PhD Thesis - Topological state sum models in four dimensions, half-twists and their applications

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Thesis submitted to the University of Nottingham for the degree of Doctor of Philosophy
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Various mathematical tools are developed with the aim of application in mathematical physics.
In the first part, a new state sum model for four-manifolds is introduced which generalises the Crane-Yetter model. It is parametrised by a pivotal functor from a spherical fusion category into a ribbon fusion category. The special case of the Crane-Yetter model for an arbitrary ribbon fusion category $\mathcal{C}$ arises when we consider the canonical inclusion $\mathcal{C} \hookrightarrow \mathcal{Z(C)}$ into the Drinfeld centre as the pivotal functor. The model is defined in terms of handle decompositions of manifolds and thus enjoys a succinct and intuitive graphical calculus, through which concrete calculations become very easy. It gives a chain-mail procedure for the Crane-Yetter model even in the case of a nonmodular category.The nonmodular Crane-Yetter model is then shown to be nontrivial: It depends at least on the fundamental group of the manifold. Relations to the Walker-Wang model and recent calculations of ground state degeneracies are established.

The second part develops the theory of involutive monoidal categories and half-twists (which are related to braided and balanced structures) further. Several gaps in the literature are closed and some missing infrastructure is developed. The main novel contribution are ``half-ribbon'' categories, which combine duals - represented by rotations in the plane by $\pi$ - with half-twists, which are represented by turns of ribbons by $\pi$ around the vertical axis. Many examples are given, and a general construction of a half-ribbon category is presented, resulting in so-called half-twisted categories.