Superconformal nets and noncommutative geometry

  • Sebastiano Carpi

    Università di Chieti-Pescara, Italy
  • Robin Hillier

    Lancaster University, UK
  • Roberto Longo

    Università di Roma “Tor Vergata”, Italy


This paper provides a further step in the program of studying superconformal nets over S1S^1 from the point of view of noncommutative geometry. For any such net A\mathcal A and any family Δ\Delta of localized endomorphisms of the even part Aγ\mathcal A^{\gamma} of A\mathcal A, we define the locally convex differentiable algebra AΔ\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 Δ\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.

Cite this article

Sebastiano Carpi, Robin Hillier, Roberto Longo, Superconformal nets and noncommutative geometry. J. Noncommut. Geom. 9 (2015), no. 2, pp. 391–445

DOI 10.4171/JNCG/196