This work constructs a cluster algebra structure within the quantum cohomology ring of a flag variety. 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. We prove that all-genus Gromov–Witten invariants of such a variety coincide with those of the flag variety.
