Dirac operators and BatalinVilkovisky quantisation in noncommutative geometryTools Nguyen, Hans (2023) Dirac operators and BatalinVilkovisky quantisation in noncommutative geometry. PhD thesis, University of Nottingham.
AbstractThe underlying theme of this thesis is noncommutative geometry, with a particular focus on Dirac operators. In the first part of the thesis, we investigate through a module theoretic approach to noncommutative Riemannian (spin) geometry how one can induce differential, Riemannian and spinorial structures from a noncommutative ambient space to an appropriate notion of a noncommutative hypersurface, thus providing a framework for constructing Dirac operators on noncommutative hypersurfaces from geometrical data on the embedding space. This is applied to the sequence $\mathbb{T}^{2}_{\theta} \hookrightarrow \mathbb{S}^{3}_{\theta} \hookrightarrow \mathbb{R}^{4}_{\theta}$ of noncommutative hypersurface embeddings. The obtained Dirac operators agree with ones found in the literature obtained by other means.
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