Quantum Modules of Semipositive Toric Varieties

TL;DR

This paper proves that for semipositive smooth projective toric varieties, the quantum module defined via quasimap invariants is naturally isomorphic to the Batyrev ring constructed from fan data, linking geometric and combinatorial structures.

Contribution

It establishes a natural isomorphism between the quantum $H^*(T)$-module and the Batyrev ring for semipositive toric varieties, connecting geometric invariants with combinatorial data.

Findings

Quantum module $QM(X)$ is isomorphic to Batyrev ring $BatM(X)$ for semipositive $X$.

Uses $2|1$-pointed quasimap invariants to define quantum modules.

Provides a bridge between geometric quantum invariants and combinatorial fan data.

Abstract

A smooth projective toric variety $X=X_\Sigma$ has a geometric quotient description $V /\!/ T$. Using $2|1$-pointed quasimap invariants, one can define a quantum $H^*(T)$-module $QM(X)$, which deforms a natural module structure given by the Kirwan map $H^*(T) \rightarrow H^*(X)$. The Batyrev ring of $X$, defined from combinatorial data of the fan $\Sigma$, 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.