#  $K3$ Elliptic Genus and
an Umbral Moonshine Module

Vassilis Anagiannis<EMAIL_ADDRESS>Institute of Physics, University of
Amsterdam, Amsterdam, the Netherlands  Miranda C. N. Cheng<EMAIL_ADDRESS>Institute of Physics, University of Amsterdam, Amsterdam, the Netherlands
Korteweg-de Vries Institute for Mathematics, University of Amsterdam,
Amsterdam, the Netherlands  Sarah M. Harrison<EMAIL_ADDRESS>Center
for the Fundamental Laws of Nature
Harvard University, Cambridge, MA 02138, USA  Department of Mathematics and
Statistics and Department of Physics, McGill University, Montreal, QC, Canada

###### Abstract

Umbral moonshine connects the symmetry groups of the 23 Niemeier lattices with
23 sets of distinguished mock modular forms. The 23 cases of umbral moonshine
have a uniform relation to symmetries of $K3$ string theories. Moreover, a
supersymmetric vertex operator algebra with Conway sporadic symmetry also
enjoys a close relation to the $K3$ elliptic genus. Inspired by the above two
relations between moonshine and $K3$ string theory, we construct a chiral CFT
by orbifolding the free theory of 24 chiral fermions and two pairs of
fermionic and bosonic ghosts. In this paper we mainly focus on the case of
umbral moonshine corresponding to the Niemeier lattice with root system given
by 6 copies of $D_{4}$ root system. This CFT then leads to the construction of
an infinite-dimensional graded module for the umbral group $G^{D_{4}^{\oplus
6}}$ whose graded characters coincide with the umbral moonshine functions. We
also comment on how one can recover all umbral moonshine functions
corresponding to the Niemeier root systems $A_{5}^{\oplus 4}D_{4}$,
$A_{7}^{\oplus 2}D_{5}^{\oplus 2}$ , $A_{11}D_{7}E_{6}$, $A_{17}E_{7}$, and
$D_{10}E_{7}^{\oplus 2}$.

###### Contents

  1. 1 Introduction
  2. 2 Moonshine and the $K3$ Elliptic Genus
    1. 2.1 Umbral Twining Genera
    2. 2.2 Conway Twining Genera
  3. 3 The Chiral CFT
    1. 3.1 The Fermions
    2. 3.2 The Ghosts
    3. 3.3 The Orbifold Theory
    4. 3.4 The Twined Characters
  4. 4 The Module for $D_{4}^{\oplus 6}$ Umbral Moonshine
    1. 4.1 The Group
    2. 4.2 The Singularities
    3. 4.3 The Mock Modular Forms
  5. 5 Discussion
  6. A Special functions
  7. B More on Ghosts
  8. C Character Tables

##  1 Introduction

The moonshine phenomenon, the study of which began with the discovery of
monstrous moonshine [1], describes an interesting connection between modular
objects and finite groups. Recent years have seen a surge of interest in
moonshine, initiated by the observation [3] that the elliptic genus of $K3$
surfaces has a close relation to the representation theory of the sporadic
finite group $M_{24}$. It was soon realised that this $M_{24}$ connection is
but one of the 23 instances of the umbral moonshine [5, 6], which associates
distinguished mock modular forms to elements of finite groups arising from the
symmetries of specific lattices.

Recall that there are 24 inequivalent positive-definite, even unimodular
lattices with rank 24. Among them, the Leech lattice $\Lambda_{\text{Leech}}$
is distinguished by the property that it has no root vectors. The other 23,
which we call the Niemeier lattices $N^{X}$, are uniquely labelled by their
root systems $X$. For each of the 23 $N^{X}$, we are interested in the finite
group $G^{X}:=\operatorname{Aut}(N^{X})/{\rm Weyl}(X)$. For each element $g\in
G^{X}$, umbral moonshine then attaches a vector-valued mock modular form
$H^{X}_{g}=(H^{X}_{g,r})$, $r\in{\mathbb{Z}}/2m$ where $m$ denotes the Coxeter
number of $X$, such that it coincides with the graded character of an
infinite-dimensional module of $G^{X}$ which we refer to as the umbral
moonshine module. More explicitly, the main conjecture of umbral moonshine
states that there exists an naturally-defined ${\mathbb{Z}}$-graded
$G^{X}$-module

$K^{X}_{r}=\bigoplus_{\begin{subarray}{c}D\in{\mathbb{Z}}_{\geq 0}\\\
D=-r^{2}\rm{mod}~{}4m\end{subarray}}K^{X}_{r,D}$

for $r\in\\{1,\dots,m\\}$ such that $H^{X}_{g}=(H^{X}_{g,r})$ for $1\leq r\leq
m$ is given by

$H^{X}_{g,r}(\tau)=-2\,q^{-{1\over
4m}}\,\delta_{r^{2},1(4m)}+\sum_{\begin{subarray}{c}D\in{\mathbb{Z}}_{\geq
0}\\\ D=-r^{2}\rm{mod}~{}4m\end{subarray}}q^{{D\over
4m}}{\rm{Tr}}_{K^{X}_{r,D}}(g)~{}$ (1.1)

where $q:=e^{2\pi i\tau}$. The other components are then determined by the
property that, when combined with the index $m$ theta functions (A.2), the
2-variable functions

$\Psi^{X}_{g}(\tau,\zeta):=\sum_{r\in{\mathbb{Z}}/2m}H^{X}_{g,r}(\tau)\theta_{m,r}(\tau,\zeta)$
(1.2)

transform under $\operatorname{\textsl{SL}}_{2}({\mathbb{Z}})$ in a way such
that they form examples of the so-called mock Jacobi foms [4]. Note that the
singular term when $\tau\to i\infty$, given by $\mp 2q^{-{1\over 4m}}$ if
$r^{2}=1\,(4m)$ and vanishes otherwise, is of lowest possible order that is
allowed by mock modularity, which means that $H^{X}_{g}$ have Fourier
coefficients that grow as slowly as possible given the mock modular properties
and are examples of so-called optimal mock Jacobi forms. This mathematical
property has led to the (unique) characterisation and classification of the
umbral moonshine mock modular forms $H^{X}_{g}$ [30, 31].

The existence of the umbral moonshine modules $K^{X}=\bigoplus_{1\leq r\leq
m}K^{X}_{r}$ has been established in [10, 11] for all 23 $X$, though their
construction is still unknown in general. To construct and to understand these
modules is arguably the most important open problem in the study of umbral
moonshine. To the best of our knowledge, a uniform construction of all 23
umbral moonshine modules, which is clearly desirable given the uniform
construction of the moonshine relation itself, is not yet in sight. In the
case of classical monstrous moonshine as well as in Conway moonshine, the
underlying modules famously possess the structure of a vertex operator algebra
(or, more loosely speaking, a chiral CFT) [2, 38]. In the umbral case, despite
the proof that $K^{X}$ has to share certain crucial features of a vertex
operator algebra [43, 44], its precise structure is not yet clear. That said,
encouraging results have been obtained for certain cases of umbral moonshine
with fairly small groups111With no more than 24 elements, they are small
compared to $|M_{24}|\sim 10^{8}$.: the cases where the Niemeier root systems
$X$ are given by $E_{8}^{\oplus 3}$ [12], $A_{6}^{\oplus 4}$ and
$A_{12}^{\oplus 2}$ [13], as well as $D_{6}^{\oplus 4}$, $D_{8}^{\oplus 3}$,
$D_{12}^{\oplus 2}$ and $D_{24}$ [7]. The construction in [12] is rather
different from those in [13, 7]: the former relies on special identities
satisfied by the mock modular forms $H^{E_{8}^{\oplus 3}}_{g,r}$ and the
latter constructs $H^{X}_{g,r}$ through the mock Jacobi forms $\Psi^{X}_{g}$
and their associated meromorphic Jacobi forms.

In this work we take a different route and exploit the relation between the
(twined) $K3$ elliptic genus and umbral and Conway moonshine which we now
explain. As mentioned before, the first case of umbral moonshine, the Mathieu
moonshine corresponding to $X=A_{1}^{\oplus 24}$, was discovered in the
context of the $K3$ elliptic genus. In [15] it was proposed that all 23 cases
of umbral moonshine, not just Mathieu moonshine, are relevant for describing
the symmetries of $K3$ sigma models. In particular, one can associate in a
uniform way, using (twined) elliptic genera of ADE singularities as one of the
main ingredients, a weak Jacobi form (of weight 0 and index 1) $\phi^{X}_{g}$
to each of the 23 root systems $X$ and each conjugacy class $[g]\in G^{X}$.
This proposal was further tested in [27] and refined in [28].

Another moonshine connection to $K3$ sigma model comes from Conway moonshine.
In [14], a generalisation of the Mukai theorem states that the physical
symmetries relevant for twining the $K3$ elliptic genus are given by 4-plane
preserving subgroups of the Conway group $\textsl{Co}_{0}$, the automorphism
group of the Leech lattice $\Lambda_{\text{Leech}}$. (Throughout the paper we
say that a subgroup of $\textsl{Co}_{0}$ or $G^{X}$ is $n$-plane preserving if
it fixes pointwise an $n$-dimensional subspace in the natural $24$-dimensional
representation, given by the corresponding lattice $\Lambda_{\text{Leech}}$ or
$N^{X}$.) This classification inspired the interesting construction that
associates to each $4$-plane preserving conjugacy class
$[g]\in\textsl{Co}_{0}$ two (possibly coinciding) weak Jacobi forms [39],
denoted $\phi_{\pm,g}$. Furthermore, it was proposed that they play the role
of twined elliptic genera of $K3$ sigma models. Inspired by the above results
and relying on various empirical evidence, in [28] (Conjecture 6) it was
conjectured that all the weak Jacobi forms arising from (the 4-plane
preserving part of) Conway and umbral moonshine are realised as $K3$ elliptic
genera twined by a supersymmetry-preserving symmetry of the sigma model at
certain points in the moduli space. Conversely, every $K3$ twined elliptic
genus (at any point in the moduli space) coincides with one of the moonshine
Jacobi forms alluded to above. Physical arguments given in [29] promote this
conjecture to a near-theorem.

In this work we mainly focus on the construction of the umbral moonshine
module for the case $X=D_{4}^{\oplus 6}$, with $G^{D_{4}^{\oplus 6}}\cong
3.S_{6}$. The first element of this construction is the following relation
between the the weak Jacobi form $\phi^{X}_{g}$ arising from umbral moonshine
and Conway moonshine. In many (but not all) cases, $\phi^{X}_{g}$ coincides
with one of the functions $\phi_{\epsilon,g^{\prime}}$ arising from Conway
moonshine. See Appendix D of [28]. In particular, for all $g\in
G^{D_{4}^{\oplus 6}}$ there is some Conway element $g^{\prime}$ and a certain
sign $\epsilon$ such that $\phi^{D_{4}^{\oplus
6}}_{g}=\phi_{\epsilon,g^{\prime}}$. It is however important to note that
$G^{D_{4}^{\oplus 6}}\cong 3.S_{6}$ is a subgroup of $\textsl{Co}_{0}$ which
is not 4-plane preserving. The second ingredient is the construction of a
chiral conformal field theory, by taking an ${\mathbb{Z}}/2$-orbifold of 24
free chiral fermions and 2 pairs of fermionic and bosonic ghost fields. When
graded by the charges of the ghost $U(1)$ current in a specific way, its
twined partition function coincides with $\phi_{\epsilon,g}$. The crucial
difference, however, is that the symmetry of this chiral theory accommodates
the full $\textsl{Co}_{0}$ without the 4-plane preserving constraints. As a
result, in some sense this chiral CFT ${\cal T}$ plays the role of a bridge
between Conway and umbral moonshine. The above two elements together with a
special property (cf. §4.3 and Conjecture 6.2 of [6]) of the $D_{4}^{\oplus
6}$ module then leads to a construction of the umbral module. We will also
comment on how one can recover all $H_{g}^{X}$ for $X=$$A_{5}^{\oplus
4}D_{4}$, $A_{7}^{\oplus 2}D_{5}^{\oplus 2}$ , $A_{11}D_{7}E_{6}$,
$A_{17}E_{7}$, and $D_{10}E_{7}^{\oplus 2}$, and some of the $H_{g}^{X}$ for
$X=A_{1}^{\oplus 24},A_{2}^{\oplus 12}$, $A_{3}^{\oplus 8}$, $A_{8}^{\oplus
3}$, $E_{4}^{\oplus 6}$ using the same ingredients.

The rest of the paper is organised as follows. In §2 we review how umbral and
Conway moonshine lead to the weak Jacobi forms $\phi^{X}_{g}$ and
$\phi_{\pm,g}$ respectively, for every $g\in G^{X}$ in the former case and
every 4-plane preserving element $g$ of $\textsl{Co}_{0}$ in the latter case.
In §3 we present the construction of the chiral conformal field theory ${\cal
T}$ and demonstrate that its graded twined partition functions coincide with
$\phi_{\pm,g}$ when making a specific choice of chemical potentials for the
ghost $U(1)$ currents. In §4 we combine these ingredients and explicitly
describe the $G^{D_{4}^{\oplus 6}}\text{--action}$ on the infinite dimensional
${\mathbb{Z}}/2$-graded vector space underlying the $D_{4}^{\oplus 6}$ case of
umbral moonshine. In §5 we describe how to recover umbral moonshine functions
for certain other cases of umbral moonshine from the twined parition functions
of ${\cal T}$ and the singularity CFTs, and comment on a few open questions.
Finally we collect details of the ghost theory and relevant data in the
appendices.

##  2 Moonshine and the $K3$ Elliptic Genus

In this section we review the construction developed in [15] and [39] of
certain weak Jacobi forms, playing the role of twined elliptic genera of $K3$
sigma models [28, 29], from umbral and Conway moonshine.

###  2.1 Umbral Twining Genera

For each case of umbral moonshine corresponding to the Niemeier lattice
$N^{X}$ there is a natural way to attach a weak Jacobi form of weight zero and
index one to each $g\in G^{X}$ [15]. This weak Jacobi form $\phi^{X}_{g}$
constitutes two parts: one is the (twined) elliptic genus of the SCFT
describing the surface singularities corresponding to the root system $X$, and
the second comes from the contribution of the umbral moonshine mock modular
forms.

We start by reviewing the first part, the singularity elliptic genus. The type
of singularities a $K3$ surface can develop are given by the so-called du Val
or Kleinein singularities, which admit an ADE classification. These
singularities are isomorphic to $\mathbb{C}^{2}/\Gamma$, where $\Gamma$ is a
finite subgroup of $SU(2)$ as in the McKay correspondence. Let $m$ denote the
Coxeter number of the corresponding root system. Recall that $\mathcal{N}=2$
superconformal minimal models (which have spectral flow symmetries) also admit
an ADE classification [23], and the central charge $c:=3\hat{c}$ is given by

$\hat{c}=1-{2\over m}.$ (2.1)

The classification of these supersymmetric minimal models stems from the
classification of modular invariant combinations of left- and right-moving
characters of affine ${\mathfrak{s}l}_{2}$, given in terms of a $2m\times 2m$
matrix, which we denote by $\Omega^{\Phi}$ for the minimal model corresponding
to the simply-laced root system $\Phi$. The explicit expression for
$\Omega^{\Phi}$ can be found in [23]. In terms of these matrices, the elliptic
genus of the super minimal model is given by [24]

$Z_{\text{minimal}}^{\Phi}(\tau,\zeta)=\sum_{r,r^{\prime}\in\mathbb{Z}/2m\mathbb{Z}}\Omega^{\Phi}_{r,r^{\prime}}\tilde{\chi}^{r}_{r^{\prime}}(\tau,\zeta)={\text{Tr}(\Omega^{\Phi}\cdot\tilde{\chi})},$
(2.2)

where $\tilde{\chi}^{r}_{s}(\tau,\zeta)$, with $|s|\leq r-1<m$, are
corresponding minimal model characters.

In [16], a 2d CFT description of type II string theory compactified on
$\mathbb{C}^{2}/\Gamma$ was proposed to be given by an
${\mathbb{Z}}_{m}$–orbifold of the corresponding supersymmetric minimal model
tensored with a non-compact CFT, and takes the form

$\left({\cal N}=2~{}\text{minimal }\otimes{\cal
N}=2~{}\left(\frac{\operatorname{\textsl{SL}}(2,{\mathbb{R}})}{U(1)}\right)_{m}~{}{\text{coset}}\right)/{{\mathbb{Z}}_{m}},$
(2.3)

where the second factor, the
$\left(\frac{\operatorname{\textsl{SL}}(2,{\mathbb{R}})}{U(1)}\right)_{m}$
supercoset model, describes the geometry of a semi-infinite cigar [25] and has
central charge

$\hat{c}=1+\frac{2}{m}.$

The spectrum of the theory contains a discrete part as well as a continuous
part; the latter exists due to the fact that the theory is non-compact and
gives a non-holomorphic (in the $\tau$-variable) contribution to the elliptic
genus of the theory [17]. Both mathematically and physically, there is a well-
defined way to isolate the holomorphic part of the elliptic genus, which we
denote by $Z_{L_{m}}$, corresponding to the contribution from the discrete
part of the spectrum. It is given by [17, 18, 19]

$Z_{L_{m}}(\tau,\zeta)=\frac{1}{2}\sum_{s=1}^{m}{\rm
Ch}^{(\tilde{R})}_{\text{massless}}(\tau,\zeta;m+2-s)+{\rm
Ch}^{(\tilde{R})}_{\text{massless}}(\tau,\zeta;s)=\frac{1}{2}\mu_{m,0}\big{(}\tau,\frac{\zeta}{m}\big{)}\frac{i\theta_{1}(\tau,\zeta)}{\eta(\tau)^{3}}.$
(2.4)

In the above equation, we make use of the Ramond character graded by
$(-1)^{F}$

${\rm Ch}^{(\tilde{R})}_{\rm
massless}(\tau,\zeta;s)=\frac{i\theta_{1}(\tau,\zeta)}{\eta^{3}(\tau)}\sum_{k\in{\mathbb{Z}}}y^{2k}q^{mk^{2}}\frac{(yq^{mk})^{\frac{s-1}{m}}}{1-yq^{mk}}$

where $s/2$ is the $U(1)$ charge of the highest weight, and the (specialized)
Appell–Lerch sum

$\mu_{m,0}(\tau,\zeta)=-\sum_{k\in\mathbb{Z}}q^{m{k}^{2}}y^{2km}\frac{1+yq^{k}}{1-yq^{k}}$
(2.5)

which is responsible for the mock modularity of $Z_{L_{m}}$.

Putting it together using the “orbifoldization formula”[26] , the (holomorphic
part of the) elliptic genus of the orbifold theory is given by

$\displaystyle{\bf EG}(\tau,\zeta;\Phi)$
$\displaystyle=\frac{1}{m}\sum_{a,b\in{\mathbb{Z}}/m{\mathbb{Z}}}q^{a^{2}}y^{2a}\,Z_{\rm
minimal}^{\Phi}(\tau,\zeta+a\tau+b)Z_{L_{m}}(\tau,\zeta+a\tau+b).$ (2.6)

See also [20, 21, 22]. In this paper we use the following definition: given
$X=\Phi_{1}\oplus\Phi_{2}\oplus\dots$ a union of simply-laced root systems
$\Phi_{i}$ with the same Coxeter number, we write ${\bf EG}(X):={\bf
EG}(\Phi_{1})+{\bf EG}(\Phi_{2})+\dots$. For instance, when $X={D_{4}}^{\oplus
6}$, we have ${\bf EG}(\tau,\zeta;{D_{4}}^{\oplus 6})=6\,{\bf
EG}(\tau,\zeta;{D_{4}})$.

Now we discuss the second part of the weak Jacobi form $\phi^{X}_{g}$, arising
from the contribution of the umbral moonshine mock modular forms. It is shown
in [15] that for each of the 23 Niemeier lattices $N^{X}$, the following
function

$\phi^{X}_{e}(\tau,\zeta):={\bf
EG}(\tau,\zeta;X)+{\theta_{1}^{2}(\tau,\zeta)\over
2\,\eta^{6}(\tau)}\left({1\over 2\pi
i}{\partial\over\partial\omega}\Psi^{X}_{e}(\tau,\omega)\right)\Big{\lvert}_{\omega=0},$
(2.7)

is always equal to ${\bf EG}(\tau,\zeta;K3)$. In other words, the above
expression gives us 23 ways to split ${\bf EG}(K3)$ into a part given by
singularity elliptic genus and a part given by umbral moonshine mock modular
forms. Moreover, for each $g\in G^{X}$ umbral moonshine gives us a natural
analogue for the second part by replacing $\Psi^{X}_{e}$ with the graded
character $\Psi^{X}_{g}$ of the umbral moonshine module. At the same time, the
explicit $G^{X}$–action on the Niemeier root system $X$ translates into an
$G^{X}$–action on the singularity CFT which preserves its superconformal
structure, and leads to a definition of its twined elliptic genus ${\bf
EG}_{g}(X):=\text{Tr}(g\dots)$. As a result, it is natural to define

$\phi_{g}^{X}(\tau,\zeta):={\bf
EG}_{g}(\tau,\zeta;X)+{\theta_{1}^{2}(\tau,\zeta)\over
2\,\eta^{6}(\tau)}\left({1\over 2\pi i}{\partial\over\partial
w}\Psi^{X}_{g}(\tau,w)\right)\Big{\lvert}_{w=0}.$ (2.8)

It can be shown that for all 23 Niemeier root systems $X$ and $g\in G^{X}$ the
above definition leads to a weak Jacobi form for certain
$\Gamma_{g}\subseteq\operatorname{\textsl{SL}}_{2}({\mathbb{Z}})$, possibly
with a non-trivial multiplier system. As mentioned in §1, when $g$ is a
4-plane preserving element, these play the role of twined elliptic genera that
are realised at certain points in the moduli space of $K3$ sigma models.

###  2.2 Conway Twining Genera

In [38] Duncan and Mack-Crane constructed a $\frac{1}{2}{\mathbb{Z}}$-graded
infinite-dimensional $\textsl{Co}_{0}$-module $V^{s\natural}$, given by the
quantum states of a chiral CFT. The starting point is a theory of 24 free real
chiral fermions, which we then orbifold by a ${\mathbb{Z}}_{2}$ which acts
with a minus sign on all 24 fermionic fields. The Neveu-Schwarz and Ramond
sectors of the original theory before orbifolding split into
$\text{A}=\text{A}^{0}\oplus\text{A}^{1}$ and
$\text{P}=\text{P}^{0}\oplus\text{P}^{1}$ respectively, where the superscripts
$0,1$ denote the invariant and anti-invariant subspaces under the orbifold
action and A, P denote the anti-periodic and periodic boundary conditions of
the fermions on the _cylinder_. Define

$V^{s\natural}:=\text{A}^{0}\oplus\text{P}^{1},~{}~{}~{}V^{s\natural}_{\text{tw}}:=\text{A}^{1}\oplus\text{P}^{0}.$
(2.9)

It was then shown that $V^{s\natural}$ (the NS sector) has the structure of a
super vertex operator algebra and $V^{s\natural}_{\text{tw}}$ (the Ramond
sector) is a canonically twisted $V^{s\natural}$-module. Moreover, both
$V^{s\natural}$ and $V^{s\natural}_{\text{tw}}$ admit a $\textsl{Co}_{0}$
action, as can be seen from identifying the 24-(complex) dimensional space in
the description of the VOA of 24 free fermions with
$\Lambda_{\text{Leech}}\otimes_{{\mathbb{Z}}}{\mathbb{C}}$. Conway moonshine
then states that the partition functions of $V^{s\natural}$ and
$V^{s\natural}_{\text{tw}}$ twined by elements of $\textsl{Co}_{0}$ are (up to
a scaling $\tau\mapsto 2\tau$) normalised Hauptmoduls of genus zero subgroups
of $\operatorname{\textsl{SL}}_{2}({\mathbb{R}})$. We refer to [38] for
further details.

Given a fixed $n$-dimensional subspace in
$\Lambda_{\text{Leech}}\otimes_{{\mathbb{Z}}}{\mathbb{R}}$, there are
different ways to build $U(1)$ currents from the fermions of
$V^{s\natural}$[40, 39]. Here we are interested in the case when a $U(1)$
current $J$ is constructed from fermions associated to a subspace of dimension
$n=4$. Fixing this $U(1)$ together with the compatibility with ${\cal N}=1$
supersymmetry breaks the symmetry of the theory from $\textsl{Co}_{0}$ to the
subgroup of $\textsl{Co}_{0}$ that preserves the given 4-plane. Conversely,
given a 4-plane preserving $G\subset\textsl{Co}_{0}$ one can construct a
$U(1)$ current $J$ such that $V^{s\natural}_{\text{tw}}$, when equipped with a
module structure for $J$ and for the ${\cal N}=1$ superconformal algebra, has
symmetry $G$. Interestingly, the $U(1)$-charged partition function of
$V^{s\natural}_{\text{tw}}$ coincides with ${\bf EG}(K3)$ (up to a sign) [39].
More generally, one can consider the $U(1)$-graded character of the twisted
Conway module twined by a 4-plane preserving element of $\textsl{Co}_{0}$:

$\phi_{g}:=-\operatorname{{tr}}_{V^{s\natural}_{\text{tw}}}\left[\mathfrak{z}~{}\hat{g}~{}y^{J_{0}}q^{L_{0}-\frac{c}{24}}\right],$
(2.10)

where $J_{0}$ is the zero mode of the aforementioned $U(1)$ current. In the
above $\hat{g}$ denotes the _lift_ of $g$ from $\text{SO}(24)$ to
$\text{Spin}(24)$, which is necessitated by the fact that the ground states in
the Ramond sector form a $4096$-dimensional spinor representation (Clifford
module) that we denote by CM, and $\mathfrak{z}$ is the lift of
$-\text{Id}\in\textsl{Co}_{0}$. Explicitly, it is given by

$\begin{split}\phi_{g}(\tau,\zeta)&=\frac{1}{2}\left[\frac{\theta_{3}(\tau,\zeta)^{2}}{\theta_{3}(\tau,0)^{2}}\frac{\eta_{-g}(\tau/2)}{\eta_{-g}(\tau)}-\frac{\theta_{4}(\tau,\zeta)^{2}}{\theta_{4}(\tau,0)^{2}}\frac{\eta_{g}(\tau/2)}{\eta_{g}(\tau)}\right.\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.-\frac{\theta_{2}(\tau,\zeta)^{2}}{\theta_{2}(\tau,0)^{2}}C_{-g}\eta_{-g}(\tau)-\frac{\theta_{1}(\tau,\zeta)^{2}}{\eta(\tau)^{6}}D_{g}\eta_{g}(\tau)\right].\end{split}$
(2.11)

The notation used above is as follows. For each $g\in\textsl{Co}_{0}$ denote
by $\lambda^{\pm 1}_{i}$ with $i=1,\dots,12$, the pairs of complex conjugate
pairs of eigenvalues of the unique non-trivial 24-dimensional representation,
arising from the natural action of
$\textsl{Co}_{0}\cong\text{Aut}(\Lambda_{\text{Leech}})$ on the Leech lattice.
Then define the quantities

$\displaystyle\begin{split}&\eta_{\pm
g}(\tau):=q\prod_{n=1}^{\infty}\prod_{i=1}^{12}\left(1\mp\lambda_{i}q^{n}\right)\left(1\mp\lambda_{i}^{-1}q^{n}\right),\\\
&C_{-g}:=\nu\prod_{i=1}^{12}\left(1+\lambda_{i}^{-1}\right)=\prod_{i=1}^{12}\left(\nu_{i}+\nu_{i}^{-1}\right)~{}.\end{split}$
(2.12)

where we have set $\nu_{i}=\lambda_{i}^{1/2}$ and
$\nu=\prod_{i=1}^{12}\nu_{i}$. Also note that
$C_{-g}=\operatorname{{tr}}_{\text{\bf CM}}\hat{g}$ and this is what
determines the branch choice of $\nu_{i}$. See [39] for more details. Now
assume that $g\in\textsl{Co}_{0}$ fixes at least a $4$-plane in the
$24$-dimensional representation, and define

$D_{g}:=\nu\sideset{}{{}^{\prime}}{\prod}_{i=1}^{12}\left(1-\lambda_{i}^{-1}\right)=\sideset{}{{}^{\prime}}{\prod}_{i=1}^{12}\left(\nu_{i}-\nu_{i}^{-1}\right).$
(2.13)

where $\sideset{}{{}^{\prime}}{\prod}$ skips two pairs of eigenvalues for
which $\lambda_{i}^{\pm 1}=1$. Notice that $D_{g}$ is non-vanishing if and
only if it fixes exactly a $4$-plane. In the latter case, $D_{g}$ is
determined up to a sign by the eigenvalues of $g$, since we are free to
exchange what we call $\lambda_{i}$ and $\lambda_{i}^{-1}$. As a result, for
exactly four-plane preserving elements there are in fact two choices of
$\phi_{g}$ depending on the choice of the sign of $D_{g}$, and we define

$\begin{split}\phi_{\epsilon,g}(\tau,\zeta)&:=\frac{1}{2}\left[\frac{\theta_{3}(\tau,\zeta)^{2}}{\theta_{3}(\tau,0)^{2}}\frac{\eta_{-g}(\tau/2)}{\eta_{-g}(\tau)}-\frac{\theta_{4}(\tau,\zeta)^{2}}{\theta_{4}(\tau,0)^{2}}\frac{\eta_{g}(\tau/2)}{\eta_{g}(\tau)}\right.\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.-\frac{\theta_{2}(\tau,\zeta)^{2}}{\theta_{2}(\tau,0)^{2}}C_{-g}\eta_{-g}(\tau)-\epsilon~{}\frac{\theta_{1}(\tau,\zeta)^{2}}{\eta(\tau)^{6}}\left|D_{g}\right|\eta_{g}(\tau)\right].\end{split}$
(2.14)

where $\epsilon=\pm 1$ encodes the sign ambiguity of $D_{g}$. In all cases, it
was shown that $\phi_{g}$ are Jacobi forms of weight $0$ and index $1$, at
some level, for every $g\in\textsl{Co}_{0}$ that fixes at least a 4-plane. As
discussed in §1, they play the role of a twined $K3$ elliptic genus in the
study of symmetries of $K3$ sigma models.

##  3 The Chiral CFT

In this section we present the construction of a chiral CFT ${\cal T}$, by
${\mathbb{Z}}_{2}$-orbifolding a free theory consisting of 12 complex chiral
fermions, 2 fermionic and 2 bosonic ghost systems. Its symmetries accommodate
the umbral group we are interested in, and its twined partition functions
reproduce (among others) the weak Jacobi forms $\phi_{g}$ reviewed in §2.

###  3.1 The Fermions

The first ingredient to build our chiral theory ${\cal T}$ is 24 real chiral
fermions $\tilde{\psi}_{1},\dots,\tilde{\psi}_{24}$, similar to the starting
point of the Conway module discussed in §2.2. Equivalently, this theory, which
we call $\mathcal{T}_{\psi}$, is given by 12 complex chiral fermions,
$\psi_{i}^{\pm}:={1\over\sqrt{2}}(\tilde{\psi}_{i}\pm i\tilde{\psi}_{i+12})$,
with the action

$\displaystyle S_{\psi}=\frac{1}{4\pi}\int
d^{2}z\sum_{i=1}^{12}\left(\psi^{+}_{i}\bar{\partial}\psi^{-}_{i}+\psi^{-}_{i}\bar{\partial}\psi^{+}_{i}\right).$
(3.1)

Their OPEs take the form

$\displaystyle\psi^{\pm}_{i}(z)\psi^{\pm}_{j}(z^{\prime})\sim\mathcal{O}(z-z^{\prime}),~{}~{}~{}\psi^{\pm}_{i}(z)\psi^{\mp}_{j}(z^{\prime})\sim\frac{\delta_{ij}}{z-z^{\prime}}.$
(3.2)

The associated Viraroso operator is given by

$\displaystyle
L^{\psi}=\sum_{n\in{\mathbb{Z}}}L^{\psi}_{n}z^{-n-2}=-\frac{1}{2}\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern
0.86108pt$\cr}}}\mathrel{\left(\psi^{+}_{i}\partial\psi^{-}_{i}+\psi^{-}_{i}\partial\psi^{+}_{i}\right)}\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr}}},$ (3.7)

with respect to which $\psi^{\pm}_{i}$ are holomorphic primary fields with
weight $1/2$. The open dots denote the regular part of the associated
expression; we refer to this as the canonical ordering. In terms of modes, it
means that the annihilators are always put to the right. By expanding the
fields in modes,

$\displaystyle\psi^{\pm}_{i}(z)=\sum_{r}\psi^{\pm}_{i,r}~{}z^{-r-\frac{1}{2}},$
(3.8)

the OPEs lead to the standard anti-commutation relations

$\displaystyle\left\\{\psi^{\pm}_{i,r}\psi^{\pm}_{j,s}\right\\}=0~{},~{}~{}~{}\left\\{\psi^{\pm}_{i,r}\psi^{\mp}_{j,s}\right\\}=\delta_{ij}\delta_{r+s,0}.$
(3.9)

The ${\operatorname{\textsl{SL}}}(2,{\mathbb{R}})$-invariant vacuum
$|0\rangle$ satisfies the usual highest weight condition

$\displaystyle\psi^{\pm}_{i,r}|0\rangle=0~{}~{}~{}\forall~{}r>0.$ (3.10)

Note that here the canonical ordering coincides with the normal ordering
corresponding to the above canonical vacuum, where the positive modes are
annihilators.

To compute the characters of the theory, consider the conformal mapping from
the complex plane to the cylinder given by $z=e^{w}$. We denote the Virasoro
zero mode on the cylinder by

$\displaystyle L^{\psi}_{\text{cyl},0}:=L^{\psi}_{0}-\frac{c_{\psi}}{24}$
(3.11)

where $c_{\psi}=12$ is the central charge. Consider general boundary
conditions parametrized by $\rho$

$\displaystyle\psi^{\pm}_{i}(w+2\pi i)=e^{\mp 2\pi i\rho}\psi^{\pm}_{i}(w).$
(3.12)

The periodic (P, $\rho=0$) and anti-periodic (A, $\rho=1/2$) cases correspond
to the usual Ramond and NS sectors. Note that the $\psi^{\pm}_{i}$ must
acquire opposite phases as in (3.12) so that the Virasoro operator remains
periodic.

The A sector Hilbert space $\mathcal{T}_{\psi,\text{A}}$ is built by acting on
the ground state $|0\rangle$ with the creation operators $\psi^{\pm}_{i,r}$
with $r\leq-1/2$, and its character is given by

$\displaystyle\chi_{\psi}^{\text{A}}(\tau):=\operatorname{{tr}}_{\mathcal{T}_{\psi,\text{A}}}\left[q^{L^{\psi}_{\text{cyl},0}}\right]=q^{-1/2}\prod_{n=1}^{\infty}\prod_{i=1}^{12}(1+q^{n-1/2})^{2}=\left(\frac{\theta_{3}(\tau,0)}{\eta(\tau)}\right)^{12}.$
(3.13)

We also define an operator $(-1)^{F}$ which has the property that it
anticommutes with all the fermionic modes, squares to the identity, and acts
trivially on $|0\rangle$. Note that $(-1)^{F}$ commutes with the Virasoro
operator and hence we can define the following character,

$\displaystyle\tilde{\chi}_{\psi}^{\text{A}}(\tau):=\operatorname{{tr}}_{\mathcal{T}_{\psi,\text{A}}}\left[(-1)^{F}q^{L^{\psi}_{\text{cyl},0}}\right]=q^{-1/2}\prod_{n=1}^{\infty}\prod_{i=1}^{12}(1-q^{n-1/2})^{2}=\left(\frac{\theta_{4}(\tau,0)}{\eta(\tau)}\right)^{12}.$
(3.14)

In the P sector $\mathcal{T}_{\psi,\text{P}}$, the ground states form a
$2^{12}$-dimensional representation of the $24$-dimensional Clifford algebra.
Explicitly, a basis can be given by the mononomials

$\displaystyle\psi^{-}_{i_{1},0}\cdots\psi^{-}_{i_{k},0}|s\rangle,$ (3.15)

where we single out $|s\rangle$ to be annihilated by all the $\psi^{+}_{i,0}$
and we require that $(-1)^{F}$ acts trivially on $|s\rangle$. The conformal
weight of the P ground states is equal to $\frac{24}{16}=\frac{3}{2}$, as each
of the complex fermions (along with its complex conjugate) contributes
$\frac{2}{16}$ due to the presence of twist fields that interpolate between
the A and P sectors. Putting things together, we obtain the following P sector
characters.

$\displaystyle\chi_{\psi}^{\text{P}}(\tau):=\operatorname{{tr}}_{\mathcal{T}_{\psi,\text{P}}}\left[q^{L^{\psi}_{\text{cyl},0}}\right]=2^{12}q\prod_{n=1}^{\infty}\prod_{i=1}^{12}(1+q^{n})^{2}=\left(\frac{\theta_{2}(\tau,0)}{\eta(\tau)}\right)^{12},$
(3.16)
$\displaystyle\tilde{\chi}_{\psi}^{\text{P}}(\tau):=\operatorname{{tr}}_{\text{P}}\left[(-1)^{F}q^{L^{\psi}_{\text{cyl},0}}\right]=\left(\frac{\theta_{1}(\tau,0)}{\eta(\tau)}\right)^{12}=0.$
(3.17)

The latter vanishes because half of the ground states have $-1$ eigenvalue
under $(-1)^{F}$, while the rest have $+1$.

Later we will consider an orbifold of ${\cal T}_{\psi}$ by a
${\mathbb{Z}}_{2}$ generated by $\xi$, which acts on the fermions by

$\displaystyle\xi\psi^{\pm}_{i}=-\psi^{\pm}_{i},$ (3.18)

and trivially on the A ground state. Note that any state with an odd (resp.
even) number of excitations is an eigenstate of $\xi$ with eigenvalue $-1$
(resp. $+1$), and hence $\xi$ acts in exactly the same way as $(-1)^{F}$ on
the quantum states in both A and P sectors. Therefore we conclude that

$\displaystyle\operatorname{{tr}}_{\mathcal{T}_{\psi,\text{A}}}\left[\xi~{}q^{L^{\psi}_{\text{cyl},0}}\right]=\tilde{\chi}_{\psi}^{\text{A}}(\tau)=\left(\frac{\theta_{4}(\tau,0)}{\eta(\tau)}\right)^{12},$
(3.19)
$\displaystyle\operatorname{{tr}}_{\mathcal{T}_{\psi,\text{P}}}\left[\xi~{}q^{L^{\psi}_{\text{cyl},0}}\right]=\tilde{\chi}_{\psi}^{\text{P}}(\tau)=0.$
(3.20)

###  3.2 The Ghosts

The next ingredients we need are the fermionic and bosonic ghost systems (see
[32], [33, 34, 35, 36, 37] and references therein for related discussions).
They are described by the action

$\displaystyle S_{\text{gh}}=\frac{1}{2\pi}\int
d^{2}z~{}\mathbf{b}\bar{\partial}\mathbf{c},$ (3.21)

where $\mathbf{b}$ and $\mathbf{c}$ are holomorphic fields of weights $h$ and
$1-h$ respectively. We focus on the cases where $h\in\frac{1}{2}{\mathbb{Z}}$.
Since there are many similarities between the fermionic and bosonic cases, we
use the boldface notation to refer to either. When we need to make the
distinction, we use $b$, $c$ to denote the fermionic ghosts and $\beta$,
$\gamma$ to denote the bosonic ghosts. We will also use a parameter $\kappa$,
which equals $+1$ for the fermionic case and $-1$ for the bosonic case.

The OPEs between the ghost fields have the form

$\displaystyle\mathbf{b}(z)\mathbf{c}(w)\sim\frac{\kappa}{z-w},~{}~{}~{}\mathbf{b}(z)\mathbf{b}(w)=\mathbf{c}(z)\mathbf{c}(w)\sim\mathcal{O}(1)$
(3.22)

and the Virasoro operator is given by

$L^{\text{gh}}=(1-h)\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern
0.86108pt$\cr}}}\mathrel{(\partial\mathbf{b})\mathbf{c}}\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern
0.86108pt$\cr}}}-h\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern
0.86108pt$\cr}}}\mathrel{\mathbf{b}(\partial\mathbf{c})}\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr}}},$ (3.23)

with respect to which $\mathbf{b}$, $\mathbf{c}$ are primary. The central
charge of the ghost system is then given by

$\displaystyle c_{\mathbf{b}\mathbf{c}}=\kappa(1-3Q^{2}),$ (3.24)

where we have introduced $Q:=\kappa(1-2h)$ for later convenience. The mode
expansions on the complex plane are

$\displaystyle\mathbf{b}(z)=\sum_{r}\mathbf{b}_{r}z^{-r-h}~{},~{}~{}~{}\mathbf{c}(z)=\sum_{r}\mathbf{c}_{r}z^{-r-(1-h)},$
(3.25)

and canonical quantization leads to the (anti)commutation relations

$\displaystyle\left\\{\mathbf{b}_{r},\mathbf{c}_{s}\right\\}_{\kappa}:=\mathbf{b}_{r}\mathbf{c}_{s}+\kappa\mathbf{c}_{s}\mathbf{b}_{r}=\kappa\delta_{r+s,0}.$
(3.26)

From (3.23) and (3.25) we see that the $SL(2,{\mathbb{R}})$-invariant vacuum
$|0\rangle$ is determined by the highest weight condition

$\displaystyle\begin{split}\mathbf{b}_{r}|0\rangle&=0~{}\forall~{}r\geq
1-h,\\\ \mathbf{c}_{r}|0\rangle&=0~{}\forall~{}r\geq h.\end{split}$ (3.27)

Consequently, the canonical ordering does not generally coincide with the
usual normal ordering. As before, we consider the $\rho$-twisted sectors for
the ghost systems corresponding to the boundary conditions

$\displaystyle\mathbf{b}(w+2\pi i)=e^{-2\pi
i\rho}\mathbf{b}(w),~{}~{}~{}\mathbf{c}(w+2\pi i)=e^{2\pi
i\rho}\mathbf{c}(w),$ (3.28)

where $w$ is the natural coordinate on the cylinder and is given by $z=e^{w}$.
The periodic case $\rho=0$ corresponds to the P sector, while the anti-
periodic case $\rho=1/2$ corresponds to the A sector222 We introduce both
sectors irrespective of the statistics of the fields, since they will both
appear when we consider the ${\mathbb{Z}}_{2}$ orbifold..

The natural ground states on the cylinder are defined as the states
annihilated by all positive modes, ${\bf b}_{r},\mathbf{c}_{r}$ with $r>0$.
Note that, for the ghost systems, these are in general different from the
$\operatorname{\textsl{SL}}(2,{\mathbb{R}})$-invariant ground state
$|0\rangle$. The corresponding energies, namely the eigenvalues of
$L^{\text{gh}}_{\text{cyl},0}=L^{\text{gh}}_{0}-c_{\mathbf{b}\mathbf{c}}/24$,
of the P and A sector ground states are given by $\frac{\kappa}{12}$ and
$-\frac{\kappa}{24}$ respectively, as is calculated in appendix B.

Another important feature of the ghost systems is that they have the following
$U(1)$ current

$\displaystyle J=-\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern
0.86108pt$\cr}}}\mathrel{\mathbf{b}\mathbf{c}}\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr}}}.$ (3.33)

In fact, as we show in appendix B, the A sector ground states for both
fermonic and bosonic ghosts are unique. The P sector has two degenerate ground
states for the fermonic ghost system due to the presence of the fermonic zero
modes $b_{0},c_{0}$, while it has a single ground state for the bosonic
system. We denote the (unique) A ground states for the fermonic (F) and
bosonic (B) ghosts by $|\Omega_{\text{A}}^{\text{F}}\rangle$ and
$|\Omega_{\text{A}}^{\text{B}}\rangle$, respectively. The (unique) P ground
state for the bosonic ghost is denoted by
$|\Omega_{\text{P}}^{\text{B}}\rangle$, and the two degenerate P ground states
for the fermionic ghost system are denoted by
$|\Omega_{\text{P},\pm}^{\text{F}}\rangle$. They are distinguished by

$\displaystyle\begin{split}b_{0}|\Omega_{\text{P},-}^{\text{F}}\rangle=0,~{}~{}~{}b_{0}|\Omega_{\text{P},+}^{\text{F}}\rangle=|\Omega_{\text{P},-}^{\text{F}}\rangle,~{}~{}~{}c_{0}|\Omega_{\text{P},+}^{\text{F}}\rangle=0,~{}~{}~{}c_{0}|\Omega_{\text{P},-}^{\text{F}}\rangle=|\Omega_{\text{P},+}^{\text{F}}\rangle.\\\
\end{split}$ (3.34)

Next we derive the characters of the ghost systems, defined by

$\displaystyle\chi_{a}^{\text{S}}(\tau,\zeta):=\operatorname{{tr}}_{\text{S}}\left[y^{J_{\text{cyl},0}}~{}q^{L^{\text{gh}}_{\text{cyl},0}}\right],$
(3.35)

where $\text{S}=\\{\text{P},\text{A}\\}$ denotes the sector and
$a=\\{\text{F},\text{B}\\}$ distinguishes between the fermionic and bosonic
ghosts, respectively. Note that the other commonly used character, defined by
$\operatorname{{tr}}_{\text{S}}\left[(-1)^{J_{\text{cyl},0}}y^{J_{cyl,0}}~{}q^{L^{\text{gh}}_{\text{cyl},0}}\right]$,
is simply given by the above by a shift $\zeta\mapsto\zeta+{1\over 2}$.

Building on the unique ground state $|\Omega_{\text{A}}^{F}\rangle$, the A
sector Hilbert space of the fermionic ghost system leads to the character

$\displaystyle\chi_{\text{F}}^{\text{A}}(\tau,\zeta)=q^{-1/24}\prod_{n=1}^{\infty}\left(1+yq^{n-1/2}\right)\left(1+y^{-1}q^{n-1/2}\right)=\frac{\theta_{3}(\tau,\zeta)}{\eta(\tau)}.$
(3.36)

Similarly, by accounting for all possible states in the Fock space created by
the negative integral modes of the ghost fields $b$, $c$ acting on both of the
ground states $|\Omega_{\text{P},\pm}^{F}\rangle$, we obtain the character

$\displaystyle\chi_{\text{F}}^{\text{P}}(\tau,\zeta)=q^{1/12}\left(y^{1/2}+y^{-1/2}\right)\prod_{n=1}^{\infty}\left(1+yq^{n}\right)\left(1+y^{-1}q^{n}\right)=\frac{\theta_{2}(\tau,\zeta)}{\eta(\tau)}.$
(3.37)

For the bosonic ghost system, the A sector character is given similarly by

$\displaystyle\chi_{\text{B}}^{\text{A}}(\tau,\zeta)=q^{1/24}\prod_{n=1}^{\infty}\left(1-yq^{n-1/2}\right)^{-1}\left(1-y^{-1}q^{n-1/2}\right)^{-1}=\frac{\eta(\tau)}{\theta_{4}(\tau,\zeta)},$
(3.38)

In the P sector, care has to be taken due to the presence of the bosonic zero
mode $\gamma_{0}$. As we will see in §3.4 (also see [35]), the contribution of
$\gamma_{0}$ can be regularised and the total character is given by

$\displaystyle\chi_{\text{B}}^{\text{P}}(\tau,\zeta)=q^{-1/12}y^{1/2}(1-y)^{-1}\prod_{n=1}^{\infty}\left(1-yq^{n}\right)^{-1}\left(1-y^{-1}q^{n}\right)^{-1}=i\frac{\eta(\tau)}{\theta_{1}(\tau,\zeta)},$
(3.39)

Finally, we would like to consider a ${\mathbb{Z}}_{2}$-orbifold of the ghost
systems, where the non-trivial group action is given by
$\xi\mathbf{b}=-\mathbf{b}$ and $\xi\mathbf{c}=-\mathbf{c}$. The resulting
characters will be related to the characters
$\chi_{a}^{\text{S}}(\tau,\zeta+1/2)$ we calculated above, since the action of
the corresponding group element $\xi$ corresponds to including the operator
$(-1)^{J_{\text{cyl},0}}$, similarly to the case of the chiral fermions
discussed in §3.1. The only nontrivial part of this implementation is the sign
of the ground state(s) under $\xi$, which is analysed in appendix B. The
results are given by

$\displaystyle\tilde{\chi}^{\text{S}}_{\text{F}}(\tau,\zeta):=\operatorname{{tr}}_{\text{S}}\left[\xi~{}y^{J_{\text{cyl},0}}q^{L^{\text{gh}}_{\text{cyl},0}}\right]=(-1)^{h-\frac{1}{2}}~{}\chi_{\text{F}}^{\text{S}}(\tau,\zeta+1/2)$
(3.40)

for the fermionic ghosts, while for the bosonic ghosts we have

$\displaystyle\tilde{\chi}^{\text{S}}_{\text{B}}(\tau,\zeta):=\operatorname{{tr}}_{\text{S}}\left[\xi~{}y^{J_{\text{cyl},0}}q^{L^{\text{gh}}_{\text{cyl},0}}\right]=(-1)^{3h+\frac{1}{2}}~{}\chi_{\text{B}}^{\text{S}}(\tau,\zeta+1/2),$
(3.41)

where S denotes either of the two sectors. Notice that all the characters we
have computed in this section do not depend on the central charge of the ghost
systems. However, we will see that by requiring the final CFT to have certain
supersymmetry we can completely fix the central charge of the ghosts systems.

###  3.3 The Orbifold Theory

After describing the basic ingredients, we now put them together and construct
the chiral CFT that will reproduce the K3 elliptic genus and its twinings. Let
$\cal{T}_{\text{B}}$ denote the theory of 2 copies of the bosonic ghost system
and the theory of 2 copies of the fermionic ghost system. We will later see
that ${\cal T}$ can be equipped with an $\mathcal{N}=4$ superconformal
symmetry for certain choices of $h_{\text{B}}$ and $h_{\text{F}}$. We will let
$h_{\text{B}}=\frac{1}{2}$ and $h_{\text{F}}=1$, corresponding to the total
central charge $c_{\mathcal{T}}=-4-2+12=6$.

We want to consider a ${\mathbb{Z}}_{2}$ orbifold of the theory

$\displaystyle\mathcal{T}^{\text{free}}=\mathcal{T}_{\text{B}}\otimes\mathcal{T}_{\text{F}}\otimes\mathcal{T}_{\psi},$
(3.42)

where ${\mathbb{Z}}_{2}=\\{1,\xi\\}$ acts on the individual components of
$\mathcal{T}^{\text{free}}$ as we have described in the previous sections.
Specifically, we want to consider

$\mathcal{T}=\left(\mathcal{T}_{\text{B},\text{A}}^{0}\otimes\mathcal{T}_{\text{F},\text{A}}^{0}\otimes\mathcal{T}_{\psi,\text{A}}^{0}\right)\oplus\left(\mathcal{T}_{\text{B},\text{P}}^{1}\otimes\mathcal{T}_{\text{F},\text{P}}^{1}\otimes\mathcal{T}_{\psi,\text{P}}^{1}\right).$
(3.43)

where the $0,1$ superscripts denote respectively the invariant and anti-
invariant part under the orbifold, in the corresponding sector denoted by the
subscript. Notice that
$\mathcal{T}_{\psi,\text{A}}^{0}\oplus\mathcal{T}_{\psi,\text{P}}^{1}$ is
isomorphic (as a VOA) to the Conway module $V^{s\natural}$ (2.9).

Introducing chemical potentials $y_{1}=e^{2\pi i\zeta_{1}}$ and $y_{2}=e^{2\pi
i\zeta_{2}}$ for the bosonic and fermionic ghosts respectively, we now define
the following partition function

$\displaystyle
Z(\tau,\zeta_{1},\zeta_{2}):=\operatorname{{tr}}_{\mathcal{T}}\left[y_{1}^{J_{\text{cyl},0}^{\text{B}}}y_{2}^{J_{\text{cyl},0}^{\text{F}}}q^{L_{0}^{\text{tot}}-\frac{6}{24}}\right],$
(3.44)

where $L_{0}^{\text{tot}}$ is the total Virasoro zero mode of the theory, and
$J_{\text{cyl},0}^{\text{B}}=J_{\text{cyl},0}^{\text{B},1}+J_{\text{cyl},0}^{\text{B},2}$,
and
$J_{\text{cyl},0}^{\text{F}}=J_{\text{cyl},0}^{\text{F},1}+J_{\text{cyl},0}^{\text{F},2}$
are the zero modes of the $U(1)$ currents of the two bosonic and two fermionic
ghosts, respectively. Using the results of the previous sections, we compute

$\begin{split}&Z(\tau,\zeta_{1},\zeta_{2})=\operatorname{{tr}}_{\mathcal{T}_{\text{A}}^{\text{free}}}\left[\frac{1}{2}(1+\xi)y_{1}^{J_{\text{cyl},0}^{\text{B}}}y_{2}^{J_{\text{cyl},0}^{\text{F}}}q^{L_{0}^{\text{tot}}-\frac{6}{24}}\right]+\operatorname{{tr}}_{\mathcal{T}_{\text{P}}^{\text{free}}}\left[\frac{1}{2}(1-\xi)y_{1}^{J_{\text{cyl},0}^{\text{B}}}y_{2}^{J_{\text{cyl},0}^{\text{F}}}q^{L_{0}^{\text{tot}}-\frac{6}{24}}\right]\\\
&=\frac{1}{2}\left[\frac{\theta_{3}(\tau,\zeta_{2})^{2}}{\theta_{4}(\tau,\zeta_{1})^{2}}\left(\frac{\theta_{3}(\tau,0)}{\eta(\tau)}\right)^{12}-\frac{\theta_{4}(\tau,\zeta_{2})^{2}}{\theta_{3}(\tau,\zeta_{1})^{2}}\left(\frac{\theta_{4}(\tau,0)}{\eta(\tau)}\right)^{12}-\frac{\theta_{2}(\tau,\zeta_{2})^{2}}{\theta_{1}(\tau,\zeta_{1})^{2}}\left(\frac{\theta_{2}(\tau,0)}{\eta(\tau)}\right)^{12}\right].\end{split}$
(3.45)

We observe that, by specializing to $\zeta_{1}=1/2$ and $\zeta_{2}:=\zeta$, we
retrieve the K3 elliptic genus in the non-standard form presented in [39]

$\displaystyle\begin{split}&\textbf{EG}(\tau,\zeta;K3)=Z\left(\tau,\frac{1}{2},\zeta\right)=\operatorname{{tr}}_{\mathcal{T}}\left[(-1)^{J_{\text{cyl},0}^{\text{B}}}~{}y^{J_{\text{cyl},0}^{\text{F}}}~{}q^{L_{0}^{\text{tot}}-\frac{6}{24}}\right]\\\
&=\frac{1}{2}\left[\frac{\theta_{3}(\tau,\zeta)^{2}}{\theta_{3}(\tau,0)^{2}}\left(\frac{\theta_{3}(\tau,0)}{\eta(\tau)}\right)^{12}-\frac{\theta_{4}(\tau,\zeta)^{2}}{\theta_{4}(\tau,0)^{2}}\left(\frac{\theta_{4}(\tau,0)}{\eta(\tau)}\right)^{12}-\frac{\theta_{2}(\tau,\zeta)^{2}}{\theta_{2}(\tau,0)^{2}}\left(\frac{\theta_{2}(\tau,0)}{\eta(\tau)}\right)^{12}\right].\end{split}$
(3.46)

In the remainder of this section we discuss the supersymmetry of
$\mathcal{T}$. We will see that an $\mathcal{N}=4$ superconformal structure is
possible when the central charge of the total theory is $c_{\cal T}=6$. In the
following sections we will not make use of this $\mathcal{N}=4$ structure.
First, one can equip the theory $\mathcal{T}_{\psi}$ of $12$ complex fermions
with the structure of $\mathcal{N}=1$ superconformal field theory [41]. To
preserve this $\mathcal{N}=1$ superconformal symmetry, the manifest Spin(24)
symmetry is broken to $\textsl{Co}_{0}$. Moreover, it is also possible equip
$\mathcal{T}_{\psi}$ with an $\mathcal{N}=4$ structure, which breaks
$\textsl{Co}_{0}$ symmetry to its 3-plane preserving subgroups depending on
the choice of the $SU(2)$ current [40]. One can also equip the whole theory
${\cal T}$, for certain choices of the ghost conformal weights, with an
$\mathcal{N}=4$ superconformal symmetry by combining the $\mathcal{N}=4$
structure of $\mathcal{T}_{\psi}$ with an $\mathcal{N}=4$ structure of the
ghost theory. In particular, for our choice $h_{\text{B}}=1/2$ and
$h_{\text{F}}=1$ the total theory has an $\mathcal{N}=4$ superconformal
algebra at $c=6$, precisely the superconformal symmetry of $K3$ non-linear
sigma models.

If the conformal weights of a pairs of $bc-\beta\gamma$ ghost system satisfy
$h_{\text{F}}=h_{\text{B}}+\tfrac{1}{2}$, there exists an $\mathcal{N}=1$
current of weight $3/2$. In our case with two pairs of $bc-\beta\gamma$
ghosts, it is given by

$G=\sum_{j=1}^{2}\left(-\frac{1}{2}\partial\beta_{j}c_{j}+\frac{2h_{\text{F}}-1}{2}\partial(\beta_{j}c_{j})-2b_{j}\gamma_{j}\right).$
(3.47)

To enhance this to $\mathcal{N}=4$, we need an $SU(2)$ subalgebra generated by
$J_{i}$ with $i=1,2,3$. One can show that such currents are given by

$J_{1}=\frac{i}{2}(\beta_{1}\gamma_{2}-\beta_{2}\gamma_{1})~{},~{}~{}~{}J_{2}=\frac{1}{2}(\beta_{1}\gamma_{2}+\beta_{2}\gamma_{1})~{},~{}~{}~{}J_{3}=\frac{1}{2}\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern
0.86108pt$\cr}}}\mathrel{\beta_{1}\gamma_{1}-\beta_{2}\gamma_{2}}\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr}}}.$ (3.48)

From now on we will choose

$h_{\text{F}}=1~{},~{}~{}h_{\text{B}}=\frac{1}{2},$ (3.49)

so that $SU(2)$ cuurent algebra is given by

$J_{i}(z)J_{j}(w)\sim\frac{\delta_{ij}~{}k_{\text{gh}}/2}{(z-w)^{2}}+\frac{i\epsilon_{ijk}J_{k}(w)}{z-w}.$
(3.50)

with $k_{\text{gh}}=-1$. Acting with the generators $J_{i}$ on $G$ (with
$h_{\text{F}}=1$) to construct the rest of the supercurrents, we get

$\begin{split}J_{i}(z)G(w)\sim-\frac{i}{2}\frac{1}{z-w}G_{i}(w).\end{split}$
(3.51)

One can check that, together with the Virasoro field

$L^{\text{gh}}=\sum_{i=1}^{2}\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern
0.86108pt$\cr}}}\mathrel{-b_{i}\partial
c_{i}+\frac{1}{2}\partial\beta_{i}\gamma_{i}-\frac{1}{2}\beta_{i}\partial\gamma_{i}}\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr}}},$ (3.52)

the fields $G$, $G_{i}$ and $J_{i}$ indeed form an $\mathcal{N}=4$ SCA with
central charge $c_{\text{gh}}=-6$ and level $k_{\text{gh}}=-1$. As in [40] we
can further define

$G_{1}^{\pm}:=\frac{1}{\sqrt{2}}(G\pm
iG_{3}),~{}~{}G_{2}^{\pm}:=\pm\frac{i}{\sqrt{2}}(G_{1}\pm iG_{2}),$ (3.53)

which transform in the representation $\mathbf{2}+\bar{\mathbf{2}}$ of
$SU(2)$, and reproduce the standard small $\mathcal{N}=4$ SCA.

We note that the supercurrents, as well as the $J_{i}$, survive the orbifold,
since they are all bilinears in the ghost fields. In the remaining part of the
paper we will however not make use of this superconformal symmetry. In
particular, we have used the grading by the ghost $U(1)$-currents instead of
the $R$-current in defining the (twined) partition function (3.44) and in
§3.4. Especially, different from the construction in [39], where the grading
is with respect to a $U(1)$ current built from a choice of four real fermions,
the symmetry groups of our theory ${\cal T}$ are not restricted to be 4-plane
preserving subgroups of $\textsl{Co}_{0}$. The choice of the central charge of
the ghost systems is in any case immaterial for the construction of our
module, since the relevant characters do not depend on it.

###  3.4 The Twined Characters

As discussed in the previous subsection, $\mathcal{T}$ has a manifest
$\text{Spin}(24)$ symmetry. We can thus consider the twined partition function

$\displaystyle
Z_{g}\left(\tau,\zeta_{1},\zeta_{2}\right):=\operatorname{{tr}}_{\mathcal{T}}\left[g~{}y_{1}^{J_{\text{cyl},0}^{\text{B}}}y_{2}^{J_{\text{cyl},0}^{\text{F}}}q^{L_{0}^{\text{tot}}-\frac{6}{24}}\right],$
(3.54)

by an element $g\in\text{Spin}(24)$, which has a manifest action on
$\mathcal{T}_{\psi}$ (cf. §2.2) and acts trivially on the ghost systems. We
now specialize to the case $g\in\textsl{Co}_{0}<\text{Spin}(24)$, i.e. to
elements that preserve the $\mathcal{N}=1$ superconformal structure of
$\mathcal{T}_{\psi}$. Specifically, the relevant characters are twined as
follows

$\displaystyle\operatorname{{tr}}_{\mathcal{T}_{\text{A}}^{\text{free}}}\left[g~{}y_{1}^{J_{\text{cyl},0}^{\text{B}}}y_{2}^{J_{\text{cyl},0}^{\text{F}}}q^{L_{0}^{\text{tot}}-\frac{6}{24}}\right]=\frac{\eta_{-g}(\tau/2)}{\eta_{-g}(\tau)}\frac{\theta_{3}(\tau,\zeta_{2})^{2}}{\theta_{3}\left(\tau,\zeta_{1}-\frac{1}{2}\right)^{2}},$
(3.55)
$\displaystyle\operatorname{{tr}}_{\mathcal{T}_{\text{A}}^{\text{free}}}\left[\xi~{}g~{}y_{1}^{J_{\text{cyl},0}^{\text{B}}}y_{2}^{J_{\text{cyl},0}^{\text{F}}}q^{L_{0}^{\text{tot}}-\frac{6}{24}}\right]=-\frac{\eta_{g}(\tau/2)}{\eta_{g}(\tau)}\frac{\theta_{4}(\tau,\zeta_{2})^{2}}{\theta_{4}\left(\tau,\zeta_{1}-\frac{1}{2}\right)^{2}},$
(3.56)
$\displaystyle\operatorname{{tr}}_{\mathcal{T}_{\text{P}}^{\text{free}}}\left[g~{}y_{1}^{J_{\text{cyl},0}^{\text{B}}}y_{2}^{J_{\text{cyl},0}^{\text{F}}}q^{L_{0}^{\text{tot}}-\frac{6}{24}}\right]=-C_{-g}\eta_{-g}(\tau)\frac{\theta_{2}(\tau,\zeta_{2})^{2}}{\theta_{2}\left(\tau,\zeta_{1}-\frac{1}{2}\right)^{2}},$
(3.57)
$\displaystyle\operatorname{{tr}}_{\mathcal{T}_{\text{P}}^{\text{free}}}\left[\xi~{}g~{}y_{1}^{J_{\text{cyl},0}^{\text{B}}}y_{2}^{J_{\text{cyl},0}^{\text{F}}}q^{L_{0}^{\text{tot}}-\frac{6}{24}}\right]=q\nu\prod_{n=1}^{\infty}\prod_{i=1}^{12}\left(1-\lambda_{i}q^{n}\right)\left(1-\lambda_{i}^{-1}q^{n-1}\right)\frac{\theta_{1}(\tau,\zeta_{2})^{2}}{\theta_{1}\left(\tau,\zeta_{1}-\frac{1}{2}\right)^{2}}.$
(3.58)

where the factors
$\theta_{i}(\tau,\zeta_{2})^{2}/\theta_{i}(\tau,\zeta_{1}-1/2)^{2}$ originate
from the ghosts contribution.

In order to make contact with K3 and the umbral module discussed in the next
section, we further specialize to a subgroup $G$ of $\textsl{Co}_{0}$, such
that each $g\in G$ generates a 4-plane preserving subgroup of
$\textsl{Co}_{0}$. Note that by requiring that $g\in G$ is $4$-plane
preserving does not imply in general that $G$ is $4$-plane preserving. For
instance, in the case $G\cong 3.S_{6}$ that is of special interest for us,
different $g\in 3.S_{6}$ do not in general fix the same $4$-plane, and thus
$3.S_{6}$ does not preserve a $4$-plane.

Finally, we specialise the fugacities of the ghost currents to the values
$\zeta_{2}=\zeta$ and $\zeta_{1}=1/2$. Note that care has to be taken when
taking the $\zeta_{1}\to 1/2$ limit in (3.58). On the one hand, the degeneracy
of A ground states in ${\cal T}_{\psi}$ and the fact that at least two of the
twelve pairs of $g$-eigenvalues are given by unity leads to a zero in the
numerator. On the other hand, the infinite degeneracy of bosonic ghost ground
states requires regularisation when taking $\zeta_{1}\to 1/2$. As a result, we
regularise the partition function by introducing an adiabatic shift in
boundary condition given by a small positive parameter $\eta$. We consider the
boundary conditions $\rho=0+\eta$ and $\rho=1/2+\eta$ as in (3.12), (3.28),
and compute the $\eta\rightarrow 0^{+}$ limit of the partition function
$Z^{\eta}_{g}\left(\tau,\frac{1}{2},\zeta\right)$ with the regulator $\eta$
present. This is straightforward for all the terms besides (3.58), which
receives the following contributions

$\displaystyle\begin{split}&\tilde{\chi}_{\text{B}}^{\text{P},\eta}(\tau,1/2)^{2}=q^{-2/12}(1-q^{\eta})^{-2}\prod_{n=1}^{\infty}\left(1-q^{n+\eta}\right)^{-2}\left(1-q^{n-\eta}\right)^{-2},\\\
&\tilde{\chi}_{\text{F}}^{\text{P},\eta}(\tau,\zeta)^{2}=-q^{2/12}\left(iy^{1/2}q^{\delta}-iy^{-1/2}\right)^{2}\prod_{n=1}^{\infty}\left(1-yq^{n+\eta}\right)^{2}\left(1-y^{-1}q^{n-\eta}\right)^{2},\\\
&\tilde{\chi}_{\psi}^{\text{P},\eta}(\tau)=q\nu\prod_{n=1}^{\infty}\left(1-q^{n-\eta}\right)^{2}\left(1-q^{n-1+\eta}\right)^{2}\prod_{i=1}^{10}\left(1-\lambda_{i}q^{n-\eta}\right)\left(1-\lambda_{i}^{-1}q^{n-1+\eta}\right).\end{split}$
(3.59)

We see that, upon multiplying the above expressions, the potentially
problematic factors $(1-q^{\eta})^{\pm 2}$ drop out and we get

$\displaystyle\begin{split}\lim_{\eta\rightarrow
0^{+}}\operatorname{{tr}}_{\mathcal{T}_{\text{P}}^{\text{free}}}\left[\xi~{}g~{}y_{1}^{J_{\text{cyl},0}^{\text{B}}}y_{2}^{J_{\text{cyl},0}^{\text{F}}}q^{L_{0}^{\text{tot}}-\frac{6}{24}}\right]&=\lim_{\eta\rightarrow
0^{+}}\tilde{\chi}_{\text{B}}^{\text{P},\eta}(\tau,1/2)^{2}~{}\tilde{\chi}_{\text{F}}^{\text{P},\eta}(\tau,\zeta)^{2}~{}\tilde{\chi}_{\psi}^{\text{P},\eta}(\tau)\\\
&=\frac{\theta_{1}(\tau,\zeta)^{2}}{\eta(\tau)^{6}}D_{g}\eta_{g}.\end{split}$
(3.60)

Putting everything together, we finally get

$\begin{split}\lim_{\eta\to
0^{+}}Z_{g}^{\eta}\left(\tau,\frac{1}{2},\zeta\right)&=\frac{1}{2}\left[\frac{\theta_{3}(\tau,\zeta)^{2}}{\theta_{3}(\tau,0)^{2}}\frac{\eta_{-g}(\tau/2)}{\eta_{-g}(\tau)}-\frac{\theta_{4}(\tau,\zeta)^{2}}{\theta_{4}(\tau,0)^{2}}\frac{\eta_{g}(\tau/2)}{\eta_{g}(\tau)}\right.\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.-\frac{\theta_{2}(\tau,\zeta)^{2}}{\theta_{2}(\tau,0)^{2}}C_{-g}\eta_{-g}(\tau)-\frac{\theta_{1}(\tau,\zeta)^{2}}{\eta(\tau)^{6}}D_{g}\eta_{g}(\tau)\right].\end{split}$
(3.61)

We observe that this agrees with $\phi_{\epsilon,g}$ as defined in (2.11) and
(2.14), the Conway twining graded by a $k=1$ $U(1)$ current. In particular,
note that the same sign ambiguity in $D_{g}$ in the twining of the Conway CFT
described in §2.2 is also present here, leading to the sign $\epsilon$ in the
definition of the twining functions. As mentioned before, a crucial difference
is that the $U(1)$ grading in the ${\cal T}$ is preserved by the $G$-action
since it is constructed out of the ghost fields which the group acts trivially
on.

##  4 The Module for $D_{4}^{\oplus 6}$ Umbral Moonshine

The goal of the section is to explain how the ingredients in the previous
sections lead to a ${\mathbb{Z}}_{2}$-graded infinite dimensional vector space
admitting a $G^{D_{4}^{\oplus 6}}$–action that underlies the $D_{4}^{\oplus
6}$ case of umbral moonshine. In particular, we will describe how the umbral
mock modular forms $H^{D_{4}^{\oplus 6}}_{g}$ for all elements $g$ of the
umbral group $G^{D_{4}^{\oplus 6}}$ are recovered from the twined partition
functions of the chiral CFT ${\cal T}$. In §4.1 we describe an explicit
construction of the group. In §4.2 we explain the action of the group on the
BPS states of 6 copies of the CFT describing a singularity of $D_{4}$ type. In
§4.3 we combine the ingredients and give expressions for $H^{D_{4}^{\oplus
6}}_{g}$ in terms of these physical ingredients.

###  4.1 The Group

For completeness, in this subsection we describe a concrete realization of the
group $3.S_{6}$, following [45]. The hexacode is the unique three-dimensional
code of length 6 over ${\mathbb{F}}_{4}$ that is Hermitian and self-dual. It
is the glue code of the Niemeier lattice $N^{D_{4}^{\oplus 6}}$ with root
system $D_{4}^{\oplus 6}$ [46], and for this reason it plays a significant
role in the case of umbral moonshine corresponding to $N^{D_{4}^{\oplus 6}}$.
Moreover, this code also plays an important role in the construction of the
largest Mathieu group $M_{24}$. Its automorphism is given by $3.A_{6}$, which
can be explicitly constructed in the following way. Write
${\mathbb{F}}_{4}=\\{0,1,\omega,\bar{\omega}\\}$ with

$\omega^{2}=\bar{\omega},~{}~{}\bar{\omega}^{2}=\omega,~{}~{}\omega^{3}=1.$

The triple cover of the alternating group $A_{6}$ can be generated by the
permutations $(1,2)(3,4)$, $(1,2)(5,6)$, $(3,4)(5,6)$, $(1,3,5)(2,4,6)$,
$(1,3)(2,4)$, as well as the composition of the permutation and multiplication
$(1,2,3){\rm diag}(1,1,1,1,\bar{\omega},\omega)$. This group acts on the 6
coordinates and in particular induces all even permutations. It also contains
the element corresponding to scalar multiplication by $\omega$ and by
$\bar{\omega}$. Hence we have constructed a group with centre
$3\cong{\mathbb{Z}}/3$ and we will call $z$ the generator of the center
corresponding to ${\rm diag}(\omega,\omega,\omega,\omega,\omega,\omega)$, the
scalar multiplication by $\omega$.

The group $3.A_{6}$ can be enlarged to $3.S_{6}$ by adjoining an extra
generator which acts on a vector in ${\mathbb{F}}_{4}^{6}$ by permuting the
last two coordinates followed by a complex conjugation:
$\omega\leftrightarrow\bar{\omega}$. This group is often referred to as the
semi-automorphism group of the hexacode, since it leaves the code invariant
but does not act linearly on it. For this reason, the group $G^{D_{4}^{\oplus
6}}:={\rm Aut}(N^{D_{4}^{\oplus 6}})/({\rm Weyl}({D_{4}}))^{\otimes 6}\cong
3.S_{6}$ is the umbral group corresponding to the corresponding case of umbral
moonshine.

From the above description, we can define a representation for the group
$G^{D_{4}^{\oplus 6}}$ given by the group homomorphism
$\epsilon:G^{D_{4}^{\oplus 6}}\to\\{1,-1\\}$, given by $\epsilon_{g}=1$ (-1)
when $g$ induces an even (odd) permutation on the 6 coordinates. In the
notation in Table 1, this is given by the irreducible character $\chi_{2}$.
This representation will play an important role in describing the umbral
module in what follows.

More generally, the action of $G^{D_{4}^{\oplus 6}}$ on ${\mathbb{F}}_{4}^{6}$
determines the umbral moonshine module for the ${D_{4}^{\oplus 6}}$ case of
umbral moonshine. For later use we will now describe this action in more
detail. Writing the natural basis of ${\mathbb{F}}_{4}^{6}$ as given by ${\bf
e}_{0}^{i}$, ${\bf e}_{1}^{i}$, ${\bf e}_{\omega}^{i}$ and ${\bf
e}_{\bar{\omega}}^{i}$ for $i=1,\dots,6$, we obtain a 24-dimensional
permutation representation of $G^{D_{4}^{\oplus 6}}$. The corresponding
24-dimensional cycle shape is denoted by $\widetilde{\Pi}_{g}$ in Table 2.
Furthermore, from the above construction of $G^{D_{4}^{\oplus 6}}$ it is clear
that the action of $G^{D_{4}^{\oplus 6}}$ does not mix ${\bf e}_{0}^{i}$ with
${\bf e}_{1}^{i}$, ${\bf e}_{\omega}^{i}$ and ${\bf e}_{\bar{\omega}}^{i}$ and
hence we arrive at a six-dimensional representation of $G^{D_{4}^{\oplus 6}}$.
The corresponding 6-dimensional cycle shape is denoted by $\bar{\Pi}_{g}$ in
Table 2, and the corresponding character denoted by $\bar{\chi}$. One has
$\bar{\chi}=\chi_{1}+\chi_{3}$ in terms of the irreducible representations
(cf. Table 1). Alternatively, one might think of the 6-dimensional
representation as spanned by the 6 vectors of the form ${\bf e}_{1}^{i}+{\bf
e}_{\omega}^{i}+{\bf e}_{\bar{\omega}}^{i}$. Similarly, we also define another
character $\chi$ by $\chi_{g}=\bar{\chi}_{g}\epsilon_{g}$. One has
$\chi=\chi_{2}+\chi_{4}$ in terms of the irreducible representations. Finally,
we have the 12-dimensional representation with basis ${\bf e}_{1}^{i}-{\bf
e}_{\omega}^{i}$ and ${\bf e}_{1}^{i}-{\bf e}_{\bar{\omega}}^{i}$ for
$i=1,\dots,6$. We denote the corresponding character by $\check{\chi}$, given
by $\check{\chi}=\chi_{14}$ in terms of the irreducible characters.

One can translate the above description of the group action on the hexacode
into an action on the root systems $D_{4}^{\oplus 6}$ in a straightforward
way. First one identifies each copy of ${\mathbb{F}}_{4}$ with a copy of
$D_{4}$ and identifies ${\bf e}_{0}$ with the central node of the dynkin
diagram and ${\bf e}_{1}$, ${\bf e}_{\omega}$ and ${\bf e}_{\bar{\omega}}$
with the three nodes connected to the central node.

###  4.2 The Singularities

As reviewed in §2.1 there are 23 different natural ways to decompose the K3
elliptic genus (and twinings thereof) into two parts, corresponding to the 23
Niemeier root systems $X$. The first part is given by the elliptic genus of
the CFTs that describes the singularities associated with $X$. The second part
is the contribution from the umbral moonshine vector-valued mock modular forms
$H^{X}$. As the umbral group $G^{X}$ naturally acts on the singularities $X$
as well as the umbral moonshine module, we can generalise the construction and
define a $g$-twined weak Jacobi form $\phi^{X}_{g}$ as in (2.8).

In this subsection we describe the construction of the twined singularity
elliptic genus $\textbf{EG}_{g}(\tau,\zeta;X)$ for $X=D_{4}^{6}$ explicitly,
for all $g\in G^{D4^{\oplus 6}}\cong 3.S_{6}$ This is expressed via (2.6) in
terms of the elliptic genus of the $D_{4}$ supersymmetric minimal model, given
by

$Z_{\text{minimal}}^{D_{4}}(\tau,\zeta)=\frac{1}{2}\operatorname{{tr}}\left(\Omega^{D_{4}}\cdot\tilde{\chi}(\tau,\zeta)\right)=\frac{\theta_{1}^{2}\left(\tau,\frac{2}{3}\zeta\right)}{\theta_{1}^{2}\left(\tau,\frac{1}{3}\zeta\right)},$
(4.1)

where the Cappelli–Itzykson–Zuber [23] omega matrix $\Omega^{D_{4}}$ is given
by

$\left(\Omega^{D_{4}}\right)_{r,r^{\prime}}=\begin{cases}2\delta_{r,r^{\prime}(12)}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{},&r=0,3\mod
6\\\
\delta_{r,r^{\prime}(12)}+\delta_{r,-r^{\prime}(6)}\delta_{r,r^{\prime}(4)},&r=1,5\mod
6~{}~{},\\\
\delta_{r,r^{\prime}(12)}+\delta_{r,-r^{\prime}(6)}\delta_{r,r^{\prime}(4)},&r=2,4\mod
6\end{cases}$ (4.2)

and $r,r^{\prime}$ are taken to be in ${\mathbb{Z}}/12$. Using the property
$\tilde{\chi}^{r}_{s}(\tau,\zeta)=-\tilde{\chi}^{-r}_{s}(\tau,\zeta)$ of the
parafermion characters, we can rewrite (4.1) as

$Z_{\text{minimal}}^{D_{4}}=\operatorname{{tr}}_{*}\left(\widehat{\Omega}^{D_{4}}\cdot\tilde{\chi}\right),$
(4.3)

where
$(\widehat{\Omega}^{D_{4}})_{r,s}=(\Omega^{D_{4}})_{r,s}-(\Omega^{D_{4}})_{r,-s}$
and is expicitly given by

$\widehat{\Omega}^{D_{4}}=\left(\begin{array}[]{ccccc}1&0&0&0&1\\\
0&0&0&0&0\\\ 0&0&2&0&0\\\ 0&0&0&0&0\\\ 1&0&0&0&1\end{array}\right)$ (4.4)

and we have used the notation $\operatorname{{tr}}_{*}$ to denote tracing over
the indices $\\{1,2,3,4,5\\}$. Then from (2.6) this gives the corresponding
singularity elliptic genus

$\mathbf{EG}(\tau,\zeta;K3)=\operatorname{{tr}}_{*}\left(\widehat{\Omega}^{D_{4}}\cdot\Xi(\tau,\zeta)\right)$
(4.5)

where we have defined

$\Xi^{r}_{s}(\tau,\zeta):=\frac{1}{6}\sum_{a,b\in{\mathbb{Z}}/6{\mathbb{Z}}}q^{a^{2}}y^{2a}\,\tilde{\chi}^{r}_{s}(\tau,\zeta+a\tau+b)Z_{L_{m}}(\tau,\zeta+a\tau+b)$
(4.6)

which has integer coefficients in the $q,y$ expansions.

Recall that the automorphism group of the $D_{4}$ root system is generated by
an order 2 element $g_{2}$ and an order 3 element $g_{3}$. The corresponding
action on the minimal model is then captured by
$Z_{\text{minimal},g_{2,3}}^{D_{4}}=\operatorname{{tr}}_{*}\left(\widehat{\Omega}^{D_{4}}_{g_{2,3}}\cdot\tilde{\chi}\right)$,
where the so-called twined Omega matrices for $D_{4}$ are given by

$\widehat{\Omega}^{D_{4}}_{g_{2}}=\left(\begin{array}[]{ccccc}1&0&0&0&-1\\\
0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\
-1&0&0&0&1\end{array}\right)~{}~{},~{}~{}~{}\widehat{\Omega}^{D_{4}}_{g_{3}}=\left(\begin{array}[]{ccccc}1&0&0&0&1\\\
0&0&0&0&0\\\ 0&0&-1&0&0\\\ 0&0&0&0&0\\\ 1&0&0&0&1\end{array}\right).$ (4.7)

From this and from the explicit description of the group action of
$G^{D_{4}^{\oplus 6}}$ on the root system $D_{4}^{\oplus 6}$ given in §4.1, we
conclude that the corresponding twined partition function for 6 copies of
$D_{4}$ singularity is given by

${\bf EG}_{g}(\tau,\zeta;D_{4}^{\oplus
6})=\operatorname{{tr}}_{*}\left(\widehat{\Omega}^{D_{4}^{\oplus
6}}_{g}\cdot\Xi\right)$ (4.8)

where the $g$-twined omega matrix $\Omega^{D_{4}^{\oplus 6}}_{g}$ is given by
the group charaters $\chi$, $\bar{\chi}$, $\check{\chi}$ discussed in §4.1 as

$\widehat{\Omega}^{D_{4}^{\oplus
6}}_{g}=\left(\begin{array}[]{ccccc}\bar{\chi}_{g}&0&0&0&\chi_{g}\\\
0&0&0&0&0\\\ 0&0&\check{\chi}_{g}&0&0\\\ 0&0&0&0&0\\\
\chi_{g}&0&0&0&\bar{\chi}_{g}\end{array}\right)$ (4.9)

for $g\in 3.S_{6}$. The above expression (4.8) makes manifest the
$G^{D_{4}^{\oplus 6}}$-supermodule underlying the singularity CFT.

###  4.3 The Mock Modular Forms

It was shown in appendix D of [28] that, for any embedding
$\iota:G^{D_{4}^{\oplus 6}}\to\textsl{Co}_{0}$ we have

$\phi_{\epsilon_{g},\iota(g)}=\phi^{D_{4}^{\oplus 6}}_{g},$ (4.10)

where we are using definition (2.14) with $\epsilon$ given by
$\epsilon_{g}:G^{D_{4}^{\oplus 6}}\rightarrow\\{1,-1\\}$, the character
defined in §4.1. The results of §3 and §4.2 amount to a construction of an
infinite-dimensional, ${\mathbb{Z}}\times{\mathbb{Z}}$-graded super-module

$W=\bigoplus_{n,\ell\in{\mathbb{Z}}}W_{n,\ell},~{}~{}W_{n,\ell}=W_{n,\ell}^{(+)}\oplus
W_{n,\ell}^{(-)}$

for $G^{D_{4}^{\oplus 6}}$, such that its graded super character, defined for
a super-module $V=V^{(+)}\oplus V^{(-)}$ as ${\rm Str}_{V}(g):={\rm
Tr}_{V^{(+)}}(g)-{\rm Tr}_{V^{(-)}}(g)$, gives

$\tilde{\phi}_{\epsilon_{g},g}(\tau,\zeta)=\sum_{n,\ell\in{\mathbb{Z}}}q^{n}y^{\ell}\,{\rm
Str}_{W_{n,\ell}}(g),$ (4.11)

where

$\tilde{\phi}_{\epsilon_{g},g}(\tau,\zeta):=\phi_{\epsilon_{g},g}(\tau,\zeta)-{\bf
EG}_{g}(\tau,\zeta;D_{4}^{\oplus 6}).$ (4.12)

Attaching the Hilbert space ${\cal H}_{\rm aux}$ of a pair of (auxilliary)
periodic bosonic ghosts, on which the group $G^{D_{4}^{\oplus 6}}$ acts
trivially, we obtain

$K_{n,\ell}=K_{n,\ell}^{(+)}\oplus K_{n,\ell}^{(-)},~{}~{}{\text{ with
}}K_{n,\ell}^{(\pm)}={\cal H}_{\rm aux}\otimes W_{n,\ell}^{(\pm)}.$ (4.13)

From the discussions above we conclude that its graded characters are given by
(cf. (3.39))

$\sum_{n,\ell\in{\mathbb{Z}}}q^{n}y^{\ell}{\rm
Str}_{K_{n,\ell}}(g)=h_{g}(\tau)$ (4.14)

where

$h_{g}(\tau):=-{\eta^{2}(\tau)\over\theta_{1}^{2}(\tau,\zeta)}\tilde{\phi}_{\epsilon_{g},g}(\tau,\zeta)$
(4.15)

Note that, from (4.10) and (2.8) we see that the function $h_{g}$ is indeed
independent of the $\zeta$-variable. This implies that ${\rm
Str}_{K_{n,\ell}}(g)=0$ for all $g\in G^{D_{4}^{\oplus 6}}$ unless $\ell=0$,
although this is not obvious from the way the virtual representation
$K_{n,\ell}$ is constructed.

Now we shall see how the umbral moonshine mock modular forms $H^{D_{4}^{\oplus
6}}_{g}=(H_{g,r})$, $r=1,3,5$, are recovered from the $G^{D_{4}^{\oplus
6}}$-supermodule $K=\bigoplus_{n,\ell\in{\mathbb{Z}}}K_{n,\ell}$. Let ${z}$ be
a non-trivial element in the center subgroup of $G^{D_{4}^{\oplus 6}}$,
$\langle z\rangle\cong{\mathbb{Z}}_{3}$. Then the umbral moonshine mock
modular forms are given by

$\displaystyle-{{\theta_{6,3}^{1}(\tau)}\over\eta^{4}(\tau)}H_{g,3}(\tau)$
$\displaystyle=\tfrac{1}{3}(2h_{g}-h_{zg}-h_{z^{2}g})(\tau)$ (4.16)
$\displaystyle-{(\theta^{1}_{6,1}+\epsilon_{g}\theta^{1}_{6,5})(\tau)\over\eta^{4}(\tau)}H_{g,1}(\tau)$
$\displaystyle=\tfrac{1}{3}(h_{g}+h_{zg}+h_{z^{2}g})(\tau)$ (4.17)
$\displaystyle-\epsilon_{g}{(\theta^{1}_{6,1}+\epsilon_{g}\theta^{1}_{6,5})(\tau)\over\eta^{4}(\tau)}H_{g,5}(\tau)$
$\displaystyle=\tfrac{1}{3}(h_{g}+h_{zg}+h_{z^{2}g})(\tau)$ (4.18)

where the unary theta functions $\theta^{1}_{6,r}(\tau)$ are defined in (A.3).

Note that despite the apparent factor of $1\over 3$, the above expressions
give a construction of $G^{D_{4}^{\oplus 6}}$-supermodules whose graded
supertraces coincide with the mock modular forms $H_{g,r}$ for $r=1,3,5$. This
is because the right-hand side of (4.16) (resp. (4.17-4.18)) has the
interpretation of projecting out the $G^{D_{4}^{\oplus 6}}$–representations
that factor through $S_{6}$ (resp. are faithful representations). One can see
this explicitly by looking at the character table (Table 1). Explicitly, let
us define the projection operator $P$ acting acting on any virtual
representation $V=\sum_{i=1}^{16}n_{i}V_{i}$ of $G^{D_{4}^{\oplus 6}}$, where
$n_{i}\in{\mathbb{Z}}$ and $V_{i}$ denotes the irreducible representation
corresponding to the character $\chi_{i}$ in Table 1, by $V\lvert
P=\sum_{i=1}^{11}n_{i}V_{i}$, and similarly $P^{\prime}:={\rm{id}}-P$ acting
as $V\lvert P^{\prime}=\sum_{i=12}^{16}n_{i}V_{i}$. Then we have333Departing
from the CFT structure, from the explicit vector space interpretation of the
prefactors in (4.19) one can define infinite-dimensional virtual (trivial)
representations $\tilde{K}_{3}=\oplus_{n}\tilde{K}_{3,n}$ such that
$H_{g,3}(\tau)=\sum_{n,n^{\prime},\ell\in{\mathbb{Z}}}q^{n+n^{\prime}}y^{\ell}\,{\rm
Str}_{\tilde{K}_{3,n}\otimes(K_{n^{\prime},\ell}\lvert P)}(g)$. Similarly, one
can define graded virtual representations $\tilde{K}_{1}$ and $\tilde{K}_{5}$
such that
$H_{g,r}(\tau)=\sum_{n,n^{\prime},\ell\in{\mathbb{Z}}}q^{n+n^{\prime}}y^{\ell}\,{\rm
Str}_{\tilde{K}_{r,n}\otimes(K_{n^{\prime},\ell}\lvert P^{\prime})}(g)$ for
$r=1,5$ .

$\displaystyle-{{\theta_{6,3}^{1}(\tau)}\over\eta^{4}(\tau)}H_{g,3}(\tau)$
$\displaystyle=\sum_{n,\ell\in{\mathbb{Z}}}q^{n}y^{\ell}\,{\rm
Str}_{K_{n,\ell}\lvert P}(g)$ (4.19)
$\displaystyle-{(\theta^{1}_{6,1}+\epsilon_{g}\theta^{1}_{6,5})(\tau)\over\eta^{4}(\tau)}H_{g,1}(\tau)$
$\displaystyle=-\epsilon_{g}{(\theta^{1}_{6,1}+\epsilon_{g}\theta^{1}_{6,5})\over\eta^{4}(\tau)}H_{g,5}(\tau)=\sum_{n,\ell\in{\mathbb{Z}}}q^{n}y^{\ell}\,{\rm
Str}_{K_{n,\ell}\lvert P^{\prime}}(g).$ (4.20)

Note that this construction makes manifest the property described in
Conjecture 6.2 in [6]. The first few coefficients of the mock modular forms
$H^{D_{4}^{\oplus 6}}_{g}$ and the corresponding $G^{D_{4}^{\oplus
6}}_{g}$–representations can be found in Appendix C.6 and D.6 of [6].

##  5 Discussion

In this paper we construct a module for the $D_{4}^{\oplus 6}$ case of umbral
moonshine. This is the first time that the module is constructed for a case of
umbral moonshine with a sizeable umbral finite group (with $|G^{D_{4}^{\oplus
6}}|\sim 10^{3}$, this group is larger than the cases of umbral moonshine
where modules have been constructed previously in [12, 13, 7], where the
groups have order dividing 24). This is also the first construction of the
umbral module which utilises the connection to symmetries of K3 string theory.
At the same time, there are clearly important open questions remaining. In the
following we discuss a few of them.

  * •

Note that our construction naturally leads to a super module for
$G^{D_{4}^{\oplus 6}}$. However, apart from the virtual representation
corresponding to the leading polar term (cf. (1.1)), the umbral module is
known to constitute of the even part. It would be nice to make this positivity
manifest.

  * •

What is the physical or geometric meaning of the chiral CFT ${\cal T}$? The
relation between the Conway CFT, which is closely related to ${\cal T}$, and a
specific K3 sigma model has been elucidated in [39, 47, 48]. It would be
interesting to understand the physical role played by the ghost systems.

  * •

An obvious question is whether one can employ a similar construction for the
other cases of umbral moonshine. Note that the chiral CFT ${\cal T}$ has
Spin(24) symmetry which preserves the fermionic and bosonic $U(1)$ ghost
currents. It is hence possible to define the regularised twined partition
function $\lim_{\eta\to 0^{+}}Z_{g}^{\eta}(\tau,\zeta_{1},\zeta_{2})$ (cf.
§3.4) for any element of any of the 23 umbral groups. To make contact with
weak Jacobi forms of the type of K3 elliptic genus, one has to specialise the
fugacity to $\zeta_{1}={1\over 2}$. However, this leads to a finite answer
only when taking $\eta\to 0^{+}$ if $g$ is 4-plane preserving. To construct
umbral moonshine modules for cases where not all group elements are 4-plane
preserving ($X=A_{1}^{\oplus 24}$, $A_{2}^{\oplus 12}$, and $A_{4}^{\oplus
6}$), one needs a construction that works with the two-elliptic-variable
functions $Z_{g}(\tau,\zeta_{1},\zeta_{2})$ directly.

  * •

Note that the contribution of the vector-valued umbral moonshine mock modular
forms $(H^{X}_{g,r})$ to the twined partition function of the theory ${\cal
T}$ is basically given by a single $q$-series ${1\over 2\pi
i}{\partial\over\partial\omega}\Psi^{X}_{e}(\tau,w)$. See (2.8). What allows
us to recover from it the individual components $H^{X}_{g,r}$ of the mock
modular forms is the following two facts. First, there are just two
independent components in the case $X=D_{4}^{\oplus 6}$, which can be taken to
be $H^{X}_{g,1}$ and $H^{X}_{g,3}$. Second, the representations underlying the
1st resp. 3rd component have the feature that they factor through $S_{6}$
resp. are faithful representations. As a result, it is possible to use the
projection operator to isolate the contributions from the two independent
components from the twined partition function of ${\cal T}$. A similar
projection property also holds for other 14 cases of umbral moonshine (cf.
Conjecture 6.3 in [6]).

In view of this, another challenge when attempting to generalise the current
construction to other cases of umbral moonshine is how to disentangle the
contributions from different components of the vector-valued umbral moonshine
mock modular forms $(H^{X}_{g,r})$ in the twined partition functions for the
cases of $X$ with many independent components. Recall that an important
feature of umbral moonshine is the “multiplicative relations” relating
$H^{X^{\prime}}_{g^{\prime}}$ and $H^{X}_{g}$ for specific pairs of Niemeier
root systems $(X,X^{\prime})$ and group elements $g\in G^{X}$ and
$g^{\prime}\in G^{X^{\prime}}$ (cf. §5.3 of [6]). As we will see in more
detail below, these relations together with the projection property enable us
to disentangle different components in the vector-valued functions $H^{X}_{g}$
in various cases.

Finally, we point out that for the cases that $g$ is a 4-plane preserving
group element of a umbral group $G^{X}$, many mock modular forms $H^{X}_{g}$
for many different $X$ and $g$ can be obtained in a similar way as discussed
in §4.

$A_{1}^{\oplus 24}$ :
Exactly the same procedure as discussed in the main part of this paper can be
used to obtain a supermodule for the group $M_{22}<G^{A_{1}^{\oplus 24}}$ that
is compatible with the $M_{24}$ moonshine, or equivalently the
$X=A_{1}^{\oplus 24}$ case of umbral moonshine.

$A_{2}^{\oplus 12}$:
An analogous procedure, using a projection operator projecting out
representations factoring through $M_{12}<G^{A_{2}^{\oplus 12}}\cong
2.M_{12}$, recover from twined parition functions (3.54) the mock modular
forms $H^{A_{2}^{\oplus 12}}_{g}$ for $g\in 2.M_{12}$ that are not in the
conjugacy classes $11AB,12A,20AB,22AB$. (Here and below we use the same naming
of the conjugacy classes as in [6].) As a result, one can construct modules
for $\tilde{G}<2.M_{12}$ compatible with the corresponding case of umbral
moonshine, for three of the maximal subgroups of $2.M_{12}$. For completeness
we list the explicit generators of these three maximal subgroups in terms of
permutation groups on 24 objects:

$\begin{split}&G_{1}=\langle(1,18,5,9,24,16)(2,6,8,11,17,20)(3,12,23,13,22,14)(4,10,19,15,21,7),\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(1,9)(2,19)(3,13(4,10)(5,24)(6,17)(7,11)(8,23)(12,22)(14,20)(15,21)(16,18)\rangle\\\
&G_{2}=\langle(1,13)(2,19)(3,9)(4,18)(5,21)(6,17)(7,11)(8,20)(10,16)(12,22)(14,23)(15,24),\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(1,5,11)(2,9,16)(3,18,8)(4,17,7)(6,19,15)(10,12,23)(13,24,20)(14,21,22),\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(1,9)(2,11)(3,13)(4,15)(5,16)(6,17)(7,19)(8,20)(10,21)(12,22)(14,23)(18,24)\rangle\\\
&G_{3}=\langle(1,12,18)(3,15,6,21,16,14)(4,17,10,5,23,13)(7,20)(8,19)(9,22,24),\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(1,11,22,24)(2,12,18,9)(3,20,10,19)(4,5)(6,23)(7,13,8,21)(14,17)(15,16)\rangle\\\
\end{split}$

$A_{3}^{\oplus 8}$, $A_{8}^{\oplus 3}$, $E_{6}^{\oplus 4}$:
Using similar analysis as above, one can recover $H^{X}_{g}$ for all elements
of $g\in\tilde{G}<G^{X}$, for $X=A_{3}^{\oplus 8}$ and $A_{8}^{\oplus 3}$. In
particular, in the $X=A_{3}^{\oplus 8}$ case we also make use of the
multiplicative relations between $H^{X}_{g}$ and
$H^{X^{\prime}}_{g^{\prime}}$, where $X^{\prime}=A_{1}^{\oplus 24}$ and
$g^{\prime}\in G^{X^{\prime}}$, and thereby obtain all $H^{X}_{g}$ except for
$g\in[8A]$. In the $X=A_{8}^{\oplus 3}$ we make use of the multiplicative
relations between $H^{X}_{g}$ and $H^{X^{\prime}}_{g^{\prime}}$, where
$X^{\prime}=A_{2}^{\oplus 12}$ and $g^{\prime}\in G^{X^{\prime}}$, and thereby
obtain all $H^{X}_{g}$ except for $g\in[3A]$ and $g\in[6A]$. In the
$X=E_{6}^{\oplus 4}$ we make use of the multiplicative relations between
$H^{X}_{g}$ and $H^{X^{\prime}}_{g^{\prime}}$, where $X^{\prime}=A_{2}^{\oplus
12}$ and $g^{\prime}\in G^{X^{\prime}}$, and thereby obtain all $H^{X}_{g}$
except for $g\in[8A]$ and $g\in[8B]$.

$A_{5}^{\oplus 4}D_{4}$, $A_{7}^{\oplus 2}D_{5}^{\oplus 2}$ ,
$A_{11}D_{7}E_{6}$, $A_{17}E_{7}$, $D_{10}E_{7}^{\oplus 2}$:
For $X=A_{5}^{\oplus 4}D_{4}$ all non-vanishing components of
$H^{X}_{g}=(H^{X}_{g,r})$, $r=1,2,\dots,5$, can be recovered from the twined
partition functions, by relating them to the umbral moonshine mock modular
forms for $X^{\prime}=D_{4}^{\oplus 6}$ that we constructed in the main part
of the paper, and for the $X^{\prime\prime}=A_{2}^{\oplus 12}$ case that we
described above. Explicitly, we have

$H^{X^{\prime}}_{g^{\prime},1}(\tau)=H^{X}_{g,1}(\tau)+H^{X}_{g,5}(\tau)$
(5.1)

for the pairs

$(g^{\prime},g)=(1A,1A/2A),(2A,2B),(2A,4A),(3B,3A/6A),(4A,8AB),$ (5.2)

and

$H^{X^{\prime}}_{g^{\prime\prime},1}(\tau)=H^{X}_{g,1}(\tau)-H^{X}_{g,5}(\tau)$
(5.3)

for the pairs
$(g^{\prime\prime},g)=(2B,1A/2A),(2B,2B),(2C,4A),(6B,3A/6A),(4B,8AB)$. For the
3rd component we make use the relation

$H^{X}_{g,3}(\tau)={1\over 2}H^{X^{\prime}}_{g^{\prime},3}(\tau)$ (5.4)

for the same pairs $(g^{\prime},g)$ as in (5.2). The even components satisfy

$H^{X}_{g,2r}=-H^{X}_{zg,2r},~{}~{}r\in{\mathbb{Z}}/3$ (5.5)

where $z$ denotes a generator of the center subgroup $\langle
z\rangle\cong{\mathbb{Z}}_{2}<G^{X}$. This forces the even components of the
vector-valued mock modular forms $H^{X}_{g}$ to vanish for elements $g$ in
conjugacy classes $2B,4A,8AB$. The rest of $H^{X}_{g,2r}$ can be recovered by
using the relation to $X^{\prime\prime}=A_{2}^{\oplus 12}$ case of umbral
moonshine:

$H^{X}_{g,2}(\tau)-H^{X}_{g,4}(\tau)=H^{X^{\prime\prime}}_{g^{\prime},2}(\tau).$
(5.6)

for the pairs $(g,g^{\prime})=(1A,2B),(3A,6C)$. Note that the two terms on the
left-hand side contribute to different powers of $q$ when regarding the whole
function as a $q$-series and the above relation is therefore enough to
determine both the 2nd and the 4th components of the mock modular forms
$H^{X}_{g}$.

Using similar analysis as above, one can recover all $H^{X}_{g}$ for all $g\in
G^{X}$, for $X=A_{7}^{\oplus 2}D_{5}^{\oplus 2}$, $A_{11}D_{7}E_{6}$,
$A_{17}E_{7}$, and $D_{10}E_{7}^{\oplus 2}$. It would be nice to construct an
explicit group action which reproduces these functions.

Acknowledgements

We are grateful to J. Duncan helpful discussions and comments on an earlier
draft. M.C. and S.H. acknowledge the kind hospitality of the Aspen Center for
Physics, which is supported by NSF grant PHY-1066293, as this was being
completed. The work of V.A. and M.C. is supported by ERC starting grant H2020
ERC StG #640159. S.H. is supported by a Harvard University Golub Fellowship in
the Physical Sciences and DOE grant DE-SC0007870.

##  Appendix A Special functions

Here we list here the definitions of the Dedekind eta function and the Jacobi
theta functions, as well as other theta functions.

$\begin{split}&\eta(\tau):=q^{1/24}\prod_{n=1}^{\infty}\left(1-q^{n}\right)\\\
&\theta_{1}(\tau,\zeta):=-i\sum_{n\in\left(\mathbb{Z}+\frac{1}{2}\right)}(-1)^{n-\frac{1}{2}}y^{n}q^{n^{2}/2}=\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=-iq^{1/8}\left(y^{1/2}-y^{-1/2}\right)\prod_{n=1}^{\infty}\left(1-q^{n}\right)\left(1-yq^{n}\right)\left(1-y^{-1}q^{n}\right)\\\
&\theta_{2}(\tau,\zeta):=\sum_{n\in\left(\mathbb{Z}+\frac{1}{2}\right)}y^{n}q^{n^{2}/2}=q^{1/8}\left(y^{1/2}+y^{-1/2}\right)\prod_{n=1}^{\infty}\left(1-q^{n}\right)\left(1+yq^{n}\right)\left(1+y^{-1}q^{n}\right)\\\
&\theta_{3}(\tau,\zeta):=\sum_{n\in\mathbb{Z}}y^{n}q^{n^{2}/2}=\prod_{n=1}^{\infty}\left(1-q^{n}\right)\left(1+yq^{n-1/2}\right)\left(1+y^{-1}q^{n-1/2}\right)\\\
&\theta_{4}(\tau,\zeta):=\sum_{n\in\mathbb{Z}}(-1)^{n}y^{n}q^{n^{2}/2}=\prod_{n=1}^{\infty}\left(1-q^{n}\right)\left(1-yq^{n-1/2}\right)\left(1-y^{-1}q^{n-1/2}\right)\\\
\end{split}$ (A.1)

where $q:=e^{2\pi i\tau}$ and $y:=e^{2\pi i\zeta}$. Given an
$m\in{\mathbb{Z}}_{>0}$ we define the index $m$ theta function:

$\theta_{m,r}(\tau,\zeta):=\sum_{k=r\,(2m)}q^{k^{2}\over 4m}y^{k}$ (A.2)

and

$\theta_{m,r}^{1}(\tau):=\Big{(}{1\over 2\pi
i}{\partial\over\partial\zeta}\theta_{m,r}(\tau,\zeta)\Big{)}\Big{\lvert}_{\zeta=0}.$
(A.3)

##  Appendix B More on Ghosts

In this appendix we discuss the ground states of the ghost systems in both the
P and A sectors as well as the action of the ${\mathbb{Z}}_{2}$ orbifold on
them, as a complement to §3.2.

### The ghost ground states

The first thing to note is that for the ghost systems the ordering
prescription generally changes when we go from the complex plane, where we use
canonical ordering, to the cylinder, where it is natural to use normal
ordering. By expressing the Virasoro zero mode in terms of the normal
ordering, a constant $B$ will appear in the following way (see [42]),

$\displaystyle
L^{\text{gh}}_{0}=\sum_{n}(-n)\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern
0.86108pt$\cr}}}\mathrel{\mathbf{b}_{n}\mathbf{c}_{-n}}\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern
0.86108pt$\cr}}}=\sum_{n}(-n)\mathrel{\mathop{:}}\mathrel{\mathbf{b}_{n}\mathbf{c}_{-n}}\mathrel{\mathop{:}}+B.$
(B.5)

We define a ground state on the cylinder as a state that is annihilated by the
normal ordered term above, so it will still have weight $B$. Thus, this state
will not generally be the $SL(2,{\mathbb{R}})$-invariant vacuum $|0\rangle$.
In order to treat both cases together, denote the ground state(s) in the A and
P sectors of either ghost system by $|\Omega_{\text{A}}\rangle$ and
$|\Omega_{\text{P}}\rangle$ respectively. The constant $B$ depends on the
central charge and the sector as

$\displaystyle\begin{split}B_{\text{A}}=-\frac{\kappa}{8}Q^{2}~{},~{}~{}B_{\text{P}}=\frac{\kappa}{8}(1-Q^{2}).\end{split}$
(B.6)

We can also compute the eigenvalues of the ground states under the Virasoro
zero mode on the cylinder
$L^{\text{gh}}_{\text{cyl},0}=L^{\text{gh}}_{0}-c_{\mathbf{b}\mathbf{c}}/24$.
We notice that the $Q$-dependence cancels and we get

$\displaystyle\begin{split}L^{\text{gh}}_{\text{cyl},0}|\Omega_{\text{A}}\rangle=-\frac{\kappa}{24}|\Omega_{\text{A}}\rangle,~{}~{}~{}L^{\text{gh}}_{\text{cyl},0}|\Omega_{\text{P}}\rangle=\frac{\kappa}{12}|\Omega_{\text{P}}\rangle,\end{split}$
(B.7)

for any value of $c_{\mathbf{b}\mathbf{c}}$. Note that all the characters we
use in this paper are defined in terms of the canonically ordered operators,
rather than the normal-ordered ones.

As we have mentioned in §3.2, the ghost systems possess a $U(1)$ current
$J=-\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern
0.86108pt$\cr}}}\mathrel{\mathbf{b}\mathbf{c}}\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr}}}$. Note that $J$ is
not a primary field, and the failure to be so is captured by the quantity $Q$:

$\displaystyle
L^{\text{gh}}(z)J(w)\sim\frac{Q}{(z-w)^{3}}+\frac{J(w)}{(z-w)^{2}}+\frac{\partial
J(w)}{z-w}.$ (B.8)

Accordingly, upon going to the cylinder the charge operator is shifted as
$J_{\text{cyl},0}=J_{0}+\frac{Q}{2}$. This $U(1)$ current can be used to
define an infinite family of primary operators [32], which will eventually be
related to the cylinder ground states. First we introduce a new (bosonic)
field $\phi$ of zero weight so that $J(z)=\kappa\partial\phi(z)$, which
results in the OPE $\phi(z)\phi(w)\sim\kappa\ln(z-w)$. We then define a vertex
operator by $V_{q}(z):=\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern
0.86108pt$\cr}}}\mathrel{e^{q\phi(z)}}\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr}}}$, which is primary
and obeys the OPEs

$\displaystyle L^{\text{gh}}(z)V_{q}(w)\sim\left[\frac{\frac{1}{2}\kappa
q(q+Q)}{(z-w)^{2}}+\frac{\partial}{z-w}\right]V_{q}(w),~{}~{}~{}J(z)V_{q}(w)\sim\frac{q}{z-w}~{}V_{q}(w).$
(B.9)

Acting on $|0\rangle$, this operator defines the state

$\displaystyle|q\rangle:=V_{q}(0)|0\rangle,$ (B.10)

which has the following weight and $U(1)$ charge

$\displaystyle\begin{split}L^{\text{gh}}_{0}|q\rangle=\frac{1}{2}\kappa
q(q+Q)|q\rangle~{},~{}~{}J_{0}|q\rangle=q|q\rangle.\end{split}$ (B.11)

These states can be regarded as vacuum states (for the Fock space) on the
plane. Note that each such $q$-vacuum is annihilated by a different set of
modes of the ghost fields, depending on the eigenvalue $q$. In particular, we
have

$\displaystyle\begin{split}&\mathbf{b}_{r}|q\rangle=0~{}~{}\forall~{}r\geq\kappa
q+1-h,\\\ &\mathbf{c}_{r}|q\rangle=0~{}~{}\forall~{}r\geq-\kappa
q+h.\end{split}$ (B.12)

These vacua belong to the periodic sector on the plane if $q\in{\mathbb{Z}}$,
and to the anti-periodic sector if
$q\in\left({\mathbb{Z}}+\frac{1}{2}\right)$. One way to see this is from
(B.12), since $r\pm h$ is should always be an integer in the periodic sector
on the plane, and half-integer in the antiperiodic sector. Also note that the
vertex operators $V_{\pm 1/2}$ interpolate between the above two sectors, and
hence can be regarded as twist fields.

The space of states on the cylinder is built by acting with the creation
operators on the ground states $|\Omega_{\text{S}}\rangle$, where
$\text{S}=\\{\text{P},\text{A}\\}$ denotes the two sectors. These are equal to
some of the $q$-vacua described above. By equating the weight $B_{\text{S}}$
and the weight of the $q$-vacua (B.11), we find that they have the following
eigenvalues under $J_{0}$:

$\displaystyle
q_{\Omega_{\text{A}}}=-\frac{1}{2}Q,~{}~{}~{}q_{\Omega_{\text{P}}}=-\frac{1}{2}(Q\mp\kappa).$
(B.13)

Accounting for the shift $J_{\text{cyl},0}=J_{0}+\frac{Q}{2}$, the
corresponding $\text{U}(1)$ charges on the cylinder are

$\displaystyle\begin{split}&J_{\text{cyl},0}|\Omega_{\text{A}}\rangle=0,\\\
&J_{\text{cyl},0}|\Omega_{\text{P}}\rangle=\pm\frac{\kappa}{2}|\Omega_{\text{P}}\rangle.\end{split}$
(B.14)

We see that in the A sector we have a single ground state, denoted
$|\Omega_{\text{A}}^{\text{F}}\rangle$ and
$|\Omega_{\text{A}}^{\text{B}}\rangle$ for the fermionic and bosonic system
respectively. In the P sector of the fermionic ghosts the zero modes
$b_{0},c_{0}$ form a Clifford algebra, which results in two degenerate ground
states with opposite $U(1)$ charges, which we denote by
$|\Omega_{\text{P},\pm}^{\text{F}}\rangle$. They obey

$\displaystyle\begin{split}b_{0}|\Omega_{\text{P},-}^{\text{F}}\rangle=0,~{}~{}~{}b_{0}|\Omega_{\text{P},+}^{\text{F}}\rangle=|\Omega_{\text{P},-}^{\text{F}}\rangle,~{}~{}~{}c_{0}|\Omega_{\text{P},+}^{\text{F}}\rangle=0,~{}~{}~{}c_{0}|\Omega_{\text{P},-}^{\text{F}}\rangle=|\Omega_{\text{P},+}^{\text{F}}\rangle.\\\
\end{split}$ (B.15)

In the bosonic case, we have to single out one of the two possible ground
states in the P sector since they do not belong in the same representation of
the $\beta,\gamma$ algebra. In other words, the zero modes
$\beta_{0},\gamma_{0}$ do not form an analogue of the Clifford algebra of
their fermionic counterparts, so one of them must be an annihilator. We choose
to use the ground state with positive charge for the torus characters that
will follow, which corresponds to $\gamma_{0}$ being a creation and
$\beta_{0}$ being an annihilation operator.

### The ghost orbifold

We now treat the ${\mathbb{Z}}_{2}$ orbifold for both fermionic and bosonic
ghosts, generated by $\xi\mathbf{b}=-\mathbf{b}$ and
$\xi\mathbf{c}=-\mathbf{c}$. In terms of the field $\phi$ introduced earlier,
the ghost fields are expressed as

$\displaystyle\begin{split}&b(z)=\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern
0.86108pt$\cr}}}\mathrel{e^{-\phi(z)}}\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern
0.86108pt$\cr}}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}c(z)=\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern
0.86108pt$\cr}}}\mathrel{e^{\phi(z)}}\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr}}},\\\
&\beta(z)=\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern
0.86108pt$\cr}}}\mathrel{e^{-{\phi(z)}}}\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern
0.86108pt$\cr}}}\partial\lambda(z),~{}~{}~{}\gamma(z)=\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern
0.86108pt$\cr}}}\mathrel{e^{\phi(z)}}\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr}}}\eta(z).\end{split}$
(B.16)

where we introduced two auxiliary fields $\eta$, $\lambda$. These form a free
fermionic ghost system by themselves, with $h_{\eta\lambda}=1$ and central
charge $c_{\eta\lambda}=-2$. They need to be introduced because the Virasoro
operator that is build out of $J$,

$\displaystyle
T_{J}=-\left(\frac{1}{2}\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern
0.86108pt$\cr}}}\mathrel{JJ}\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr}}}-\frac{1}{2}Q\partial
J\right),$ (B.21)

is not enough to describe the bosonic ghosts [32]. In particular, there is a
“residual” Virasoro operator $T_{-2}$ that needs to be added, so that the
total ghost Virasoro operator is given by $L=T_{J}+T_{-2}$, where $T_{-2}$ is
precisely the Virasoro operator of the fermionic ghosts $\eta$, $\lambda$.

One can implement the action of $\xi$ on the “bosonized” form of the ghost
fields (B.16) in both cases by setting

$\displaystyle\xi\phi=\phi+(2k+1)\pi
i,~{}~{}k\in{\mathbb{Z}},~{}~{}~{}~{}~{}~{}~{}~{}\xi\eta=\eta,~{}~{}~{}~{}~{}\xi\lambda=\lambda,$
(B.22)

Consequently, for the vacuum state $|q\rangle$ with $q\in{\mathbb{Z}}$ we get

$\displaystyle\xi|q\rangle=\xi\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern
0.86108pt$\cr}}}\mathrel{e^{q\phi(0)}}\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern
0.86108pt$\cr}}}|0\rangle=\left\\{\begin{array}[]{cc}+|q\rangle,&q~{}\text{even}\\\
-|q\rangle,&q~{}\text{odd}\end{array}\right.,$ (B.29)

while for $q\in{\mathbb{Z}}+\frac{1}{2}$ we have

$\displaystyle\xi|q\rangle=\xi\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern
0.86108pt$\cr}}}\mathrel{e^{q\phi(0)}}\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern
0.86108pt$\cr}}}|0\rangle=\left\\{\begin{array}[]{cc}+i(-1)^{k}|q\rangle,&(q-1/2)~{}~{}\text{even}\\\
-i(-1)^{k}|q\rangle,&(q-1/2)~{}~{}\text{odd}\end{array}\right..$ (B.36)

For convenience, we will choose $k=0$ without loss of generality.

As we have seen, the ground states on the cylinder correspond to certain
states $|q\rangle$ on the plane, with charge $q$ under $J_{0}$. Specifically,
for the ground states $|\Omega_{\text{A}}^{\text{F}}\rangle$ and
$|\Omega_{\text{P},\pm}^{\text{F}}\rangle$ in the corresponding sectors of the
fermionic system, we have already seen that

$\displaystyle|\Omega^{\text{F}}_{\text{A}}\rangle=\left|-\frac{Q}{2}\right\rangle,~{}~{}~{}|\Omega_{\text{P},\pm}^{\text{F}}\rangle=\left|-\frac{1}{2}(Q\mp
1)\right\rangle.$ (B.37)

From (B.29) and (B.36) we have that

$\displaystyle\xi|\Omega^{\text{F}}_{\text{A}}\rangle=e^{\pi
i\left(h-\frac{1}{2}\right)}|\Omega^{\text{F}}_{\text{A}}\rangle,~{}~{}~{}~{}~{}\xi|\Omega^{\text{F}}_{\text{P},\pm}\rangle=\pm
e^{\pi ih}|\Omega^{\text{F}}_{\text{P},\pm}\rangle.$ (B.38)

Combining the $\xi$ action on the ground states and the oscillators, we get

$\displaystyle\begin{split}&\tilde{\chi}^{\text{A}}_{\text{F}}(\tau,\zeta):=\operatorname{{tr}}_{\text{A}}\left[\xi~{}y^{J_{\text{cyl},0}}q^{L^{\text{gh}}_{\text{cyl},0}}\right]\\\
&~{}~{}~{}~{}~{}~{}~{}=e^{\pi
i\left(h-\frac{1}{2}\right)}q^{-\frac{1}{24}}\prod_{n=1}^{\infty}\left(1-yq^{n-1/2}\right)\left(1-y^{-1}q^{n-1/2}\right)=(-1)^{h-\frac{1}{2}}~{}\chi_{\text{F}}^{\text{A}}(\tau,\zeta+1/2),\end{split}$
(B.39)
$\displaystyle\begin{split}&\tilde{\chi}^{\text{P}}_{\text{F}}(\tau,\zeta):=\operatorname{{tr}}_{\text{P}}\left[\xi~{}y^{J_{\text{cyl},0}}q^{L^{\text{gh}}_{\text{cyl},0}}\right]\\\
&~{}~{}~{}~{}~{}~{}~{}=e^{\pi
ih}q^{\frac{1}{12}}\left(y^{1/2}-y^{-1/2}\right)\prod_{n=1}^{\infty}\left(1-yq^{n}\right)\left(1-y^{-1}q^{n}\right)=(-1)^{h-\frac{1}{2}}~{}\chi_{\text{F}}^{\text{P}}(\tau,\zeta+1/2),\end{split}$
(B.40)

Similarly, on the bosonic system ground states

$\displaystyle|\Omega^{\text{B}}_{\text{A}}\rangle=\left|-\frac{Q}{2}\right\rangle,~{}~{}~{}|\Omega^{\text{B}}_{\text{P}}\rangle=\left|-\frac{1}{2}(Q-1)\right\rangle,$
(B.41)

the orbifold acts as

$\displaystyle\xi|\Omega^{\text{B}}_{\text{A}}\rangle=e^{\pi
i\left(-h+\frac{1}{2}\right)}|\Omega^{\text{B}}_{\text{A}}\rangle,~{}~{}~{}~{}~{}\xi|\Omega^{\text{B}}_{\text{P}}\rangle=e^{\pi
i\left(-h+1\right)}|\Omega^{\text{B}}_{\text{P}}\rangle.$ (B.42)

Thus we calculate

$\displaystyle\begin{split}&\tilde{\chi}^{\text{A}}_{\text{B}}(\tau,\zeta):=\operatorname{{tr}}_{\text{A}}\left[\xi~{}y^{J_{\text{cyl},0}}q^{L^{\text{gh}}_{\text{cyl},0}}\right]\\\
&~{}~{}~{}=e^{\pi
i\left(-h+\frac{1}{2}\right)}q^{1/24}\prod_{n=1}^{\infty}\left(1+yq^{n-1/2}\right)^{-1}\left(1+y^{-1}q^{n-1/2}\right)^{-1}=(-1)^{3h+\frac{1}{2}}~{}\chi_{\text{B}}^{\text{A}}(\tau,\zeta+1/2),\end{split}$
(B.43)
$\displaystyle\begin{split}&\tilde{\chi}^{\text{P}}_{\text{B}}(\tau,\zeta):=\operatorname{{tr}}_{\text{P}}\left[\xi~{}y^{J_{\text{cyl},0}}q^{L^{\text{gh}}_{\text{cyl},0}}\right]\\\
&~{}~{}~{}=e^{\pi
i\left(-h+1\right)}q^{-1/12}y^{1/2}(1+y)^{-1}\prod_{n=1}^{\infty}\left(1+yq^{n}\right)^{-1}\left(1+y^{-1}q^{n}\right)^{-1}=(-1)^{3h+\frac{1}{2}}~{}\chi_{\text{B}}^{\text{P}}(\tau,\zeta+1/2).\end{split}$
(B.44)

Also note that all traces over the ghost Hilbert space involve defining a dual
Fock space: corresponding to each in-state
$|x\rangle=\prod_{i}\mathbf{c}_{-r_{i}}\prod_{j}\mathbf{b}_{-s_{j}}|q\rangle$
there us an out-state $\langle y|=\langle
q^{\prime}|\left(\prod_{i}\mathbf{c}_{-r_{i}}\prod_{j}\mathbf{b}_{-s_{j}}\right)^{t}$
such that their inner product is $\langle y|x\rangle=1$. Due to the charge
asymmetry [32], namely $J_{0}^{t}=-J_{0}-Q$, the dual to the vertex state
$|q\rangle$ is $\langle-q-Q|:=\langle
0|\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern
0.86108pt$\cr}}}\mathrel{e^{(-q-Q)\phi(z)}}\mathrel{\vbox{\offinterlineskip\halign{#\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr\kern 1.5pt\cr$\kern
3.01385pt\vbox{\hbox{\circle{1.5}}}\kern 0.86108pt$\cr}}}$, while the duals of
the oscillator modes are $\mathbf{c}_{-r}^{t}=\mathbf{b}_{r}$ and
$\mathbf{b}_{-r}^{t}=\mathbf{c}_{r}$. The latter is compatible with
transposing the (anti-) commutation relations
$\\{\mathbf{b}_{r},\mathbf{c}_{-r}\\}_{\kappa}=\kappa$, and we have have
$L_{0}^{t}=L_{0}$.

##  Appendix C Character Tables

Table 1: Character table of ${G}^{{X}}\simeq 3.{\textsl{Sym}}_{6}$, ${X}=D_{4}^{6}$ $[g]$ | FS | 1A | 3A | 2A | 6A | 3B | 3C | 4A | 12A | 5A | 15A | 15B | 2B | 2C | 4B | 6B | 6C
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$[g^{2}]$ |  | 1A | 3A | 1A | 3A | 3B | 3C | 2A | 6A | 5A | 15A | 15B | 1A | 1A | 2A | 3B | 3C
$[g^{3}]$ |  | 1A | 1A | 2A | 2A | 1A | 1A | 4A | 4A | 5A | 5A | 5A | 2B | 2C | 4B | 2B | 2C
$[g^{5}]$ |  | 1A | 3A | 2A | 6A | 3B | 3C | 4A | 12A | 1A | 3A | 3A | 2B | 2C | 4B | 6B | 6C
${\chi}_{1}$ | $+$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$
${\chi}_{2}$ | $+$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$
${\chi}_{3}$ | $+$ | $5$ | $5$ | $1$ | $1$ | $2$ | $-1$ | $-1$ | $-1$ | $0$ | $0$ | $0$ | $3$ | $-1$ | $1$ | $0$ | $-1$
${\chi}_{4}$ | $+$ | $5$ | $5$ | $1$ | $1$ | $2$ | $-1$ | $-1$ | $-1$ | $0$ | $0$ | $0$ | $-3$ | $1$ | $-1$ | $0$ | $1$
${\chi}_{5}$ | $+$ | $5$ | $5$ | $1$ | $1$ | $-1$ | $2$ | $-1$ | $-1$ | $0$ | $0$ | $0$ | $-1$ | $3$ | $1$ | $-1$ | $0$
${\chi}_{6}$ | $+$ | $5$ | $5$ | $1$ | $1$ | $-1$ | $2$ | $-1$ | $-1$ | $0$ | $0$ | $0$ | $1$ | $-3$ | $-1$ | $1$ | $0$
${\chi}_{7}$ | $+$ | $16$ | $16$ | $0$ | $0$ | $-2$ | $-2$ | $0$ | $0$ | $1$ | $1$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$
${\chi}_{8}$ | $+$ | $9$ | $9$ | $1$ | $1$ | $0$ | $0$ | $1$ | $1$ | $-1$ | $-1$ | $-1$ | $3$ | $3$ | $-1$ | $0$ | $0$
${\chi}_{9}$ | $+$ | $9$ | $9$ | $1$ | $1$ | $0$ | $0$ | $1$ | $1$ | $-1$ | $-1$ | $-1$ | $-3$ | $-3$ | $1$ | $0$ | $0$
${\chi}_{10}$ | $+$ | $10$ | $10$ | $-2$ | $-2$ | $1$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $2$ | $-2$ | $0$ | $-1$ | $1$
${\chi}_{11}$ | $+$ | $10$ | $10$ | $-2$ | $-2$ | $1$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $-2$ | $2$ | $0$ | $1$ | $-1$
${\chi}_{12}$ | $\circ$ | $6$ | $-3$ | $-2$ | $1$ | $0$ | $0$ | $2$ | $-1$ | $1$ | $b_{15}$ | $\overline{b_{15}}$ | $0$ | $0$ | $0$ | $0$ | $0$
${\chi}_{13}$ | $\circ$ | $6$ | $-3$ | $-2$ | $1$ | $0$ | $0$ | $2$ | $-1$ | $1$ | $\overline{b_{15}}$ | $b_{15}$ | $0$ | $0$ | $0$ | $0$ | $0$
${\chi}_{14}$ | $+$ | $12$ | $-6$ | $4$ | $-2$ | $0$ | $0$ | $0$ | $0$ | $2$ | $-1$ | $-1$ | $0$ | $0$ | $0$ | $0$ | $0$
${\chi}_{15}$ | $+$ | $18$ | $-9$ | $2$ | $-1$ | $0$ | $0$ | $2$ | $-1$ | $-2$ | $1$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$
${\chi}_{16}$ | $+$ | $30$ | $-15$ | $-2$ | $1$ | $0$ | $0$ | $-2$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
Table 2: Twisted Euler characters and Frame shapes at $\ell=6+3$, ${X}=D_{4}^{6}$ $[g]$ | 1A | 3A | 2A | 6A | 3B | 3C | 4A | 12A | 5A | 15AB | 2B | 2C | 4B | 6B | 6C
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$n_{g}|h_{g}$ | $1|1$ | $1|3$ | $2|1$ | $2|3$ | $3|1$ | $3|3$ | $4|2$ | $4|6$ | $5|1$ | $5|3$ | $2|1$ | $2|2$ | $4|1$ | $6|1$ | $6|6$
$\bar{\chi}_{g}$ | $6$ | $6$ | $2$ | $2$ | $3$ | $0$ | $0$ | $0$ | $1$ | $1$ | $4$ | $0$ | $2$ | $1$ | $0$
${\chi}_{g}$ | $6$ | $6$ | $2$ | $2$ | $3$ | $0$ | $0$ | $0$ | $1$ | $1$ | $-4$ | $0$ | $-2$ | $-1$ | $0$
$\check{\chi}_{g}$ | $12$ | $-6$ | $4$ | $-2$ | $0$ | $0$ | $0$ | $0$ | $2$ | $-1$ | $0$ | $0$ | $0$ | $0$ | $0$
$\bar{\Pi}_{g}$ | $1^{6}$ | $1^{6}$ | $1^{2}2^{2}$ | $1^{2}2^{2}$ | $1^{3}3^{1}$ | $3^{2}$ | $2^{1}4^{1}$ | $2^{1}4^{1}$ | $1^{1}5^{1}$ | $1^{1}5^{1}$ | $1^{4}2^{1}$ | $2^{3}$ | $1^{2}4^{1}$ | $1^{1}2^{1}3^{1}$ | $6^{1}$
$\widetilde{\Pi}_{g}$ | $1^{24}$ | $1^{6}3^{6}$ | $1^{8}2^{8}$ | $1^{2}2^{2}3^{2}6^{2}$ | $1^{6}3^{6}$ | $3^{8}$ | $2^{4}4^{4}$ | $2^{1}4^{1}6^{1}12^{1}$ | $1^{4}5^{4}$ | $1^{1}3^{1}5^{1}15^{1}$ | $1^{2}2^{8}$ | $2^{12}$ | $1^{4}2^{2}4^{4}$ | $1^{2}2^{2}3^{2}6^{2}$ | $6^{4}$

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