Spectral Triples and the Super-Virasoro Algebra

Abstract

We construct infinite dimensional spectral triples associated with representations of the super-Virasoro algebra. In particular the irreducible, unitary positive energy representation of the Ramond algebra with central charge c and minimal lowest weight h = c/24 is graded and gives rise to a net of even θ-summable spectral triples with non-zero Fredholm index. The irreducible unitary positive energy representations of the Neveu-Schwarz algebra give rise to nets of even θ-summable generalised spectral triples where there is no Dirac operator but only a superderivation.

Superconformal nets and noncommutative geometry

“…One of those issues is the study of higher degree of supersymmetry, i.e., super-Virasoro algebras involving further odd fields apart from G. This can be done for arbitrary degree, but the first and already very interesting case with new emerging structures is the N = 2 super-Virasoro algebra (in contrast to the usual N = 1 super-Virasoro algebra investigated in the present paper and in [CHKL10,CKL08]). The corresponding nets, their representations and extensions were studied by us together with Y. Kawahigashi and F. Xu in [CHKLX].…”

“…Actually, we have a family of cocycles since we may perform this construction for every local algebra A(I), I ∈ I, as well as for nice global algebras like the universal C*-or von Neumann algebra. While we kept this generality in [CHKL10], we shall recognize below that the local algebras are not sufficient for our task and we have to choose a global one in Definition 4.8.…”

Super-KMS Functionals for Graded-Local Conformal Nets

“…(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.…”

“…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.…”

Conformal Nets and KK-Theory

“…However, in view of the striking success of KK-theory on the one hand (e.g. [CMR07] for an overview) and the noncommutative geometrization program for (super-) conformal nets and their representations [Lon01,CKL08,CHKL10] on the other hand, we feel that there are good reasons to give a closer look at this subject, as it could possibly reveal a lot of potential for further investigations.…”