Quantum Modules of Semipositive Toric Varieties

Abstract

A smooth projective toric variety $X=X_{\hat{I}}\pounds$ has a geometric quotient description $V//T$. Using $2|1$-pointed quasimap invariants, one can define a quantum $H^{\ast }(T)$-module $QM(X)$, which deforms a natural module structure given by the Kirwan map $H^{\ast }(T)\to H^{\ast }(X)$. The Batyrev ring of $X$, defined from combinatorial data of the fan $\hat{I}\pounds$, has its natural module structure given by the quotient of a polynomial ring, say BatM$(X)$. In this paper, we prove that $QM(X)$ and BatM$(X)$ are naturally isomorphic when $X$ is semipositive.