Dirac operators and Batalin-Vilkovisky quantisation in noncommutative geometry - Nguyen, Hans (2023)
The 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 $\mathbb{T}^{2}_{θ} \hookrightarrow \mathbb{S}^{3}_{θ} \hookrightarrow \mathbb{R}^{4}_{θ}$ of noncommutative hypersurface embeddings. The obtained Dirac operators agree with ones found in the literature obtained by other means.