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For $X$ a smooth projective variety, the quantum cohomology ring $QH^*(X)$ isa deformation of the usual cohomology ring $H^*(X)$, where the productstructure is modified to incorporate quantum corrections. These correctionterms are defined using Gromov-Witten invariants. When $X$ is toric with thegeometric 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 anew deformation using quasimap invariants with a light point. This defines$H^*(T)$-module structure on $H^*(X)$ through a modified version of the WDVVequations. Using the Atiyah-Bott localization theorem, we explicitly computethis structure for the Hirzebruch surface of type 2. We conjecture that thisnew quantum module structure is isomorphic to the natural module structure ofthe Batyrev ring for a semipositive toric variety.

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