Topological state sum models in four dimensions, half-twists 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 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 fu...
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| Format: | Thesis (University of Nottingham only) |
| Language: | English |
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2017
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| Online Access: | https://eprints.nottingham.ac.uk/41720/ |
| _version_ | 1848796339694993408 |
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| author | Bärenz, Manuel |
| author_facet | Bärenz, Manuel |
| author_sort | Bärenz, Manuel |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T19:46:25Z |
| format | Thesis (University of Nottingham only) |
| id | nottingham-41720 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-14T19:46:25Z |
| publishDate | 2017 |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-417202025-02-28T13:43:41Z https://eprints.nottingham.ac.uk/41720/ Topological state sum models in four dimensions, half-twists and their applications Bärenz, Manuel 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. 2017-07-12 Thesis (University of Nottingham only) NonPeerReviewed application/pdf en arr https://eprints.nottingham.ac.uk/41720/8/Thesis_1.pdf Bärenz, Manuel (2017) Topological state sum models in four dimensions, half-twists and their applications. PhD thesis, University of Nottingham. Topological quantum field theory ; state sum models ; quantum gravity ; category theory |
| spellingShingle | Topological quantum field theory ; state sum models ; quantum gravity ; category theory Bärenz, Manuel Topological state sum models in four dimensions, half-twists and their applications |
| title | Topological state sum models in four dimensions, half-twists and their applications |
| title_full | Topological state sum models in four dimensions, half-twists and their applications |
| title_fullStr | Topological state sum models in four dimensions, half-twists and their applications |
| title_full_unstemmed | Topological state sum models in four dimensions, half-twists and their applications |
| title_short | Topological state sum models in four dimensions, half-twists and their applications |
| title_sort | topological state sum models in four dimensions, half-twists and their applications |
| topic | Topological quantum field theory ; state sum models ; quantum gravity ; category theory |
| url | https://eprints.nottingham.ac.uk/41720/ |