State sums and geometry
In this thesis I review the definition of topological quantum field theories through state sums on triangulated manifolds. I describe the construction of state sum invariants of 3-manifolds from a graphical calculus and show how to evaluate the invariants as boundary amplitudes. I review how to defi...
| Main Author: | |
|---|---|
| Format: | Thesis (University of Nottingham only) |
| Language: | English |
| Published: |
2011
|
| Online Access: | https://eprints.nottingham.ac.uk/11803/ |
| _version_ | 1848791362838724608 |
|---|---|
| author | Hellmann, Frank |
| author_facet | Hellmann, Frank |
| author_sort | Hellmann, Frank |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | In this thesis I review the definition of topological quantum field theories through state sums on triangulated manifolds. I describe the construction of state sum invariants of 3-manifolds from a graphical calculus and show how to evaluate the invariants as boundary amplitudes. I review how to define such a graphical calculus through SU(2) representation theory. I then review various geometricity results for the representation theory of SU(2), Spin(4) and SL(2,C), and define coherent boundary manifolds for state sums based on these representations. I derive the asymptotic geometry of the SU(2) based Ponzano-Regge invariant in three dimensions, and the SU(2) based Ooguri models amplitude in four dimensions. As a corollary to the latter results I derive the asymptotic behaviour of various recently proposed spin foam models motivated from the Plebanski formulation of general relativity. Finally the asymptotic geometry of the SL(2,C) based model is derived. |
| first_indexed | 2025-11-14T18:27:19Z |
| format | Thesis (University of Nottingham only) |
| id | nottingham-11803 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-14T18:27:19Z |
| publishDate | 2011 |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-118032025-02-28T11:15:42Z https://eprints.nottingham.ac.uk/11803/ State sums and geometry Hellmann, Frank In this thesis I review the definition of topological quantum field theories through state sums on triangulated manifolds. I describe the construction of state sum invariants of 3-manifolds from a graphical calculus and show how to evaluate the invariants as boundary amplitudes. I review how to define such a graphical calculus through SU(2) representation theory. I then review various geometricity results for the representation theory of SU(2), Spin(4) and SL(2,C), and define coherent boundary manifolds for state sums based on these representations. I derive the asymptotic geometry of the SU(2) based Ponzano-Regge invariant in three dimensions, and the SU(2) based Ooguri models amplitude in four dimensions. As a corollary to the latter results I derive the asymptotic behaviour of various recently proposed spin foam models motivated from the Plebanski formulation of general relativity. Finally the asymptotic geometry of the SL(2,C) based model is derived. 2011-07-13 Thesis (University of Nottingham only) NonPeerReviewed application/pdf en arr https://eprints.nottingham.ac.uk/11803/1/Thesis-arx.pdf Hellmann, Frank (2011) State sums and geometry. PhD thesis, University of Nottingham. |
| spellingShingle | Hellmann, Frank State sums and geometry |
| title | State sums and geometry |
| title_full | State sums and geometry |
| title_fullStr | State sums and geometry |
| title_full_unstemmed | State sums and geometry |
| title_short | State sums and geometry |
| title_sort | state sums and geometry |
| url | https://eprints.nottingham.ac.uk/11803/ |