Integral forms in vertex operator algebras, a survey

TL;DR

This survey reviews recent developments in the study of integral forms within vertex operator algebras, highlighting key theoretical advances and open questions in the field.

Contribution

It provides a comprehensive overview of the current state of research on integral forms in VOAs, summarizing recent progress and identifying future directions.

Findings

Summarizes recent progress in integral forms of VOAs

Highlights key techniques and results in the field

Identifies open problems and future research directions

Abstract

We give a brief survey of recent work on integral forms in vertex operator algebras (VOAs).

Equations

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$${\hat{H}}^0(⟨g⟩,K):=K^g/(1+g+g^2)K.$$

$$0<(1+\gamma +{\gamma }^2)d<d^{\gamma }<d,\text{ dimensions 0, 7, 14, 28.}$$

Full text

1 Introduction

The definition of VOA is too long to give here. We refer the reader to a standard reference for VOA theory: [8], definition p.244. Short version: $V=\bigoplus_{n\geq 0}V_{n}$, a graded vector space in characteristic 0 with each $dim(V_{i})$ finite; vacuum element ${\bf 1}\in V_{0}$, Virasoro element $\omega\in V_{2}$; a linear monomorphism $Y:V\rightarrow End(V)[[z^{-1},z]]$, written $Y(v,z)=\sum_{n\in{\mathbb{Z}}}v_{n}z^{-n-1}$.

For each $n\in{\mathbb{Z}}$, there is a product $a,b\mapsto a_{n}b$ (meaning the endomorphism $a_{n}$ applied to $b$), giving a ring $(V,n^{th})$.

A vertex algebra (VA) is a generalization of VOA to graded modules over a commutative rings of scalars. It has a vacuum element but not necessarily a Virasoro element. Our examples will be over finite fields or the integers.

Definition 1.1**.**

An integral form in an algebra in characteristic 0 is the ${\mathbb{Z}}$-span of a basis which is closed under the product.

Example 1.2**.**

(1) $Mat_{n\times n}({\mathbb{Z}})\leq Mat_{n\times n}({\mathbb{C}})$; (2) In a simple finite dimensional complex Lie algebra, the ${\mathbb{Z}}$-span of a Chevalley basis is an integral form.

Example 1.3**.**

For an even integral lattice $L$, there is a lattice type VOA $V_{L}$ which, as a graded vector space, has shape $\SS(\hat{H})\otimes{\mathbb{C}}[L]$. Here, $\SS$ means symmetric algebra of a vector space, $H:={\mathbb{C}}\otimes_{{\mathbb{Z}}}L$ and ${\hat{H}}$ means $H_{1}\oplus H_{2}\oplus\dots$ where $H_{k}$ is a copy of $H$ declared to have degree $k$. Finally ${\mathbb{C}}[L]$ is the group algebra of the abelian group $L$, with basis $e^{\alpha}$, for $\alpha\in L$.

Example 1.4**.**

The Moonshine VOA ${V^{\natural}}$ is a twisted version of $V_{\Lambda}$, where $\Lambda$ is the Leech lattice (rank 24, determinant 1, minimum norm 4). It has $Aut({V^{\natural}})\cong{\mathbb{M}}$, the Monster.

Definition 1.5**.**

An integral form $R$ in a vertex operator algebra $V=\bigoplus_{n\in{\mathbb{Z}}}V_{n}$ with a nondegenerate symmetric bilinear form is the ${\mathbb{Z}}$-span of a basis which is closed under all the VOA products and for all $n$, $R\cap V_{n}$ is an integral form of the vector space $V_{n}$; also $R$ must contain the vacuum element and a positive integer multiple of the Virasoro element.

So, an integral form in a VOA is a vertex algebra over the ring of integers.

Chongying Dong and RLG [6, 7] studied the following question. Given a finite group $G$ in $Aut(V)$, is there an integral form in $V$ which is stable under $G$? We have some general sufficient conditions.

Our main applications: (1) for lattice type VOAs, ($L$ of rank $r$) there is a $G$-invariant integral form where $G$ has the form $2^{r}.O(L)$. (One description: it is generated as a vertex algebra by the ${\mathbb{Z}}$-span of the components of $Y(e^{\alpha},z){\bf 1}$ for $\alpha\in L$.)

(2) For the Moonshine VOA ${V^{\natural}}$, we proved that there is a Monster-invariant integral form. (This is created by a kind of averaging, and is not (yet) described explicitly.)

The recent preprint of Carnahan [3] proves existence of an integral form in which every graded component has determinant 1. This form is not given explicitly.

Remark 1.6*.*

We learned after our proof was written, that in the 80s, Borcherds had asserted the existence of an integral form for lattice type VOAs. Borcherds also observed that there is a Monster-invariant ${\mathbb{Z}}[\frac{1}{2}]$-form in ${V^{\natural}}$ but claimed nothing about a form over ${\mathbb{Z}}$.

If $J$ is an integral form in a VOA, it inherits a symmetric bilinear form from the VOA. We say $J$ is lattice integral if $\langle x,y\rangle\in{\mathbb{Z}}$ for all $x,y\in J$. It is unclear when the restriction of the form to $J$ is integral-valued (or can be multiplied by a scalar to become integral valued).

Dong and RLG gave two sufficient conditions to prove lattice integrality. (1) We showed that it is integral valued whenever the integral form $J$ is generated by quasi-primary vectors (in VOA theory, this means vectors annihilated by a certain operator $L(1)$). This criterion applies to the Monster-invariant form we built earlier. (2) We gave an averaging-type argument.

2 Classical lattice type VOAs

This section represents joint work with Ching Hung Lam [10, 11]. Consider the case of lattice type VOA where $L$ is a root lattice of type ADE, and let $V:=V_{L}$. Then $(V_{1},0^{th})$ is a copy of the Lie algebra associated to $L$. *If $J$ is our integral form, $J\cap V_{1}$ is the ${\mathbb{Z}}$-lattice spanned by a Chevalley basis! So, this $J$ is spanned as an abelian group by a set of elements which generalizes “Chevalley basis”. * It turns out that a Chevalley group can be defined on $V_{L}$ with the standard generators $x_{r}(\pm 1)$ fixing the integral form.

We can also take any commutative associative ring $R$ and form $R\otimes J$, a vertex algebra over $R$, called the classical VA of type $L$ over $R$. We also get an action of the Chevalley group of type $L$ over $R$ on $R\otimes J$ as VA automorphisms.

When $R$ is a field we get all Chevalley groups of types ADE (with graph outer automorphisms) as full automorphism groups of these VAs over $R$. We also defined VA over $R$ for types BCGF. This gives the Steinberg variations (twisted Chevalley groups) acting on VAs and being essentially the full automorphism groups.

We would like to find a series of VAs whose automorphism groups are essentially the Ree and Suzuki groups but have not (yet) done so.

Remark 2.1*.*

Our construction gives infinite dimensional graded modules for each Chevalley group and Steinberg variation over its field of definition. These modules may be a good opportunity for study of representation theory ($Ext$, indecomposables, etc. ).

3 The degree 2 component of a VOA

Given a VOA $V=\oplus_{i\geq 0}V_{i}$, the $k$-th product gives a bilinear mao $V_{i}\times V_{j}{\longrightarrow}V_{i+j-k-1}$. So, $V_{n}$ under the $n^{th}$ product is a finite dimensional algebra, denoted $(V_{n},(n{-}1)^{th})$.

In addition, (a) if $dim(V_{0})=1$, $(V_{1},0^{th})$ is a Lie algebra; (b) if $dim(V_{0})=1$ and $dim(V_{1})=0$, then $(V_{2},1^{st})$ is a commutative algebra with a symmetric, associative form $(ab,c)=(a,bc)$. Algebras as in (b) are sometimes called *Griess algebras. *

There are many examples of algebras (b) with finite automorphism groups, e.g., [13, 14], [4, 5]. The 196884-dimension algebra used to construct the Monster occurs this way in the Moonshine VOA $V^{\sharp}$.

Now suppose $dim(V_{0})=1$ and that $e\in V_{2}$ is a conformal vector of central charge $\frac{1}{2}$ and that the subVOA generated by $e$ is simple. Miyamoto showed that $e$ gives $t_{e}\in Aut(V)$ of order 1 or 2 (called a Miyamoto involution).

In the special case of a dihedral VOA (generated by a pair of such conformal vectors $e,f$), the degree 2 algebra has integral forms. Those which are maximal integral forms and invariant under the dihedral group $\langle t_{e},t_{f}\rangle$ were classified in the thesis of Greg Simon (U Michigan, 2016). For the nine types of dihedral VOAs (classified by Sakuma [12]; they correspond to nodes of the extended $E_{8}$-diagram, displayed below), there is just one maximal invariant form in all cases but $2A$, in which case there are three.

[TABLE]

This classification of maximal invariant forms in $V_{2}$ does not (yet) extend to invariant forms in the entire dihedral VOA.

4 Modular Moonshine of Borcherds and Ryba

Borcherds and Ryba wrote several articles [2, 15] about Modular Moonshine (positive characteristic) which imitated the story of the Monster and the graded representation ${V^{\natural}}$ and modular forms, but for smaller sporadic groups.

They discussed an interesting case. In ${\mathbb{M}}$, take $g$ a $3C$-element; then $C(g)=\langle g\rangle\times S$, where $S\cong F_{3}$ a sporadic simple group of order $2^{15}3^{10}5^{3}7^{2}13\cdot 19\cdot 31$ (Thompson’s group).

Borcherds and Ryba used $K$, Borcherds’s ${\mathbb{Z}}[\frac{1}{2}]$-form in ${V^{\natural}}$, then considered its 0th Tate cohomology group

[TABLE]

This inherits structure to make a VA over ${\mathbb{F}}_{3}$. It looked like the classical $E_{8}$ type VA over ${\mathbb{F}}_{3}$, but nonzero terms occur only in degrees 0, 3, 6 , . . . and have respective dimensions $1,248,\dots$, just like for the genuine $E_{8}$ VA in degrees 0, 1, 2, . . .. There was no obvious isomorphism (which triples degree of the grading) between these two VAs over ${\mathbb{F}}_{3}$.

To prove existence of an isomorphism, Lam and RLG adapted a covering idea of Frohardt-Griess [9], which is illustrated in the following example.

Example 4.1**.**

$F$ algebraically closed field of char.3. The Lie algebra $a_{2}(F)$ has a 1-dimensional central ideal, $Z$. While $Aut(a_{2}(F))$ is $PGL(2,F){:}2$, $Aut(a_{2}(F)/Z)\cong G_{2}(F)$. Our proof takes the Lie algebra $d:=d_{4}(F)$ and graph automorphism $\gamma$ of order 3, then considers

[TABLE]

The group $G_{2}(F)$ acts on each subobject and on the 7-dimensional quotient Lie algebra $d^{\gamma}/(1+\gamma+\gamma^{2})d$. One can see inside $d$ (look at the long roots) a copy of $a_{2}(F)$ which maps onto $d^{\gamma}/(1+\gamma+\gamma^{2})d$; the image is isomorphic to $a_{2}(F)/Z$. Therefore $Aut(a_{2}(F)/Z)$ contains a copy of $G_{2}(F)$. This containment is equality.

Lam and RLG took $M$, the standard integral form for $V_{E_{8}}$, and a sublattice $M^{\prime}$ of $K$ (integral form in ${V^{\natural}}$) which “covered” the Tate cohomology group $K^{g}/(1+g+g^{2})K$. (Roughly, $M^{\prime}$ is the ${\mathbb{Z}}$-span of the image of $M$ under a map suggested by $x\mapsto x\otimes x\otimes x$ for $x\in EE_{8}\cong\sqrt{2}E_{8}$; think of the containment of lattices $EE_{8}\perp EE_{8}\perp EE_{8}\leq\Lambda$, the Leech lattice). This led to an isomorphism.

An application of the Borcherds-Ryba theory is a new proof that the group $F_{3}$ of Thompson embeds in $E_{8}(3)$ (first proof 1974, by Thompson and P. Smith, used a study of Dempwolff decompositions and computer work). This VA viewpoint gives a nontrivial homomorphism of $C(g)/\langle g\rangle\cong F_{3}$ into the group $E_{8}(3)$ without knowing much about the structure of $C(g)$.

My web site contains files of certain articles in the reference list.