2}$ and
$\chi_{3}$ in Table 3). Then the assignment $g\mapsto
I\otimes\varrho(g)\,\otimes\mathrel{\raisebox{0.0pt}{\rotatebox{180.0}{$\\!\\!\varrho$}}}(g)$
defines actions of $G^{(14+7)}$ on $W^{(\frac{7}{2})}$ and
$W^{(\frac{7}{2})}_{\mathrm{tw}}$. Set $(-1)^{F}:=(-I)\otimes(-I)\otimes I$,
and let $J_{\mathfrak{e}}(0)$ denote the coefficient of $z^{-1}$ in
$Y_{\mathrm{tw}}(\jmath_{\mathfrak{e}},z)$. Let $J(0)$ be the coefficient of
$z^{-1}$ in $Y_{\mathrm{tw}}(\jmath^{(\frac{7}{2})},z)$, and let $L(0)$ be the
coefficient of $z^{-2}$ in $Y_{\mathrm{tw}}(\omega^{(\frac{7}{2})},z)$. For
$g\in G^{(14+7)}$ consider the formal series
$\widetilde{\psi}^{(\frac{7}{2})}_{g}\in{\mathbb{C}}[y][[y^{-1}]][[q]]$
defined by

$\displaystyle\widetilde{\psi}^{(\frac{7}{2})}_{g}:=-2\operatorname{tr}(gJ_{\mathfrak{e}}(0)(-1)^{F}y^{J(0)}q^{L(0)}|W^{(\frac{7}{2})}_{\mathrm{tw}}).$
(3.7)

######  Theorem 3.2.

For $g\in G^{(14+7)}$ the series $\widetilde{\psi}^{(\frac{7}{2})}_{g}$ is the
expansion of $\psi^{(\frac{7}{2})}_{g}$ in the domain $0<-\Im(z)<\Im(\tau)$.

###### Proof.

Let $g\in G^{(14+7)}$. Then $g$ acts as a multiplication by a scalar,
$\lambda$ say, on $\mathfrak{a}^{+}$, and we may write
$\\{\mathrel{\raisebox{0.0pt}{\rotatebox{180.0}{$\\!\lambda$}}}_{1},\mathrel{\raisebox{0.0pt}{\rotatebox{180.0}{$\\!\lambda$}}}_{2}\\}$
for the eigenvalues for its action on $\mathfrak{b}^{+}$. As in §3.1 we have

$\displaystyle\widetilde{\psi}^{(\frac{7}{2})}_{g}=2y^{-\frac{1}{2}}\prod_{n>0}\frac{(1-q^{n})^{2}(1-\bar{\lambda}y^{-3}q^{n-1})(1-\lambda
y^{3}q^{n})}{\prod_{j=1}^{2}(1-\bar{\mathrel{\raisebox{0.0pt}{\rotatebox{180.0}{$\\!\lambda$}}}}_{j}y^{-1}q^{n-1})(1-\mathrel{\raisebox{0.0pt}{\rotatebox{180.0}{$\\!\lambda$}}}_{j}\\!yq^{n})},$
(3.8)

and this series converges in the domain $0<-\Im(z)<\Im(\tau)$ upon
substitution of $\operatorname{e}(\tau)$ for $q$ and $\operatorname{e}(z)$ for
$y$. To check that the right-hand side of (3.8) agrees with the meromorphic
Jacobi form $\psi^{(\frac{7}{2})}_{g}$ when viewed as a function of $\tau$ and
$z$ we again perform a case by case check, using the explicit expressions

$\displaystyle\begin{split}\psi^{(\frac{7}{2})}_{1A}(\tau,z)&:=2i\eta(\tau)^{3}{\theta_{1}(\tau,3z)}{\theta_{1}(\tau,z)^{-2}},\\\
\psi^{(\frac{7}{2})}_{2A}(\tau,z)&:=2i\eta(\tau)^{3}{\theta_{2}(\tau,3z)}{\theta_{1}(\tau,z)^{-1}\theta_{2}(\tau,z)^{-1}},\\\
\psi^{(\frac{7}{2})}_{3A}(\tau,z)&:=-2i\eta(3\tau)\theta_{1}(\tau,z)\theta_{1}(\tau,3z)\theta_{1}(3\tau,3z)^{-1},\end{split}$
(3.9)

reproduced here from §B.3.15 of [6], and the values of $\lambda$ and
$\mathrel{\raisebox{0.0pt}{\rotatebox{180.0}{$\\!\lambda$}}}_{j}$ in Table 4.
∎

Table 4: Eigenvalues for $\ell=14+7$ $[g]$ | $\lambda$ | $\\{\mathrel{\raisebox{0.0pt}{\rotatebox{180.0}{$\\!\lambda$}}}_{j}\\}$
---|---|---
1A | $1$ | $\\{1,1\\}$
2A | $-1$ | $\\{1,-1\\}$
3A | $1$ | $\\{\omega,\omega^{2}\\}$

###  3.3 Lambency $22+11$

Let $\mathfrak{e}$ and $\mathfrak{a}$ be $2$-dimensional complex vector spaces
equipped with non-degenerate symmetric bilinear forms, and let $\mathfrak{b}$
and $\mathfrak{b}^{\prime}$ be $2$-dimensional complex vector spaces equipped
with non-degenerate anti-symmetric bilinear forms. Fix polarisations
$\mathfrak{e}=\mathfrak{e}^{+}\oplus\mathfrak{e}^{-}$,
$\mathfrak{a}=\mathfrak{a}^{+}\oplus\mathfrak{a}^{-}$,
$\mathfrak{b}=\mathfrak{b}^{+}\oplus\mathfrak{b}^{-}$ and
$\mathfrak{b}^{\prime}={\mathfrak{b}^{\prime}}^{+}\oplus{\mathfrak{b}^{\prime}}^{-}$,
and let $\\{e^{\pm}\\}$, $\\{a^{\pm}\\}$, $\\{b^{\pm}\\}$ and
$\\{{b^{\prime}}^{\pm}\\}$ be bases for $\mathfrak{e}^{\pm}$,
$\mathfrak{a}^{\pm}$, $\mathfrak{b}^{\pm}$ and
${\mathfrak{b}^{\prime}}^{\pm}$, respectively, such that
${\langle}e^{-},e^{+}{\rangle}={\langle}a^{-},a^{+}{\rangle}=\langle\\!\langle
b^{-},b^{+}{\rangle}=\langle\\!\langle{b^{\prime}}^{-},{b^{\prime}}^{+}{\rangle}=1$.

Define a super vertex operator algebra $W^{(\frac{11}{2})}$, and a canonically
twisted $W^{(\frac{11}{2})}$-module $W^{(\frac{11}{2})}_{\mathrm{tw}}$ by
setting

$\displaystyle\begin{split}W^{(\frac{11}{2})}&:=A(\mathfrak{e})\otimes
A(\mathfrak{a})\,\otimes\mathrel{\raisebox{0.0pt}{\rotatebox{180.0}{$\\!\\!A$}}}(\mathfrak{b})\otimes\mathrel{\raisebox{0.0pt}{\rotatebox{180.0}{$\\!\\!A$}}}(\mathfrak{b}^{\prime}),\\\
W^{(\frac{11}{2})}_{\mathrm{tw}}&:=A(\mathfrak{e})_{\mathrm{tw}}\otimes
A(\mathfrak{a})_{\mathrm{tw}}\,\otimes\mathrel{\raisebox{0.0pt}{\rotatebox{180.0}{$\\!\\!A$}}}(\mathfrak{b})_{\mathrm{tw}}\otimes\mathrel{\raisebox{0.0pt}{\rotatebox{180.0}{$\\!\\!A$}}}(\mathfrak{b}^{\prime})_{\mathrm{tw}}.\end{split}$
(3.10)

Equip $W^{(\frac{11}{2})}$ with the usual tensor product Virasoro element,
$\omega^{(\frac{11}{2})}$, set
$\jmath_{\mathfrak{e}}:=\jmath\otimes\mathbf{v}\otimes\mathbf{v}\otimes\mathbf{v}$,
and set

$\displaystyle\jmath^{(\frac{11}{2})}:=4\mathbf{v}\otimes\jmath\otimes\mathbf{v}\otimes\mathbf{v}+\mathbf{v}\otimes\mathbf{v}\otimes\jmath\otimes\mathbf{v}+2\mathbf{v}\otimes\mathbf{v}\otimes\mathbf{v}\otimes\jmath.$
(3.11)

Then
${\textsl{GL}}(\mathfrak{e}^{+})\otimes{\textsl{GL}}(\mathfrak{a}^{+})\otimes{\textsl{GL}}(\mathfrak{b}^{+})\otimes{\textsl{GL}}({\mathfrak{b}^{\prime}}^{+})$
acts naturally on $W^{(\frac{11}{2})}$ and $W^{(\frac{11}{2})}_{\mathrm{tw}}$,
respecting the super vertex operator algebra module structures and preserving
the bigradings.

The umbral group $G^{(22+11)}$ is cyclic of order $2$ according to §2.4 of
[3]. Define an action of $G^{(22+11)}$ on $W^{(\frac{11}{2})}$ and
$W^{(\frac{11}{2})}_{\mathrm{tw}}$ by mapping the non-trivial element to
$I\otimes I\otimes(-I)\otimes(-I)$. Similar to §§3.1,3.2 we set
$(-1)^{F}:=(-I)\otimes(-I)\otimes I\otimes I$, let $J_{\mathfrak{e}}(0)$
denote the coefficient of $z^{-1}$ in
$Y_{\mathrm{tw}}(\jmath_{\mathfrak{e}},z)$, let $J(0)$ be the coefficient of
$z^{-1}$ in $Y_{\mathrm{tw}}(\jmath^{(\frac{11}{2})},z)$, and let $L(0)$ be
the coefficient of $z^{-2}$ in $Y_{\mathrm{tw}}(\omega^{(\frac{11}{2})},z)$.
Then to $g\in G^{(\frac{11}{2})}$ we assign the formal series

$\displaystyle\widetilde{\psi}^{(\frac{11}{2})}_{g}:=-2\operatorname{tr}(gJ_{\mathfrak{e}}(0)(-1)^{F}y^{J(0)}q^{L(0)}|W^{(\frac{11}{2})}_{\mathrm{tw}}).$
(3.12)

######  Theorem 3.3.

For $g\in G^{(22+11)}$ the series $\widetilde{\psi}^{(\frac{11}{2})}_{g}$ is
the expansion of $\psi^{(\frac{11}{2})}_{g}$ in the domain
$0<-\Im(z)<\Im(\tau)$.

###### Proof.

We have

$\displaystyle\widetilde{\psi}^{(\frac{11}{2})}_{g}=2y^{-\frac{1}{2}}\prod_{n>0}\frac{(1-q^{n})^{2}(1-y^{-4}q^{n-1})(1-y^{4}q^{n})}{(1\mp
y^{-1}q^{n-1})(1\mp yq^{n})(1\mp y^{-2}q^{n-1})(1\mp y^{2}q^{n})},$ (3.13)

where the signs in the denominator are minus for $g=e$, and plus for $g$ the
non-trivial element of $G^{(22+11)}$. In the former case the right hand side
of (3.13) is the expansion of

$\displaystyle
2i\eta(\tau)^{3}\theta_{1}(\tau,4z)\theta_{1}(\tau,z)^{-1}\theta_{1}(\tau,2z)^{-1}$
(3.14)

in the domain $0<-\Im(z)<\Im(\tau)$, and (3.14) is exactly the expression for
$\psi^{(\frac{11}{2})}_{1A}$ that appears in §B.3.19 of [6]. In the latter
case (3.13) gives the expansion of

$\displaystyle-2i\eta(\tau)^{3}\theta_{1}(\tau,4z)\theta_{2}(\tau,z)^{-1}\theta_{2}(\tau,2z)^{-1},$
(3.15)

which is the expression for $\psi^{(\frac{11}{2})}_{2A}$ that appears in
§B.3.19 of [6]. The proof is complete. ∎

###  3.4 Lambency $46+23$

The vertex operator superalgebra we use to realise the
$\psi^{(\frac{23}{2})}_{g}$ for $\ell=46+23$ is exactly the same as for
$\ell=22+11$, but has a different bigrading. That is, we set
$W^{(\frac{23}{2})}:=W^{(\frac{11}{2})}$ and
$W^{(\frac{23}{2})}_{\mathrm{tw}}:=W^{(\frac{11}{2})}_{\mathrm{tw}}$, take
$\jmath_{\mathfrak{e}}$ as in §3.3, and also define
$\omega^{(\frac{23}{2})}:=\omega^{(\frac{11}{2})}$, but set

$\displaystyle\jmath^{(\frac{23}{2})}:=6\mathbf{v}\otimes\jmath\otimes\mathbf{v}\otimes\mathbf{v}+2\mathbf{v}\otimes\mathbf{v}\otimes\jmath\otimes\mathbf{v}+3\mathbf{v}\otimes\mathbf{v}\otimes\mathbf{v}\otimes\jmath.$
(3.16)

The group $G^{(46+23)}$ is in fact trivial according to §2.4 of [3], so we
only aim to realise a single function $\psi^{(\frac{23}{2})}_{1A}$. We define
$(-1)^{F}$, $J_{\mathfrak{e}}(0)$ and $L(0)$ exactly as in §3.3, but let
$J(0)$ be the coefficient of $z^{-1}$ in
$Y_{\mathrm{tw}}(\jmath^{(\frac{23}{2})},z)$. We consider the formal series

$\displaystyle\widetilde{\psi}^{(\frac{23}{2})}_{e}:=-2\operatorname{tr}(J_{\mathfrak{e}}(0)(-1)^{F}y^{J(0)}q^{L(0)}|W^{(\frac{23}{2})}_{\mathrm{tw}}).$
(3.17)

######  Theorem 3.4.

The series $\widetilde{\psi}^{(\frac{23}{2})}_{e}$ is the expansion of
$\psi^{(\frac{23}{2})}_{1A}$ in the domain $0<-\Im(z)<\Im(\tau)$.

###### Proof.

From §B.3.23 of [6] we have
$\psi^{(\frac{23}{2})}_{1A}=2i\eta(\tau)^{3}\theta_{1}(\tau,6z)\theta_{1}(\tau,2z)^{-1}\theta_{1}(\tau,3z)^{-1}$,
which can be written as an infinite product,

$\displaystyle\psi^{(\frac{23}{2})}_{1A}=2y^{-\frac{1}{2}}\prod_{n>0}\frac{(1-q^{n})^{2}(1-y^{-6}q^{n-1})(1-y^{6}q^{n})}{(1-y^{-2}q^{n-1})(1-y^{2}q^{n})(1-y^{-3}q^{n-1})(1-y^{3}q^{n})}.$
(3.18)

On the other hand $\widetilde{\psi}^{(\frac{23}{2})}_{e}$ is the series we
obtain by replacing $(1-X)^{-1}$ with $\sum_{k\geq 0}X^{k}$ in (3.18), and
converges in the given domain. This proves the claim. ∎

##  4 Discussion

In this section we discuss a few features and a possible interpretation of the
pure D-type umbral moonshine module constructions presented in §3.

Relation to ${\cal N}=2$ Superconformal Algebra

Note that all the meromorphic Jacobi forms $\psi^{({M\over 4})}_{g}$ discussed
in §3 have the property that $\psi^{({M\over
4})}_{g}(\Psi_{1,-\frac{1}{2}})^{-1}$ is a weight 0 weak (holomorphic) Jacobi
form for $\operatorname{\textsl{SL}}_{2}({\mathbb{Z}})$ of index ${M\over
4}+{1\over 2}$, where

$\Psi_{1,-{1\over 2}}(\tau,z):=i{\eta^{3}(\tau)\over\theta_{1}(\tau,z)}$ (4.1)

(cf. (3.5)). Then the splitting of meromorphic Jacobi forms into polar and
finite parts as described by Theorem 2.4 corresponds precisely to the
decomposition of the weight 0 weak Jacobi form $\psi^{({M\over
4})}_{g}(\Psi_{1,-\frac{1}{2}})^{-1}$ into characters of the ${\cal N}=2$
superconformal algebra. In particular, the universal factor
$(\Psi_{1,-\frac{1}{2}})^{-1}$ multiplying the polar and finite parts give the
contributions from the massless and massive representations (respectively) of
the ${\cal N}=2$ superconformal algebra of central charge $c=6({M\over
4}+{1\over 2})$. We refer to §7 of [19] for details.

This is reminiscent of the situation in the case $X=A_{1}^{\oplus 24}$,
$\ell=2$ of umbral moonshine (cf. [3]), namely Mathieu moonshine for $M_{24}$.
There, analogous to the other 22 cases of umbral moonshine, we can regard the
weight $\frac{1}{2}$ mock modular form $H^{(2)}_{g}$ for all $g\in M_{24}$ as
arising from splitting a certain weight 1 index 2 meromorphic Jacobi form
$\psi^{(2)}_{g}$ into its polar and finite parts (see §3.4 of [3]).
Alternatively, we can regard $H^{(2)}_{g}$ as arising from decomposing a
weight 0 index 1 weak Jacobi form $\psi^{(2)}_{g}(\Psi_{1,1})^{-1}$ into
characters of the ${\cal N}=4$ superconformal algebra at central charge $6$
(see §4.3 of [3]), where

$\Psi_{1,1}(\tau,z):=i{\eta^{3}(\tau)\theta_{1}(\tau,2z)\over\theta_{1}(\tau,z)^{2}}.$
(4.2)

Famously, for the identity class $g=e$, the weak Jacobi form
$\psi^{(2)}_{g}(\Psi_{1,1})^{-1}$ equals the elliptic genus of K3 surfaces,
and its ${\cal N}=4$ decomposition is precisely the context in which this
$M_{24}$ moonshine was initially discovered [1].

Relation to Pure A-Type Umbral Moonshine Modules

For the other cases of pure A-type, corresponding to the Niemeier root systems
$A_{2}^{\oplus 12}$, $A_{3}^{\oplus 8}$, $A_{4}^{\oplus 6}$, $A_{6}^{\oplus
4}$, $A_{8}^{\oplus 3}$, $A_{12}^{\oplus 2}$ and $A_{24}$, there is for
general group elements $g$ no weight 0 weak Jacobi form related to
$\psi^{(\ell)}_{g}$ via a multiplication of $(\Psi_{1,1})^{-1}$ due to the
presence of poles not just at lattice points
$z\in{\mathbb{Z}}\tau+{\mathbb{Z}}$ but also at $2$-torsion points
$z\in{\mathbb{Z}}\tau+{\mathbb{Z}}+{1\over 2}$. (Note the contrast with the
$\psi^{(\frac{M}{4})}_{g}$ in §3, whose poles are restricted to lattice points
for all $g$.) However, for all the pure A-type cases $A_{\ell-1}^{\oplus d}$
where $d={24\over\ell-1}$, the function $\psi^{(\ell)}_{e}(\Psi_{1,1})^{-1}$
is a weight 0 weak Jacobi form for
$\operatorname{\textsl{SL}}_{2}({\mathbb{Z}})$ corresponding to the identity
class.

As a result, we see that the graded dimension of the pure A-type and pure
D-type umbral moonshine modules, captured by weight 1 meromorphic Jacobi forms
at integral and half-integral indices respectively, are both related to weak
Jacobi forms of weight 0. In fact, the pairs of pure D-type and pure A-type
cases of umbral moonshine listed in Table 5 give rise to the same weak Jacobi
forms. Precisely, if $M$ is as in Table 5 and $M^{\prime}:=\frac{1}{2}M+1$
then $\psi^{(\frac{M}{4})}_{e}$ is the weight 1 index $\frac{M}{4}$
meromorphic Jacobi form that represents the $D_{M^{\prime}}^{d^{\prime}}$ case
of umbral moonshine (cf. §2.5), where $M^{\prime}d^{\prime}=24$, and the
corresponding pure A-type case of umbral moonshine is
$A_{M^{\prime\prime}-1}^{d^{\prime\prime}}$, where
$M^{\prime\prime}=(M^{\prime})^{\prime}=\frac{1}{4}M+\frac{3}{2}$ and
$(M^{\prime\prime}-1)d^{\prime\prime}=24$. The corresponding meromorphic
Jacobi forms are related by

$\psi^{({M\over
4})}_{e}={\theta_{1}(\tau,z)\over\theta_{1}(\tau,2z)}\psi^{(M^{\prime\prime})}_{e}.$
(4.3)

In other words, in these four cases (and also for $M=6$, which is not
considered in this work) the weight 0 weak Jacobi forms $\psi^{({M\over
4})}_{e}(\Psi_{1,-\frac{1}{2}})^{-1}=\psi^{(M^{\prime\prime})}_{e}(\Psi_{1,1})^{-1}$
can be interpreted as encoding graded dimensions for either a pure D-type or a
pure A-type moonshine module, the former constructed in the present paper and
the latter constructed (in whole for $M\in\\{22,26\\}$ but in part for
$M\in\\{10,14\\}$) in [14].

Table 5: The A–D Correspondence $M$ | D-type | A-type
---|---|---
$10$ | $D_{6}^{\oplus 4}$ | $A_{3}^{\oplus 8}$
$14$ | $D_{8}^{\oplus 3}$ | $A_{4}^{\oplus 6}$
$22$ | $D_{12}^{\oplus 2}$ | $A_{6}^{\oplus 4}$
$46$ | $D_{24}$ | $A_{12}^{\oplus 2}$

Possible Interpretation of the Module

Note that in each of the four pure D-type cases of umbral moonshine discussed
in this paper, the corresponding module construction in §3 can be interpreted
as given by $d$ pairs of $bc$-$\beta\gamma$ systems, where $d=3$ for
$\ell=10+5$ (cf. §3.1), and $d=2$ in the remaining cases (cf. §§3.2-3.4). This
is reminiscent of the construction of the chiral de Rham complex [20], which
defines a sheaf of superconformal vertex operator superalgebras over a
Calabi–Yau manifold of complex dimension $d$. In that construction, each local
section of the chiral de Rham complex is given by $d$ pairs of
$bc$-$\beta\gamma$ systems, and has the structure of a so-called rank $d$
topological ${\cal N}=2$ superconformal algebra. In particular the OPE of the
stress-energy tensor with itself gives a vanishing central charge. Note
however that our choices of $U(1)$ charges for the different $bc$ and
$\beta\gamma$ systems do not preserve this topological ${\cal N}=2$ structure.
Nonetheless it would be interesting to explore whether this close connection
to the chiral de Rham complex, present also in [14], indeed reflects a
hitherto unnoticed physical aspect to umbral moonshine.

## Acknowledgements

We thank Andrew O’Desky for discussions on closely related topics. The work of
M.C. was supported by ERC starting grant H2020 ERC StG #640159. J.D.
acknowledges support from the Simons Foundation (#316779), and the U.S.
National Science Foundation (DMS 1203162, DMS 1601306).

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