Spectral flow and the Atiyah-Patodi-Singer index theorem

TL;DR

This paper derives a formula for spectral flow of twisted Dirac operators on odd-dimensional spin manifolds, linking it to geometric invariants and providing a new proof of a scalar curvature rigidity theorem.

Contribution

It generalizes Getzler's spectral flow formula using Atiyah-Patodi-Singer index theorem techniques, applicable to both even and odd dimensions.

Findings

Derived a spectral flow formula involving $$-form, Chern character, and $$-invariants.

Provided a simple proof of Llarull's scalar curvature rigidity theorem.

Extended the applicability of spectral flow analysis to a broader class of manifolds.

Abstract

We establish a formula for the spectral flow of a smooth family of twisted Dirac operators on a closed odd-dimensional Riemannian spin manifold, generalizing a result by Getzler. The spectral flow is expressed in terms of the $\hat{A}$-form of the manifold, the odd Chern character form of the family of connections, and the $\xi$-invariants of the initial and final operators. Our proof is based on a reduction to the Atiyah-Patodi-Singer index theorem for manifolds with boundary, which provides a conceptually very simple approach to the problem. As an application, we give a proof of Llarull's rigidity theorem for scalar curvature of strictly convex hypersurfaces in Euclidean space which works the same in even and odd dimensions.