Mirror symmetric Gamma conjecture for toric GIT quotients via Fourier transform
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917049649856512 |
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| author | Aleshkin, Konstantin Fang, Bohan Wang, Junxiao |
| author_facet | Aleshkin, Konstantin Fang, Bohan Wang, Junxiao |
| contents | Let $\mathcal X=[(\mathbb C^r\setminus Z)/G]$ be a toric Fano orbifold. We compute the Fourier transform of the $G$-equivariant quantum cohomology central charge of any $G$-equivariant line bundle on $\mathbb C^r$ with respect to certain choice of parameters. This gives the quantum cohomology central charge of the corresponding line bundle on $\mathcal X$, while in the oscillatory integral expression it becomes the oscillatory integral in the mirror Landau-Ginzburg mirror of $\mathcal X$. Moving these parameters to real numbers simultaneously deforms the integration cycle to the mirror Lagrangian cycle of that line bundle. This computation produces a new proof the mirror symmetric Gamma conjecture for $\mathcal X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_14222 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mirror symmetric Gamma conjecture for toric GIT quotients via Fourier transform Aleshkin, Konstantin Fang, Bohan Wang, Junxiao Algebraic Geometry Mathematical Physics Symplectic Geometry Let $\mathcal X=[(\mathbb C^r\setminus Z)/G]$ be a toric Fano orbifold. We compute the Fourier transform of the $G$-equivariant quantum cohomology central charge of any $G$-equivariant line bundle on $\mathbb C^r$ with respect to certain choice of parameters. This gives the quantum cohomology central charge of the corresponding line bundle on $\mathcal X$, while in the oscillatory integral expression it becomes the oscillatory integral in the mirror Landau-Ginzburg mirror of $\mathcal X$. Moving these parameters to real numbers simultaneously deforms the integration cycle to the mirror Lagrangian cycle of that line bundle. This computation produces a new proof the mirror symmetric Gamma conjecture for $\mathcal X$. |
| title | Mirror symmetric Gamma conjecture for toric GIT quotients via Fourier transform |
| topic | Algebraic Geometry Mathematical Physics Symplectic Geometry |
| url | https://arxiv.org/abs/2501.14222 |