Codes, Lattices, and Vertex Operator Algebras – A Comprehensive Study

Question

At the end of "Notes on Chapter 1" in the Preface to the Third Edition of Sphere packings, lattices and groups, Conway and Sloane write the following:

Finally, we cannot resist calling attention to the remark of Frenkel, Lepowsky and Meurman, that vertex operator algebras (or conformal field theories) are to lattices as lattices are to codes.

I would like to understand better what the precise analogy is that is being made here.

Through my attempts to read Frenkel, Lepowsky and Meurman's book, I am aware of the story about how the "exceptional" objects,

Golay code —> Leech lattice —> Moonshine module,

form a hierarchy with increasingly large symmetry groups,

Mathieu group M₂₄ —> Conway group Co₁ —> Monster group,

and how this hierarchy led to the conjecture that uniqueness results for the Golay code and Leech lattice carry over to a uniqueness property of the Moonshine module. Frenkel, Lepowsky and Meurman speak of many analogies between the theories of codes, lattices, and vertex operator algebras. I have some understanding of the connections between codes and lattices, but so far very little understanding of vertex operator algebras and of their connection with lattices (despite having a bit of relevant physics background in conformal field theory).

My questions are

1. Are the parallels alluded to above a peculiar feature of these exceptional structures, or something that holds more generally?

2. Is there a "baby example" one can look at of these correspondences – something based on smaller and more elementary objects?

Answer 1

I think the analogy you describe cannot be made precise with our current technology. For example, the word "functor" doesn't seem to have made an appearance yet in this context.

If you have a code, there are methods to construct lattices using it, but some of the constructions (like the Leech lattice) require special properties of the code. If you have a lattice, there are methods to construct vertex operator algebras using it (e.g., the lattice VOA), but some of the constructions, like orbifolds, depend on properties of the lattice, like the existence of automorphisms of certain orders. In the case of the moonshine module, we need the $-1$ automorphism of Leech, which isn't particularly special for lattices. Conjecturally (see work of Dong, Mason, and Montague 1994-95), we could use any fixed-point free automorphism of Leech to get an isomorphic VOA, and that is somewhat more special.

One class of "baby examples" that arise is when you take the root lattice of a simple (or more generally, reductive) algebraic group. This lattice has an action of the Weyl group. The lattice vertex operator algebra of this lattice has an action of the corresponding Kac-Moody Lie algebra (more generally, I think the centrally extended loop group acts). This is one of the more natural ways to construct $E_8$ from its lattice.

I'm afraid I'm not qualified to describe good baby examples of the transition from codes to lattices.