Cluster algebras and quantum cohomology rings: A-type
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916773513658368 |
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| author | He, Weiqiang Zhang, Yingchun |
| author_facet | He, Weiqiang Zhang, Yingchun |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_00394 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cluster algebras and quantum cohomology rings: A-type He, Weiqiang Zhang, Yingchun Algebraic Geometry Representation Theory 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. |
| title | Cluster algebras and quantum cohomology rings: A-type |
| topic | Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2501.00394 |