Cluster algebra and quasimap quantum cohomology
Yingchun Zhang, Zijun Zhou
TL;DR
The paper develops an explicit abelianization framework for the $\mathsf{T}$-equivariant quasimap quantum cohomology $\operatorname{QH}_\mathsf{T}(X)$ of GIT quotients, showing that its ring structure can be presented as a quotient of $H^*_\mathsf{T}(pt) \otimes H^*_K(pt)^\mathsf{W}$ by an ideal generated from abelianized $I$-function relations. It then specializes to quiver varieties, deriving quantum multiplication relations among tautological Chern classes and establishing cluster-algebra realization: for oriented-acyclic quivers there is an injective map from the cluster algebra to the polynomial part of $\operatorname{QH}_\mathsf{T}(X)$, with cluster variables expressed via truncated Chern quotients $\delta_t(\cdot,\cdot)$. The framework connects to quiver flag varieties, providing rigidity results and a comparison with ordinary quantum cohomology, including a GW-poly cluster structure. Collectively, the work reveals a robust bridge between cluster algebras and quasimap quantum cohomology, yielding new computational tools and structural insights with potential applications in mirror symmetry and representation theory.
Abstract
We apply the abelianization technique to obtain an explicit ring presentation for the quasimap quantum cohomology of GIT quotients. As an application, for quiver varieties associated with oriented-acyclic quivers, we establish a cluster algebra structure on their equivariant quasimap quantum cohomology rings.