2026-09-19T19:39:50Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/1450872022-08-30T03:35:03Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Hopkins, Michael J.
Wilson, Araminta Amabel
Massachusetts Institute of Technology. Department of Mathematics
2022-08-29T16:31:57Z
2022-08-29T16:31:57Z
2022-05
2022-06-07T15:34:08.185Z
https://hdl.handle.net/1721.1/145087
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.
Massachusetts Institute of Technology
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Genera via Deformation Theory and Supersymmetric Mechanics
Thesis
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Doctoral
Doctor of Philosophy
Publication
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<Abstract>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.</Abstract>
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