Boundaries, spectral triples and K-homology
(2019) In Journal of Noncommutative Geometry 13(2). p.407-472- 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◃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... (More) - 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. (Less)
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https://lup.lub.lu.se/record/a2b8adab-8dd6-40e0-a83f-43b2fdb7d33d
- author
- Goffeng, Carl Henrik Tryggve Magnus LU ; Forsyth, Iain ; Rennie, Adam and Mesland, Bram
- publishing date
- 2019
- type
- Contribution to journal
- publication status
- published
- keywords
- spectral triple, manifold-with-boundary, K-homology
- in
- Journal of Noncommutative Geometry
- volume
- 13
- issue
- 2
- pages
- 407 - 472
- publisher
- European Mathematical Society Publishing House
- external identifiers
-
- scopus:85070505143
- ISSN
- 1661-6960
- DOI
- 10.4171/JNCG/331
- language
- English
- LU publication?
- no
- id
- a2b8adab-8dd6-40e0-a83f-43b2fdb7d33d
- date added to LUP
- 2021-03-12 11:58:18
- date last changed
- 2022-04-19 05:01:46
@article{a2b8adab-8dd6-40e0-a83f-43b2fdb7d33d,
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◃A. Examples include manifolds with boundary, manifolds with conical singularities, dimension drop algebras, θ-deformations and Cuntz–Pimsner algebras of vector bundles.<br/><br/>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<br/>-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.<br/><br/>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.<br/><br/>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.}},
author = {{Goffeng, Carl Henrik Tryggve Magnus and Forsyth, Iain and Rennie, Adam and Mesland, Bram}},
issn = {{1661-6960}},
keywords = {{spectral triple; manifold-with-boundary; K-homology}},
language = {{eng}},
number = {{2}},
pages = {{407--472}},
publisher = {{European Mathematical Society Publishing House}},
series = {{Journal of Noncommutative Geometry}},
title = {{Boundaries, spectral triples and K-homology}},
url = {{http://dx.doi.org/10.4171/JNCG/331}},
doi = {{10.4171/JNCG/331}},
volume = {{13}},
year = {{2019}},
}