Boundary value problems with Atiyah-Patodi-Singer type conditions and spectral triples

TL;DR

This paper investigates boundary value problems with Atiyah-Patodi-Singer conditions, analyzing their properties and constructing spectral triples for manifolds with boundary, extending noncommutative geometry tools to these settings.

Contribution

It develops a framework for elliptic pseudodifferential operators with APS conditions and constructs spectral triples for manifolds with boundary of arbitrary dimension.

Findings

Ellipticity and Fredholm properties are established for APS-type boundary conditions.

Spectral triples are constructed for manifolds with boundary, generalizing noncommutative geometry.

The algebra closure coincides with continuous functions constant on boundary components.

Abstract

We study realizations of pseudodifferential operators acting on sections of vector-bundles on a smooth, compact manifold with boundary, subject to conditions of Atiyah-Patodi-Singer type. Ellipticity and Fredholm property, compositions, adjoints and self-adjointness of such realizations are discussed. We construct regular spectral triples $(\mathcal{A},\mathcal{H},\mathcal{D})$ for manifolds with boundary of arbitrary dimension, where $\mathcal{H}$ is the space of square integrable sections. Starting out from Dirac operators with APS-conditions, these triples are even in case of even dimensional manifolds; we show that the closure of $\mathcal{A}$ in $\mathscr{L}(\mathcal{H})$ coincides with the continuous functions on the manifold being constant on each connected component of the boundary.