Dirac operators and Batalin-Vilkovisky quantisation in noncommutative geometryTools Nguyen, Hans (2023) Dirac operators and Batalin-Vilkovisky 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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