Computations in equivariant Gromov-Witten theory of GKM spaces
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917072644079616 |
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| author | Holmes, Daniel Muratore, Giosuè |
| author_facet | Holmes, Daniel Muratore, Giosuè |
| contents | We study equivariant Gromov-Witten invariants and quantum cohomology in GKM theory. Building on the localization formula, we prove that the resulting expression is independent of the choice of compatible connection, and provide an equivalent formulation without auxiliary choices. Motivated by this theoretical refinement, we develop a software package, $\mathtt{GKMtools . jl}$, that implements the computation of equivariant GW invariants and quantum products directly from the GKM graph. We apply our framework to several geometric settings: Calabi-Yau rank two vector bundles on the projective line, where we obtain a new proof for a recent connection to Donaldson-Thomas theory of Kronecker quivers; twisted flag manifolds, which give symplectic but non-algebraic examples of GKM spaces; realizability questions for abstract GKM graphs; classical enumerative problems involving curves in hyperplanes; and quantum Schubert calculus for smooth Schubert varieties. These results demonstrate both the theoretical flexibility of GKM methods and the effectiveness of computational tools in exploring new phenomena. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_07562 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Computations in equivariant Gromov-Witten theory of GKM spaces Holmes, Daniel Muratore, Giosuè Algebraic Geometry Symplectic Geometry 14L30, 14N35, 53D20 (Primary), 53D45, 14-04, 14N10 (Secondary) We study equivariant Gromov-Witten invariants and quantum cohomology in GKM theory. Building on the localization formula, we prove that the resulting expression is independent of the choice of compatible connection, and provide an equivalent formulation without auxiliary choices. Motivated by this theoretical refinement, we develop a software package, $\mathtt{GKMtools . jl}$, that implements the computation of equivariant GW invariants and quantum products directly from the GKM graph. We apply our framework to several geometric settings: Calabi-Yau rank two vector bundles on the projective line, where we obtain a new proof for a recent connection to Donaldson-Thomas theory of Kronecker quivers; twisted flag manifolds, which give symplectic but non-algebraic examples of GKM spaces; realizability questions for abstract GKM graphs; classical enumerative problems involving curves in hyperplanes; and quantum Schubert calculus for smooth Schubert varieties. These results demonstrate both the theoretical flexibility of GKM methods and the effectiveness of computational tools in exploring new phenomena. |
| title | Computations in equivariant Gromov-Witten theory of GKM spaces |
| topic | Algebraic Geometry Symplectic Geometry 14L30, 14N35, 53D20 (Primary), 53D45, 14-04, 14N10 (Secondary) |
| url | https://arxiv.org/abs/2509.07562 |