Spectral triples and the super-Virasoro algebra

TL;DR

This paper constructs spectral triples linked to super-Virasoro algebra representations, revealing new geometric structures and index properties in conformal field theory models.

Contribution

It introduces a novel method to associate spectral triples with super-Virasoro algebra representations, including cases without Dirac operators.

Findings

Spectral triples with non-zero Fredholm index for Ramond algebra representations

Generalized spectral triples without Dirac operators for Neveu-Schwarz algebra

Establishment of a geometric framework for superconformal algebras

Abstract

We construct infinite dimensional spectral triples associated with representations of the super-Virasoro algebra. In particular the irreducible, unitary positive energy representation of the Ramond algebra with central charge c and minimal lowest weight h=c/24 is graded and gives rise to a net of even theta-summable spectral triples with non-zero Fredholm index. The irreducible unitary positive energy representations of the Neveu-Schwarz algebra give rise to nets of even theta-summable generalised spectral triples where there is no Dirac operator but only a superderivation.