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On Reconstruction Theorems in Noncommutative Riemannian Geometry


Ćaćić, Branimir Josip (2013) On Reconstruction Theorems in Noncommutative Riemannian Geometry. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/WRT2-7630.

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We present a novel account of the theory of commutative spectral triples and their two closest noncommutative generalisations, almost-commutative spectral triples and toric noncommutative manifolds, with a focus on reconstruction theorems, viz, abstract, functional-analytic characterisations of global-analytically defined classes of spectral triples. We begin by reinterpreting Connes's reconstruction theorem for commutative spectral triples as a complete noncommutative-geometric characterisation of Dirac-type operators on compact oriented Riemannian manifolds, and in the process clarify folklore concerning stability of properties of spectral triples under suitable perturbation of the Dirac operator. Next, we apply this reinterpretation of the commutative reconstruction theorem to obtain a reconstruction theorem for almost-commutative spectral triples. In particular, we propose a revised, manifestly global-analytic definition of almost-commutative spectral triple, and, as an application of this global-analytic perspective, obtain a general result relating the spectral action on the total space of a finite normal compact oriented Riemannian cover to that on the base space. Throughout, we discuss the relevant refinements of these definitions and results to the case of real commutative and almost-commutative spectral triples. Finally, we outline progess towards a reconstruction theorem for toric noncommutative manifolds.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Noncommutative geometry, spectral triple, Dirac-type operator, spectral action
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Marcolli, Matilde
Thesis Committee:
  • Marcolli, Matilde (chair)
  • Makarov, Nikolai G.
  • Markovic, Vladimir
  • Venselaar, Jan Jitse
Defense Date:17 May 2013
Record Number:CaltechTHESIS:05242013-033054707
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Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:7749
Deposited By: Branimir Cacic
Deposited On:29 May 2013 20:29
Last Modified:28 Oct 2021 19:02

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