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Mirror symmetric Gamma conjecture for toric GIT quotients via Fourier transform (2501.14222v1)
Published 24 Jan 2025 in math.AG, math-ph, math.MP, and math.SG
Abstract: Let $\mathcal X=[(\mathbb Cr\setminus Z)/G]$ 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 $\mathbb Cr$ 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 $\mathcal X$.