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 := d dt + A (the abstract analogue of a Riemannian Dirac operator) and D := 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 the Fredholm index of the operator D equipped with APS boundary conditions is equal to the spectral flow of the family A(t) .

Locations

• Operators and Matrices - View - PDF
• arXiv (Cornell University) - View - PDF