Genera via Deformation Theory and Supersymmetric Mechanics
We study naturally occurring genera (i.e. cobordism invariants) from the deformation theory in- spired by supersymmetric quantum mechanics. First, we construct a canonical deformation quantization for symplectic supermanifolds. This gives a novel proof of the super-analogue of Fedosov quantization....
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| Format: | Thesis |
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Massachusetts Institute of Technology
2022
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| Online Access: | https://hdl.handle.net/1721.1/145087 |
| _version_ | 1811083329043169280 |
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| author | Wilson, Araminta Amabel |
| author2 | Hopkins, Michael J. |
| author_facet | Hopkins, Michael J. Wilson, Araminta Amabel |
| author_sort | Wilson, Araminta Amabel |
| collection | MIT |
| description | We study naturally occurring genera (i.e. cobordism invariants) from the deformation theory in- spired by supersymmetric quantum mechanics. First, we construct a canonical deformation quantization for symplectic supermanifolds. This gives a novel proof of the super-analogue of Fedosov quantization. Our proof uses the formalism of Gelfand-Kazhdan descent, whose foundations we establish in the super-symplectic setting.
In the second part of this thesis, we prove a super-version of Nest-Tsygan’s algebraic index theorem, generalizing work of Engeli. This work is inspired by the appearance of the same genera in three related stories: index theory, trace methods in deformation theory, and partition functions in quantum field theory. Using the trace methodology, we compute the genus appearing in the story for supersymmetric quantum mechanics. This involves investigating supertraces on Weyl-Clifford algebras and deformations of symplectic supermanifolds. |
| first_indexed | 2024-09-23T12:31:19Z |
| format | Thesis |
| id | mit-1721.1/145087 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T12:31:19Z |
| publishDate | 2022 |
| publisher | Massachusetts Institute of Technology |
| record_format | dspace |
| spelling | mit-1721.1/1450872022-08-30T03:35:03Z Genera via Deformation Theory and Supersymmetric Mechanics Wilson, Araminta Amabel Hopkins, Michael J. Massachusetts Institute of Technology. Department of Mathematics We study naturally occurring genera (i.e. cobordism invariants) from the deformation theory in- spired by supersymmetric quantum mechanics. First, we construct a canonical deformation quantization for symplectic supermanifolds. This gives a novel proof of the super-analogue of Fedosov quantization. Our proof uses the formalism of Gelfand-Kazhdan descent, whose foundations we establish in the super-symplectic setting. In the second part of this thesis, we prove a super-version of Nest-Tsygan’s algebraic index theorem, generalizing work of Engeli. This work is inspired by the appearance of the same genera in three related stories: index theory, trace methods in deformation theory, and partition functions in quantum field theory. Using the trace methodology, we compute the genus appearing in the story for supersymmetric quantum mechanics. This involves investigating supertraces on Weyl-Clifford algebras and deformations of symplectic supermanifolds. Ph.D. 2022-08-29T16:31:57Z 2022-08-29T16:31:57Z 2022-05 2022-06-07T15:34:08.185Z Thesis https://hdl.handle.net/1721.1/145087 In Copyright - Educational Use Permitted Copyright MIT http://rightsstatements.org/page/InC-EDU/1.0/ application/pdf Massachusetts Institute of Technology |
| spellingShingle | Wilson, Araminta Amabel Genera via Deformation Theory and Supersymmetric Mechanics |
| title | Genera via Deformation Theory and Supersymmetric Mechanics |
| title_full | Genera via Deformation Theory and Supersymmetric Mechanics |
| title_fullStr | Genera via Deformation Theory and Supersymmetric Mechanics |
| title_full_unstemmed | Genera via Deformation Theory and Supersymmetric Mechanics |
| title_short | Genera via Deformation Theory and Supersymmetric Mechanics |
| title_sort | genera via deformation theory and supersymmetric mechanics |
| url | https://hdl.handle.net/1721.1/145087 |
| work_keys_str_mv | AT wilsonaramintaamabel generaviadeformationtheoryandsupersymmetricmechanics |