Boundaries, spectral triples and K-homology

TL;DR

This paper generalizes spectral triples to relative spectral triples, introduces a Clifford normal for doubling, and explores boundary spectral triples in non-commutative geometry, with applications to manifolds with boundary and singularities.

Contribution

It extends spectral triples to a relative setting, introduces the Clifford normal for boundary constructions, and connects these to $K$-homology boundary maps in non-commutative geometry.

Findings

Relative spectral triples include manifolds with boundary and singularities.

The Clifford normal enables doubling into spectral triples and defines boundary data.

Conditions are provided for boundary triples to represent $K$-homological boundaries.

Abstract

This paper extends the notion of a spectral triple to a relative spectral triple, an unbounded analogue of a relative Fredholm module for an ideal $J\triangleleft A$. Examples include manifolds with boundary, manifolds with conical singularities, dimension drop algebras, $\theta$-deformations and Cuntz-Pimsner algebras of vector bundles. The bounded transform of a relative spectral triple is a relative Fredholm module, making the image of a relative spectral triple under the boundary mapping in $K$-homology easy to compute. We introduce an additional operator called a Clifford normal with which a relative spectral triple can be doubled into a spectral triple. The Clifford normal also provides a boundary Hilbert space, a representation of the quotient algebra, a boundary Dirac operator and an analogue of the Calderon projection. In the examples this data does assemble to give a boundary spectral triple, though we can not prove this in general. When we do obtain a boundary spectral triple, we provide sufficient conditions for the boundary triple to represent the $K$-homological boundary. Thus we abstract the proof of Baum-Douglas-Taylor's "boundary of Dirac is Dirac on the boundary" theorem into the realm of non-commutative geometry.