Title:

Topological state sum models in four dimensions, halftwists and their applications

Various mathematical tools are developed with the aim of application in mathematical physics. In the first part, a new state sum model for fourmanifolds is introduced which generalises the CraneYetter model. It is parametrised by a pivotal functor from a spherical fusion category into a ribbon fusion category. The special case of the CraneYetter model for an arbitrary ribbon fusion category C arises when we consider the canonical inclusion C↪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 chainmail procedure for the CraneYetter model even in the case of a nonmodular category. The nonmodular CraneYetter model is then shown to be nontrivial: It depends at least on the fundamental group of the manifold. Relations to the WalkerWang model and recent calculations of ground state degeneracies are established. The second part develops the theory of involutive monoidal categories and halftwists (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 ``halfribbon'' categories, which combine duals  represented by rotations in the plane by π  with halftwists, which are represented by turns of ribbons by π around the vertical axis. Many examples are given, and a general construction of a halfribbon category is presented, resulting in socalled halftwisted categories.
