Self-Dual Vertex Operator Superalgebras of Large Minimal Weight

TL;DR

This paper establishes new upper bounds on the minimal weights of self-dual vertex operator superalgebras and explores their properties, especially for central charges up to 48, including cases with N=1 supersymmetry and connections to the monster group.

Contribution

It proves a general upper bound for minimal weights of self-dual vertex operator superalgebras and provides improved estimates for central charges up to 48, also analyzing supersymmetric cases and symmetry group actions.

Findings

Established upper bound mu <= [c/24] + 1 for minimal weight.

Derived improved estimates for c <= 48.

Showed the monster group cannot act on certain extremal superalgebras at c=48.

Abstract

The new general upper bound mu <= [c/24] + 1 for the minimal weight mu of a self-dual vertex operator superalgebra of central charge c different from 47/2 is proven. For central charges c <= 48, further improved estimates are given and examples of with large minimal weight are discussed. We also study the case of vertex operator superalgebras with N=1 supersymmetry which was first considered by Witten in connection with three-dimensional quantum gravity. The upper bound mu^* <= (1/2)[c/12]+1/2 for the minimal superconformal weight is obtained for c different from 47/2. In addition, we show that it is impossible that the monster sporadic group acts on an extremal self-dual N=1 supersymmetric vertex operator superalgebra of central charge 48 in a way proposed by Witten if certain standard assumptions about orbifold constructions hold. The same statement holds for extremal self-dual vertex operator algebras of central charge 48.