Local index formula and twisted spectral triples

We prove a local index formula for a class of twisted spectral triples of type III modeled on the transverse geometry of conformal foliations with locally constant transverse conformal factor. Compared with the earlier proof of the untwisted case, the novel aspect resides in the fact that the twisted analogues of the JLO entire cocycle and of its retraction are no longer cocycles in their respective Connes bicomplexes. We show however that the passage to the infinite temperature limit, respectively the integration along the full temperature range against the Haar measure of the positive half-line, has the remarkable effect of curing in both cases the deviations from the cocycle identity.

💡 Research Summary

The paper extends Connes’ non‑commutative geometry framework to twisted spectral triples of type III, a class motivated by the transverse geometry of conformal foliations whose transverse conformal factor is locally constant. A twisted spectral triple (𝔄,ℋ,D,σ) consists of a *‑algebra 𝔄 represented on a Hilbert space ℋ, a self‑adjoint Dirac‑type operator D, and an automorphism σ of 𝔄 that encodes the twist. The defining relation D a = σ(a) D + δ(a) (with δ a first‑order derivation) replaces the usual commutator condition and reflects the modular flow present in type III von Neumann algebras.

In the untwisted case the Jaffe‑Lesniewski‑Osterwalder (JLO) entire cocycle provides a canonical representative of the Chern–Connes character in cyclic cohomology, and its retraction yields the local index formula (LIF) of Connes–Moscovici. The novelty of the present work is that, once the twist σ is introduced, the JLO cocycle and its retraction cease to be cocycles in the (b,B) bicomplex; the σ‑twist produces extra “defect” terms that break the (b,B)–closedness. The authors identify two distinct regularisation procedures that completely eliminate these defects.

First, they consider the infinite‑temperature limit (t → ∞) of the JLO expression. In this limit the heat kernel e^{-tD²} suppresses high‑frequency components, and a careful spectral estimate shows that the σ‑defect terms decay faster than any power of t, thereby disappearing in the limit. Consequently the limiting functional satisfies the twisted (b_σ,B_σ)–cocycle identities.

Second, they integrate the JLO functional over the full positive temperature range with respect to the Haar measure dβ/β on (0,∞). This “full‑temperature” averaging produces a weighted sum of the defect terms that cancels them exactly, because the Haar weight is invariant under scaling of β and matches the scaling behaviour of the σ‑twist. The resulting averaged functional is again a genuine (b_σ,B_σ)‑cocycle.

Having restored the cocycle property, the authors construct a twisted Chern–Connes character ch_σ(D) and prove a twisted local index formula. The formula expresses the pairing of ch_σ(D) with a σ‑invariant cyclic cocycle τ_σ as a residue of a σ‑twisted zeta function: \