Equivariant quantum connections in positive characteristic
In this thesis, we apply techniques from symplectic Gromov--Witten theory to study the equivariant quantum connections in positive characteristic. The main examples of interest arise from symplectic resolutions. We introduce equivariant generalizations of the quantum Steenrod operations of Fukaya, p...
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| Format: | Thesis |
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Massachusetts Institute of Technology
2024
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| Online Access: | https://hdl.handle.net/1721.1/155417 https://orcid.org/0000-0002-7022-8735 |
| _version_ | 1811068414047813632 |
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| author | Lee, Jae Hee |
| author2 | Seidel, Paul |
| author_facet | Seidel, Paul Lee, Jae Hee |
| author_sort | Lee, Jae Hee |
| collection | MIT |
| description | In this thesis, we apply techniques from symplectic Gromov--Witten theory to study the equivariant quantum connections in positive characteristic. The main examples of interest arise from symplectic resolutions. We introduce equivariant generalizations of the quantum Steenrod operations of Fukaya, provide nontrivial computations in the example of the cotangent bundle of the projective line, and explore the relationship with Varchenko's construction of mod p solutions to the quantum differential equation. We then prove the compatibility of the equivariant quantum Steenrod operations with the quantum differential and difference connections. As a consequence, we obtain an identification of our operations for divisor classes with the p-curvature of the quantum connection in a wide range of examples. |
| first_indexed | 2024-09-23T07:55:38Z |
| format | Thesis |
| id | mit-1721.1/155417 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T07:55:38Z |
| publishDate | 2024 |
| publisher | Massachusetts Institute of Technology |
| record_format | dspace |
| spelling | mit-1721.1/1554172024-06-28T04:06:07Z Equivariant quantum connections in positive characteristic Lee, Jae Hee Seidel, Paul Massachusetts Institute of Technology. Department of Mathematics In this thesis, we apply techniques from symplectic Gromov--Witten theory to study the equivariant quantum connections in positive characteristic. The main examples of interest arise from symplectic resolutions. We introduce equivariant generalizations of the quantum Steenrod operations of Fukaya, provide nontrivial computations in the example of the cotangent bundle of the projective line, and explore the relationship with Varchenko's construction of mod p solutions to the quantum differential equation. We then prove the compatibility of the equivariant quantum Steenrod operations with the quantum differential and difference connections. As a consequence, we obtain an identification of our operations for divisor classes with the p-curvature of the quantum connection in a wide range of examples. Ph.D. 2024-06-27T19:51:59Z 2024-06-27T19:51:59Z 2024-05 2024-05-15T16:20:35.961Z Thesis https://hdl.handle.net/1721.1/155417 https://orcid.org/0000-0002-7022-8735 Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) Copyright retained by author(s) https://creativecommons.org/licenses/by-nc-nd/4.0/ application/pdf Massachusetts Institute of Technology |
| spellingShingle | Lee, Jae Hee Equivariant quantum connections in positive characteristic |
| title | Equivariant quantum connections in positive characteristic |
| title_full | Equivariant quantum connections in positive characteristic |
| title_fullStr | Equivariant quantum connections in positive characteristic |
| title_full_unstemmed | Equivariant quantum connections in positive characteristic |
| title_short | Equivariant quantum connections in positive characteristic |
| title_sort | equivariant quantum connections in positive characteristic |
| url | https://hdl.handle.net/1721.1/155417 https://orcid.org/0000-0002-7022-8735 |
| work_keys_str_mv | AT leejaehee equivariantquantumconnectionsinpositivecharacteristic |