Superconformal nets and noncommutative geometry

This paper provides a further step in the program of studying superconformal nets over S¹ from the point of view of noncommutative geometry. For any such net A and any family Δ of localized endomorphisms of the even part A^γ of A, we define the locally convex differentiable algebra U<inf>Δ</inf> 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 Δ 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.