Superconformal nets and noncommutative geometry

Abstract

This paper provides a further step in the program of studying superconformal nets over $S^1$ from the point of view of noncommutative geometry. For any such net $\mathcal{A}$ and any family $\Delta$ of localized endomorphisms of the even part ${\mathcal{A}}^{\gamma }$ of $\mathcal{A}$, we define the locally convex differentiable algebra ${\mathfrak{A}}_{\Delta }$ 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.