van Ekeren, Möller, Scheithauer on holomorphic orbifolds

There aren’t many blog posts about vertex operator algebras, so I thought I’d help fill this gap by mentioning a substantial advance by Jethro van Ekeren, Sven Möller, and Nils Scheithauer that appeared on the ArXiv last month. The most important feature is that this paper resolves several folklore conjectures that have been around since near the beginning of vertex operator algebra theory. This was good for me, since I was able to use some of these results to prove the Generalized Moonshine Conjecture much more quickly than I had expected. I won’t say much about moonshine here, as I think it deserves its own post.

I briefly discussed vertex operator algebras in my earlier post on generalized moonshine. While an ordinary commutative ring has a multiplication structure A \otimes A \to A, a vertex operator algebra (or VOA) has a “meromorphic” version V \otimes V \to V((z)), and there is an integer grading on the underlying vector space that is compatible with the powers of z in a straightforward way.

I won’t say much about VOAs in general, but rather, I will consider those that satisfy some of the following nice properties:
Rational: Any V-module is a direct sum of irreducibles.
Holomorphic: Any V-module is a direct sum of copies of V.
C_2 cofinite: This is a rather technical-sounding condition that ends up being equivalent to a lot of natural representation-theoretic finiteness properties, like “every representation is a direct sum of generalized eigenspaces for the energy operator L(0)”.
It is conjectured that every rational VOA is C_2 cofinite.

As usual, when we have a collection of nice objects, we may want to classify them, or at least find ways of building new ones and discovering invariants and constraints.

Some basic invariants are the central charge c (a complex number), and the character of a module M, given by the graded dimension T_M(\tau) = Tr(q^{L(0)-c/24}|M), where the grading is given by the energy operator L(0), and we view the power series as a function on the complex upper half plane using q = \exp(2\pi i \tau). One of the first general results for “nice” VOAs is Zhu’s 1996 proof that if V is rational and C_2 cofinite, then the characters of irreducible V-modules form a vector-valued modular form for a finite dimensional representation of SL_2(\mathbb{Z}). Furthermore, he showed that in this case, the central charge c is a rational number, and if V is holomorphic, then c is a nonnegative integer divisible by 8.

Dong and Mason classified the holomorphic C_2 cofinite VOAs of central charge 8 and 16 – there is one isomorphism class for central charge 8, and 2 isomorphism classes for central charge 16. All three are given by a lattice VOA construction. In general, if you are given an even unimodular positive definite lattice (which only exists in dimension divisible by 8), you get a a holomorphic C_2 cofinite VOA from it, so the central charge 8 object comes from the E_8 lattice, and the central charge 16 objects come from the E_8 \times E_8 and D_{16}^* lattices. Central charge 24 is at a sweet spot of difficulty, where Schellekens did a long calculation in 1993 and conjectured the existence of 71 isomorphism types. Central charge 32 is more or less impossible, since lattices alone give over 10^9 types.

For central charge 24, because the L(0) eigenspace V_1 with eigenvalue 1 is naturally a Lie algebra, the proposed isomorphism types are labeled by finite dimensional Lie algebras. Schellekens’s list is basically

1. The monster VOA, with V_1 = 0.
2. The Leech lattice VOA, with V_1 commutative of dimension 24.
3. 69 extensions of rational Kac-Moody VOAs by suitable modules (here the Lie algebras are products of simple Lie algebras and in particular noncommutative).

As far as existence is concerned, 23 of the 69 come from lattices, known as the Niemeier lattices. An additional 14 come from Z/2 orbifolds of lattices. Another 18 come from a “framed VOA” construction, given by adjoining modules to a tensor product of Ising models according to some codes (Lam, Shimakura, and Yamauchi are the main names here). The remaining 12 are more difficult, and after this recent paper, there are 2 that have not been constructed. There are only a few cases where uniqueness is known, such as the Leech lattice VOA. The V_1 = 0 case is wide open, and perhaps the worst for uniqueness, since there isn’t any Lie algebra structure to work with.

One of the results of van Ekeren, Möller, and Scheithauer was a reconstruction of Schellekens list, i.e., eliminating other choices of Lie algebras from possibility. This was desirable, since the original paper was quite sketchy in places and didn’t have proofs. A second result was a collection of new examples, in particular nearly filling out this list of 69. They did this by solving an old problem, namely the construction of holomorphic orbifolds. The idea is the following: Given a holomorphic C_2 cofinite VOA V, and a finite order automorphism g, take the fixed-point subalgebra V^g, and take a direct sum with some V^g-modules not in V to get something new. In fact, the desired V^g-modules were more or less known – there is a notion of g-twisted V-module V(g), and one takes the submodules of all V(g^i) fixed by a suitable lift of g. To show that this even makes sense requires substantial development of the theory.

First, the existence and uniqueness of irreducible g-twisted V-modules V(g) was a nonconstructive theorem of Dong, Li, and Mason in 2000. Then, to get a multiplication operation on the component V^g-modules, one first shows that irreducible V^g-modules have a nice tensor structure (in particular, are simple currents), so that the space of suitable multiplication maps is highly constrained. This requires recent major theorems of Miyamoto (V^g is rational and C_2 cofinite – 2013), and Huang (if V is rational and C_2 cofinite, then Rep(V) is a modular tensor category and the Verlinde formula holds – 2008). By some clever applications of the Verlinde formula, van Ekeren, Möller, and Scheithauer showed that once we have simple currents with suitable L(0)-eigenvalues, the homological obstruction to a well-behaved multiplication vanishes, and one gets a holomorphic VOA.

The intermediate results that I found most useful for my own purposes were:
1. assembly of an abelian intertwining algebra (a generalization of VOA where the commutativity of multiplication is allowed some monodromy) from all irreducible V^g-modules.
2. the explicit description of the SL_2(\mathbb{Z}) action on the characters of irreducible V^g-modules. This also solves a conjecture of Dong, Li, and Mason concerning the graded dimension of twisted modules.

In particular, if g has order n, then the simple currents are arranged into a central extension 0 \to \mathbb{Z}/n\mathbb{Z} \to A \to \mathbb{Z}/n\mathbb{Z} \to 0, where the kernel is given by an action of g, and the image is the twisting on modules. The group A is also equipped with a canonical \mathbb{Q}/\mathbb{Z}-valued quadratic form. One obtains an A-graded abelian intertwining algebra with monodromy determined by the quadratic form (up to a certain coboundary), and the SL_2(\mathbb{Z}) action is by the corresponding Weil representation (up to the c/24 correction).