A reconstruction theorem for almost-commutative spectral triples
arXiv:1101.5908 · doi:10.1007/s11005-011-0534-5
Abstract
We propose an expansion of the definition of almost-commutative spectral triple that accommodates non-trivial fibrations and is stable under inner fluctuation of the metric, and then prove a reconstruction theorem for almost-commutative spectral triples under this definition as a simple consequence of Connes's reconstruction theorem for commutative spectral triples. Along the way, we weaken the orientability hypothesis in the reconstruction theorem for commutative spectral triples, and following Chakraborty and Mathai, prove a number of results concerning the stability of properties of spectral triples under suitable perturbation of the Dirac operator.
AMS-LaTeX, 19 pp. V4: Updated version incorporating the erratum of June 2012, correcting the weak orientability axiom in the definition of commutative spectral triple, stengthening Lemma A.10 to cover the odd-dimensional case and the proof of Corollary 2.19 to accommodate the corrected weak orientability axiom
References in corpus (7)
- A Lorentzian version of the non-commutative geometry of the standard model of particle physics
- Noncommutative Geometry and the standard model with neutrino mixing
- On a Classification of Irreducible Almost-Commutative Geometries IV
- The noncommutative geometry of Yang-Mills fields
- Moduli spaces of Dirac operators for finite spectral triples
- On a Classification of Irreducible Almost-Commutative Geometries V
- The geometry of determinant line bundles in noncommutative geometry
Cited by in corpus (7)
- On globally non-trivial almost-commutative manifolds
- On stochastic generation of ultrametrics in high-dimension Euclidean spaces
- Gauge theory on noncommutative Riemannian principal bundles
- A reconstruction theorem for Connes-Landi deformations of commutative spectral triples
- Real structures on almost-commutative spectral triples
- Spin geometry of the rational noncommutative torus
- Factorization of Dirac operators on toric noncommutative manifolds