Connes-Moscovici Residue Cocycle For Some Dirac-Type Operators

TL;DR

This paper computes the Connes-Moscovici residue cocycle for certain Dirac-type operators using heat kernel asymptotics and Getzler calculus, extending the understanding of local index formulas in noncommutative geometry.

Contribution

It introduces a modified Getzler calculus to explicitly compute the residue cocycle for Bismut's Dirac-type operators deformed by a 3-form, including non-closed cases.

Findings

Computed the residue cocycle for Bismut's Dirac operators using heat kernel asymptotics.

Extended calculations to low-dimensional cases with non-closed 3-forms.

Clarified the relationship between the residue cocycle and heat kernel asymptotics.

Abstract

The residue cocycle associated to a suitable spectral triple is the key component of the Connes-Moscovici local index theorem in noncommutative geometry. We review the relationship between the residue cocycle and heat kernel asymptotics. We use a modified version of the Getzler calculus to compute the cocycle for a class of Dirac-type operators introduced by Bismut, obtained by deforming a Dirac operator by a closed 3-form B. We also compute the cocycle in low-dimensions when the 3-form B is not closed.