Higher Spectral Flow for Dirac Operators with Local Boundary Conditions

TL;DR

This paper develops a formula for the higher spectral flow of a family of Dirac operators with local boundary conditions on fibered manifolds, generalizing previous results to the case of parameterized families.

Contribution

It extends the higher spectral flow formula to families of Dirac operators with local boundary conditions, broadening the scope of spectral flow analysis.

Findings

Established a formula for higher spectral flow in the families case.

Generalized previous results by Gorokhovsky and Lesch.

Applicable to fiberwise twisted Dirac operators with boundary conditions.

Abstract

We consider a gauge invariant one parameter family of families of fiberwise twisted Dirac type operators on a fiberation with the typical fiber an even dimensional compact manifold with boundary, i.e., a family $\{D_u\}, u\in [0,1]$ with $D_1=gD_0g^{-1}$ for a suitable unitary automorphism $g$ of the twisted bundle. Suppose all the operators $D_u$ are imposed with a certain \emph{local elliptic} boundary condition $F$ and $D_{u,F}$ is the self-adjoint extension of $D_u$. We establish a formula for the higher spectral flow of $\{D_{u,F}\}$, $u\in[0,1]$. Our result generalizes a recent result of Gorokhovsky and Lesch to the families case.