A Quantum $H^*(T)$-module via Quasimap Invariants

For $X$ a smooth projective variety, the quantum cohomology ring $QH^*(X)$ is a deformation of the usual cohomology ring $H^*(X)$, where the product structure is modified to incorporate quantum corrections. These correction terms are defined using Gromov-Witten invariants. When $X$ is toric with the geometric quotient description $V /\!/ T$, the cohomology ring $H^*(V /\!/T)$ also has the structure of a quantum $H^*(T)$-module. In this paper, we give a new deformation using quasimap invariants with a light point. This defines $H^*(T)$-module structure on $H^*(X)$ through a modified version of the WDVV equations. Using the Atiyah-Bott localization theorem, we explicitly compute this structure for the Hirzebruch surface of type 2. We conjecture that this new quantum module structure is isomorphic to the natural module structure of the Batyrev ring for a semipositive toric variety.