A reconstruction theorem for almost-commutative spectral triples

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.Comment: 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 axio

Dirac operators, gauge systems and quantisation

Contains fulltext : 129492.pdf (publisher's version ) (Open Access)Radboud Universiteit Nijmegen, 11 september 2014Promotor : Landsman, N.P. Co-promotor : Suijlekom, W.D. van197 p

Quantization commutes with singular reduction: Cotangent bundles of compact Lie groups

Contains fulltext : 207671.pdf (preprint version ) (Open Access)We analyze the 'quantization commutes with reduction' problem (first studied in physics by Dirac, and known in the mathematical literature also as the Guillemin-Sternberg Conjecture) for the conjugate action of a compact connected Lie group G on its own cotangent bundle T*G. This example is interesting because the momentum map is not proper and the ensuing symplectic (or Marsden-Weinstein quotient) T*G//AdG is typically singular. In the spirit of (modern) geometric quantization, our quantization of T*G (with its standard Kahler structure) is defined as the kernel of the Dolbeault-Dirac operator (or, equivalently, the spin(C)-Dirac operator) twisted by the pre-quantum line bundle. We show that this quantization of T*G reproduces the Hilbert space found earlier by Hall (2002) using geometric quantization based on a holomorphic polarization. We then define the quantization of the singular quotient T*G//AdG as the kernel of the twisted Dolbeault-Dirac operator on the principal stratum, and show that quantization commutes with reduction in the sense that either way one obtains the same Hilbert space L-2(T)(W(G,T))