A Self-Dual Integral Form of the Moonshine Module

A Self-Dual Integral Form of the Moonshine Module††This paper is a
contribution to the Special Issue on Moonshine and String Theory. The full
collection is available at https://www.emis.de/journals/SIGMA/moonshine.html

Scott CARNAHAN S. Carnahan University of Tsukuba, Japan
<EMAIL_ADDRESS>http://www.math.tsukuba.ac.jp/~carnahan/

Received February 13, 2018, in final form April 06, 2019; Published online
April 19, 2019

We construct a self-dual integral form of the moonshine vertex operator
algebra, and show that it has symmetries given by the Fischer–Griess monster
simple group. The existence of this form resolves the last remaining open
assumption in the proof of the modular moonshine conjecture by Borcherds and
Ryba. As a corollary, we find that Griess’s original 196884-dimensional
representation of the monster admits a positive-definite self-dual integral
form with monster symmetry.

moonshine; vertex operator algebra; orbifold; integral form

17B69; 11F22; 20C10; 20C20; 20C34

###### Contents

  1. 1 Introduction
  2. 2 Cyclic orbifolds over subrings of the complex numbers
    1. 2.1 Descent and gluing for finite projective modules and bilinear forms
    2. 2.2 Vertex operator algebras over commutative rings
    3. 2.3 Abelian intertwining algebras over subrings of complex numbers
    4. 2.4 The standard form for a lattice vertex operator algebra
    5. 2.5 The cyclic orbifold construction
  3. 3 Forms of the monster vertex operator algebra
    1. 3.1 The Abe–Lam–Yamada method
    2. 3.2 Monster symmetry
    3. 3.3 Comparison of monster-symmetric forms
    4. 3.4 Descent of self-dual forms
  4. References

##  1 Introduction

In this paper, we construct self-dual $R$-forms of the moonshine module vertex
operator algebra [27] over various commutative rings $R$, culminating in the
universal case where $R$ is the ring of rational integers $\mathbb{Z}$. For
the $\mathbb{Z}$-form, we show that the vertex operator algebra has monster
symmetry, and is self-dual with respect to an invariant bilinear form that
respects the monster symmetry. Base change then gives us self-dual monster-
symmetric vertex operator algebras over any commutative ring. Our construction
yields the final step in the affirmative resolution of Ryba’s modular
moonshine conjecture.

Monstrous moonshine began in the 1970s with McKay’s observation that the
number 196884, namely the $q^{1}$ coefficient in Klein’s modular
$j$-invariant, is one more than the dimension of the smallest faithful complex
representation of the monster simple group $\mathbb{M}$. Similar observations
by Thompson for higher-order coefficients [44] led to what is now called the
McKay–Thompson conjecture, asserting the existence of a natural graded
faithful complex representation $V=\bigoplus_{n\geq 0}V_{n}$ of the monster
with graded dimension given by

$\displaystyle\sum_{n\geq 0}\dim V_{n}q^{n-1}=J(\tau)=j(\tau)-744.$

The McKay–Thompson conjecture was resolved by the construction of the
moonshine module $V^{\natural}$ in [26]. This construction was then refined in
[27] so that $V^{\natural}$ is not just a graded vector space, but a vertex
operator algebra whose automorphism group is precisely the monster. The vertex
operator algebra structure on $V^{\natural}$ turned out to be essential to
Borcherds’s resolution of the monstrous moonshine conjecture [3] for
$V^{\natural}$. This conjecture [13] is a substantial refinement of the
McKay–Thompson conjecture, given by replacing graded dimensions with graded
traces of elements of $\mathbb{M}$, and asserts that the resulting functions
are Hauptmoduln for certain genus zero groups commensurable to ${\rm
SL}_{2}(\mathbb{Z})$.

It is natural to ask whether $V^{\natural}$ can be defined over smaller
subrings of $\mathbb{C}$, and the first progress on this question was given at
the end of [27], where a monster-symmetric self-dual $\mathbb{Q}$-form is
constructed. In [6], the authors show that the $\mathbb{Q}$-form can be
defined over $\mathbb{Z}[1/2]$, but find that inversion of 2 is essential to
the construction. Finally, an integral form of $V^{\natural}$ with monster
symmetry was constructed in [18], but part of the construction involves
passing to a monster-invariant sublattice to obtain the full symmetry, and
this process generally destroys any control over self-duality. In particular,
the inner product may be quite degenerate after reduction modulo a prime. A
similar question concerns forms of the 196884-dimensional algebra $V_{2}$,
which is essentially what Griess used in his initial construction of the
monster [29]. Conway sketches a construction due to Norton of a monster-
symmetric integral form of the 196884-dimensional subalgebra $V_{2}$ in [11],
and expresses hope that it may be self-dual. However, the question of
extending such a form to the rest of $V^{\natural}$ has not been addressed.
Indeed, remarks in [6] suggest that this form would be incompatible with the
form that we construct in this paper. We show in this paper that both
questions have the best possible answers: there is a positive definite self-
dual integral form of $V^{\natural}$ with monster symmetry, and its weight 2
subspace is a positive definite self-dual integral form of Griess’s algebra
(up to a minor change in the multiplication on the Virasoro vector, as is
pointed out in [27]), with monster symmetry.

The main motivation for this paper comes from the modular moonshine
conjecture, which arose out of experimental observations by Ryba in [42] that
suggested the existence of a mod $p$ version of moonshine. Modular moonshine
asserted the existence of certain vertex algebras over $\mathbb{F}_{p}$
attached to elements $g$ of $\mathbb{M}$ in class $p$A such that Brauer traces
of $p$-regular centralizing elements $h$ are given by certain McKay–Thompson
series, i.e., graded traces on $V^{\natural}$. This conjecture, which was
somewhat strengthened in [6], was essentially resolved in [6] and [4], up to
some assumptions about the existence of integral versions of various
constructions that were known over $\mathbb{R}$ or $\mathbb{C}$. Some of these
assumptions, such as a description of root spaces of a $\mathbb{Z}_{p}$-form
of the monster Lie algebra, were resolved in [5]. The remaining problem was
that the conjecture asserts the existence of a self-dual $\mathbb{Z}$-form of
$V^{\natural}$ with $\mathbb{M}$-symmetry. Since such a form was not
immediately available, the authors of [6] noted that for the original
conjecture, one only needs for each prime $p$ dividing the order of
$\mathbb{M}$, a self-dual form over the ring $\mathbb{Z}_{p}$ of $p$-adic
integers. If a self-dual $\mathbb{Z}_{p}$-form (or more generally, a form over
any $p$-adic integer ring) with monster symmetry exists, the arguments given
in [6] and [4] (combined with some technical details resolved in [5]) yield a
resolution of the conjecture. For odd primes $p$, it therefore suffices to
produce a self-dual $\mathbb{Z}[1/2]$-form, and in [6], the authors showed
that the construction in [27] in fact yields such a form. For the case $p=2$,
the problem has remained unambiguously open until now.

In this paper, we give a construction of a self-dual integral form of
$V^{\natural}$ with monster symmetry, which resolves the last open assumption
in modular moonshine. The construction requires several very recent
developments in the theory of vertex operator algebras. The most notable of
these is the cyclic orbifold construction given in [45]. However, despite the
dependence on recent technology, there is an effortless quality to our
construction, like we are getting this object for free. In particular, the
argument requires a surprisingly small amount of group theoretic input,
essentially all compressed into some information about maximal subgroups of
$\mathbb{M}$ in the proof of Lemma 3.7. Aside from this input, the main
ingredients are:

  1. 1.

The existence and uniqueness of self-dual $R$-forms of abelian intertwining
algebras, from generating subalgebras. This is basically given by extending
some well-known skew-symmetry and associativity properties of vertex algebras
to the abelian intertwining algebra setting.

  2. 2.

A very recent method for analyzing $V^{\natural}$ by applying multiple cyclic
orbifolds to the Leech lattice vertex operator algebra, due to Abe, Lam, and
Yamada. Together with the previous tool, this lets us extract self-dual forms
of $V^{\natural}$ over cyclotomic S-integer rings.

  3. 3.

Faithfully flat descent. This technique is more than a half-century old and
has broad applications, but does not seem to get much attention in the vertex
operator algebra world, where the base ring is almost always $\mathbb{C}$.

While modular moonshine is now resolved, there are still many related open
questions remaining. For example, most of the questions listed at the end of
[6] remain unresolved, and the question of what happens in the case $g$ has
composite order seems particularly natural.

##  2 Cyclic orbifolds over subrings of the complex numbers

The main goal of this section is to show that the cyclic orbifold
construction, as worked out in [45], is definable over distinguished subrings
of $\mathbb{C}$. For this purpose, we introduce some commutative algebra
technology.

For any complex number $r$, we write $e(r)$ to denote the normalized
exponential ${\rm e}^{2\pi{\rm i}r}$. We will also use the following notation
conventions, some of which may be unfamiliar to non-specialists in group
schemes (see, e.g., [15]):

  1. 1.

For any commutative ring $R$ and any $n\in\mathbb{Z}_{\geq 1}$, we write
$\mu_{n}(R)$ to denote the group $\\{r\in R\,|\,r^{n}=1\\}$. The functor
$\mu_{n}$ is represented by the affine scheme
$\operatorname{Spec}\mathbb{Z}[x]/\big{(}x^{n}-1\big{)}$.

  2. 2.

For any commutative ring $R$, we write $\mathbb{G}_{m}(R)$ to denote the group
$R^{\times}$ of invertible elements. The functor $\mathbb{G}_{m}$ is
represented by the affine scheme $\operatorname{Spec}\mathbb{Z}[x,y]/(xy-1)$,
which is also written as the spectrum of the ring
$\mathbb{Z}\big{[}x,x^{-1}\big{]}$ of Laurent polynomials.

  3. 3.

Given commutative group schemes $G$ and $H$, we write
$\operatorname{\underline{Hom}}(G,H)$ for the functor that takes a ring $R$ to
the group of natural transformations $G\times\operatorname{Spec}R\to
H\times\operatorname{Spec}R$ over $\operatorname{Spec}R$. In this paper, we
will only consider the case $G$ is a torus or a lattice of rank $n$, and $H$
is either $\mu_{n}$ or $\mathbb{G}_{m}$. In particular,
$D(\mathbb{Z}^{n})=\operatorname{\underline{Hom}}(\mathbb{Z}^{n},\mathbb{G}_{m})\cong\mathbb{G}_{m}^{n}$,
and
$D(\mathbb{G}_{m}^{n})=\operatorname{\underline{Hom}}(\mathbb{G}_{m}^{n},\mathbb{G}_{m})\cong\mathbb{Z}^{n}$.

###  2.1 Descent and gluing for finite projective modules and bilinear forms

######  Definition 2.1.

Let $R$ be a commutative ring. We say that an $R$-module $M$ is faithfully
flat if for any 3-term complex $M_{1}\to M_{2}\to M_{3}$ of $R$-modules, the
following conditions are equivalent:

  1. 1)

$M_{1}\to M_{2}\to M_{3}$ is exact,

  2. 2)

$M\otimes_{R}M_{1}\to M\otimes_{R}M_{2}\to M\otimes_{R}M_{3}$ is exact.

Given a homomorphism $f\colon R\to S$ of commutative rings, we say $f$ is
faithfully flat if $S$ is faithfully flat as an $R$-module under the induced
action.

See [43, Section 00H9] for a brief overview of basic properties of flat and
faithfully flat modules and ring maps.

######  Definition 2.2.

Let $f\colon R\to S$ be a homomorphism of commutative rings. A descent datum
for modules with respect to $f$ is a pair $(M,\phi)$, where $M$ is an
$S$-module, and $\phi\colon M\otimes_{R}S\to S\otimes_{R}M$ is an
$S\otimes_{R}S$-module isomorphism satisfying the “cocycle condition”, i.e.,
that the following diagram commutes:

$\textstyle{M\otimes_{R}S\otimes_{R}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{0,1}=\phi\otimes
1}$$\scriptstyle{\phi_{0,2}}$$\textstyle{S\otimes_{R}M\otimes_{R}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{1,2}=1\otimes\phi}$$\textstyle{S\otimes_{R}S\otimes_{R}M,}$

where, if we write $\phi(m\otimes 1)=\sum s_{i}\otimes m_{i}$, then the
$S\otimes_{R}S\otimes_{R}S$-module homomorphisms $\phi_{i,j}$ are defined by

$\displaystyle\phi_{0,1}(m\otimes 1\otimes 1)=\sum s_{i}\otimes m_{i}\otimes
1,$ $\displaystyle\phi_{0,2}(m\otimes 1\otimes 1)=\sum s_{i}\otimes 1\otimes
m_{i},$ $\displaystyle\phi_{1,2}(1\otimes m\otimes 1)=\sum 1\otimes
s_{i}\otimes m_{i}.$

A morphism of descent data from $(M,\phi)$ to $(M^{\prime},\phi^{\prime})$ is
an $S$-module homomorphism $\psi\colon M\to M^{\prime}$ such that the
following diagram commutes:

$\textstyle{M\otimes_{R}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{\psi\otimes
1}$$\textstyle{S\otimes_{R}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\otimes\psi}$$\textstyle{M^{\prime}\otimes_{R}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{\prime}}$$\textstyle{S\otimes_{R}M^{\prime}.}$

######  Theorem 2.3.

Let $f\colon R\to S$ be a homomorphism of commutative rings. If $f$ is
faithfully flat, then the category of $R$-modules is equivalent to the
category of descent data for modules with respect to $f$. In particular, the
functor taking an $R$-module $M$ to the $S$-module $M\otimes_{R}S$ equipped
with descent datum $\phi((m\otimes s)\otimes s^{\prime})=s\otimes(m\otimes
s^{\prime})$ has quasi-inverse given by $(M,\phi)\mapsto\tilde{M}=\\{m\in
M\,|\,1\otimes m=\phi(m\otimes 1)\\}$.

###### Proof.

See [43, Proposition 023N] or Theorem 4.21 of [46] for freely available
expositions, or Theorem 1 of [30] for the original reference. ∎

We note that faithful flatness can be replaced by the weaker condition of
“universally injective for modules”, namely that $f\otimes{\rm id}_{M}\colon
M=R\otimes_{R}M\to S\otimes_{R}M$ is injective for all $R$-modules $M$. The
fact that this condition is necessary and sufficient for effectiveness of
descent of modules was noted without proof in [41] and proved in [39].

######  Definition 2.4.

Given a homomorphism $f\colon R\to S$ of commutative rings, a bilinear map
with descent datum with respect to $f$ is a quadruple $(\kappa\colon M\times
N\to T,\phi_{M},\phi_{N},\phi_{T})$, where $M$, $N$, $T$ are $S$-modules,
$\kappa$ is an $S$-bilinear map, and $\phi_{M}\colon M\otimes_{R}S\to
S\otimes_{R}M$, $\phi_{N}\colon N\otimes_{R}S\to S\otimes_{R}N$,
$\phi_{T}\colon T\otimes_{R}S\to S\otimes_{R}T$ are descent data for modules
with respect to $f$, such that the following diagram of $S\otimes_{R}S$-module
homomorphisms commutes:

$\textstyle{(M\otimes_{R}S)\otimes_{S\otimes_{R}S}(N\otimes_{R}S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{M}\otimes\phi_{N}}$$\scriptstyle{\bar{\kappa}\otimes
1}$$\textstyle{(S\otimes_{R}M)\otimes_{S\otimes_{R}S}(S\otimes_{R}N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\otimes\bar{\kappa}}$$\textstyle{T\otimes_{R}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{T}}$$\textstyle{S\otimes
T.}$

Here, $\bar{\kappa}\colon M\otimes N\to T$ is the $S$-module map canonically
attached to $\kappa$.

######  Lemma 2.5.

Given a faithfully flat homomorphism $f\colon R\to S$ of commutative rings,
any bilinear map with descent datum with respect to $f$ is equivalent to the
base change of a unique bilinear map of $R$-modules. That is, given
$(\kappa\colon M\times N\to T,\phi_{M},\phi_{N},\phi_{T})$, there is a unique
$R$-linear map $\tilde{\kappa}\colon\tilde{M}\otimes\tilde{N}\to\tilde{T}$
such that the following diagram commutes:

$\textstyle{\big{(}\tilde{M}\otimes_{R}S\big{)}\otimes_{S}\big{(}\tilde{N}\otimes_{R}S\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\kappa}\otimes
1}$$\scriptstyle{\psi_{M}\otimes\psi_{N}}$$\textstyle{\tilde{T}\otimes_{R}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{T}}$$\textstyle{M\otimes
N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\kappa}}$$\textstyle{T.}$

Here, $\psi_{M}\colon\tilde{M}\otimes_{R}S\to M$,
$\psi_{N}\colon\tilde{N}\otimes_{R}S\to N$,
$\psi_{T}\colon\tilde{T}\otimes_{R}S\to T$ are the $S$-module isomorphisms
induced by the descent data.

###### Proof.

This amounts to applying the equivalence in Theorem 2.3 to the $S$-module map
$\bar{\kappa}$. ∎

######  Proposition 2.6.

Let $f\colon R\to S$ be a faithfully flat homomorphism of commutative rings,
and let $M$ be an $R$-module. Then $M$ is a finite projective $($equivalently,
flat and finitely presented$)$ $R$-module if and only if $M\otimes_{R}S$ is a
finite projective $S$-module.

###### Proof.

This is given by [31, Proposition 2.5.2]. ∎

######  Definition 2.7.

Given a commutative ring $R$, we define the groupoid of finite projective
$R$-modules with bilinear form by setting:

  1. 1.

Objects are given by pairs $(M,\kappa)$, where $M$ is a finite projective
$R$-module, and $\kappa\colon M\times_{R}M\to R$ is an $R$-bilinear map.

  2. 2.

Morphisms $(M,\kappa)\to(M^{\prime},\kappa^{\prime})$ are $R$-module
isomorphisms $\psi\colon M\to M^{\prime}$ such that

$\displaystyle\kappa^{\prime}(\psi(m_{1}),\psi(m_{2}))=\kappa(m_{1},m_{2})\qquad\text{for
all}\quad m_{1},m_{2}\in M.$

Given a homomorphism $f\colon R\to S$ of commutative rings, a descent datum
for finite projective modules with bilinear form with respect to $f$ is a
triple $(M,\kappa,\phi)$, where $(M,\phi)$ is a finite projective $S$-module
with descent datum with respect to $f$, and $(\kappa,\phi,\phi,{\rm id}_{S})$
is a bilinear map with descent datum with respect to $f$. An isomorphism of
descent data from $(M,\kappa,\phi)$ to
$(M^{\prime},\kappa^{\prime},\phi^{\prime})$ is an $S$-module isomorphism
$\psi\colon M\to M^{\prime}$ such that
$\kappa(m_{1},m_{2})=\kappa^{\prime}(\psi(m_{1}),\psi(m_{2}))$ for all
$m_{1},m_{2}\in M$, and the following diagram commutes:

$\textstyle{M\otimes_{R}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{\psi\otimes
1}$$\textstyle{S\otimes_{R}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\otimes\psi}$$\textstyle{M^{\prime}\otimes_{R}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{\prime}}$$\textstyle{S\otimes_{R}M^{\prime}.}$

A bilinear form $\kappa$ on a finite projective $R$-module is called non-
singular if it induces an $R$-module isomorphism
$M\to\operatorname{Hom}_{R}(M,R)$ under the canonical adjunction isomorphism
$\operatorname{Hom}_{R}(M\otimes_{R}M,R)\to\operatorname{Hom}_{R}(M,\operatorname{Hom}_{R}(M,R))$.

######  Theorem 2.8.

Let $f\colon R\to S$ be a homomorphism of commutative rings. If $f$ is
faithfully flat, then the groupoid of finite projective $R$-modules with
bilinear form is equivalent to the groupoid of descent data for finite
projective modules with bilinear form with respect to $f$. Furthermore, the
subgroupoid of non-singular forms is preserved under this equivalence.

###### Proof.

Given a descent datum $(M,\kappa,\phi)$ for finite projective modules with
bilinear form with respect to $f$, by Theorem 2.3 and Proposition 2.6, there
exists a finite projective $R$-module $\tilde{M}$ such that base change yields
the original descent datum for modules, and by Lemma 2.5, there is a unique
$R$-bilinear form $\tilde{\kappa}$ whose base-change to $S$ yields the descent
datum $(\kappa,\phi,\phi,{\rm id}_{S})$. By Theorem 2.3, isomorphisms $\psi$
of descent data are in natural bijection with isomorphisms $\tilde{\psi}$ of
$R$-modules that preserve $\tilde{\kappa}$.

We now consider the condition that a form is non-singular. Given an $R$-module
map $\bar{\kappa}\colon M\to\operatorname{Hom}_{R}(M,R)$ that corresponds to a
bilinear form $\kappa$, base change along $f$ yields an $S$-module map
$\bar{\kappa}\otimes 1\colon
M\otimes_{R}S\to\operatorname{Hom}_{R}(M,R)\otimes_{R}S$. Because $M$ is
finite projective, the canonical map
$\mathrm{can}\colon\operatorname{Hom}_{R}(M,R)\otimes_{R}S\to\operatorname{Hom}_{S}(M\otimes_{R}S,S)$
is an isomorphism. We then have a commutative diagram

$\textstyle{M\otimes_{R}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\kappa}\otimes
1}$$\scriptstyle{\overline{\kappa\otimes
1}}$$\textstyle{\operatorname{Hom}_{R}(M,R)\otimes_{R}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{can}}$$\textstyle{\operatorname{Hom}_{S}(M\otimes_{R}S,S),}$

where $\overline{\kappa\otimes 1}$ is the $S$-module homomorphism that
naturally corresponds to the $S$-valued bilinear form $\kappa\otimes 1$ on
$M\otimes_{R}S$. This form is non-singular if and only if $\bar{\kappa}\otimes
1$ is an isomorphism. By faithfully flat descent, $\bar{\kappa}\otimes 1$ is
an isomorphism if and only if $\bar{\kappa}$ is an isomorphism, i.e., if and
only if $\kappa$ is non-singular. ∎

We consider a method of gluing that is well-suited to the objects we obtain in
this paper.

######  Definition 2.9.

Suppose we are given a diagram $R_{1}\to R_{3}\leftarrow R_{2}$ of commutative
rings. A gluing datum for modules over this diagram is a triple
$(M_{1},M_{2},f)$, where

  1. 1)

$M_{1}$ is an $R_{1}$-module,

  2. 2)

$M_{2}$ is an $R_{2}$-module, and

  3. 3)

$f\colon
M_{1}\otimes_{R_{1}}R_{3}\overset{\sim}{\to}M_{2}\otimes_{R_{2}}R_{3}$ is an
$R_{3}$-module isomorphism.

If $P$ is a property of modules over a commutative ring, we say that a gluing
datum for modules satisfies property $P$ if the component modules satisfy
property $P$. A morphism of gluing data from $(M_{1},M_{2},f)$ to
$(M^{\prime}_{1},M^{\prime}_{2},f^{\prime})$ is a pair $(g_{1}\colon M_{1}\to
M^{\prime}_{1},\,g_{2}\colon M_{2}\to M^{\prime}_{2})$ of module maps such
that the following diagram commutes:

$\textstyle{M_{1}\otimes_{R_{1}}R_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g_{1}\otimes{\rm
id}}$$\textstyle{M_{2}\otimes_{R_{2}}R_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{2}\otimes{\rm
id}}$$\textstyle{M^{\prime}_{1}\otimes_{R_{1}}R_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{M^{\prime}_{2}\otimes_{R_{2}}R_{3}.}$

We first note that gluing of quasi-coherent sheaves is effective for Zariski
open covers.

######  Lemma 2.10.

Let $R$ be a commutative ring, and let $a,b\in R$ be coprime elements, i.e.,
the ideal generated by $a$ and $b$ is all of $R$. Let $R_{a}$ and $R_{b}$ be
the respective localizations, i.e., such that $\operatorname{Spec}R_{a}$ and
$\operatorname{Spec}R_{b}$ form a Zariski open cover of
$\operatorname{Spec}R$. Then base change induces an equivalence between the
category of $R$-modules and the category of gluing data for modules over the
diagram $R_{a}\to R_{ab}\leftarrow R_{b}$. This restricts to an equivalence
between the category of finite projective $R$-modules and the category of
finite projective gluing data, and a corresponding equivalence for groupoids
of finite projective modules with non-singular bilinear form.

###### Proof.

The first claim amounts to faithfully flat descent for the Zariski open cover
$R\to R_{a}\oplus R_{b}$. All of the components of a descent datum for this
cover are given in the gluing datum, because we have the identifications
$R_{a}\otimes_{R}R_{a}=R_{a}$ and $R_{b}\otimes_{R}R_{b}=R_{b}$. A detailed
proof can be found at [43, Lemma 00EQ]. The remaining claims follow from the
corresponding descent results in the previous section. ∎

We wish to consider situations where the rings in a gluing datum are not
necessarily Zariski localizations of $R$. In particular, we will consider the
case where the diagram of rings has the form $R_{1}\to
R_{1}\otimes_{R}R_{2}\leftarrow R_{2}$, where $i_{1}\colon R\to R_{1}$ and
$i_{2}\colon R\to R_{2}$ are faithfully flat.

######  Definition 2.11.

Suppose $i_{1}\colon R\to R_{1}$ and $i_{2}\colon R\to R_{2}$ are faithfully
flat. A gluing datum for the diagram $R_{1}\to R_{1}\otimes_{R}R_{2}\leftarrow
R_{2}$ is strict if there is some $R_{1}\otimes_{R}R_{2}$-module $M_{3}$ such
that $M_{1}$ and $M_{2}$ are contained in $M_{3}$ as an
$R_{1}\otimes_{R}R$-submodule and an $R\otimes_{R}R_{2}$-submodule, such that
extension of scalars gives the identifications $M_{1}\otimes_{R}R_{2}\to
M_{3}$ and $R_{1}\otimes_{R}M_{2}\to M_{3}$.

######  Lemma 2.12.

Suppose $i_{1}\colon R\to R_{1}$ and $i_{2}\colon R\to R_{2}$ are faithfully
flat. Then, any gluing datum $(M_{1},M_{2},f)$ for the diagram $R_{1}\to
R_{1}\otimes_{R}R_{2}\leftarrow R_{2}$ is isomorphic to a strict gluing datum
$\big{(}M_{1},f^{-1}(R\otimes_{R}M_{2}),{\rm
id}_{M_{1}\otimes_{R}R_{2}}\big{)}$. In particular, there is an equivalence
between the category of gluing data for the diagram $R_{1}\to
R_{1}\otimes_{R}R_{2}\leftarrow R_{2}$ and the full subcategory of strict
gluing data.

###### Proof.

Because $i_{1}$ is faithfully flat (in particular, universally injective for
modules), $R\otimes_{R}R_{2}$ is a subring of $R_{1}\otimes_{R}R_{2}$. Because
$f^{-1}$ is an $R_{1}\otimes_{R}R_{2}$-module isomorphism,
$f^{-1}(R\otimes_{R}M_{2})$ is an $R\otimes_{R}R_{2}$-submodule of
$M_{1}\otimes_{R}R_{2}$, and transporting the action of $R_{2}$ through the
canonical isomorphism $R_{2}\overset{\sim}{\to}R\otimes_{R}R_{2}$, we obtain
an $R_{2}$-module isomorphism $g_{2}\colon f^{-1}(R\otimes_{R}M_{2})\to
M_{2}$. It is straightforward to see that extending scalars along $i_{1}$
yields the map $f$, once we make the identification
$R_{1}\otimes_{R}f^{-1}(R\otimes_{R}M_{2})=M_{1}\otimes_{R}R_{2}$, by
$r\otimes f^{-1}(1\otimes x)=f^{-1}(r\otimes x)$. Thus, the pair $({\rm
id},g_{2})\colon\big{(}M_{1},f^{-1}(R\otimes_{R}M_{2}),{\rm
id}_{M_{1}\otimes_{R}R_{2}}\big{)}\to(M_{1},M_{2},f)$ induces a commutative
diagram:

$\textstyle{M_{1}\otimes_{R}R_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\scriptstyle{{\rm
id}}$$\textstyle{M_{1}\otimes_{R}R_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{M_{1}\otimes_{R}R_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{R_{1}\otimes_{R}M_{2},}$

and is therefore an isomorphism of gluing data.

We therefore have a full subcategory that spans all isomorphism classes of
objects, so it is equivalent to the category of gluing data. ∎

######  Lemma 2.13.

Let $i_{1}\colon R\to R_{1}$ and $i_{2}\colon R\to R_{2}$ be faithfully flat
homomorphisms of commutative rings. Then base change induces an equivalence
between the category of $R$-modules $($with morphisms given by arbitrary
$R$-module maps$)$ and the category of gluing data for modules over the
diagram $R_{1}\to R_{1}\otimes_{R}R_{2}\leftarrow R_{2}$. This restricts to an
equivalence between the category of finite projective $R$-modules and the
category of finite projective gluing data, and a corresponding equivalence for
groupoids of finite projective modules with non-singular bilinear form.

###### Proof.

By Lemma 2.12, it suffices to consider strict gluing data instead of all
gluing data. Let $F$ be the base change functor that takes $R$-modules to
strict gluing data. We first define a quasi-inverse functor $G$. Let
$(M_{1},M_{2},{\rm id})$ be a strict gluing datum. Thus, we assume $M_{1}$ is
an $R_{1}\otimes_{R}R$-submodule of some $R_{1}\otimes_{R}R_{2}$-module
$M_{3}$, and similarly for the $R\otimes_{R}R_{2}$-module $M_{2}$. From
$M_{2}$, we have a canonical descent datum on $M_{3}$, i.e., an
$(R_{1}\otimes_{R}R_{2})\otimes_{R_{2}}(R_{1}\otimes_{R}R_{2})$-module
isomorphism

$\displaystyle\phi_{2}\colon\
M_{3}\otimes_{R_{2}}(R_{1}\otimes_{R}R_{2})\to(R_{1}\otimes_{R}R_{2})\otimes_{R_{2}}M_{3},$

such that $M_{2}$ is the set of elements $x\in M_{3}$ satisfying
$\phi_{2}(x\otimes(1\otimes 1))=(1\otimes 1)\otimes x$. Similarly, there is a
descent datum

$\displaystyle\phi_{1}\colon\
M_{3}\otimes_{R_{1}}(R_{1}\otimes_{R}R_{2})\to(R_{1}\otimes_{R}R_{2})\otimes_{R_{1}}M_{3},$

such that $M_{1}$ is the set of elements $x\in M_{3}$ satisfying
$\phi_{1}(x\otimes(1\otimes 1))=(1\otimes 1)\otimes x$. We set
$G(M_{1},M_{2},{\rm id})$ to be the $R$-module that is the intersection of
$M_{1}$ and $M_{2}$ in $M_{3}$, or equivalently, the set of elements $x\in
M_{3}$ satisfying both $\phi_{1}(x\otimes(1\otimes 1))=(1\otimes 1)\otimes
x\in(R_{1}\otimes_{R}R_{2})\otimes_{R_{1}}M_{3}$ and
$\phi_{2}(x\otimes(1\otimes 1))=(1\otimes 1)\otimes
x\in(R_{1}\otimes_{R}R_{2})\otimes_{R_{2}}M_{3}$. Given a morphism of strict
gluing data, we obtain a homomorphism of $R$-modules on the intersections, so
$G$ is a functor.

We first check that the composite $G\circ F$ is isomorphic to identity on
$R$-modules. Because $i_{1}$ and $i_{2}$ are faithfully flat, they are
universally injective for modules, so any $R$-module $M$ injects into both
$M\otimes_{R}R_{1}$ and $M\otimes_{R}R_{2}$. The intersection in
$M\otimes_{R}R_{1}\otimes_{R}R_{2}$ is then $M\otimes_{R}R\otimes_{R}R\cong
M$, and we have our isomorphism.

We define a natural transformation from $F\circ G$ to identity by sending any
object $(M_{1},M_{2},{\rm id})$ to the pair $(g_{1},g_{2})$, where
$g_{1}\colon R_{1}\otimes_{R}G(M_{1},M_{2},{\rm id})\to M_{1}$ and
$g_{2}\colon R_{2}\otimes_{R}G(M_{1},M_{2},{\rm id})\to M_{2}$ are given by
restricting the action maps $R_{1}\otimes_{R}M_{1}\to M_{1}$ and
$R_{2}\otimes_{R}M_{2}\to M_{2}$. To show that $F$ is an equivalence, it
remains to show that $g_{1}$ and $g_{2}$ are isomorphisms of $R_{1}$-modules
and $R_{2}$-modules, respectively. The base change of $g_{1}$ along $i_{2}$ is
given by the restriction
$g_{3}\colon(R_{1}\otimes_{R}R_{2})\otimes_{R}G(M_{1},M_{2},{\rm id})\to
M_{3}$ of the action map on $M_{3}$, and since $i_{2}$ is faithfully flat,
this map is injective (resp. surjective) if and only if $g_{1}$ is also. The
base change of $g_{2}$ along $i_{1}$ is also given by the restricted action
map, so it suffices to show that at least one of these three maps ($g_{1}$,
$g_{2}$, and $g_{3}$) is injective and at least one is surjective.

To prove injectivity, we consider the following diagram:

$\textstyle{R_{1}\otimes_{R}G(M_{1},M_{2},{\rm
id})\otimes_{R}R_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
id}\otimes\iota\otimes{\rm id}}$$\scriptstyle{g_{1}\otimes{\rm
id}}$$\textstyle{R_{1}\otimes_{R}M_{1}\otimes_{R}R_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{f}$$\textstyle{(R_{1}\otimes_{R}R_{2})\otimes_{R_{2}}(M_{1}\otimes_{R}R_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{2}^{-1}}$$\textstyle{M_{1}\otimes_{R}R_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
id}\otimes i_{1}\otimes{\rm
id}}$$\textstyle{M_{1}\otimes_{R}R_{1}\otimes_{R}R_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{(M_{1}\otimes_{R}R_{2})\otimes_{R_{2}}(R_{1}\otimes_{R}R_{2}),}$

where $\iota\colon G(M_{1},M_{2},{\rm id})\to M_{1}$ is the inclusion, and $f$
is the unique $R_{1}\otimes_{R}R_{1}\otimes_{R}R_{2}$-module isomorphism that
makes the square on the right commute. The horizontal arrows on the left are
injective homomorphisms, so to prove injectivity of $g_{1}$, it suffices to
check commutativity of the left square. Along the lower left path, we have

$\displaystyle({\rm id}\otimes i_{1}\otimes{\rm id})\circ(g_{1}\otimes{\rm
id})(r_{1}\otimes m\otimes r_{2})=r_{1}m\otimes 1\otimes r_{2},$

which is then sent to $(r_{1}m\otimes 1)\otimes(1\otimes r_{2})$. Along the
upper right path, we note that for any $m\in G(M_{1},M_{2},{\rm id})$,
$f(1\otimes m\otimes 1)$ is identified with $\phi_{2}^{-1}((1\otimes
1)\otimes(m\otimes 1))=(m\otimes 1)\otimes(1\otimes 1)$, so is equal to
$m\otimes 1\otimes 1$. Thus, we have

$\displaystyle f\circ({\rm id}\otimes\iota\otimes{\rm id})(r_{1}\otimes
m\otimes r_{2})=f(r_{1}\otimes m\otimes r_{2})=f((r_{1}\otimes 1\otimes
r_{2})(1\otimes m\otimes 1))$ $\displaystyle\qquad{}=(r_{1}\otimes 1\otimes
r_{2})f(1\otimes m\otimes 1)=(r_{1}\otimes 1\otimes r_{2})(m\otimes 1\otimes
1)=r_{1}m\otimes 1\otimes r_{2}.$

This proves $g_{1}\otimes{\rm id}=g_{3}$ is injective.

We reduce the question of surjectivity to the case that $R$ is a local ring as
follows. By Claim 5 of [43, Lemma 00HN], a map $M\to M^{\prime}$ of
$R$-modules (for any commutative ring $R$) is surjective if and only if for
each prime ideal $P$ of $R$, the localized map $M_{P}\to M^{\prime}_{P}$ of
$R_{P}$-modules is surjective. Each $R_{P}$ is a local ring, so we shall
assume $R$ is a local ring for the remainder of this proof.

Let $x\in M_{1}$, so $x\otimes 1\in M_{1}\otimes_{R}R_{2}$, and write

$\displaystyle\phi_{2}((x\otimes 1)\otimes(1\otimes 1)=\sum_{i}(s_{i}\otimes
t_{i})\otimes(x_{i}\otimes
1)\in(R_{1}\otimes_{R}R_{2})\otimes_{R_{2}}(M_{1}\otimes_{R}R_{2}).$

Note that we may move multipliers in $R_{2}$ across the middle tensor product,
and this is why we may write $1$ in the rightmost factor. For each $x_{i}\in
M_{1}$, we write

$\displaystyle\phi_{2}((x_{i}\otimes 1)\otimes(1\otimes
1))=\sum_{j}(s_{ij}\otimes t_{ij})\otimes(y_{ij}\otimes 1).$

Then, because $\phi_{2}$ satisfies the cocycle condition, we have

$\displaystyle\sum_{i}(s_{i}\otimes t_{i})\otimes(1\otimes
1)\otimes(x_{i}\otimes 1)$ $\displaystyle=\sum_{i,j}(s_{i}\otimes
t_{i})\otimes(s_{ij}\otimes t_{ij})\otimes(y_{ij}\otimes 1)$
$\displaystyle=\sum_{i}(s_{i}\otimes t_{i})\otimes\phi_{2}((x_{i}\otimes
1)\otimes(1\otimes 1))$

as elements of
$(R_{1}\otimes_{R}R_{2})\otimes_{R_{2}}(R_{1}\otimes_{R}R_{2})\otimes_{R_{2}}(R_{1}\otimes_{R}R_{2})$.
We now use our assumption that $R$ is a local ring, and in particular, the
property that any finitely generated submodule of an $R$-flat module is
$R$-free. This property is not explicitly stated in Proposition 3.G of [37],
which asserts that all finitely generated $R$-flat modules are $R$-free when
$R$ is local, but Matsumura’s proof of the proposition yields this stronger
result with no substantial change. For the case at hand, we take the $R$-free
submodules of $R_{1}$ and $R_{2}$ generated by the finite sets $\\{s_{i}\\}$
and $\\{t_{i}\\}$, respectively. By choosing bases and performing a suitable
rearrangement, we may assume the elements $s_{i}$ and $t_{i}$ in the sum
$\sum_{i}(s_{i}\otimes t_{i})\otimes(x_{i}\otimes 1)$ are basis elements of
the free submodules, and that the summands are $R$-linearly independent. Then,
the cocycle condition implies

$\displaystyle\phi_{2}((x_{i}\otimes 1)\otimes(1\otimes 1))=(1\otimes
1)\otimes(x_{i}\otimes 1)$

for all $i$, and in particular, the elements $x_{i}$ lie in
$G(M_{1},M_{2},{\rm id})$. Because $\phi_{2}$ is an
$(R_{1}\otimes_{R}R_{2})\otimes_{R_{2}}(R_{1}\otimes_{R}R_{2})$-module
isomorphism, we can then write

$\displaystyle\phi_{2}^{-1}((s_{i}\otimes t_{i})\otimes(x_{i}\otimes 1))$
$\displaystyle=((s_{i}\otimes t_{i})\otimes(1\otimes
1))\cdot\phi_{2}^{-1}((1\otimes 1)\otimes(x_{i}\otimes 1))$
$\displaystyle=((s_{i}\otimes t_{i})\otimes(1\otimes 1))\cdot((x_{i}\otimes
1)\otimes(1\otimes 1))$ $\displaystyle=(s_{i}x_{i}\otimes
t_{i})\otimes(1\otimes 1).$

Summing over $i$, we find that $x\otimes 1=\sum_{i}s_{i}x_{i}\otimes t_{i}$,
so the image of $g_{3}$ contains $M_{1}$. Because $M_{1}$ is an $R_{1}$-form
of $M_{3}$, the restricted action map
$(R_{1}\otimes_{R}R_{2})\otimes_{R}M_{1}\to M_{3}$ is surjective. The
compatibility between iterated actions and ring multiplication then implies
$g_{3}$ is surjective.

Summing up, we have proved that $g_{3}$ is an isomorphism, so $g_{1}$ and
$g_{2}$ are isomorphisms. We therefore have a natural isomorphism from $F\circ
G$ to identity, so we conclude that $F$ is an equivalence of categories.

The claims about finite projective modules and bilinear forms then follow from
essentially the same argument as in Theorem 2.8. ∎

###  2.2 Vertex operator algebras over commutative rings

Vertex algebras over commutative rings were first defined in [2]. Since all of
the examples we consider will be conformal with strong finiteness conditions,
we will mainly consider vertex operator algebras, roughly following the
treatment in [25].

######  Definition 2.14.

A vertex algebra over a commutative ring $R$ is an $R$-module $V$ equipped
with a distinguished vector $\mathbf{1}\in V$ and a left multiplication map
$Y\colon V\to(\operatorname{End}V)\big{[}\big{[}z,z^{-1}\big{]}\big{]}$,
written $Y(a,z)=\sum\limits_{n\in\mathbb{Z}}a_{n}z^{-n-1}$ satisfying the
following conditions:

  1. 1.

For any $a,b\in V$, $a_{n}b=0$ for $n$ sufficiently large. Equivalently, $Y$
defines a multiplication map $V\otimes_{R}V\to V((z))$.

  2. 2.

$Y(\mathbf{1},z)={\rm id}_{V}z^{0}$ and $Y(a,z)\mathbf{1}\in a+zV[[z]]$.

  3. 3.

For any $r,s,t\in\mathbb{Z}$, and any $u,v,w\in V$,

$\displaystyle\sum_{i\geq 0}\binom{r}{i}(u_{t+i}v)_{r+s-i}w=\sum_{i\geq
0}(-1)^{i}\binom{t}{i}\big{(}u_{r+t-i}(v_{s+i}w)-(-1)^{t}v_{s+t-i}(u_{r+i}w)\big{)}.$

Equivalently, the Jacobi identity holds:

$\displaystyle
x^{-1}\delta\left(\frac{y-z}{x}\right)Y(a,y)Y(b,z)-x^{-1}\delta\left(\frac{z-y}{-x}\right)Y(b,z)Y(a,y)$
$\displaystyle\qquad{}=z^{-1}\delta\left(\frac{y-x}{z}\right)Y(Y(a,x)b,z),$

where $\delta(z)=\sum\limits_{n\in\mathbb{Z}}z^{n}$, and
$\delta\left(\frac{y-z}{x}\right)$ is expanded as a formal power series with
non-negative powers of $z$, i.e., as
$\sum\limits_{n\in\mathbb{Z},m\in\mathbb{Z}_{\geq
0}}(-1)^{m}\binom{n}{m}x^{-n}y^{n-m}z^{m}$.

A vertex algebra homomorphism
$(V,\mathbf{1}_{V},Y_{V})\to(W,\mathbf{1}_{W},Y_{W})$ is an $R$-module
homomorphism $\phi\colon V\to W$ satisfying
$\phi(\mathbf{1}_{V})=\mathbf{1}_{W}$ and $\phi(u_{n}v)=\phi(u)_{n}\phi(v)$
for all $u,v\in V$ and all $n\in\mathbb{Z}$.

######  Remark 2.15.

Mason notes in [36] that a vertex algebra over $R$ is the same thing as a
vertex algebra $V$ over $\mathbb{Z}$ equipped with a vertex algebra
homomorphism $R\to V$, where $R$ is viewed as a commutative vertex algebra
with product $Y(a,z)b=a_{-1}bz^{0}=ab$.

######  Definition 2.16.

Let $f\colon R\to S$ be a homomorphism of commutative rings. A descent datum
for vertex algebras with respect to $f$ is a pair $(V,\phi)$, where $V$ is a
vertex algebra over $S$, and $\phi\colon V\otimes_{R}S\to S\otimes_{R}V$ is an
isomorphism of vertex algebras over $S\otimes_{R}S$ such that the cocycle
condition holds, i.e., the following diagram of isomorphisms of vertex
algebras over $S\otimes_{R}S\otimes_{R}S$ commutes:

$\textstyle{V\otimes_{R}S\otimes_{R}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{01}=\phi\otimes
1}$$\scriptstyle{\phi_{02}}$$\textstyle{S\otimes_{R}V\otimes_{R}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{12}=1\otimes\phi}$$\textstyle{S\otimes_{R}S\otimes_{R}V.}$

Here, the maps $\phi_{i,j}$ are defined as in Definition 2.2. A morphism of
descent data from $(V,\phi)$ to $(V^{\prime},\phi^{\prime})$ is a homomorphism
$\psi\colon V\to V^{\prime}$ of vertex algebras over $S$ such that the
following diagram commutes:

$\textstyle{V\otimes_{R}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{\psi\otimes
1}$$\textstyle{S\otimes_{R}V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\otimes\psi}$$\textstyle{V^{\prime}\otimes_{R}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{\prime}}$$\textstyle{S\otimes_{R}V^{\prime}.}$

######  Proposition 2.17.

Faithfully flat descent of vertex algebras is effective. That is, if we are
given a faithfully flat ring homomorphism $R\to S$, then there is an
equivalence between the category of vertex algebras $\tilde{V}$ over $R$ and
the category of vertex algebras $V$ over $S$ equipped with descent data
$\phi$, given in one way by
$\tilde{V}\mapsto(\tilde{V}\otimes_{R}S,\phi((v\otimes s)\otimes
s^{\prime})=s\otimes(v\otimes s^{\prime}))$ and the other way by
$(V,\phi)\mapsto\tilde{V}=\\{v\in V\,|\,1\otimes v=\phi(v\otimes 1)\\}$.

###### Proof.

We briefly explain why this follows from the effectiveness of faithfully flat
descent of modules. Essentially, faithfully flat descent for modules with
additional structure holds as long as that structure is defined by
homomorphisms of modules, with conditions given by commutativity of diagrams.
Here the additional structure is given by a distinguished map $S\to V$ induced
by the unit vector, and the “$z^{-n-1}$-coefficient maps” $\cdot_{n}\colon
V\otimes_{S}V\to V$. The vertex algebra axioms, such as the Jacobi identity,
may be interpreted as equality of certain composites of such maps.

Now, suppose we are given a descent datum $(V,\phi)$ for vertex algebras with
respect to $f$. The isomorphism $\phi$ induces a descent datum for the
underlying $S$-module, so faithfully flat descent for modules, given as
Theorem 2.3, yields an $R$-module $\tilde{V}$ and a distinguished $S$-module
isomorphism $\psi\colon\tilde{V}\otimes_{R}S\cong V$.

For each integer $n$, we have the “$z^{-n-1}$-coefficient map”
$\cdot_{n}\colon V\otimes_{S}V\to V$, which we may now rewrite as
$\cdot_{n}\colon\big{(}\tilde{V}\otimes_{R}\tilde{V}\big{)}\otimes_{R}S\to\tilde{V}\otimes_{R}S$
using $\psi$. To show that this is the base change of a map
$\tilde{\cdot}_{n}\colon\tilde{V}\otimes_{R}\tilde{V}\to\tilde{V}$ along $f$,
it is necessary and sufficient to show that $\cdot_{n}$ is a morphism of
descent data for modules with respect to $f$.

By our assumption that $\phi$ is a descent datum for vertex algebras, the
following diagram commutes:

$\textstyle{\big{(}\tilde{V}\otimes_{R}\tilde{V}\big{)}\otimes_{R}S\otimes_{R}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi\otimes\phi}$$\scriptstyle{\cdot_{n}\otimes
1}$$\textstyle{S\otimes_{R}\big{(}\tilde{V}\otimes_{R}\tilde{V}\big{)}\otimes_{R}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\otimes\cdot_{n}}$$\textstyle{\tilde{V}\otimes_{R}S\otimes_{R}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{S\otimes_{R}\tilde{V}\otimes_{R}S,}$

so we have a map of descent data. Here, the map $\phi\otimes\phi$ along the
top uses an identification
$\big{(}\tilde{V}\otimes_{R}\tilde{V}\big{)}\otimes_{R}S\otimes_{R}S\cong(V\otimes_{R}S)\otimes_{S}(V\otimes_{R}S)$
induced by $\psi$.

Essentially the same argument shows that the unit map $S\to V$ that takes $1$
to $\mathbf{1}$ is the base change of an $R$-module map $R\to\tilde{V}$. The
remaining vertex algebra conditions then follow from the fact that base-change
along $f$ yields a formulas that hold for $V$. ∎

Given a vertex algebra $V$ over $R$ and a vertex algebra $W$ over $S$, we say
that $V$ is an $R$-form of $W$ with respect to a ring homomorphism $\phi\colon
R\to S$ if $V\otimes_{R,\phi}S$ is isomorphic to $W$ as a vertex algebra over
$S$. If $\phi$ is implicitly fixed, we will simply say that $V$ is an $R$-form
of $W$. Thus, the previous proposition amounts to the claim that the
construction of an $R$-form with respect to a faithfully flat map is
equivalent to the construction of a descent datum.

######  Definition 2.18.

Let $c\in R$. A conformal vertex algebra over $R$ with half central charge $c$
is a vertex algebra over $R$ equipped with a $\mathbb{Z}$-grading
$V=\bigoplus_{n\in\mathbb{Z}}V_{n}$ and a distinguished vector $\omega\in
V_{2}$ such that:

  1. 1.

The coefficients of $Y(\omega,z)=\sum\limits_{k\in\mathbb{Z}}L_{k}z^{-k-2}$
satisfy the Virasoro relations at half central charge $c$, i.e.,

$\displaystyle[L_{m},L_{n}]=(m-n)L_{m+n}+c\binom{m+1}{3}\delta_{m+n,0}{\rm
id}_{V}.$

  2. 2.

The product structure is homogeneous: $u_{k}v\in V_{m+n-k-1}$ for $u\in
V_{m}$, $v\in V_{n}$.

  3. 3.

$L_{0}v=nv$ for all $v\in V_{n}$.

  4. 4.

$L_{-1}v$ is the $z^{1}$-term in $Y(v,z)\mathbf{1}$.

A vertex operator algebra over $R$ with half central charge $c$ is a conformal
vertex algebra such that all $V_{n}$ are finite projective $R$-modules, and
$V_{n}=0$ for $n\ll 0$.

######  Remark 2.19.

Our definition of vertex operator algebra over $R$ differs slightly from the
definition in [25]. We remove the condition that the base ring $R$ be an
integral domain in which 2 is invertible, and we replace the condition that
each $V_{n}$ be a finite free $R$-module with the condition that each $V_{n}$
be a finite projective $R$-module. The first change is necessary for us to
consider the case $R=\mathbb{Z}$, and the second change is necessary in order
for faithfully flat descent of vertex operator algebras to be effective in
general (see Proposition 2.23). That is, free modules don’t necessarily glue
to form free modules, and standard examples come from locally trivial but
nontrivial vector bundles. This distinction doesn’t affect the strength of our
final result, since $\mathbb{Z}$ is a principal ideal domain.

######  Remark 2.20.

It may be reasonable someday to change the definition of “vertex operator
algebra over $R$” to require an action of the smooth integral form
$U^{+}(Vir)$ of the universal enveloping algebra of Virasoro constructed in
Section 5 of [5]. The self-dual integral form of $V^{\natural}$ constructed in
this paper admits such an action by virtue of the same being true of
$(V_{\Lambda})_{\mathbb{Z}}$. Furthermore, the proof of Modular Moonshine for
primes greater than 13 in [4] assumes the existence of such an action.
However, this condition is at the moment too difficult to verify in general to
be particularly useful.

######  Definition 2.21.

A vertex algebra homomorphism from $V$ to $W$ is an $R$-linear map $\phi\colon
V\to W$ satisfying $\phi(\mathbf{1}_{V})=\mathbf{1}_{W}$ and
$\phi(Y_{V}(u,z)v)=Y_{W}(\phi(u),z)\phi(v)$ for all $u,v\in V$. A vertex
operator algebra homomorphism from $V$ to $W$ is a vertex algebra homomorphism
$\phi\colon V\to W$ that preserves the $\mathbb{Z}$-grading, and sends
$\omega_{V}$ to $\omega_{W}$.

######  Remark 2.22.

The condition that $\phi(\omega_{V})=\omega_{W}$ is sometimes reserved for a
special class of homomorphisms, using terms like “strong” or “strict” or
“conformal” homomorphism in the literature. This is because it is often
fruitful to consider vertex algebra maps between vertex operator algebras of
different central charge. We will not need to consider such maps in this
paper.

######  Proposition 2.23.

Faithfully flat descent of vertex operator algebras is effective. That is, if
we are given a faithfully flat ring homomorphism $R\to S$, then there is an
equivalence between the category of vertex operator algebras $\bar{V}$ over
$R$ and the category of vertex operator algebras $V$ over $S$ equipped with
descent data $\phi\colon V\otimes_{R}S\to S\otimes_{R}V$, given in one way by
$\bar{V}\mapsto(\bar{V}\otimes_{R}S,$ $\phi((v\otimes s)\otimes
s^{\prime})=s\otimes(v\otimes s^{\prime}))$ and the other way by
$(V,\phi)\mapsto\bar{V}=\\{v\in V|1\otimes v=\phi(v\otimes 1)\\}$.

###### Proof.

By Proposition 2.17, faithfully flat descent for vertex algebras is effective.
It then suffices to show that the conformal vector $\omega$ descends, and that
the finite projective property of modules descends. The first assertion is
clear from the fact that $\phi$ sends conformal vectors to conformal vectors.
The second assertion is given in [43, Proposition 058S] (or [31, Proposition
2.5.2]). ∎

######  Remark 2.24.

This result allows us to consider vertex algebras and vertex operator algebras
over a broad class of geometric objects, including all schemes and all
algebraic stacks.

######  Definition 2.25.

A Möbius structure on a vertex algebra over $R$ is an integer grading,
together with an action of the Kostant integral form
$U(\mathfrak{sl}_{2})_{\mathbb{Z}}=\big{\langle}L_{-1}^{(n)},\binom{L_{0}}{n},L_{1}^{(n)}\big{\rangle}_{n\geq
1}$ on the underlying $R$-module, such that the following conditions hold:

  1. 1.

$Y(u,z)\mathbf{1}=\sum\limits_{n\geq 0}L_{-1}^{(n)}uz^{n}$ for all $u\in V$.

  2. 2.

If $u$ has weight $k$, then $\binom{L_{0}}{n}u=\binom{k}{n}u$.

  3. 3.

The subalgebra generated by $\\{L_{1}^{(n)}\\}_{n\geq 1}$ acts locally
nilpotently on $V$.

  4. 4.

For all $i\in\\{-1,0,1\\}$,
$L_{i}Y(u,z)v-Y(u,z)(L_{i}v)=\sum\limits_{j=0}^{i+1}\binom{i+1}{j}z^{j}Y(L_{i-j}u,v)$.

We note that in the embedding of $U(\mathfrak{sl}_{2})_{\mathbb{Z}}$ in
$U(\mathfrak{sl}_{2})$, we have $L_{i}^{(n)}=\frac{L_{i}^{n}}{n!}$.

######  Definition 2.26.

Let $V$ be a vertex algebra with Möbius structure, and let $u\in V$ have
weight $k$. For each integer $n$, we define the operator
$(u_{n})^{*}=(-1)^{k}\sum\limits_{i\geq
0}\big{(}L_{1}^{(i)}u\big{)}_{2k-i-n-2}$. An invariant bilinear form on a
vertex algebra $V$ with Möbius structure is a symmetric bilinear form
$(\,,\,)\colon V\times V\to R$ such that

$\displaystyle(u_{n}v,w)=(v,(u_{n})^{*}w)$

for all homogeneous vectors $u$ and all vectors $v$ and $w$. That is,
$(u_{n})^{*}$ is the adjoint of the operator $u_{n}$. We say that a vertex
operator algebra is self-dual with respect to an invariant bilinear form if
each graded piece $V_{n}$ is self-dual as a finite projective $R$-module.

######  Remark 2.27.

If $R$ contains $\mathbb{Z}$ as a subring, then we may write the inner product
formula equivalently as $(Y(u,z)v,w)=\big{(}v,Y\big{(}{\rm
e}^{zL_{1}}\big{(}{-}z^{-2}\big{)}^{L_{0}}u,z^{-1}\big{)}w\big{)}$. Otherwise,
we must replace $L_{0}$ with an integer indicator of weight.

######  Proposition 2.28.

Faithfully flat descent is effective for both Möbius structure and invariant
bilinear forms, on both vertex algebras and vertex operator algebras.
Furthermore, the self-dual property for vertex operator algebras with
invariant bilinear forms descends effectively.

###### Proof.

For Möbius structure, the generating operators descend as module maps, and
automatically satisfy the defining relations. For invariant bilinear forms, it
suffices to use descent of module maps $V\otimes_{R}V\to R$ as in Lemma 2.5.
For the self-dual property, we may reduce to the case of self-duality of the
finite projective modules $V_{n}$, and the isomorphism property of the map
$V_{n}\to\operatorname{Hom}_{R}(V_{n},R)$ descends effectively as in Theorem
2.8. ∎

We use this to describe gluing explicitly.

######  Definition 2.29.

Suppose we are given the following diagram of commutative ring homomorphisms:
$R_{1}\to R_{3}\leftarrow R_{2}$. A gluing datum for vertex operator algebras
over this diagram is a triple $\big{(}V^{1},V^{2},f\big{)}$, where

  1. 1.

$V^{1}$, $V^{2}$ are vertex operator algebras over $R_{1}$ and $R_{2}$,
respectively.

  2. 2.

$f\colon V^{1}\otimes_{R_{1}}R_{3}\to V^{2}\otimes_{R_{1}}R_{3}$ is an
isomorphism of vertex operator algebras over $R_{3}$.

A gluing datum is Möbius and self-dual if the corresponding vertex operator
algebras are Möbius and self-dual, and the map $f$ preserves the Möbius
structure and bilinear form. A morphism of gluing data from
$\big{(}V^{1},V^{2},f\big{)}$ to
$\big{(}V^{\prime,1},V^{\prime,2},f^{\prime}\big{)}$ is a pair
$\big{(}g_{1}\colon V^{1}\to V^{\prime,1},g_{2}\colon V^{2}\to
V^{\prime,2}\big{)}$ of vertex algebra homomorphisms such that the following
diagram commutes:

$\textstyle{V^{1}\otimes_{R_{1}}R_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g_{1}\otimes{\rm
id}}$$\textstyle{V^{2}\otimes_{R_{2}}R_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{2}\otimes{\rm
id}}$$\textstyle{V^{\prime,1}\otimes_{R_{1}}R_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{V^{\prime,2}\otimes_{R_{2}}R_{3}.}$

######  Lemma 2.30.

Let $R$ be a commutative ring, and let $a,b\in R$ be coprime elements, i.e.,
the ideal generated by $a$ and $b$ is all of $R$. Let $R_{a}$ and $R_{b}$ be
the respective localizations, i.e., such that $\operatorname{Spec}R_{a}$ and
$\operatorname{Spec}R_{b}$ form a Zariski open cover of
$\operatorname{Spec}R$. Then base change induces an equivalence between the
category of vertex operator algebras over $R$ and the category of gluing data
for vertex operator algebras over the diagram $R_{a}\to R_{ab}\leftarrow
R_{b}$. This also yields equivalences between the corresponding categories of
self-dual Möbius objects.

###### Proof.

We obtain a quasi-inverse functor by applying Lemma 2.10 to obtain the
underlying modules and structure maps. ∎

######  Lemma 2.31.

Let $R\to R_{1}$ and $R\to R_{2}$ be faithfully flat homomorphisms of
commutative rings. Then tensor product induces an equivalence between the
category of vertex operator algebras over $R$ and the category of gluing data
for vertex operator algebras over the diagram $R_{1}\to
R_{1}\otimes_{R}R_{2}\leftarrow R_{2}$. This also yields equivalences between
the corresponding categories of self-dual Möbius objects.

###### Proof.

We first note that faithful flatness implies the tensor product functor
preserves the finite projective property of graded pieces, so it takes vertex
operator algebras over $R$ to gluing data for vertex operator algebras over
the diagram $R_{1}\to R_{1}\otimes_{R}R_{2}\leftarrow R_{2}$. Similarly,
tensor product preserves the self-dual Möbius structure. We obtain the quasi-
inverse functor by applying Lemma 2.13 to obtain the underlying modules and
the structure maps. It remains to check that the conformal vertex algebra over
$R$ that we obtain is in fact a vertex operator algebra over $R$, and this
follows from Proposition 2.23. For self-dual Möbius structure, the claim
follows from Proposition 2.28. ∎

######  Lemma 2.32.

Let $R\to R_{1}$ and $R\to R_{2}$ be homomorphisms of commutative rings of the
form given in either Lemma 2.30 or Lemma 2.31. Let $V^{1}$ and $V^{2}$ be
vertex operator algebras over $R_{1}$ and $R_{2}$, respectively, such that
$V^{1}\otimes_{R}R_{2}$ and $V^{2}\otimes_{R}R_{1}$ are isomorphic as vertex
operator algebras over $R_{1}\otimes_{R}R_{2}$. Any choice of gluing datum
induces injective group homomorphisms
$\operatorname{Aut}_{R_{1}}V^{1}\to\operatorname{Aut}_{R_{1}\otimes_{R}R_{2}}\big{(}V^{1}\otimes_{R}R_{2}\big{)}$
and
$\operatorname{Aut}_{R_{2}}V^{2}\to\operatorname{Aut}_{R_{1}\otimes_{R}R_{2}}\big{(}V^{1}\otimes_{R}R_{2}\big{)}$,
and if the corresponding double coset space is a singleton, then there is a
unique isomorphism type of vertex operator algebra $V$ over $R$ such that
$V^{1}\cong V\otimes_{R}R_{1}$ and $V^{2}\cong V\otimes_{R}R_{2}$. In
particular, if the inclusions are isomorphisms, then $V$ is unique, and the
natural inclusions of $\operatorname{Aut}_{R}V$ into
$\operatorname{Aut}_{R_{1}}V^{1}$ and $\operatorname{Aut}_{R_{2}}V^{2}$ are
isomorphisms. The corresponding statement also holds for Möbius self-dual
vertex operator algebras.

###### Proof.

There is a natural action of
$\operatorname{Aut}_{R_{1}}V^{1}\times\operatorname{Aut}_{R_{2}}V^{2}$ on the
set of gluing data by isomorphisms, given by

$\displaystyle f\mapsto(g_{2}\otimes{\rm id})\circ f\circ(g_{1}\otimes{\rm
id})^{-1}=f^{\prime},$

because such isomorphisms are defined by the condition that the following
diagram commutes:

$\textstyle{V^{1}\otimes_{R_{1}}R_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g_{1}\otimes{\rm
id}}$$\textstyle{V^{2}\otimes_{R_{2}}R_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{2}\otimes{\rm
id}}$$\textstyle{V^{1}\otimes_{R_{1}}R_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{V^{2}\otimes_{R_{2}}R_{3}.}$

Because isomorphisms of gluing data correspond bijectively to isomorphisms of
vertex operator algebras over $R$, we get a unique vertex operator algebra
over $R$ if and only if the set of orbits is a singleton. For any two choices
of gluing data $f$ and $f^{\prime}$, there exists a unique
$h\in\operatorname{Aut}_{R_{1}\otimes_{R}R_{2}}\big{(}V^{1}\otimes_{R}R_{2}\big{)}$
such that $h\circ f=f^{\prime}$. This establishes a non-canonical bijection
between gluing data and elements of
$\operatorname{Aut}_{R_{1}\otimes_{R}R_{2}}\big{(}V^{1}\otimes_{R}R_{2}\big{)}$.
Fixing some $f$ as a base point induces inclusion maps
$\phi_{1}\colon\operatorname{Aut}_{R_{1}}V^{1}\to\operatorname{Aut}_{R_{1}\otimes_{R}R_{2}}\big{(}V^{1}\otimes_{R}R_{2}\big{)}$
and
$\phi_{2}\colon\operatorname{Aut}_{R_{2}}V^{2}\to\operatorname{Aut}_{R_{1}\otimes_{R}R_{2}}\big{(}V^{1}\otimes_{R}R_{2}\big{)}$,
and identifies the action with $h\mapsto\phi_{2}(g_{2})h\phi_{1}(g_{1})^{-1}$.
This identifies orbits with double cosets in
$\phi_{2}\big{(}\operatorname{Aut}_{R_{2}}V^{2}\big{)}\backslash\operatorname{Aut}_{R_{1}\otimes_{R}R_{2}}\big{(}V^{1}\otimes_{R}R_{2}\big{)}/\phi_{1}\big{(}\operatorname{Aut}_{R_{1}}V^{1}\big{)}$.

For the special case that $\phi_{1}$ and $\phi_{2}$ are isomorphisms, there is
clearly a single double coset, hence a single isomorphism type of vertex
operator algebra $V$ over $R$. Furthermore, each
$R_{1}\otimes_{R}R_{2}$-automorphism of $\big{(}V^{1}\otimes_{R}R_{2}\big{)}$
descends to an $R$-automorphism of $V$, by effective descent of morphisms
given by the equivalences of categories in Lemmas 2.30 and 2.31 ∎

The following fact is rather elementary, but we will use it a lot.

######  Lemma 2.33.

Let $V$ be a vertex operator algebra over a ring $R$ with Möbius structure and
an invariant bilinear form, and suppose $V$ is self-dual. Let $g$ be an
automorphism of order $n\in\mathbb{Z}_{\geq 1}$ that preserves the bilinear
form. If $n$ is invertible in $R$ and $\mu_{n}(R)\cong\mathbb{Z}/n\mathbb{Z}$,
then $V$ is isomorphic to the direct sum of its $g$-eigenspaces
$\bigoplus_{\zeta\in\mu_{n}(R)}V^{g=\zeta}$, as an $R$-module with bilinear
form. Furthermore, each eigenspace is a module for the fixed-point vertex
operator subalgebra $V^{g=1}$, and the bilinear form induces a homogeneous
perfect pairing between $V^{g=\zeta}$ and $V^{g=1/\zeta}$ for all
$\zeta\in\mu_{n}(R)$.

###### Proof.

The hypotheses on $R$ imply any action of $\mathbb{Z}/n\mathbb{Z}$ on an
$R$-module gives a unique decomposition into eigenspaces for $n$-th roots of
unity. For $V$, the decomposition yields modules for the fixed-point vertex
operator subalgebra. Self-duality of $V$ implies the eigenspaces with
reciprocal eigenvalues are paired. ∎

###  2.3 Abelian intertwining algebras over subrings of complex numbers

We suspect that with enough work, one may extend the definition of abelian
intertwining algebra, introduced in [19], to allow definition over arbitrary
commutative rings with suitable divisibility properties and equipped with
distinguished roots of unity. However, the results in this paper only require
us to consider base rings $R$ that lie inside $\mathbb{C}$, and the formalism
becomes somewhat more manageable in this case.

Recall from [35] that if we are given abelian groups $(A,+)$ and $(B,\cdot)$,
an abelian 3-cocycle on $A$ with coefficients in $B$ is a pair $(F\colon
A\times A\times A\to B,\Omega\colon A\times A\to B)$ of functions satisfying
the Eilenberg–MacLane cocycle conditions:

  1. 1)

$F(i,j,k)\cdot F(i,j+k,\ell)\cdot F(j,k,\ell)=F(i+j,k,\ell)\cdot
F(i,j,k+\ell)$ for all $i,j,k,\ell\in A$,

  2. 2)

$F(i,j,k)^{-1}\cdot\Omega(i,j+k)\cdot F(j,k,i)^{-1}=\Omega(i,j)\cdot
F(j,i,k)^{-1}\cdot\Omega(i,k)$,

  3. 3)

$F(i,j,k)\cdot\Omega(i+j,k)\cdot F(k,i,j)=\Omega(j,k)\cdot
F(i,k,j)\cdot\Omega(i,k)$,

and the restriction of $\Omega$ to the diagonal $i\mapsto\Omega(i,i)$ gives a
bijection from abelian cohomology classes to $B$-valued quadratic functions on
$A$. We say that an abelian 3-cocycle is normalized if
$F(i,j,0)=F(i,0,k)=F(0,j,k)=1$ and $\Omega(i,0)=\Omega(0,j)=1$ for all
$i,j,k\in A$, and we note that any abelian cohomology class admits a
normalized representative.

######  Definition 2.34.

Let $N\in\mathbb{Z}_{\geq 1}$, let $R$ be a commutative subring of
$\mathbb{C}$ containing $\frac{1}{N}$ and $e(1/2N)$. Let $A$ be an abelian
group, let $(F,\Omega)$ be a normalized abelian 3-cocycle on $A$ with
coefficients in $R^{\times}$, and assume $\Omega(a,a)^{N}=1$ for all $a\in A$.
We then let $q_{\Omega}$ be the unique
$\frac{1}{N}\mathbb{Z}/\mathbb{Z}$-valued quadratic form on $A$ such that
$\Omega(a,a)=e(q_{\Omega}(a))$, and let $b_{\Omega}\colon A\times
A\to\frac{1}{N}\mathbb{Z}$ be any fixed function that reduces to the bilinear
form attached to $q_{\Omega}$, i.e., $b_{\Omega}(a,b)\equiv
q_{\Omega}(a+b)-q_{\Omega}(a)-q_{\Omega}(b)\mod\mathbb{Z}$. Then an abelian
intertwining algebra over $R$ of level $N$ and half central charge $c\in R$
associated to the data $(A,F,\Omega)$ is an $R$-module $V$ equipped with

  1. 1)

a $\frac{1}{N}\mathbb{Z}\times A$-grading
$V=\bigoplus_{n\in\frac{1}{N}\mathbb{Z}}V_{n}=\bigoplus_{i\in
A}V^{i}=\bigoplus_{n\in\frac{1}{N}\mathbb{Z},i\in A}V_{n}^{i}$,

  2. 2)

a left-multiplication operation $Y\colon
V\to(\operatorname{End}V)\big{[}\big{[}z^{1/N},z^{-1/N}\big{]}\big{]}$ that we
expand as $Y(a,z)$ $=\sum\limits_{n\in\frac{1}{N}\mathbb{Z}}a_{n}z^{-n-1}$,
and

  3. 3)

distinguished vectors $\mathbf{1}\in V_{0}^{0}$ and $\omega\in V_{2}^{0}$,

satisfying the following conditions for any $i,j,k\in A$, $a\in V^{i}$, $b\in
V^{j}$, $u\in V^{k}$, and $n\in\frac{1}{N}\mathbb{Z}$:

  1. 1)

$a_{n}b\in V^{i+j}$,

  2. 2)

$a_{n}b=0$ for $n$ sufficiently large,

  3. 3)

$Y(\mathbf{1},z)a=a$,

  4. 4)

$Y(a,z)\mathbf{1}\in a+zV[[z]]$,

  5. 5)

the coefficients of $Y(\omega,z)=\sum\limits_{k\in\mathbb{Z}}L_{k}z^{-k-2}$
satisfy the Virasoro relations at half central charge $c$,

  6. 6)

$L_{0}v=nv$ for all $v\in V_{n}$,

  7. 7)

$\frac{{\rm d}}{{\rm d}z}Y(a,z)=Y(L_{-1}a,z)$ for all $a\in V$,

  8. 8)

$Y(a,z)b=\sum\limits_{k\in b_{\Omega}(i,j)+\mathbb{Z}}a_{k}bz^{-k-1}$,

  9. 9)

the Jacobi identity:

$\displaystyle
x^{-1}\left(\frac{y-z}{x}\right)^{b_{\Omega}(i,j)}\delta\left(\frac{y-z}{x}\right)Y(a,y)Y(b,z)u$
$\displaystyle\qquad{}-B(i,j,k)x^{-1}\left(\frac{z-y}{-x}\right)^{b_{\Omega}(i,j)}\delta\left(\frac{z-y}{-x}\right)Y(b,z)Y(a,y)u$
$\displaystyle\qquad\quad{}=F(i,j,k)z^{-1}\delta\left(\frac{y-x}{z}\right)Y(Y(a,x)b,z)\left(\frac{y-x}{z}\right)^{-b_{\Omega}(i,k)}u,$

where $B(i,j,k)=\frac{\Omega(i,j)F(i,j,k)}{F(j,i,k)}$, and $(-x)^{k/N}$ is
interpreted as $e(k/2N)x^{k/N}$.

We say an abelian intertwining algebra $V$ is well-graded if each piece
$V_{n}^{i}$ is a finite projective $R$-module. A Möbius structure on $V$ is an
$A$-homogeneous action of
$U(\mathfrak{sl}_{2})_{\mathbb{Z}}=\big{\langle}L_{-1}^{(n)},\binom{L_{0}}{n},L_{1}^{(n)}\big{\rangle}_{n\geq
1}$ on the underlying $R$-module, such that the following conditions hold:

  1. 1)

$Y(u,z)\mathbf{1}=\sum\limits_{n\geq 0}L_{-1}^{(n)}uz^{n}$ for all $u\in V$,

  2. 2)

if $u$ has weight $k$, then $\binom{L_{0}}{n}u=\binom{k}{n}u$,

  3. 3)

the subalgebra generated by $\\{L_{1}^{(n)}\\}_{n\geq 1}$ acts locally
nilpotently on $V$,

  4. 4)

for all $i\in\\{-1,0,1\\}$,
$L_{i}Y(u,z)v-Y(u,z)(L_{i}v)=\sum\limits_{j=0}^{i+1}\binom{i+1}{j}z^{j}Y(L_{i-j}u,v)$.

We will need the following skew-symmetry property, which can be (more or less)
found in the proof of Proposition 2.6 of [28].

######  Lemma 2.35.

If $V$ is an abelian intertwining operator algebra, then $V$ satisfies the
following skew-symmetry property:

$\displaystyle Y(a,y)b=\Omega(i,j){\rm e}^{yL_{-1}}Y\big{(}b,{\rm e}^{\pi{\rm
i}}y\big{)}a$

for $a\in V^{i}$ and $b\in V^{j}$. In particular, for any $n\in
b_{\Omega}(i,j)+\mathbb{Z}$, $a_{n}b$ is an $R$-linear combination of terms of
the form $b^{\prime}_{k}a^{\prime}$ for $b^{\prime}\in V^{j}$ and
$a^{\prime}\in V^{i}$.

###### Proof.

We consider the Jacobi identity with vectors $a$, $b$, $\mathbf{1}$.
Multiplying both sides by $x^{b_{\Omega}(i,j)+N}$ for some
$N\in\mathbb{Z}_{>0}$ and taking $\operatorname{Res}_{x}$ yields:

$\displaystyle(y-z)^{b_{\Omega}(i,j)+N}Y(a,y)Y(b,z)\mathbf{1}-B(i,j,0){\rm
e}^{(-\pi{\rm
i})(b_{\Omega}(i,j)+N)}(z-y)^{b_{\Omega}(i,j)+N}Y(b,z)Y(a,y)\mathbf{1}$

on the left side, and some power series with powers of $x$ uniformly bounded
below times $Y(Y(a,x)b,z)$ on the right. When $N$ is sufficiently large, the
right side therefore vanishes. Because our cocycle is normalized, we have the
following equality in $V^{i+j}\big{[}\big{[}y^{\pm 1},z^{\pm
1}\big{]}\big{]}$:

$\displaystyle(y-z)^{b_{\Omega}(i,j)+N}Y(a,y){\rm
e}^{zL_{-1}}b=\Omega(i,j){\rm e}^{(-\pi{\rm
i})(b_{\Omega}(i,j)+N)}(z-y)^{b_{\Omega}(i,j)+N}Y(b,z){\rm e}^{yL_{-1}}a.$

This is essentially a version of the “operator-valued rational function”
statement in Remark 12.31 of [19]. The left side has no negative powers of
$z$, and the right side has no negative powers of $y$, so this is an equality
in $V^{i+j}[[y,z]]$. By the exponential translation formula, we find that

$\displaystyle(y-z)^{b_{\Omega}(i,j)+N}Y(a,y){\rm
e}^{zL_{-1}}b=\Omega(i,j){\rm e}^{(-\pi{\rm
i})(b_{\Omega}(i,j)+N)}(z-y)^{b_{\Omega}(i,j)+N}{\rm e}^{yL_{-1}}Y(b,z-y)a.$

Because this is an equality in $V^{i+j}[[y,z]]$, we may set $z=0$ to get the
answer we want, after substituting ${\rm e}^{\pi{\rm i}}y+z$ for $z-y$. ∎

We also need an associativity result, adapted from Lemma 3.12 of [33]. Li
credits [20] with this result, and I suspect he is referring to an argument
leading up to Remark 3.3 in that paper.

######  Lemma 2.36.

Let $a\in V^{i}$, $b\in V^{j}$, $u\in V^{k}$ be homogeneous elements of an
abelian intertwining algebra. Let $r\in b_{\Omega}(i,j+k)+\mathbb{Z}$ and
$s\in b_{\Omega}(j,k)+\mathbb{Z}$. Let $l\in b_{\Omega}(i,k)+\mathbb{Z}_{\geq
0}$ be such that $z^{l}Y(a,z)u\in V[[z]]$, and let $m\in\mathbb{Z}_{\geq 0}$
be such that $z^{m+s}Y(b,z)u\in V[[z]]$. Then

$\displaystyle a_{r}b_{s}u=F(i,j,k)\sum_{t=0}^{m}\sum_{n\in\mathbb{Z}_{\geq
0}}\binom{r-l}{t}\binom{l}{n}(a_{r-l-t+n}b)_{s+l+t-n}u.$

In particular $($since this is always a finite sum$)$, $a_{r}b_{s}u$ is an
$R$-linear combination of elements of the form $(a_{p}b)_{q}u$.

###### Proof.

By Proposition 2.5 of [34] (which is essentially the part of Remark 12.31 in
[19] that applies to “generalized rationality of iterates”), the following
associativity rule holds:

$\displaystyle(x+z)^{l}Y(a,x+z)Y(b,z)u=F(i,j,k)(z+x)^{l}Y(Y(a,x)b,z)u$

as elements of $z^{-m-s}x^{-b_{\Omega}(i,j)-N}V^{i+j+k}[[z,x]]$ for some
$N\in\mathbb{Z}$. To extract $a_{r}b_{s}u$ from the left side, we rewrite:

$\displaystyle a_{r}b_{s}u$
$\displaystyle=\operatorname{Res}_{y}\operatorname{Res}_{z}y^{r}z^{s}Y(a,y)Y(b,z)u$
$\displaystyle=\operatorname{Res}_{x}\operatorname{Res}_{y}\operatorname{Res}_{z}x^{-1}\delta\left(\frac{y-z}{x}\right)y^{r}z^{s}Y(a,y)Y(b,z)u$
$\displaystyle=\operatorname{Res}_{x}\operatorname{Res}_{y}\operatorname{Res}_{z}y^{-1}\delta\left(\frac{x+z}{y}\right)y^{r}z^{s}Y(a,y)Y(b,z)u$
$\displaystyle=\operatorname{Res}_{x}\operatorname{Res}_{y}\operatorname{Res}_{z}y^{-1}\delta\left(\frac{x+z}{y}\right)(x+z)^{r}z^{s}Y(a,x+z)Y(b,z)u$
$\displaystyle=\operatorname{Res}_{x}\operatorname{Res}_{z}(x+z)^{r}z^{s}Y(a,x+z)Y(b,z)u.$

Combining this with the associativity rule, we find that

$\displaystyle\begin{split}&a_{r}b_{s}u=\operatorname{Res}_{x}\operatorname{Res}_{z}(x+z)^{r-l}z^{s}\big{[}(x+z)^{l}Y(a,x+z)Y(b,z)u\big{]}\\\
&\hphantom{a_{r}b_{s}u}{}=F(i,j,k)\operatorname{Res}_{x}\operatorname{Res}_{z}(x+z)^{r-l}z^{s}\big{[}(z+x)^{l}Y(Y(a,x)b,z)u\big{]}.\end{split}$

The powers of $z$ in the last expression are bounded below, so the power
series $(x+z)^{r-l}$ only contributes finitely many terms to the residue with
respect to $x$. That is, we may truncate it as

$\displaystyle p(x,z)=\sum_{t=0}^{m}\binom{r-l}{t}x^{r-l-t}z^{t}.$

We then obtain

$\displaystyle
a_{r}b_{s}u=F(i,j,k)\operatorname{Res}_{x}\operatorname{Res}_{z}p(x,z)z^{s}(z+x)^{l}Y(Y(a,x)b,z)u,$

which immediately yields the answer we want. ∎

######  Definition 2.37.

Let $V$ be a Möbius abelian intertwining algebra over $R$ associated to the
data $(A,F,\Omega)$. Then an invariant bilinear form on $V$ is an inner
product such that $(u_{n}v,w)=(v,e(k/2)\sum\limits_{i\geq
0}\big{(}L_{1}^{(i)}u\big{)}_{2k-i-n-2}w)$ whenever $u$ has weight $k$, and
$(u,v)=0$ if $u\in V^{a}$ and $v\in V^{b}$ for $a+b\neq 0$.

######  Lemma 2.38.

Let $V$ be a well-graded Möbius abelian intertwining algebra over $R$
associated to the data $(A,F,\Omega)$, and suppose $V$ admits an invariant
bilinear form. Then this form is self-dual if and only if for each $a\in A$
the form induces $V^{0}$-module isomorphisms $V^{a}\cong(V^{-a})^{\vee}$.

###### Proof.

Any invariant form induces a $V^{0}$-module map $\phi_{a}\colon
V^{a}\to(V^{-a})^{\vee}$ by setting $\phi_{a}(v)(w)=(v,w)$, because
compatibility with the $V^{0}$-action follows from the defining relation
applied to homogeneous vectors $u\in V^{0}$. By the orthogonality of $V^{a}$
and $V^{b}$ for $a+b\neq 0$, self-duality is then equivalent to $\phi_{a}$
being an isomorphism for all $a$. ∎

The following two results will form a primary engine behind our construction.

######  Proposition 2.39.

Let $R$ be a subring of $\mathbb{C}$ containing $1/N$ and ${\rm e}^{2\pi{\rm
i}/N}$. Let $V$ be an abelian intertwining algebra over $\mathbb{C}$
associated to the data $(A,F,\Omega)$, equipped with an invariant bilinear
form. Suppose the following properties hold:

  1. $1.$

$V$ is self-dual with respect to the invariant form.

  2. $2.$

Each $V^{a}$ is an irreducible $V^{0}$-module.

  3. $3.$

$V$ is generated by abelian intertwining subalgebras
$\big{\\{}V^{A_{i}}\big{\\}}_{i\in I}$, where $A_{i}$ range over a set of
subgroups of $A$ that generate $A$, and $V^{A_{i}}=\bigoplus_{a\in
A_{i}}V^{a}$.

  4. $4.$

We are given Möbius $R$-forms $V^{A_{i}}_{R}$ of each $V^{A_{i}}$, which
coincide on pairwise intersections of the subalgebras $V^{A_{i}}$.

Then the abelian intertwining subalgebra $V_{R}$ of $V$ over $R$ generated by
$\big{\\{}V^{A_{i}}_{R}\big{\\}}_{i\in I}$ is a Möbius $R$-form $V_{R}$ of
$V$, i.e., $V_{R}\otimes_{R}\mathbb{C}\cong V$.

###### Proof.

By induction, it suffices to consider the case where $I=\\{1,2\\}$, and
$A_{1}\cap A_{2}=\\{0\\}$. As it happens, this is the only case that we will
use in this paper.

We define $V_{R}$ to be the sub-$R$-module of $V$ generated by products of
elements in $V^{A_{i}}_{R}$. We then define $V^{a}_{R}$ to be the part of
$V_{R}$ of degree $a$. In the course of this proof, we will show that if $a\in
A_{i}$, then $V^{a}_{R}$ is equal to the degree $a$ part of $V^{A_{i}}_{R}$,
but before we prove this, we will write $\big{(}V^{A_{i}}_{R}\big{)}^{a}$ for
the latter space to distinguish them.

To show that $V_{R}$ is an $R$-form of $V$, we fix $a\in A$ and consider the
base change map $\phi\colon V^{a}_{R}\otimes_{R}\mathbb{C}\to V^{a}$. It
suffices to show that $\phi$ is an isomorphism. The map $\phi$ is surjective,
because we may write any element of $V^{a}$ as a $\mathbb{C}$-linear
combination of products of elements of $V^{A_{i}}$, and these elements in turn
are $\mathbb{C}$-linear combinations of elements of $V^{A_{i}}_{R}$.

To show that $\phi$ is injective, it suffices to show that any $R$-linearly
independent set in $V^{a}_{R}$ is $\mathbb{C}$-linearly independent. We first
show that any element of $V_{R}$ is an $R$-linear combination of products
$u_{r}v$, where $u\in V^{A_{1}}_{R}$ and $v\in V^{A_{2}}_{R}$. By skew-
symmetry as in Lemma 2.35, $v_{s}u$ is an $R$-linear combination of elements
of the desired form, so we may switch $A_{1}$ and $A_{2}$. By induction on
word length, it suffices to show that any length 3 words of the form
$a_{r}b_{s}w$ and $(a_{r}b)_{s}w$ have this form, where $a,b,w\in
V^{A_{1}}_{R}\cup V^{A_{2}}_{R}$. Again by skew-symmetry, we do not need to
consider the case $(a_{r}b)_{s}w$ separately. Under this reduction, if $b$ and
$w$ are both in either $V^{A_{1}}_{R}$ or $V^{A_{2}}_{R}$, then there is
nothing to show. We therefore are reduced to considering $a_{r}b_{s}w$ in the
following cases:

  1. 1)

$a,b\in V^{A_{1}}_{R}$ and $w\in V^{A_{2}}_{R}$,

  2. 2)

$b\in V^{A_{1}}_{R}$ and $a,w\in V^{A_{2}}_{R}$.

These cases are taken to each other by applying skew-symmetry to $b$ and $w$
and switching $A_{1}$ with $A_{2}$. It therefore suffices to consider the
first case, which is handled by Lemma 2.36. Thus, $V_{R}$ is $R$-spanned by
products of the form $u_{r}v$, where $u\in V^{A_{1}}_{R}$ and $v\in
V^{A_{2}}_{R}$. It follows immediately that
$V^{a}_{R}=\big{(}V^{A_{i}}_{R}\big{)}^{a}$ whenever $a\in A_{i}$.

To show that any $R$-linearly independent set in $V^{a}_{R}$ is
$\mathbb{C}$-linearly independent, we consider the contrapositive. Take a
finite subset
$\big{\\{}u^{1,1}_{r_{1}}u^{2,1},\ldots,u^{1,k}_{r_{k}}u^{2,k}\big{\\}}\subset
V^{a}_{R}$, and suppose
$c_{1}u^{1,1}_{r_{1}}u^{2,1}+\cdots+c_{k}u^{1,k}_{r_{k}}u^{2,k}=0$ for some
coefficients $c_{1},\ldots,c_{k}\in\mathbb{C}$, not all zero. We want to show
that $r_{1}u^{1,1}_{r_{1}}u^{2,1}+\cdots+r_{k}u^{1,k}_{r_{k}}u^{2,k}=0$ for
some $r_{1},\ldots,r_{k}\in R$, not all zero. We may assume that this set of
vectors is minimal with respect to $\mathbb{C}$-linear dependence, and in
particular, that
$\big{\\{}u^{1,1}_{r_{1}}u^{2,1},\ldots,u^{1,k-1}_{r_{k-1}}u^{2,k-1}\big{\\}}$
is a $\mathbb{C}$-linearly independent set. By rescaling, we may assume

$\displaystyle
u^{1,k}_{r_{k}}u^{2,k}=c_{1}u^{1,1}_{r_{1}}u^{2,1}+\cdots+c_{k-1}u^{1,k-1}_{r_{k-1}}u^{2,k-1}$

for uniquely defined $c_{1},\ldots,c_{k-1}\in\mathbb{C}$.

Proposition 11.9 of [19] asserts that if $w^{1}$ and $w^{2}$ are nonzero
elements of irreducible modules $M_{1}$ and $M_{2}$ of a generalized vertex
algebra, and $\mathcal{Y}$ is an intertwining operator, then
$\mathcal{Y}\big{(}w^{1},z\big{)}w^{2}=0$ implies $\mathcal{Y}$ is identically
zero. Here, our hypothesis is that $V^{a}$ and $V^{-a_{1}}$ are irreducible
$V^{0}$-modules, and the self-duality of the invariant bilinear form on $V$
implies the multiplication operation is nonzero. Thus, for any nonzero
$v^{1}\in V^{-a_{1}}_{R}$, $Y\big{(}v^{1},z\big{)}$ is an injection to
$z^{s}V^{a_{2}}((z))$ for some $s\in\mathbb{Q}$. Thus,

$\displaystyle\big{\\{}Y\big{(}v^{1},z\big{)}u^{1,1}_{r_{1}}u^{2,1},\ldots,Y\big{(}v^{1},z\big{)}u^{1,k-1}_{r_{k-1}}u^{2,k-1}\big{\\}}$

is a $\mathbb{C}$-linearly independent set in $z^{s}V^{a_{2}}((z))$, so

$\displaystyle
Y\big{(}v^{1},z\big{)}u^{1,k}_{r_{k}}u^{2,k}=b_{1}Y\big{(}v^{1},z\big{)}u^{1,1}_{r_{1}}u^{2,1}+\cdots+b_{k-1}Y\big{(}v^{1},z\big{)}u^{1,k-1}_{r_{k-1}}u^{2,k-1}$

holds if and only if $b_{i}=c_{i}$ for all $i\in\\{1,\ldots,k-1\\}$. However,
both $Y\big{(}v^{1},z\big{)}u^{1,k}_{r_{k}}u^{2,k}$ and each summand
$Y\big{(}v^{1},z\big{)}u^{1,i}_{r_{i}}u^{2,i}$ lie in
$z^{s}V^{a_{2}}_{R}((z))$, and by hypothesis, this space is an $R$-form of
some subspace of $z^{s}V^{a_{2}}((z))$. We conclude that all $c_{i}$ lie in
the fraction field of $R$. By clearing denominators, we see that there is some
nonzero $x\in R$ such that

$\displaystyle
xc_{1}u^{1,1}_{r_{1}}u^{2,1}+\cdots+xc_{k-1}u^{1,k-1}_{r_{k-1}}u^{2,k-1}-xu^{1,k}_{r_{k}}u^{2,k}=0$

and in particular, the set
$\big{\\{}u^{1,1}_{r_{1}}u^{2,1},\ldots,u^{1,k}_{r_{k}}u^{2,k}\big{\\}}$ is
$R$-linearly dependent. Thus, $\phi$ is injective, and
$V_{R}\otimes_{R}\mathbb{C}\cong V$ as abelian intertwining algebras.

The Möbius structure on $V_{R}$ follows from the formula describing the action
of $L_{1}^{(n)}$ on products of generators. ∎

We now refine the previous result, showing that $V_{R}$ is self-dual and well-
graded.

######  Proposition 2.40.

Let $R$ be a subring of $\mathbb{C}$ containing $1/N$ and ${\rm e}^{2\pi{\rm
i}/N}$. Let $V$ be an abelian intertwining algebra over $\mathbb{C}$
associated to the data $(A,F,\Omega)$, equipped with an invariant bilinear
form. Suppose the following properties hold:

  1. $1.$

$V$ is self-dual with respect to the invariant form.

  2. $2.$

Each $V^{a}$ is an irreducible $V^{0}$-module.

  3. $3.$

$V$ is generated by abelian intertwining subalgebras
$\big{\\{}V^{A_{i}}\big{\\}}_{i\in I}$, where $A_{i}$ range over a set of
subgroups of $A$ that generate $A$, and $V^{A_{i}}=\bigoplus_{a\in
A_{i}}V^{a}$.

  4. $4.$

We are given Möbius $R$-forms $V^{A_{i}}_{R}$ of each $V^{A_{i}}$ such that
the restriction of the invariant bilinear form takes values in $R$, and
$V^{A_{i}}_{R}$ is self-dual with respect to this bilinear form.

  5. $5.$

The $R$-forms and invariant bilinear forms coincide on pairwise intersections
of the subalgebras $V^{A_{i}}$.

Then, the following holds:

  1. $1.$

The invariant bilinear form on $V$ restricts to an $R$-valued invariant
bilinear form on the $R$-form $V_{R}$ of $V$ given in Proposition 2.39.

  2. $2.$

$V_{R}$ is self-dual with respect to this bilinear form.

  3. $3.$

$V_{R}$ is the unique $R$-form of $V$ whose intersection with $V^{A_{i}}$ is
$V^{A_{i}}_{R}$, such that the invariant bilinear form on $V$ restricts to an
$R$-valued invariant bilinear form.

  4. $4.$

$V_{R}$ is well-graded as an $R$-module, i.e., the graded components
$(V_{R})^{a}_{n}$ are finite projective $R$-modules.

###### Proof.

We begin by considering the invariant inner product. For $u^{1}_{r}u^{2}\in
V^{a}_{R}$ and $v^{1}_{s}v^{2}\in V^{-a}_{R}$, we apply Lemma 2.36 to see that
the inner product is given by

$\displaystyle\big{(}v^{1}_{s}v^{2},u^{1}_{r}u^{2}\big{)}=\bigg{(}v^{2},e(k/2)\sum_{i\geq
0}\big{(}L_{1}^{(i)}v^{1}\big{)}_{2k-i-s-2}u^{1}_{r}u^{2}\bigg{)}$
$\displaystyle\hphantom{\big{(}v^{1}_{s}v^{2},u^{1}_{r}u^{2}\big{)}}{}=e(k/2)\sum_{i\geq
0}\sum_{t=0}^{m}\sum_{n\in\mathbb{Z}_{\geq 0}}\binom{2k-i-s-2-l}{t}\times$
$\displaystyle\hphantom{\big{(}v^{1}_{s}v^{2},u^{1}_{r}u^{2}\big{)}=}{}\times\binom{l}{n}\big{(}v^{2},\big{(}\big{(}L_{1}^{(i)}v^{1}\big{)}_{2k-i-s-2-l-t+n}u^{1}\big{)}_{r+l+t-n}u^{2}\big{)},$

which is an $R$-linear combination of inner products of vectors in
$V^{A_{2}}_{R}$. By hypothesis, this is an element of $R$, so the inner
product on $V_{R}$ is $R$-valued. We therefore find that for each $a\in A$ and
$r\in\mathbb{Q}$, there is a canonical map
$\big{(}V^{a}_{R}\big{)}_{r}\to\operatorname{Hom}_{R}\big{(}\big{(}V^{-a}_{R}\big{)}_{r},R\big{)}$
of $R$-modules induced by the inner product, and by self-duality of the base
change to $\mathbb{C}$, this map is injective. To show that $V_{R}$ is self-
dual, it suffices to show that this map is surjective.

Let $f\colon\big{(}V^{-a}_{R}\big{)}_{r}\to R$ be an $R$-module map. Because
$V$ is self-dual, $f$ is induced by taking the inner product with a unique
element $u\in V^{a}_{r}$. We shall show that
$u\in\big{(}V^{a}_{R}\big{)}_{r}$.

By precomposing $f$ with any $\big{(}v^{1}_{t}\big{)}^{*}$ for homogeneous
$v^{1}\in V^{-a_{1}}_{R}$, we obtain an $R$-linear map from some homogeneous
piece $\big{(}V^{-a_{2}}_{R}\big{)}_{r+\operatorname{wt}(v^{1})-t-1}$ to $R$.
By self-duality of $V^{A_{2}}_{R}$, this is necessarily given by the inner
product with a homogeneous vector
$v^{\prime}\in\big{(}V^{a_{2}}_{R}\big{)}_{r+\operatorname{wt}(v^{1})-t-1}$.
That is,

$\displaystyle\big{(}v^{1}_{t}u,-\big{)}=\big{(}u,\big{(}v^{1}_{t}\big{)}^{*}-\big{)}=f\big{(}\big{(}v^{1}_{t}\big{)}^{*}-\big{)}=(v^{\prime},-).$

In other words, all $v^{1}_{t}$ operators take $u$ to elements of
$V^{a_{2}}_{R}$. This implies that for any $v^{1}\in V^{-a_{1}}_{R}$,
$Y\big{(}v^{1},z\big{)}u\in z^{s}V^{a_{2}}_{R}((z))$ for some
$s\in\mathbb{Q}$. By the skew-symmetry Lemma 2.35, $Y(u,z)$ takes elements of
$V^{-a_{1}}_{R}$ to elements of $z^{s}V^{a_{2}}_{R}((z))$.

Because $V^{A_{1}}_{R}$ is self-dual, we may present $\mathbf{1}$ as a finite
$R$-linear combination of products of the form $v^{1}_{k}u^{1}$, where
$u^{1}\in V^{a_{1}}_{R}$ and $v^{1}\in V^{-a_{1}}_{R}$. Then
$u_{-1}\mathbf{1}=u$ is a finite $R$-linear combination of products of the
form $u_{-1}v^{1}_{k}u^{1}$. By Lemma 2.36, this is an $R$-linear combination
of products of the form $\big{(}u_{s}v^{1}\big{)}_{t}u^{1}$. Since each
$u_{s}v^{1}\in V^{a_{2}}_{R}$, we find that $u\in V^{a}_{R}$. Thus, the inner
product on $V_{R}$ is self-dual.

Self-duality implies each of the graded pieces $(V^{a}_{R})_{r}$ are finite
projective $R$-modules, because finite projective $R$-modules are precisely
the dualizable $R$-modules. That is, $V_{R}$ is well-graded as an $R$-module.
Furthermore (as an anonymous referee has pointed out), self-duality of $V_{R}$
implies uniqueness of $R$-forms containing the subalgebras $V^{A_{i}}_{R}$
with $R$-valued inner product. This is because $V_{R}$ is generated by those
subalgebras, so any other such form must strictly contain $V_{R}$,
contradicting self-duality of $V_{R}$. ∎

######  Corollary 2.41.

Let $A$ be a finite abelian group, and let $A_{1}$, $A_{2}$ be subgroups that
generate $A$. Let $n$ be the exponent of $A$, and let $R$ be a subring of
$\mathbb{C}$ containing $1/n$ and $e(1/2n)$. Let $V$ be a simple vertex
operator algebra over $\mathbb{C}$, and suppose $V$ admits an $A$-grading
$($i.e., given by an action of the Pontryagin dual group $A^{*})$. If we are
given self-dual $R$-forms $V_{R}^{1}$ and $V_{R}^{2}$ of the vertex operator
subalgebras of $V$ given by the parts graded by $A_{1}$ and $A_{2}$, such that
the $A_{1}\cap A_{2}$-graded subalgebras of $V_{1}$ and $V_{2}$ are
isomorphic, then there exists a unique $R$-form of $V$ whose $A_{1}$-graded
subalgebra is $V_{1}$ and whose $A_{2}$-graded subalgebra is $V_{2}$.

###### Proof.

By Theorem 3 of [23] our assumption that $V$ is simple implies the graded
pieces of $V$ are simple $V^{0}$-modules. Then the hypotheses of Proposition
2.40 are satisfied, where $(F,\Omega)$ is trivial. ∎

###  2.4 The standard form for a lattice vertex operator algebra

A description of an integral form for a lattice vertex algebra is given in the
original paper [2] where vertex algebras are defined, and more properties are
established in [5]. We will follow the treatments in [18] and [38], because
the proofs are somewhat more detailed.

Let $L$ be an even integral lattice, i.e., a finite rank free abelian group
equipped with a $\mathbb{Z}$-valued bilinear form that is even on the
diagonal. Then there is a double cover $\hat{L}$, written as a nontrivial
central extension by $\langle\kappa\rangle\cong\\{\pm 1\\}$, that is unique up
to non-unique isomorphism. Choosing lifts $\\{e_{a}\\}_{a\in L}$ of lattice
vectors, the cocycle defining the central extension is determined up to
equivalence by the signs relating $e_{a}e_{b}$ to $e_{b}e_{a}$, and in our
case it is by $e_{a}e_{b}=(-1)^{(a,b)}e_{b}e_{a}$. The twisted group ring
$\mathbb{C}\\{L\\}$ is then the quotient of $\mathbb{C}\big{[}\hat{L}\big{]}$
by the ideal $(\kappa+1)$, and we write $\iota(e_{a})$ for the image of
$e_{a}$. The vertex algebra $V_{L}$ is given by the tensor product of
$\mathbb{C}\\{L\\}$ with a Heisenberg vertex algebra $M(1)$ which we will not
describe further.

If we choose a basis $\\{\gamma_{1},\ldots,\gamma_{d}\\}$ of $L$, then for any
$\alpha\in L^{\vee}$, we set

$\displaystyle
E^{-}(-\alpha,z)=\exp\left(\sum_{n>0}\frac{\alpha(-n)}{n}z^{n}\right)=\sum_{n\geq
0}s_{\alpha,n}z^{n}.$

Note that $E^{-}(-\alpha,z)E^{-}(\alpha,z)=1$ for all $\alpha\in L$, i.e.,
$\sum\limits_{i=0}^{n}s_{\alpha,i}s_{-\alpha,n-i}=\delta_{n,0}$ for all $n\geq
0$. Define $(V_{L})_{\mathbb{Z}}$ to be the $\mathbb{Z}$-span of
$s_{\alpha_{1},n_{1}}\cdots s_{\alpha_{k},n_{k}}{\rm e}^{\alpha}$ for
$\alpha_{i}\in\\{\gamma_{1},\ldots,\gamma_{d}\\}$, $n_{1}\geq\cdots\geq
n_{k}$, $k\geq 0$, and $\alpha\in L$. Here, ${\rm e}^{\alpha}$ denotes the
image of $\iota(e_{\alpha})$ for some lift $e_{\alpha}$ – this choice of
notation has an ambiguous sign, but in this paper we will not do calculations
where the choice of sign is important. By Proposition 3.6 of [18],
$(V_{L})_{\mathbb{Z}}$ is an integral form of $V_{L}$, generated by ${\rm
e}^{\pm\gamma_{i}}$, and if $L$ is positive definite and unimodular, then
$(V_{L})_{\mathbb{Z}}$ is a direct sum of positive definite unimodular
lattices under its usual invariant bilinear form. By Proposition 5.8 of [38],
there is a conformal element $\omega$ in an integral form for $V_{L}$ if and
only if $L$ is unimodular, and the central charge is equal to the rank.

######  Definition 2.42.

Let $L$ be a positive definite even unimodular lattice, and let $R$ be a
commutative ring. We call the base change
$(V_{L})_{\mathbb{Z}}\otimes_{\mathbb{Z}}R$ the standard $R$-form of $V_{L}$,
and denote it by $(V_{L})_{R}$.

######  Lemma 2.43.

Let $L$ be a positive definite even unimodular lattice, and let $R$ be a
commutative ring. The standard $R$-form $(V_{L})_{R}$ is a Möbius vertex
operator algebra over $R$, and admits an invariant bilinear form for which it
is self-dual.

###### Proof.

The Möbius claim is a special case of Lemma 5.6 of [5], and in fact Borcherds
proves the claim for an action of an integral form of the universal enveloping
algebra of the Virasoro algebra. The existence of the bilinear form is
asserted in [2], and self-duality is proved as Proposition 3.6 of [18]. ∎

We describe some finer details of automorphisms.

######  Proposition 2.44.

Let $L$ be a positive definite even unimodular lattice of rank $d$ with no
roots $($i.e., no vectors of norm $2)$. The contravariant functor that sends
an affine scheme $\operatorname{Spec}R$ to the automorphism group of
$(V_{L})_{R}$ is represented by a finite type affine group scheme
$\operatorname{\underline{Aut}}(V_{L})_{\mathbb{Z}}$ over $\mathbb{Z}$. This
group scheme has the form $TO\big{(}\hat{L}\big{)}$, where $T$ is a normal
subgroup scheme that is a split torus over $\mathbb{Z}$, and
$O\big{(}\hat{L}\big{)}$ is the finite flat group scheme of isometries of the
double cover of $L$, and lies in a canonical exact sequence

$\displaystyle 1\to\operatorname{\underline{Hom}}(L,\mu_{2})\to
O\big{(}\hat{L}\big{)}\to O(L)\to 1.$

The torus $T$ is given as the diagonalizable group
$D(L)=\operatorname{Spec}\mathbb{Z}[L]$, isomorphic to $\mathbb{G}_{m}^{d}$.
The intersection between $T$ and $O\big{(}\hat{L}\big{)}$ is
$\operatorname{\underline{Hom}}(L,\mu_{2})$, which is isomorphic to the
constant group scheme $\\{\pm 1\\}^{d}$ over any field of characteristic not
equal to $2$.

###### Proof.

In the beginning of Section 2 of [17], we have a description of the
automorphism group of any finitely generated vertex operator algebra $V$ over
$\mathbb{C}$ as a closed subgroup of ${\rm GL}(U)$ for
$U=\bigoplus_{m=0}^{k}V_{m}$ a generating subspace. The arguments given there
extend straightforwardly to any finitely generated vertex operator algebra
over a commutative ring in a way that commutes with base change, so the
automorphism functor is represented by a finite type affine group scheme.

Let $\mathfrak{t}_{R}$ denote the weight 1 subspace of $(V_{L})_{R}$. By our
assumption that $L$ has no roots, $\mathfrak{t}$ is isomorphic to $L\otimes R$
as an $R$-module with inner product given by $(a,b)\mathbf{1}=a_{1}b$. By [2],
$\mathfrak{t}_{R}$ has a canonical Lie algebra structure given by
$[a,b]=a_{0}b$, and in this case it is abelian. Consider an automorphism
$\sigma$ of $(V_{L})_{R}$. Because $\sigma$ respects the weight grading, it
acts as an $R$-linear isometry on $\mathfrak{t}_{R}$ that induces an isometry
on the $L$-grading of $(V_{L})_{R}$. Thus, the isometry on $\mathfrak{t}_{R}$
is defined over $\mathbb{Z}$, i.e., the restriction is given by some
$\bar{\sigma}\in O(L)$.

The group $O\big{(}\hat{L}\big{)}$ acts by permutations on the set
$\\{\epsilon{\rm e}^{\alpha}\\}_{\epsilon\in\mu_{2}(R),\alpha\in L}$, and this
action induces vertex operator algebra automorphisms on $(V_{L})_{R}$ that act
by $O(L)$ on the $L$-grading. There exists a lift $\tau$ of $\bar{\sigma}$ to
$O\big{(}\hat{L}\big{)}$, so $\psi=\sigma\tau^{-1}$ is an automorphism of
$(V_{L})_{R}$ that fixes $\mathfrak{t}_{R}$ pointwise and hence fixes the
$L$-grading. Because the generators ${\rm e}^{\pm\gamma_{i}}$ are minimal
weight in their respective $L$-graded components, $\psi$ must act on these
generators by scalars. Furthermore, Lemma 2.5 in [24] (which is proved for the
case $R=\mathbb{C}$ but extends without change) implies this $\psi$ is
necessarily an element of the torus $\operatorname{Hom}(L,R^{\times})$, which
is the group of $R$-points of $D(L)$. The $L$-grading given by assigning
degree $\alpha$ to ${\rm e}^{\alpha}$ endows $(V_{L})_{\mathbb{Z}}$ with a
$\mathbb{Z}[L]$-comodule structure, so the automorphism group contains the
rank $d$ split torus $D(L)$. We conclude that
$\operatorname{\underline{Aut}}(V_{L})_{\mathbb{Z}}=TO\big{(}\hat{L}\big{)}$.

Because $T$ acts trivially on $\mathfrak{t}_{R}$, it is clear that the
intersection of $T(R)$ with $O\big{(}\hat{L}\big{)}$ is the preimage of the
identity element of $O(L)$, and this is precisely
$\operatorname{Hom}(L,\mu_{2}(R))$. ∎

######  Lemma 2.45.

Let $L$ be a positive definite even unimodular lattice of rank $d$ with no
roots, and let $\bar{g}$ and $\bar{h}$ be commuting automorphisms of $L$. Then
for any lifts $\hat{g}$ of $\bar{g}$ and $\hat{h}$ of $\bar{h}$ to
$\operatorname{Aut}(V_{L})_{R}$, we have
$\hat{g}\hat{h}=c_{\hat{g},\hat{h}}\hat{h}\hat{g}$ for some
$c_{\hat{g},\hat{h}}\in T(R)$. Furthermore, if $\tilde{g}=\gamma\hat{g}$ and
$\tilde{h}=\delta\hat{h}$ are different lifts, where $\gamma,\delta\in T(R)$,
then

$\displaystyle
c_{\tilde{g},\tilde{h}}=\frac{\bar{g}\cdot\delta}{\delta}\frac{\gamma}{\bar{h}\cdot\gamma}c_{\hat{g},\hat{h}},$

where $\bar{g}\cdot\delta$ denotes the image of $\delta$ under the canonical
action of $\operatorname{Aut}L$ on $T(R)=\operatorname{Hom}(L,R^{\times})$.

###### Proof.

The first claim is straightforward from the description of the automorphism
group of $(V_{L})_{R}$ given in Proposition 2.44. The second claim follows
from a short calculation, essentially using the fact that
$\hat{g}\delta=(\bar{g}\cdot\delta)\hat{g}$. ∎

######  Lemma 2.46.

Let $L$ be a positive definite even unimodular lattice of rank $d$ with no
roots, let $\bar{g}$ be a fixed-point free automorphism of $L$ such that all
nontrivial powers are also fixed-point free, and let $n$ be its order. Then
the set of automorphisms $g$ of $(V_{L})_{R}$ that map to $\bar{g}$ in $O(L)$
is a torsor under $T(R)$, and each such lift has order $n$. Given any pair
$\hat{g}$ and $\tilde{g}$ of lifts of $\bar{g}$ to
$\operatorname{Aut}(V_{L})_{R}$, there exists an extension $R^{\prime}$ of $R$
given by adjoining finitely many roots of units, such that $\tilde{g}$ and
$\hat{g}$ are conjugate in $\operatorname{Aut}(V_{L})_{R^{\prime}}$, in fact
by an element of $T(R^{\prime})$. In particular, if $R^{\times}$ is
$n$-divisible $($i.e., each unit has an $n$th root$)$, then all lifts of
$\bar{g}$ are conjugate in $\operatorname{Aut}(V_{L})_{R}$, and if $R$ is a
subring of $\mathbb{C}$ containing $e(1/2n)$, then all lifts of $\bar{g}$ in
$\operatorname{Aut}(V_{L})_{\mathbb{Z}}$ are conjugate in
$\operatorname{Aut}(V_{L})_{R}$. Finally, if $R$ is a subring of $\mathbb{C}$
containing $\frac{1}{n}$ and $e(1/n)$, then for any fixed lift $g$ of
$\bar{g}$, $(V_{L})_{R}$ splits into a direct sum of irreducible
$(V_{L})_{R}^{g}$-modules
$\big{\\{}(V_{L})_{R}^{g=e(k/n)}\big{\\}}_{k=0}^{n-1}$, and the invariant
bilinear form induces a homogeneous perfect pairing between
$(V_{L})_{R}^{g=e(k/n)}$ and $(V_{L})_{R}^{g=e(-k/n)}$.

###### Proof.

The parametrization of lifts of $\bar{g}$ by a $T(R)$-torsor is given in
Proposition 2.44. To show that each lift has order $n$, we first show that
there is a lift to $O\big{(}\hat{L}\big{)}$ with order $n$. The obstruction to
the existence of an order $n$ lift of any order $n$ automorphism is described
in [3, Lemma 12.1] in the case of the Leech lattice, but the argument applies
in general. Namely, if $n$ is odd, there is no obstruction, and if $n$ is
even, there is an obstruction if and only if $\big{(}\bar{g}^{n/2}v,v\big{)}$
is an odd integer for some $v\in L$. Since we assume $\bar{g}^{n/2}$ is a
fixed-point free element of order 2, all of its eigenvalues are $-1$, so
$\big{(}\bar{g}^{n/2}v,v\big{)}=(-v,v)\in 2\mathbb{Z}$, and the obstruction
vanishes.

For the general problem of conjugation, we consider lifts $\hat{g}$ and
$\tilde{g}$ of $\bar{g}$, and note that they satisfy $\gamma\hat{g}=\tilde{g}$
for a unique $\gamma\in T(R)$. If there is some $\delta\in T(R^{\prime})$ such
that $\delta^{-1}\hat{g}\delta=\tilde{g}$ for some extension $R^{\prime}$ of
$R$, then by Lemma 2.45, we have $\delta^{-1}(\bar{g}\cdot\delta)=\gamma$. We
note that by the identification $T(R)=\operatorname{Hom}(L,R^{\times})$, we
may define $\gamma$ by the values in $R^{\times}$ it takes on a basis of $L$.
Then, if we extend $R$ to $R^{\prime}$ by adjoining $n$-th roots of those
values, we find that $\gamma$ has an $n$-th root $\gamma^{\prime}\in
T(R^{\prime})$. Then by setting

$\displaystyle\delta=\gamma^{\prime}(\bar{g}\cdot\gamma^{\prime})^{2}\cdots\big{(}\bar{g}^{n-1}\cdot\gamma^{\prime}\big{)}^{n}\in
T(R^{\prime}),$

we find that $\delta^{-1}(\bar{g}\cdot\delta)=(\gamma^{\prime})^{n}=\gamma$.
Thus,

$\displaystyle\delta^{-1}\hat{g}\delta=\delta^{-1}(\bar{g}\cdot\delta)\hat{g}=\gamma\hat{g}=\tilde{g},$

so $\hat{g}$ and $\tilde{g}$ are conjugate by an element of $T(R^{\prime})$.

For the claim about $n$-divisible $R^{\times}$, the identification
$T(R)=\operatorname{Hom}(L,R^{\times})$ implies $T(R)$ is also $n$-divisible.
For the claim about subrings of $\mathbb{C}$, we note that
$\operatorname{Aut}(V_{L})_{\mathbb{Z}}=O\big{(}\hat{L}\big{)}$, so any
discrepancy $\gamma$ of lifts of $\bar{g}$ necessarily lies in
$T(\mathbb{Z})=\operatorname{Hom}(L,\pm 1)$. Thus, if $R$ contains $e(1/2n)$,
then $T(R)$ contains all $n$-th roots of elements of $T(\mathbb{Z})$.

The last claim follows immediately from Lemma 2.33. ∎

######  Proposition 2.47.

Let $\Lambda$ be the Leech lattice, i.e., the unique positive definite even
unimodular lattice of rank $24$ with no roots, and let $\bar{g}$ be a fixed-
point free automorphism of prime order $p$ $($therefore, an element in one of
the classes $2a$, $3a$, $5a$, $7a$, $13a$ according to the notation of [7]).
Let $R$ be a subring of $\mathbb{C}$ that contains $e(1/p)$, and let $g$ be a
lift of $\bar{g}$ to $(V_{\Lambda})_{R}$. Then
$C_{\operatorname{Aut}(V_{\Lambda})_{R}}(g)\cong
C_{\operatorname{Aut}V_{\Lambda}}(g)$, and in particular, has the form
$p^{24/(p-1)}.C_{Co_{0}}(\bar{g})$.

###### Proof.

It suffices to show that any automorphism of $(V_{\Lambda})_{\mathbb{C}}$ that
commutes with $g$ also preserves the standard $R$-form, viewed as an
$R$-submodule. From the description of the automorphism group given in
Proposition 2.44, it is clear that both the $\mathbb{C}$-form and the $R$-form
have the $O\big{(}\hat{\Lambda}\big{)}$ part of the automorphism group in
common, so it suffices to show that all centralizing complex elements in the
torus $T=D(\Lambda)$ are defined over $R$.

We claim that the split torus $T$ equivariantly decomposes under $\bar{g}$ as
a direct sum of $\frac{24}{p-1}$ copies of the torus $\mathbb{G}_{m}^{p-1}$
with $\bar{g}$ acting on points as

$\displaystyle(a_{1},\ldots,a_{p-1})\mapsto\left(\frac{1}{a_{1}\cdots
a_{p-1}},a_{1},\ldots,a_{p-2}\right).$

To show this, we may use the fact that $D$ gives an involutive anti-
equivalence between the category of split tori and the category of free
abelian groups of finite rank (see, e.g., [15, Exposé VIII, Section 1]). The
claim then follows from the classification of indecomposable $\mathbb{Z}$-free
$\mathbb{Z}[\mathbb{Z}/p\mathbb{Z}]$-modules given in [16] (see also [14,
Theorem 74.3]), and in particular, the fact that only one isomorphism type is
fixed-point free.

Any centralizing $\mathbb{C}$-point $(a_{1},\ldots,a_{p-1})$ in the torus
$\mathbb{G}_{m}^{p-1}$ is fixed by this action of $\bar{g}$, so the
coordinates satisfy

$\displaystyle a_{1}=\cdots=a_{p-1}=\frac{1}{a_{1}\cdots
a_{p-1}}\in\mu_{p}(\mathbb{C}).$

However, $R$ contains a full set of $p$-th roots of unity by our hypothesis,
so all of the centralizing complex points in $T$ are defined over $R$. ∎

###  2.5 The cyclic orbifold construction

We recall that if $V$ is a simple, $C_{2}$-cofinite, holomorphic vertex
operator algebra $V$, and $g$ is an automorphism of finite order $n$, then by
Theorem 10.3 of [21], there is a unique $g$-twisted $V$-module, up to
isomorphism (which we will call $V(g)$), and its $L(0)$-spectrum lies in some
coset of $\frac{1}{n}\mathbb{Z}$ in $\mathbb{Q}$. We say that $g$ is anomaly-
free if this coset is $\frac{1}{n}\mathbb{Z}$, and we say that $g$ is
anomalous otherwise.

The construction that makes this paper possible is the following: By Theorem
5.15 of [45], if $V$ is a simple, $C_{2}$-cofinite, holomorphic vertex
operator algebra $V$ of CFT type, and $g$ is an automorphism of finite order
$n$, such that the nontrivial irreducible twisted modules $V(g^{i})$ have
strictly positive $L(0)$-spectrum, then there is some
$t\in\mathbb{Z}/n\mathbb{Z}$ (uniquely determined by the property that the
$L(0)$-spectrum of $V(g)$ lies in $\frac{t}{n^{2}}+\frac{1}{n}\mathbb{Z}$) and
an abelian intertwining algebra structure on
${}^{g}V=\bigoplus_{i=0}^{n-1}V(g^{i})$, graded by an abelian group $D$ that
lies in an exact sequence $0\to\mathbb{Z}/n\mathbb{Z}\to
D\to\mathbb{Z}/n\mathbb{Z}\to 0$, with addition law determined by the “add
with carry” 2-cocycle

$\displaystyle c_{2t}(i,k)=\begin{cases}0,&i_{n}+k_{n}<n,\\\
2t,&i_{n}+k_{n}\geq n,\end{cases}$

where the notation $i_{n}$ denotes the unique representative of
$i\in\mathbb{Z}/n\mathbb{Z}$ in $\\{0,\ldots,n-1\\}$. By loc. cit. Proposition
5.13, the quadratic form $q_{\Delta}$ on $D$ given by conformal weights is
isomorphic to the discriminant form on the even lattice with Gram matrix
$\left(\begin{smallmatrix}-2t_{n}&n\\\ n&0\end{smallmatrix}\right)$.

Furthermore, by Theorem 5.16, if $t=0$ (i.e., $g$ is anomaly-free), then the
abelian intertwining algebra ${}^{g}V$ is naturally graded by
$D=\mathbb{Z}/n\mathbb{Z}\times\mathbb{Z}/n\mathbb{Z}$, such that $V$ is the
sum of the degree $(0,i)$ pieces, and that there is a simple $C_{2}$-cofinite,
holomorphic vertex operator algebra $V/g$ of CFT type given by the sum of the
degree $(j,0)$ pieces, for $0\leq j<n$. The natural
$\mathbb{Z}/n\mathbb{Z}$-grading from this decomposition endows $V/g$ with a
canonical automorphism $g^{*}$ whose order is equal to $|g|$, such that
$(V/g)/g^{*}\cong V$ and $g^{**}=g$. More generally, they showed that if $H$
is any order $n$ subgroup of
$\mathbb{Z}/n\mathbb{Z}\times\mathbb{Z}/n\mathbb{Z}$ that is isotropic with
respect to $q_{\Omega}$, then $\bigoplus_{a\in H}({}^{g}V)^{a}$ is a
holomorphic $C_{2}$-cofinite vertex operator algebra of CFT type.

######  Proposition 2.48.

Let $V$ be a holomorphic $C_{2}$-cofinite vertex operator algebra of CFT type,
and let $g$ be an anomaly-free automorphism of order $n$. Suppose both $V$ and
$V/g$ admit $R$-forms $V_{R}$ and $(V/g)_{R}$ for some subring
$R\subset\mathbb{C}$ containing $1/n$ and $e(1/2n)$, such that the $R$-forms
coincide in $V\cap V/g$ in ${}^{g}V$, and both $g$ and $g^{*}$ are
automorphisms of the respective $R$-forms. Suppose further that both $V_{R}$
and $(V/g)_{R}$ admit $R$-invariant bilinear forms for which they are self-
dual, and that coincide on their intersection in ${}^{g}V$, and assume that
$g$ and $g^{*}$ preserve the bilinear form. Let $G$ be the automorphism group
of $V_{R}$ and let $G^{*}$ be the automorphism group of $(V/g)_{R}$. Then:

  1. $1.$

The abelian intertwining algebra ${}^{g}V$ has a unique $R$-form
$({}^{g}V)_{R}$ with a bilinear form extending those on $V_{R}$ and
$(V/g)_{R}$, and it is self-dual with respect to this form.

  2. $2.$

The group of homogeneous automorphisms of $({}^{g}V)_{R}$ is equal to a
central extension of $C_{G}(g)$ by $\langle g^{*}\rangle$, and also a central
extension of $C_{G^{*}}(g^{*})$ by $\langle g\rangle$.

  3. $3.$

For any divisor $d$ of $n$,
$\bigoplus_{j=0}^{n/d-1}\bigoplus_{i=0}^{d-1}({}^{g}V)_{R}^{dj,(n/d)i}$ is an
$R$-form of $V/g^{d}$ that is self-dual with respect to the induced invariant
inner product.

###### Proof.

The first claim follows immediately from Proposition 2.40.

For the second claim, we note that restriction yields the following
commutative diagram:

$\textstyle{\operatorname{Aut}({}^{g}V)_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{\operatorname{Aut}V_{R}}(g)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{\operatorname{Aut}(V/g)_{R}}(g^{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Aut}\big{(}V_{R}^{g}\big{)},}$

where the centralizers of $g$ (resp. $g^{*}$) are precisely the groups of
automorphisms that are compatible with the grading by eigenspaces for $g$
(resp. $g^{*}$). It suffices to show that the maps out of
$\operatorname{Aut}({}^{g}V)_{R}$ are surjective with cyclic central kernel of
order $|g|$, and the argument given in the proof of Proposition 2.5.2 of [8]
works here with minimal change.

For the third claim, the fact that
$\bigoplus_{j=0}^{n/d-1}\bigoplus_{i=0}^{d-1}{}^{g}V^{dj,(n/d)i}$ is a
holomorphic vertex operator algebra follows from the fact that the group of
degrees in question is isotropic of order $n$, and the identification with
$V/g^{d}$ is straightforward. The fact that we have an $R$-form that is self-
dual follows from the corresponding claims for $({}^{g}V)_{R}$. ∎

##  3 Forms of the monster vertex operator algebra

We now use the tools from the previous section to construct $R$-forms of
$V^{\natural}$, as $R$ ranges over some cyclotomic $S$-integer rings.

###  3.1 The Abe–Lam–Yamada method

In [1], several constructions of $V^{\natural}$ were given by cyclic orbifolds
of odd prime order $p$ on $V_{\Lambda}$, and analyzed using cyclic orbifolds
of order $2p$ in order to produce a comparison with the original order 2
orbifold construction of [27]. We will apply a similar method to produce
actions of the monster on orbifolds over various rings.

To be specific, we consider cyclic orbifolds of $V_{\Lambda}$ with respect to
lifts of fixed-point free isometries of $\Lambda$, such that those whose order
is prime yield $V^{\natural}$, and those whose order is a product of two
primes yield $V_{\Lambda}$. The key is that we may use Proposition 2.40 to
produce self-dual $R$-forms of abelian intertwining algebras from those
orbifolds yielding $V_{\Lambda}$, and this automatically yields self-dual
$R$-forms for $V^{\natural}$ by restriction.

We summarize the information about cyclic orbifolds over $\mathbb{C}$ that we
need.

######  Lemma 3.1.

Let $P_{0}=\\{2,3,5,7,13\\}$. Then:

  1. $1.$

$P_{0}$ is the set of primes $p$ such that there exists a fixed-point free
automorphism of the Leech lattice of order $p$.

  2. $2.$

For each $p\in P_{0}$, there is a unique conjugacy class $[\bar{g}_{p}]$ of
fixed-point free automorphisms in $Co_{0}$ of order $p$, and there exists a
unique conjugacy class $[g_{p}]$ of automorphisms of $V_{\Lambda}$ lifting
$[\bar{g}_{p}]$. For any representative element $g_{p}$, the order of $g_{p}$
is $p$, and we have an isomorphism $V_{\Lambda}/g_{p}\cong V^{\natural}$ of
vertex operator algebras over $\mathbb{C}$.

  3. $3.$

Given a pair $p_{1}$, $p_{2}$ of distinct elements of $P_{0}$, if there exists
an automorphism of $\Lambda$ of order $p_{1}p_{2}$, then there is a unique
algebraic conjugacy class $[\bar{g}_{p_{1}p_{2}}]$ of automorphisms of
$\Lambda$ such that $\bar{g}_{p_{1}p_{2}}^{p_{1}}\in[\bar{g}_{p_{2}}]$ and
$\bar{g}_{p_{1}p_{2}}^{p_{2}}\in[\bar{g}_{p_{1}}]$. When such an automorphism
exists, it is fixed-point free, and there exists a unique algebraic conjugacy
class $[g_{p_{1}p_{2}}]$ of automorphisms of $V_{\Lambda}$ lifting
$[\bar{g}_{p_{1}p_{2}}]$. For any representative element $g_{p_{1}p_{2}}$, we
have an isomorphism $V_{\Lambda}/g_{p_{1}p_{2}}\cong V_{\Lambda}$ of vertex
operator algebras over $\mathbb{C}$.

###### Proof.

All of the claims about automorphisms of $\Lambda$ can be checked by
examination of characters and power maps of $Co_{0}$ in [7]. As it happens,
each $[\bar{g}_{p}]$ is labeled $p$a with frame shape $1^{-24/p-1}p^{24/p-1}$,
and each $[\bar{g}_{p_{1}p_{2}}]$ is labeled $p_{1}p_{2}$a with frame shape
$1^{k}p_{1}^{-k}p_{2}^{-k}(p_{1}p_{2})^{k}$, with the exception of order 39,
where (39a, 39b) is an algebraically conjugate pair satisfying our criteria.
We note that the inadmissible pairs are those satisfying
$p_{1}p_{2}\in\\{65,91\\}$.

The claims about existence and uniqueness of lifts of automorphisms to
$V_{\Lambda}$ follow from Section 4.2 of [32], and a short version of the
argument is Proposition 2.1 in [9]. The identification of prime order cyclic
orbifolds with $V^{\natural}$ follows from the main construction of [27] for
$p=2$, Theorem 1.1 of [10] for $p=3$, and Theorem 4.4 in [1] for $p=5,7,13$.
The identification of order $p_{1}p_{2}$ cyclic orbifolds with $V_{\Lambda}$
follows from Theorem 4.1 in [1] for $p_{1}p_{2}\in\\{6,10,14,26\\}$, and for
the others, the result follows from essentially the same argument: it suffices
to show that the weight 1 subspace of the irreducible twisted module
$V_{\Lambda}(g_{p_{1}p_{2}})$ has dimension $\frac{24}{(p_{1}-1)(p_{2}-1)}$,
and one can do this by manipulating the frame shape. ∎

We note that for $p\in\\{3,5,7,13\\}$, these orbifold constructions were
conjectured in [27], and partially worked out in [22] and [40].

We now consider forms over rings. Note that in this section, we are not
claiming that any particular $R$-form of $V^{\natural}$ necessarily carries
monster symmetry.

######  Lemma 3.2.

Let $P_{0}=\\{2,3,5,7,13\\}$, let $p$ and $q$ be distinct elements of $P_{0}$
such that $pq\not\in\\{65,91\\}$, and let $R=\mathbb{Z}[1/pq,e(1/2pq)]$. Let
$g$ be an automorphism of $V_{\Lambda}$ in the class $[g_{pq}]$ described in
Lemma 3.1 $($namely the fixed-point free class $pq\mathrm{a}$ in [7]) such
that $g$ preserves $(V_{\Lambda})_{\mathbb{Z}}$ $($such $g$ exists by Lemma
2.46). Then ${}^{g}V_{\Lambda}$ has a unique self-dual $R$-form that restricts
to the standard $R$-forms on $V_{\Lambda}$ and $(V_{\Lambda})/g\cong
V_{\Lambda}$.

###### Proof.

Consider the decomposition $\bigoplus_{i=0}^{pq-1}V^{i}_{R}$ of the standard
$R$-form of $V_{\Lambda}$ into eigen-$R$-modules for $g$. This decomposition
preserves self-duality by Lemma 2.33. By Lemma 3.1, the abelian intertwining
subalgebras $\bigoplus_{i=0}^{pq-1}({}^{g}V_{\Lambda})^{i,0}$ and
$\bigoplus_{j=0}^{pq-1}({}^{g}V_{\Lambda})^{0,j}$ are isomorphic to
$V_{\Lambda}$ under its decomposition $\bigoplus_{i=0}^{pq-1}V^{i}$ into
eigenspaces for $g$. Fix embeddings $\phi,\psi\colon
V_{\Lambda}\to{}^{g}V_{\Lambda}$ such that $\phi|_{V^{i}}$ is an isomorphism
to $V^{i,0}$ and $\psi|_{V^{j}}$ is an isomorphism $V^{0,j}$, and
$\phi|_{V^{0}}=\psi|_{V^{0}}$. These embeddings are unique up to composition
with automorphisms of $V_{\Lambda}$ that commute with $g$. Then restriction to
$(V_{\Lambda})_{R}$ yields embeddings that satisfy the hypotheses of
Proposition 2.40. Thus, there is a unique $R$-form $({}^{g}V_{\Lambda})_{R}$
for ${}^{g}V_{\Lambda}$ that extends the $R$-forms given by $\phi$ and $\psi$,
and by the uniqueness of $\phi$ and $\psi$, it is the unique $R$-form that
restricts to the standard $R$-form on the two copies of $V_{\Lambda}$.
Furthermore, the proposition asserts that $({}^{g}V_{\Lambda})_{R}$ admits a
unique invariant bilinear form that extends the form on each copy of
$V_{\Lambda}$, and $({}^{g}V_{\Lambda})_{R}$ is self-dual under this form. ∎

######  Proposition 3.3.

With notation as in Lemma 3.2, the abelian intertwining subalgebras

$\displaystyle\bigoplus_{i=0}^{q-1}\bigoplus_{j=0}^{p-1}({}^{g}V_{\Lambda})_{R}^{pi,qj}\qquad\text{and}\qquad\bigoplus_{i=0}^{p-1}\bigoplus_{j=0}^{q-1}({}^{g}V_{\Lambda})_{R}^{qi,pj}$

of $({}^{g}V_{\Lambda})_{R}$ are isomorphic self-dual $R$-forms of
$V^{\natural}$.

###### Proof.

By Lemma 3.1, the two abelian intertwining subalgebras
$\bigoplus_{i=0}^{q-1}\bigoplus_{j=0}^{p-1}({}^{g}V_{\Lambda})^{pi,qj}$ and
$\bigoplus_{i=0}^{p-1}\bigoplus_{j=0}^{q-1}({}^{g}V_{\Lambda})^{qi,pj}$ are
both isomorphic to $V^{\natural}$. Thus, the abelian intertwining subalgebras
$\bigoplus_{i=0}^{q-1}\bigoplus_{j=0}^{p-1}({}^{g}V_{\Lambda})_{R}^{pi,qj}$
and
$\bigoplus_{i=0}^{p-1}\bigoplus_{j=0}^{q-1}({}^{g}V_{\Lambda})_{R}^{qi,pj}$
are $R$-forms for $V^{\natural}$ that are self-dual under the induced
invariant bilinear form.

We now consider the automorphism of ${}^{g}V_{\Lambda}$ given by switching
coordinates, i.e., sending
$({}^{g}V_{\Lambda})^{i,j}\mapsto({}^{g}V_{\Lambda})^{j,i}$. Explicitly, the
map is defined by taking the composite isomorphisms

$\displaystyle({}^{g}V_{\Lambda})^{i,0}\overset{\phi}{\leftarrow}V^{i}\overset{\psi}{\to}({}^{g}V_{\Lambda})^{0,i}$

and their inverses, and extending uniquely to ${}^{g}V_{\Lambda}$ by the fact
that ${}^{g}V_{\Lambda}$ is generated by the images of $\phi$ and $\psi$. This
automorphism restricts to an automorphism of the $R$-form
$({}^{g}V_{\Lambda})_{R}$, and it transports the two $R$-forms of
$V^{\natural}$ to each other. Thus, the two $R$-forms of $V^{\natural}$ are
isomorphic. ∎

######  Definition 3.4.

We write $V^{\natural}[1/pq,e(1/2pq)]$ to denote the $R$-form of
$V^{\natural}$ given in Proposition 3.3.

######  Remark 3.5.

Because $R$ is a subring of $\mathbb{C}$ containing $e(1/2pq)$, Lemma 2.46
implies $g$ is unique up to conjugation. Thus, the formation of
$V^{\natural}[1/pq,e(1/2pq)]$ does not depend on our choice of lift $g$.

The following lemma implies the automorphism $g^{*}$ is compatible with the
cyclic orbifold duals of $V_{\Lambda}$ arising from $g^{p}$ and $g^{q}$, in
the sense that $(g^{p})^{*}=(g^{*})^{p}$ and $(g^{q})^{*}=(g^{*})^{q}$.

######  Lemma 3.6.

With notation as in Lemma 3.2, there are unique self-dual $R$-forms of
${}^{g^{p}}V_{\Lambda}$ and ${}^{g^{q}}V_{\Lambda}$ such that:

  1. $1)$

${}^{g^{p}}V_{\Lambda}^{i,0}=V^{\natural}[1/pq,e(1/2pq)]^{(g^{*})^{p}=e(i/q)}$
and
${}^{g^{q}}V_{\Lambda}^{i,0}=V^{\natural}[1/pq,e(1/2pq)]^{(g^{*})^{q}=e(i/p)}$
for all $i\in\mathbb{Z}$,

  2. $2)$

${}^{g^{p}}V_{\Lambda}^{0,j}=(V_{\Lambda})_{R}^{g^{p}=e(j/q)}$ and
${}^{g^{q}}V_{\Lambda}^{0,j}=(V_{\Lambda})_{R}^{g^{q}=e(j/p)}$ for all
$j\in\mathbb{Z}$.

These $R$-forms naturally embed into $({}^{g}V_{\Lambda})_{R}$ by decomposing
into $g$-eigenspaces.

###### Proof.

Existence follows from applying Proposition 2.40 to the embeddings of
$(V_{\Lambda})_{R}$ and $V^{\natural}[1/pq,e(1/2pq)]$ into
${}^{g^{p}}V_{\Lambda}$ and ${}^{g^{q}}V_{\Lambda}$. Uniqueness follows from
the first claim of Proposition 2.48. The abelian intertwining subalgebras
$\bigoplus_{i=0}^{q-1}\bigoplus_{j=0}^{pq-1}({}^{g}V_{\Lambda})_{R}^{pi,j}$
and
$\bigoplus_{i=0}^{p-1}\bigoplus_{j=0}^{pq-1}({}^{g}V_{\Lambda})_{R}^{qi,j}$ of
$({}^{g}V_{\Lambda})_{R}$ are self-dual $R$-forms of ${}^{g^{p}}V_{\Lambda}$
and ${}^{g^{q}}V_{\Lambda}$, equipped with decompositions into
$g$-eigenspaces. ∎

###  3.2 Monster symmetry

We now consider symmetries of these $R$-forms of $V^{\natural}$, where once
again $R=\mathbb{Z}[1/pq,e(1/2pq)]$. The next lemma is where we use recent
developments in finite group theory. I suspect less powerful results can yield
the same answer, and I welcome any insights from specialists in finite group
theory.

######  Lemma 3.7.

Let $P_{0}=\\{2,3,5,7,13\\}$, let $p$ and $q$ be distinct elements of $P_{0}$
such that $pq\not\in\\{65,91\\}$. Let $g_{p}$ and $g_{q}$ be elements of
$\mathbb{M}$ in classes $pB$ and $qB$, respectively $($i.e., the unique non-
Fricke classes of those orders$)$. Then any subgroup of the monster simple
group $\mathbb{M}$ that contains $C_{\mathbb{M}}(g_{p})$ and
$C_{\mathbb{M}}(g_{q})$ is $\mathbb{M}$ itself.

###### Proof.

This follows from known constraints on the maximal subgroups of $\mathbb{M}$,
e.g., given in [48]. The important point is that for $p\in\\{2,3,5\\}$,
$C_{\mathbb{M}}(g_{p})$ is contained in only one isomorphism type of maximal
subgroup of $\mathbb{M}$. However, for each prime $q$ under consideration,
$C_{\mathbb{M}}(g_{q})$ contains the Sylow $q$-subgroup of $\mathbb{M}$, so it
suffices to check that the order of the maximal subgroup containing
$C_{\mathbb{M}}(g_{p})$ has insufficient $q$-valuation. ∎

######  Lemma 3.8.

The $R$-form $\big{(}{}^{g^{p}}V_{\Lambda}\big{)}_{R}$ given in Lemma 3.6 has
automorphism group given by a central extension of
$C_{\operatorname{Aut}V_{\Lambda}}(g^{p})$ by $\langle(g^{*})^{p}\rangle$.
Furthermore, the abelian intertwining subalgebra $V^{\natural}[1/pq,e(1/2pq)]$
admits a faithful action of $C_{\mathbb{M}}((g^{*})^{p})$. The same claims
also hold with $p$ and $q$ switched.

###### Proof.

The second claim of Proposition 2.48 gives the description of the automorphism
group of $\big{(}{}^{g^{p}}V_{\Lambda}\big{)}_{R}$ as a central extension of
$C_{\operatorname{Aut}(V_{\Lambda})_{R}}(g^{p})$ by
$\langle(g^{*})^{p}\rangle$, but in Proposition 2.47, we identify this
centralizer with $C_{\operatorname{Aut}V_{\Lambda}}(g^{p})$. By the cyclic
orbifold correspondence for prime-order orbifolds given in [1], the
restriction of this action to $V^{\natural}[1/pq,e(1/2pq)]$ induces a
surjection to $C_{\mathbb{M}}((g^{*})^{p})$, with kernel generated by $g^{p}$.
∎

######  Theorem 3.9.

The embedded $R$-form $V^{\natural}[1/pq,e(1/2pq)]$ of $V^{\natural}$ is
preserved by the action of $\mathbb{M}$ given in [27]. In particular, the
automorphism group of $V^{\natural}[1/pq,e(1/2pq)]$ is the monster simple
group $\mathbb{M}$.

###### Proof.

By Lemma 3.8, $V^{\natural}[1/pq,e(1/2pq)]$ admits faithful actions of
$C_{\mathbb{M}}(g_{p})$ and $C_{\mathbb{M}}(g_{q})$. Base change to
$\mathbb{C}$ yields $V^{\natural}$, whose automorphism group is $\mathbb{M}$,
so we obtain embeddings of these groups in $\mathbb{M}$. However, by Lemma
3.7, these subgroups generate $\mathbb{M}$. We conclude that the action of
$\mathbb{M}$ on $V^{\natural}$ preserves the $R$-form. ∎

######  Corollary 3.10 (weak modular moonshine).

The modular moonshine conjecture holds in $($a slight weakening of$)$ its
original form given in [42]. That is, for each prime $p$ dividing the order of
the monster, and each element $g$ in class $p$A in the monster, there is a
vertex algebra $V^{p}$ over $\mathbb{F}_{p^{n}}$ for some $n$ $($where $n=1$
is asserted in the original statement$)$ equipped with an action of
$C_{\mathbb{M}}(g)$ such that the graded Brauer character
$\sum\operatorname{Tr}(h|V^{p})$ of a $p$-regular element $h\in
C_{\mathbb{M}}(g)$ is equal to the McKay–Thompson series
$T_{gh}(\tau)=\sum\operatorname{Tr}(gh|V^{\natural})$.

###### Proof.

Following [6], we may take $V^{p}$ to be the Tate cohomology group
$\hat{H}^{0}(g,V)$ for $V$ a form of $V^{\natural}$ defined over a $p$-adic
integer ring. Then this is a special case of the modular moonshine conjecture
proved in [6] and [4], under some assumptions that were not known to be true
at the time. The assumption about homogeneous pieces of the Lie algebra
$\mathfrak{m}\otimes\mathbb{Z}_{p}$ was proved as Theorem 7.1 of [5] by
applying an integral enhancement of the no-ghost theorem. The last remaining
open assumption is the existence of a suitable $\mathbb{Z}[1/3]$-form of
$V^{\natural}$. However, for our weakened version of Theorem 5.2 of [6], it
suffices to have a form of $V^{\natural}$ with monster symmetry defined over a
2-adic integer ring, i.e., a construction that does not involve division by 2,
that decomposes into $2A$-modules that are submodules of a corresponding form
of $V_{\Lambda}$. Theorem 3.9 gives 4 separate constructions, by setting
$pq\in\\{15,21,35,39\\}$, and the decomposition condition follows from the
discussion in Section 5 of [6]. In particular, the existence of a normalizing
${\rm SL}_{2}(\mathbb{F}_{3})$ follows from the computation of normalizers of
elementary abelian subgroups of the monster in [47]. The resulting 2-adic
forms are defined over unramified extensions of $\mathbb{Z}_{2}$ of degree
$4$, $6$, $12$, and $12$, respectively. Reduction mod $2$ then implies the
existence of a suitable vertex algebra $V^{2}$ defined over
$\mathbb{F}_{2^{n}}$ for some $n\leq 4$. ∎

######  Remark 3.11.

It is possible that the method given in Section 5 of [6] to reduce the
conjectural $\mathbb{Z}[1/3,e(1/3)]$-form to a $\mathbb{Z}[1/3]$-form can work
here to remove the superfluous roots of unity. However, we will obtain
stronger results without needing this method.

###  3.3 Comparison of monster-symmetric forms

Now that we have $R$-forms of $V^{\natural}$ with monster-symmetric self-dual
invariant bilinear forms for various $R$, we will glue them. In this section,
we will describe the glue.

######  Lemma 3.12.

Let $p$, $q$, $r$ be distinct elements in $P_{0}$, and let $g\in p$B be an
anomaly-free non-Fricke element of order $p$. Assume
$pq,pr\not\in\\{65,91\\}$. Let $R=\mathbb{Z}[1/pqr,e(1/2pqr)]$. Then
$\big{(}V^{\natural}[1/pq,e(1/2pq)]\big{)}^{g}_{R}\cong\big{(}V^{\natural}[1/pr,e(1/2pr)]\big{)}^{g}_{R}$.

###### Proof.

By Lemma 3.6, both sides are isomorphic to $(V_{\Lambda})_{R}^{g^{*}}$. ∎

######  Lemma 3.13.

Let $p\in\\{2,3\\}$. Then there exists a subgroup $H_{p}\subset\mathbb{M}$
such that

  1. $1.$

$H_{p}$ is isomorphic to $(\mathbb{Z}/p\mathbb{Z})^{2}$.

  2. $2.$

All non-identity elements in $H_{p}$ lie in the conjugacy class $p$B, i.e.,
they are non-Fricke.

###### Proof.

The existence of $p$B-pure non-cyclic elementary abelian subgroups can be
extracted from the table “maximal $p$-local subgroups” in [12]. ∎

######  Proposition 3.14.

Let $p\in\\{2,3\\}$, and let $q,r$ be distinct elements of
$P_{0}\setminus\\{p\\}$. Let $R=\mathbb{Z}[1/pqr,e(1/2pqr)]$. Then

$\displaystyle
V^{\natural}[1/pq,e(1/2pq)]\otimes_{\mathbb{Z}[1/pq,e(1/2pq)]}R\cong
V^{\natural}[1/pr,e(1/2pr)]\otimes_{\mathbb{Z}[1/pr,e(1/2pr)]}R.$

###### Proof.

Let $H_{p}$ be a group of the form given in Lemma 3.13, and let $g\in
H_{p}\setminus\\{1\\}$ be a nontrivial element.

Decomposing $V_{pq}=\big{(}V^{\natural}[1/pq,e(1/2pq)]\big{)}_{R}$ and
$V_{pr}=\big{(}V^{\natural}[1/pr,e(1/2pr)]\big{)}_{R}$ into eigenmodules for
$H_{p}$, we obtain $V_{pq}^{i,j}$ and $V_{pr}^{i,j}$, for
$i,j\in\\{0,\ldots,p-1\\}$. Because $H_{p}$ is generated by $p$B-elements, we
have isomorphisms:

$\displaystyle\bigoplus_{i=0}^{p-1}V_{pq}^{0,i}\cong
V_{pq}^{g}\cong\bigoplus_{i=0}^{p-1}V_{pq}^{i,0},\qquad\bigoplus_{i=0}^{p-1}V_{pr}^{0,i}\cong
V_{pr}^{g}\cong\bigoplus_{i=0}^{p-1}V_{pr}^{i,0}.$

Lemma 3.12 yields $V_{pq}^{g}\cong V_{pr}^{g}$, so all of the vertex operator
subalgebras are isomorphic. Both
$V^{\natural}[1/pq,e(1/2pq)]\otimes_{\mathbb{Z}[1/pq,e(1/2pq)]}R$ and
$V^{\natural}[1/pr,e(1/2pr)]\otimes_{\mathbb{Z}[1/pr,e(1/2pr)]}R$ are
generated by these vertex operator subalgebras, so by the uniqueness claim of
Corollary 2.41, the two $R$-forms are isomorphic. ∎

######  Proposition 3.15.

Let $p$, $q$, $r$ be distinct elements of $P_{0}$, such that $pq$ and $pr$ are
not elements of $\\{65,91\\}$. Let $R=\mathbb{Z}[1/pqr,e(1/2pqr)]$. Then

$\displaystyle
V^{\natural}[1/pq,e(1/2pq)]\otimes_{\mathbb{Z}[1/pq,e(1/2pq)]}R\cong
V^{\natural}[1/pr,e(1/2pr)]\otimes_{\mathbb{Z}[1/pr,e(1/2pr)]}R.$

###### Proof.

We first note that if $p\in\\{2,3\\}$, then this follows from Proposition
3.14. Otherwise, we may assume $q$ is the smallest prime among $p$, $q$, $r$,
and therefore, that $q\in\\{2,3\\}$. Let $g,h\in\mathbb{M}$ lie in classes
$pB$ and $rB$, respectively. By Lemma 3.12, we have isomorphisms

$\displaystyle\big{(}V^{\natural}[1/pq,e(1/2pq)]\big{)}^{g}_{R}\cong\big{(}V^{\natural}[1/pr,e(1/2pr)]\big{)}^{g}_{R},$

and

$\displaystyle\big{(}V^{\natural}[1/qr,e(1/2qr)]\big{)}^{h}_{R}\cong\big{(}V^{\natural}[1/pr,e(1/2pr)]\big{)}^{h}_{R}.$

However, by Proposition 3.14, we have an isomorphism

$\displaystyle\big{(}V^{\natural}[1/pq,e(1/2pq)]\big{)}_{R}\cong\big{(}V^{\natural}[1/qr,e(1/2qr)]\big{)}_{R},$

hence an isomorphism

$\displaystyle\big{(}V^{\natural}[1/pq,e(1/2pq)]\big{)}^{h}_{R}\cong\big{(}V^{\natural}[1/pr,e(1/2pr)]\big{)}^{h}_{R}.$

Both $\big{(}V^{\natural}[1/pq,e(1/2pq)]\big{)}_{R}$ and
$\big{(}V^{\natural}[1/pr,e(1/2pr)]\big{)}_{R}$ are $R$-forms of abelian
intertwining subalgebras of ${}^{gh}V_{\Lambda}$ generated by isomorphic
$R$-forms of $\big{(}V^{\natural}\big{)}^{g}$ and
$\big{(}V^{\natural}\big{)}^{h}$, so by the uniqueness claim of Corollary
2.41, they are therefore isomorphic. ∎

######  Remark 3.16.

An alternative proof of the previous proposition can be given by using the
same technique as in Proposition 3.14, because there are $p$B-pure non-cyclic
elementary abelian subgroups of $\mathbb{M}$ for all $p\in P_{0}$ (see [47]).

###  3.4 Descent of self-dual forms

Before we start gluing forms of $V^{\natural}$, we show that the output of
gluing is unique and has monster symmetry.

######  Lemma 3.17.

Let $i_{1}\colon R\to R_{1}$, $i_{2}\colon R\to R_{2}$ be homomorphisms of
subrings of $\mathbb{C}$, such that $R_{1}\otimes_{R}R_{2}$ is also a subring
of $\mathbb{C}$, and suppose either:

  1. $1)$

$R_{1}$ and $R_{2}$ are Zariski localizations of $R$ with respect to a coprime
pair of elements, or

  2. $2)$

$i_{1}$ and $i_{2}$ are faithfully flat.

Suppose we have a gluing datum $\big{(}V^{1},V^{2},f\big{)}$ of self-dual
forms of $V^{\natural}$ over the diagram $R_{1}\to
R_{1}\otimes_{R}R_{2}\leftarrow R_{2}$, and suppose both forms have monster
symmetry. Then there is a unique form $V$ of $V^{\natural}$ over $R$ such that
both $V^{1}$ and $V^{2}$ are base-changes of $V$. Furthermore, this form has
monster symmetry.

###### Proof.

Because the double quotient $\mathbb{M}\backslash\mathbb{M}/\mathbb{M}$ is a
singleton, Lemma 2.32 asserts that we get a unique isomorphism type of
$R$-form, and the same result also shows that this form has monster symmetry.
∎

######  Lemma 3.18.

Let $n\in\mathbb{Z}_{\geq 1}$. Suppose we have a commutative diagram

---
$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T}$

of inclusions of commutative subrings of $\mathbb{C}$, where all maps are
either

  1. $1)$

faithfully flat, and $R_{1}\cap\cdots\cap R_{n}=R$, or

  2. $2)$

Zariski localizations, forming a Zariski open cover of $\operatorname{Spec}R$.

Suppose we are given a self-dual $T$-form $V^{\natural}_{T}$ of $V^{\natural}$
with $\mathbb{M}$-symmetry, and for each $i\in\\{1,\ldots,n\\}$, we are given
a self-dual $R_{i}$-form $V^{\natural}_{R_{i}}$ of $V^{\natural}$ with
$\mathbb{M}$-symmetry, together with an isomorphism
$V^{\natural}_{R_{i}}\otimes_{R_{i}}T\overset{\sim}{\to}V^{\natural}_{T}$.
Then there exists a self-dual $R$-form $V^{\natural}_{R}$ of $V^{\natural}$,
unique up to isomorphism, such that base change along the diagram of
inclusions yields the original diagram of forms. In particular, for each
$i\in\\{1,\ldots,n\\}$, we have $V^{\natural}_{R_{i}}\cong
V^{\natural}_{R}\otimes_{R}R_{i}$.

###### Proof.

The case $n=1$ is trivial, and the case $n=2$ is covered in Lemma 3.17. When
$n\geq 3$, we apply induction, assuming existence and uniqueness for all
smaller collections of rings. Then, we may reduce the question to the case
$n=3$ by partitioning $\\{1,\ldots,n\\}$ into three nonempty sets $X$, $Y$,
$Z$, and introducing subrings $R_{X}=\bigcap_{i\in X}R_{i}$,
$R_{Y}=\bigcap_{i\in Y}R_{i}$, and $R_{Z}=\bigcap_{i\in Z}R_{i}$ of $T$. By
our induction assumption, we have unique self-dual forms $V^{\natural}_{X}$,
$V^{\natural}_{Y}$, $V^{\natural}_{Z}$ of $V^{\natural}$ over $R_{X}$,
$R_{Y}$, $R_{Z}$ satisfying the expected tensor product compatibility. It
suffices to show that gluing to make an $R_{X}\cap R_{Y}$-form
$V^{\natural}_{XY}$ followed by gluing with $V^{\natural}_{Z}$ yields an
$R$-form $V^{\natural}_{XY,Z}$ that is isomorphic to the $R$-form
$V^{\natural}_{X,YZ}$ we get by forming an $R_{Y}\cap R_{Z}$-form
$V^{\natural}_{YZ}$ followed by gluing with $V^{\natural}_{X}$.

By uniqueness of pairwise gluing, to obtain the isomorphism
$V^{\natural}_{XY,Z}\cong V^{\natural}_{X,YZ}$ it suffices to show that
$V^{\natural}_{XY,Z}\otimes_{R}R_{X}\cong V^{\natural}_{X}$ and
$V^{\natural}_{XY,Z}\otimes_{R}(R_{Y}\cap R_{Z})\cong V^{\natural}_{YZ}$. The
first isomorphism follows from the fact that the formation of
$V^{\natural}_{XY,Z}$ yields isomorphisms

$\displaystyle V^{\natural}_{X}\cong V^{\natural}_{XY}\otimes_{R_{X}\cap
R_{Y}}R_{X}\cong V^{\natural}_{XY,Z}\otimes_{R}(R_{X}\cap
R_{Y})\otimes_{R_{X}\cap R_{Y}}R_{X}\cong
V^{\natural}_{XY,Z}\otimes_{R}R_{X}.$

A similar argument then yields analogous expansions of $V^{\natural}_{Y}$ and
$V^{\natural}_{Z}$. The second isomorphism then follows from the uniqueness
statement of Lemma 3.17. To elaborate, both
$V^{\natural}_{XY,Z}\otimes_{R}(R_{Y}\cap R_{Z})$ and $V^{\natural}_{YZ}$ are
self-dual $R_{Y}\cap R_{Z}$-forms satisfying the property that tensor product
with $R_{Y}$ and $R_{Z}$ yield $V^{\natural}_{Y}$ and $V^{\natural}_{Z}$,
respectively. They are therefore isomorphic over $R_{Y}\cap R_{Z}$. ∎

######  Lemma 3.19.

Let $p$, $q$, $r$ be distinct elements of $P_{0}$, such that $pq$ and $pr$ are
not in $\\{65,91\\}$. Then, there exists a unique self-dual
$\mathbb{Z}[1/pqr,e(1/2p)]$-form $V^{\natural}[1/pqr,e(1/2p)]$ of
$V^{\natural}$ such that:

$\displaystyle
V^{\natural}[1/pqr,e(1/2p)]\otimes_{\mathbb{Z}[1/pqr,e(1/2p)]}\mathbb{Z}[1/pqr,e(1/2pq)]$
$\displaystyle\qquad{}\cong
V^{\natural}[1/pq,e(1/2pq)]\otimes_{\mathbb{Z}[1/pq,e(1/2pq)]}\mathbb{Z}[1/pqr,e(1/2pq)],$
$\displaystyle
V^{\natural}[1/pqr,e(1/2p)]\otimes_{\mathbb{Z}[1/pqr,e(1/2p)]}\mathbb{Z}[1/pqr,e(1/2pr)]$
$\displaystyle\qquad{}\cong
V^{\natural}[1/pr,e(1/2pr)]\otimes_{\mathbb{Z}[1/pr,e(1/2pr)]}\mathbb{Z}[1/pqr,e(1/2pr)].$

Furthermore, this form has monster symmetry.

###### Proof.

By Proposition 3.15, we have a gluing datum for the diagram
$\mathbb{Z}[1/pqr,e(1/2pq)]\to\mathbb{Z}[1/pqr,e(1/2pqr)]\leftarrow\mathbb{Z}[1/pqr,e(1/2pr)]$
in self-dual vertex operator algebras. By Lemma 3.18 we obtain a unique self-
dual vertex operator algebra over $\mathbb{Z}[1/pqr,e(1/2p)]$ such that tensor
product yields the input vertex operator algebras, and furthermore, this
vertex operator algebra has monster symmetry. ∎

######  Lemma 3.20.

Let $p$, $q$, $r$, $\ell$ be distinct elements of $P_{0}$ such that
$pq,pr,p\ell\not\in\\{65,91\\}$. Then there exists a unique self-dual
$\mathbb{Z}[1/p,e(1/2p)]$-form $V^{\natural}[1/p,e(1/2p)]$ of $V^{\natural}$
such that:

$\displaystyle