Superconformal nets and noncommutative geometry

TL;DR

This paper advances the study of superconformal nets over S^1 by integrating noncommutative geometric methods, defining spectral triples, and analyzing their cyclic cocycles to encode representation theory.

Contribution

It introduces a framework linking superconformal nets with noncommutative geometry, including the construction of spectral triples and cyclic cocycles for localized endomorphisms.

Findings

Spectral triples associated to superconformal nets are nontrivial.

Cohomology classes of cyclic cocycles distinguish inequivalent endomorphisms.

The approach encodes parts of the net's representation theory in noncommutative geometric terms.

Abstract

This paper provides a further step in our program of studying superconformal nets over S^1 from the point of view of noncommutative geometry. For any such net A and any family Delta of localized endomorphisms of the even part A^gamma of A, we define the locally convex differentiable algebra A_Delta with respect to a natural Dirac operator coming from supersymmetry. Having determined its structure and properties, we study the family of spectral triples and JLO entire cyclic cocycles associated to elements in Delta and show that they are nontrivial and that the cohomology classes of the cocycles corresponding to inequivalent endomorphisms can be separated through their even or odd index pairing with K-theory in various cases. We illustrate some of those cases in detail with superconformal nets associated to well-known CFT models, namely super-current algebra nets and super-Virasoro nets. All in all, the result allows us to encode parts of the representation theory of the net in terms of noncommutative geometry.