Getzler rescaling via adiabatic deformation and a renormalized local index formula

TL;DR

This paper establishes a local index theorem for Dirac operators on manifolds with Lie structures at infinity, utilizing a renormalized supertrace and a rescaling technique akin to Getzler's method.

Contribution

It introduces a new local index formula for Dirac operators on Lie manifolds using adiabatic deformation and a renormalized supertrace, extending classical index theory.

Findings

Proves a local index theorem for Lie manifolds.

Develops a heat kernel expansion coefficient calculation via deformation.

Connects the index formula with the adiabatic groupoid framework.

Abstract

We prove a local index theorem of Atiyah-Singer type for Dirac operators on manifolds with a Lie structure at infinity (Lie manifolds for short). With the help of a renormalized supertrace, defined on a suitable class of regularizing operators, the proof of the index theorem relies on a rescaling technique similar in spirit to Getzler's rescaling. With a given Lie manifold we associate an appropriate integrating Lie groupoid. We then describe the heat kernel of a geometric Dirac operator via a functional calculus with values in the convolution algebra of sections of a rescaled bundle over the adiabatic groupoid. Finally, we calculate the right coefficient in the heat kernel expansion by deforming the Dirac operator into a polynomial coefficient operator over the rescaled bundle and applying the Lichnerowicz theorem to the fibers of the groupoid and the Lie manifold.