Cluster algebras and quantum cohomology rings: A-type

TL;DR

This paper constructs a cluster algebra structure within the quantum cohomology ring of A-type quiver varieties and explores their relation to Seiberg duality and Gromov-Witten invariants.

Contribution

It establishes an injective homomorphism from A-type cluster algebras to quantum cohomology rings and proposes a framework for general quivers with potential.

Findings

Injective homomorphism from A_n-type cluster algebra to quantum cohomology ring.

Framework for cluster algebra construction in general quiver varieties.

Proof that Gromov-Witten invariants are invariant under Seiberg duality for A-type quivers.

Abstract

We construct a cluster algebra structure within the quantum cohomology ring of a quiver variety associated with an $A$-type quiver. Specifically, let $Fl:=Fl(N_1,\ldots,N_{n+1})$ denote a partial flag variety of length $n$, and $QH_S^*(Fl)[t]:=QH_S^*(Fl)\otimes \mathbb C[t]$ be its equivariant quantum cohomology ring extended by a formal variable $t$, regarded as a $\mathbb Q$-algebra. We establish an injective $\mathbb Q$-algebra homomorphism from the $A_n$-type cluster algebra to the algebra $QH_S^*(Fl)[t]$. Furthermore, for a general quiver with potential, we propose a framework for constructing a homomorphism from the associated cluster algebra to the quantum cohomology ring of the corresponding quiver variety. The second main result addresses the conjecture of all-genus Seiberg duality for $A_n$-type quivers. For any quiver with potential mutation-equivalent to an $A_n$-type quiver, we consider the associated variety defined as the critical locus of the potential function. We prove that all-genus Gromov-Witten invariants of such a variety coincide with those of the flag variety.