Equivariant Spectral Flow and Equivariant $\eta$-invariants on Manifolds With Boundary

TL;DR

This paper investigates equivariant spectral invariants and eta-invariants for Dirac operators on manifolds with boundary under group actions, establishing key equalities and relations among these invariants.

Contribution

It introduces new relations between equivariant spectral flow, Maslov indices, and eta-invariants, extending spectral flow formulas to equivariant settings.

Findings

Proves equality between equivariant winding numbers, spectral flow, and Maslov indices.

Establishes a relation between equivariant eta-invariants and Maslov triple indices.

Extends Getzler's spectral flow formula to equivariant cases.

Abstract

In this article, we study several closely related invariants associated to Dirac operators on odd-dimensional manifolds with boundary with an action of the compact group $H$ of isometries. In particular, the equality between equivariant winding numbers, equivariant spectral flow, and equivariant Maslov indices is established. We also study equivariant $\eta$-invariants which play a fundamental role in the equivariant analog of Getzler's spectral flow formula. As a consequence, we establish a relation between equivariant $\eta$-invariants and equivariant Maslov triple indices in the splitting of manifolds.