| Summary: | 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 $\mathbb{T}^{2}_{\theta} \hookrightarrow \mathbb{S}^{3}_{\theta} \hookrightarrow \mathbb{R}^{4}_{\theta}$ of noncommutative hypersurface embeddings. The obtained Dirac operators agree with ones found in the literature obtained by other means.
The second part of the thesis deals with BV quantisation of finite dimensional noncommutative field theories. The modern formulation of the BV formalism of Costello and Gwilliam is adapted to fit our setting. The formalism is illustrated through the computation of correlation functions for scalar field theories and Chern-Simons theories on the fuzzy $2$-sphere. The techniques are generalised to accommodate theories with symmetries encoded by triangular Hopf algebras. We use this to compute correlation functions for braided scalar field theories on the fuzzy $2$-torus. The BV formalism is also used to study gauge-theoretic aspects of dynamical fuzzy spectral triple models of quantum gravity. Perturbations around the trivial Dirac operator $D_{0} = 0$ and an example of perturbations around a non-trivial Dirac operator $D_{0} \neq 0$ in the quartic $(0,1)$-model are investigated. From our analysis, we conclude that the gauge-theoretical effects on the correlation functions depend strongly on the amount of gauge symmetry that is broken by the background Dirac operator one chooses to perturb around.
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