Noncommutative geometry, conformal geometry, and the local equivariant index theorem

Abstract

We prove a local index formula in conformal geometry by computing the Connes-Chern character for the conformal Dirac (twisted) spectral triple recently constructed by Connes-Moscovici. Following an observation of Moscovici, the computation reduces to the computation of the CM cocycle of an equivariant Dirac (ordinary) spectral triple. This computation is obtained as a straightforward consequence of a new proof of the local equivariant index theorem of Patodi, Donelly-Patodi and Gilkey. This proof is obtained by combining Getzler's rescaling with an equivariant version of Greiner's approach to the heat kernel asymptotic. It is believed that this approach should hold in various other geometric settings. On the way we give a geometric description of the index map of a twisted spectral in terms of (twisted) connections on finitely generated projective modules.

Superseded by arXiv:1310.6131, arXiv:1411.3701, and arXiv:1411.3703

References in corpus (3)

• Type III and spectral triples
• The Gauss-Bonnet Theorem for the noncommutative two torus
• Local index formula and twisted spectral triples

Cited by in corpus (6)

• Noncommutative Geometry and Conformal Geometry. III. Vafa-Witten Inequality and Poincaré Duality
• Volterra calculus, local equivariant family index theorem and equivariant eta forms
• On the Scalar Curvature for the Noncommutative Four Torus
• The Greiner's approach to heat kernel asymptotics and the variation formulas for the equivariant Ray-Singer metric
• The Noncommutative Infinitesimal Equivariant Index Formula
• The noncommutative family Atiyah-Patodi-Singer index theorem