# Superconformal nets and noncommutative geometry

April 15, 2013

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.

Keywords:

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