Higher genus Gromov-Witten theory of \(\mathsf{Hilb}^n(\mathbb{C}^2)\) and \(\mathsf{CohFTs}\) associated to local curves.(English)Zbl 1461.14076
Summary: We study the higher genus equivariant Gromov-Witten theory of the Hilbert scheme of \(n\) points of \(\mathbb{C}^2\). Since the equivariant quantum cohomology, computed byA. Okounkov andR. Pandharipande [Invent. Math. 179, No. 3, 523–557 (2010;Zbl 1198.14054)], is semisimple, the higher genus theory is determined by an \(\mathsf{R}\)-matrix via the Givental-Teleman classification of Cohomological Field Theories (CohFTs). We uniquely specify the required \(\mathsf{R}\)-matrix by explicit data in degree 0. As a consequence, we lift the basic triangle of equivalences relating the equivariant quantum cohomology of the Hilbert scheme \(\mathsf{Hilb}^n(\mathbb{C}^2)\) and the Gromov-Witten/Donaldson-Thomas correspondence for 3-fold theories of local curves to a triangle of equivalences in all higher genera. The proof uses the analytic continuation of the fundamental solution of the QDE of the Hilbert scheme of points determined byA. Okounkov andR. Pandharipande [Transform. Groups 15, No. 4, 965–982 (2010;Zbl 1213.14108)]. The GW/DT edge of the triangle in higher genus concerns new CohFTs defined by varying the 3-fold local curve in the moduli space of stable curves. The equivariant orbifold Gromov-Witten theory of the symmetric product \(\mathsf{Sym}^n(\mathbb{C}^2)\) is also shown to be equivalent to the theories of the triangle in all genera. The result establishes a complete case of the crepant resolution conjecture [J. Bryan andT. Graber, Proc. Symp. Pure Math. 80, 23–42 (2009;Zbl 1198.14053);T. Coates et al., Geom. Topol. 13, No. 5, 2675–2744 (2009;Zbl 1184.53086);T. Coates andY. Ruan, Ann. Inst. Fourier 63, No. 2, 431–478 (2013;Zbl 1275.53083)].
MSC:
| 14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |
| 53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |
Cite
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