2024/12/31 by Weiqiang He, Yingchun Zhang, He, Weiqiang +1
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2501.00394
openalex publication_date 2024/12/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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(N1,…,Nn+1) denote a partial flag variety of length n, and QHS^*(Fl)[t]:=QHS^*(Fl)⊗ \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 An-type cluster algebra to the algebra QHS^*(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 An-type quivers. For any quiver with potential mutation-equivalent to an An-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.