An index formula for families of end-periodic Dirac operators

TL;DR

This paper develops a new index formula for families of Dirac operators on end-periodic manifolds, utilizing a transgression formula for the Bismut superconnection and introducing an end-periodic eta form.

Contribution

It introduces a novel index formula involving an end-periodic eta form, generalizing existing eta invariants and forms for end-periodic Dirac operators.

Findings

Derived a transgression formula for the renormalized Chern character.

Established an index formula involving the new end-periodic eta form.

Unified previous eta invariants within a broader framework.

Abstract

We derive a transgression formula for the renormalized Chern character of the Bismut superconnection in the context of end-periodic fiber bundles and families of end-periodic Clifford modules. The transgression is expressed in terms of the Fourier-Laplace transform of the Bismut superconnection using the renormalized supertrace of Mrowka-Ruberman-Saveliev. Consequently, we establish an index formula for families of Dirac operators on end-periodic manifolds. The index formula involves a new ``end-periodic eta form'' which generalizes both the Bismut-Cheeger eta form and the end-periodic eta invariant of Mrowka-Ruberman-Saveliev.