• Login
• Admin

Bärenz, Manuel (2017) Topological state sum models in four dimensions, half-twists and their applications. PhD thesis, University of Nottingham.

PDF (Thesis - as examined) - Repository staff only - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader Download (960kB) Request Thesis

Abstract

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 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 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 π - with half-twists, which are represented by turns of ribbons by π 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.

Item Type:	Thesis (University of Nottingham only) (PhD)
Supervisors:	Barrett, John W. Krasnov, K.
Keywords:	Topological quantum field theory ; state sum models ; quantum gravity ; category theory
Subjects:	Q Science > QA Mathematics > QA611 Topology Q Science > QC Physics > QC170 Atomic physics. Constitution and properties of matter
Faculties/Schools:	UK Campuses > Faculty of Science > School of Mathematical Sciences
Item ID:	41720
Depositing User:	Bärenz, Manuel
Date Deposited:	12 Jul 2017 04:40
Last Modified:	13 Oct 2017 01:15
URI:	https://eprints.nottingham.ac.uk/id/eprint/41720

Actions (Archive Staff Only)

Edit View