Structure and Classification of Superconformal Nets

TL;DR

This paper analyzes the structure of superconformal nets, classifies their irreducible extensions, and explores their representations, indices, and modular properties, especially for central charges less than 3/2.

Contribution

It provides a comprehensive classification of all superconformal nets with central charge less than 3/2 and characterizes their representations and indices.

Findings

Classification of all irreducible Fermi extensions of super-Virasoro nets in the discrete series

Establishment of a formula relating Fredholm index of supercharge and Jones index

Verification of modularity for nets with central charge in the discrete series

Abstract

We study the general structure of Fermi conformal nets of von Neumann algebras on the circle, consider a class of topological representations, the general representations, that we characterize as Neveu-Schwarz or Ramond representations, in particular a Jones index can be associated with each of them. We then consider a supersymmetric general representation associated with a Fermi modular net and give a formula involving the Fredholm index of the supercharge operator and the Jones index. We then consider the net associated with the super-Virasoro algebra and discuss its structure. If the central charge c belongs to the discrete series, this net is modular by the work of F. Xu and we get an example where our setting is verified by considering the Ramond irreducible representation with lowest weight c/24. We classify all the irreducible Fermi extensions of any super-Virasoro net in the discrete series, thus providing a classification of all superconformal nets with central charge less than 3/2.