The APS-index and the spectral flow

Abstract

We study the Atiyah-Patodi-Singer (APS) index, and its equality to the spectral flow, in an abstract, functional analytic setting. More precisely, we consider a (suitably continuous or differentiable) family of self-adjoint Fredholm operators $A(t)$ on a Hilbert space, parametrised by $t$ in a finite interval. We then consider two different operators, namely $D:=\frac{d}{dt}+A$ (the abstract analogue of a Riemannian Dirac operator) and $D:=\frac{d}{dt}-iA$ (the abstract analogue of a Lorentzian Dirac operator). The latter case is inspired by a recent index theorem by Bär and Strohmaier (Amer.\ J.\ Math. 141 (2019), 1421--1455) for a Lorentzian Dirac operator equipped with APS boundary conditions. In both cases, we prove that Fredholm index of the operator $D$ equipped with APS boundary conditions is equal to the spectral flow of the family $A(t)$.

References in corpus (2)

• A noncommutative Atiyah-Patodi-Singer index theorem in KK-theory
• Families of spectral triples and foliations of space(time)

Cited by in corpus (2)

• Local Index Theory for Lorentzian Manifolds
• An index theorem on asymptotically static spacetimes with compact Cauchy surface