A Quantum $H^*(T)$-module via Quasimap Invariants
Jae Hwang Lee
TL;DR
The paper introduces a novel deformation of quantum cohomology for smooth projective toric varieties via $2|1$-pointed quasimaps, yielding a quantum $H^*(T)$-module structure on $H^*(X_\Sigma)$. It constructs a global zero-locus model and uses localized Euler classes to define the virtual fundamental class, establishing a WDVV-type Splitting Lemma to prove associativity. The Hirzebruch surface $\mathbb{F}_2$ is computed explicitly using Atiyah–Bott localization, producing concrete two-pointed invariants and demonstrating that the resulting quantum module aligns with the Batyrev ring as a module. A conjecture is formulated that, for any semipositive toric variety $V // T$, the quantum $H^*(T)$-module structure coincides with the natural Batyrev module, supported by the $\mathbb{F}_2$ case. Collectively, the work provides a geometric realization of Batyrev rings via $2|1$-quasimap invariants and showcases computational techniques that leverage localization and boundary-splitting arguments in toric settings.
Abstract
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.