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 A and any family ∆ of localized endomorphisms of the even part A γ of A, we define the locally convex differentiable algebra A ∆ 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 ∆ 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 supercurrent 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.

Super-KMS Functionals for Graded-Local Conformal Nets

“…Then according to (B.7) δ Q I satisfies (3.5) on D ∞ . Applying then Borel functional calculus, almost literally as in the proof of [CHL13,Prop.6.4] but with Q I instead of Q and J(f ) U(1)-currents instead of G-currents, we obtain (3.6) on A 0 ∩ A(I) ⊂ dom(δ Q I ), which was to be proved. In particular, in analogy to [CHKL10,Prop.5.3], δ Q I does not depend on the actual choice of ϕ I ∈ C ∞ c (R) as long as ϕ I ↾ I = 1.…”

“…(1) Notice that the situation is completely different for completely rational graded-local conformal nets over the circle S 1 and sKMS functionals with respect to the periodic rotation action, like those treated in [CHKL10,CHL13,CHKLX13]. In that case, there are in fact bounded though nonpositive sKMS functionals on the universal C*-algebra of the net (in its universal locally normal representation) as explained in Section 4.…”

“…(3) Can we formulate a general classification of super-KMS functionals for graded translationcovariant nets, in analogy to [CLTW12a,CLTW12b]? (4) Do super-KMS functionals for graded translation-covariant nets give rise to generalized homotopy-invariant entire cyclic cocycles and geometric invariants of the original net, generalizing [CHL13]?…”

“…With a suitable and natural definition of the underlying global algebra for those nets, we shall see that there bounded super-KMS functionals do exist and they are actually all combinations of super-Gibbs functionals. Hence to them one can apply the constructions in [JLW89,Kas89], and in special cases this has in fact been exploited in the recent articles [CHL13] and [CHKLX13].…”

Loop groups and noncommutative geometry

“…Let henceforth e R be the projection onto an arbitrary but fixed one-dimensional subspace of H π R ,0,+ . For a more expanded summary about the CAR algebra and Ramond representations with the present notation, we refer to [16,Sect.6] and for details and proofs to [1,7] together with [49,Sect.5.3].…”