Quantum cohomology of variations of GIT quotients and flips

TL;DR

This paper establishes a decomposition theorem for the quantum cohomology of GIT quotient variations, revealing how quantum $D$-modules split across wall-crossings and applying to local models of flips in birational geometry.

Contribution

It introduces a new decomposition theorem for quantum cohomology in the context of GIT variations and flips, extending understanding of quantum invariants in birational transformations.

Findings

Quantum $D$-module of $X_-$ decomposes into that of $X_+$ and the wall $S$.

Decomposition applies to local models of standard flips.

Provides tools for analyzing quantum cohomology across GIT wall-crossings.

Abstract

We prove a decomposition theorem for the quantum cohomology of variations of GIT quotients. More precisely, for any reductive group $G$ and a simple $G$-VGIT wall-crossing $X_- \dashrightarrow X_+$ with a wall $S$, we show that the quantum $D$-module of $X_-$ can be decomposed into a direct sum of that of $X_+$ and copies of that of $S$. As an application, we obtain a decomposition theorem for the quantum cohomology of local models of standard flips in birational geometry.