Publication Details

Forsyth, I., Goffeng, M., Mesland, B. & Rennie, A. (2019). Boundaries, spectral triples and K-homology. Journal of Noncommutative Geometry, 13 (2), 407-472.


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◃A. Examples include manifolds with boundary, manifolds with conical singularities, dimension drop algebras, θ-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.

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