Selbstduale Vertexoperatorsuperalgebren und das Babymonster (Self-dual Vertex Operator Super Algebras and the Baby Monster)

TL;DR

This paper explores self-dual vertex operator superalgebras, constructs a new example related to the Baby Monster group, and introduces extremal VOAs and SVOAs, highlighting their existence at specific central charges and their analogies with lattices and codes.

Contribution

It constructs the shorter Moonshine module, a self-dual SVOA of central charge 23.5, and introduces the concept of extremal VOAs and SVOAs with existence results at various central charges.

Findings

Construction of the shorter Moonshine module for the Baby Monster

Existence of extremal VOAs at specific central charges

Existence of extremal SVOAs only at certain central charges

Abstract

We investigate self-dual vertex operator algebras (VOAs) and super algebras (SVOAs). Using the genus one correlation functions, it is shown that self-dual SVOAs exist only for half-integral central charges. It is described how self-dual SVOAs can be constructed from self-dual VOAs of larger central charge. The analogy with integral lattices and binary codes is emphasized. One main result is the construction of the shorter Moonshine module, a self-dual SVOA of central charge 23.5 on which the Baby monster - the second largest sporadic simple group - acts by automorphisms. The shorter Moonshine module has the character q^(-47/48)*(1+ 4371q^(3/2)+ 96256q^2+ 1143745q^(5/2) +...) and is the "shorter cousin" of the Moonshine module. Its lattice and code analog are the shorter Leech lattice and shorter Golay code. We conjecture that the shorter Moonshine module is the unique SVOA with this character. The final chapter introduces the notion of extremal VOAs and SVOAs. These are self-dual (S)VOAs with character having the same first few coefficients as the vacuum representation of the Virasoro algebra of the same central charge. We show that extremal VOAs exist at least for the central charges 8, 16, 24, 32, 40 and that extremal SVOAs exist only for the central charges c=0.5, 1, ..., 7.5, 8, 12, 14, 15, 15.5, 23.5 and 24. Examples for c=24 (resp. 23.5) are the (shorter) Moonshine module. Again, our results are similar to results known for codes and lattices.