Superconnections and Index Theory

TL;DR

This paper develops index theory for Dirac operators coupled with superconnections, establishing local index theorems, eta-invariants, an APS-theorem, and geometric determinant line bundles with explicit curvature and holonomy calculations.

Contribution

It introduces a new framework for index theory involving superconnections, extending classical results to this broader context.

Findings

Proved a local index theorem for Dirac operators with superconnections

Established an APS-theorem for eta-invariants in this setting

Constructed and analyzed a geometric determinant line bundle

Abstract

We investigate index theory in the context of Dirac operators coupled to superconnections. In particular, we prove a local index theorem for such operators, and for families of such operators. We investigate eta-invariants and prove an APS-theorem, and construct a geometric determinant line bundle for families of such operators, computing its curvature and holonomy in terms of familiar index theoretic quantities.