Quantum cohomology of variations of GIT quotients and flips

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 $X_{-}⇢X_{+}$ 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.