Fourier analysis of equivariant quantum cohomology
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908499695370240 |
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| author | Iritani, Hiroshi |
| author_facet | Iritani, Hiroshi |
| contents | Equivariant quantum cohomology possesses the structure of a difference module by shift operators (Seidel representation) of equivariant parameters. Teleman's conjecture suggests that shift operators and equivariant parameters acting on QH_T(X) should be identified, respectively, with the Novikov variables and the quantum connection of the GIT quotient X//T. This can be interpreted as a form of Fourier duality between equivariant quantum cohomology (D-module) of X and quantum cohomology (D-module) of the GIT quotient X//T.
We introduce the notion of "quantum volume," derived from Givental's path integral over the Floer fundamental cycle, and present a conjectural Fourier duality relationship between the T-equivariant quantum volume of X and the quantum volume of X//T. We also explore the "reduction conjecture," developed in collaboration with Fumihiko Sanda, which expresses the I-function of X//T as a discrete Fourier transform of the equivariant J-function of X. Furthermore, we demonstrate how to use Fourier analysis of equivariant quantum cohomology to observe toric mirror symmetry and prove a decomposition of quantum cohomology D-modules of projective bundles or blowups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_18849 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fourier analysis of equivariant quantum cohomology Iritani, Hiroshi Algebraic Geometry High Energy Physics - Theory Symplectic Geometry Equivariant quantum cohomology possesses the structure of a difference module by shift operators (Seidel representation) of equivariant parameters. Teleman's conjecture suggests that shift operators and equivariant parameters acting on QH_T(X) should be identified, respectively, with the Novikov variables and the quantum connection of the GIT quotient X//T. This can be interpreted as a form of Fourier duality between equivariant quantum cohomology (D-module) of X and quantum cohomology (D-module) of the GIT quotient X//T. We introduce the notion of "quantum volume," derived from Givental's path integral over the Floer fundamental cycle, and present a conjectural Fourier duality relationship between the T-equivariant quantum volume of X and the quantum volume of X//T. We also explore the "reduction conjecture," developed in collaboration with Fumihiko Sanda, which expresses the I-function of X//T as a discrete Fourier transform of the equivariant J-function of X. Furthermore, we demonstrate how to use Fourier analysis of equivariant quantum cohomology to observe toric mirror symmetry and prove a decomposition of quantum cohomology D-modules of projective bundles or blowups. |
| title | Fourier analysis of equivariant quantum cohomology |
| topic | Algebraic Geometry High Energy Physics - Theory Symplectic Geometry |
| url | https://arxiv.org/abs/2501.18849 |