Spectral triples and manifolds with boundary

TL;DR

This paper explores the construction of spectral triples on manifolds with boundary within noncommutative geometry, demonstrating the absence of tadpoles for Dirac operators with chiral boundary conditions.

Contribution

It introduces a method to construct spectral triples with boundary conditions and proves the non-existence of tadpoles in this setting.

Findings

Spectral triples can be constructed for manifolds with boundary.

Dirac operators with chiral boundary conditions have no tadpoles.

The approach extends noncommutative geometry to boundary cases.

Abstract

We investigate manifolds with boundary in noncommutative geometry. Spectral triples associated to a symmetric differential operator and a local boundary condition are constructed. For a classical Dirac operator with a chiral boundary condition, we show that there is no tadpole.