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 by Okounkov and Pandharipande [Invent. Math. 179 (2010), 523–557], 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 by Okounkov and Pandharipande [Transform. Groups 15 (2010), 965–982]. 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 [Bryan and Graber, Algebraic Geometry–Seattle 2005, Part 1, Proceedings of Symposia in Pure Mathematics, 80 (American Mathematical Society, Providence, RI, 2009), 23–42; Coates et al., Geom. Topol. 13 (2009), 2675–2744; Coates & Ruan, Ann. Inst. Fourier (Grenoble) 63 (2013), 431–478].

中文翻译：

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高格罗莫夫属——局部曲线的维滕理论并与局部曲线相关

我们研究 Hilbert 方案的更高属等变 Gromov-Witten 理论 $n$ 的点 $\mathbb{C}^{2}$ . 由于Okounkov和Pandharipande计算的等变量子上同调[发明。数学。 179(2010), 523–557]，是半简单的，高等属理论由 $\mathsf{R}$ -矩阵通过上同调场论 (CohFTs) 的 Givental-Teleman 分类。我们唯一指定所需的 $\mathsf{R}$ - 以度为单位的显式数据矩阵 $0$ . 因此，我们提升了与希尔伯特方案的等变量子上同调相关的基本等价三角形 $\mathsf{Hilb}^{n}(\mathbb{C}^{2})$ 以及 Gromov-Witten/Donaldson-Thomas 对应的局部曲线的 3 倍理论与所有高等属中的等价三角形。证明使用由 Okounkov 和 Pandharipande 确定的点的希尔伯特方案的 QDE 基本解的解析延拓 [转变。团体 15(2010), 965–982]。较高属三角形的 GW/DT 边缘涉及通过改变稳定曲线模空间中的 3 倍局部曲线定义的新 CohFT。对称积的等变轨道 Gromov-Witten 理论 $\mathsf{符号}^{n}(\mathbb{C}^{2})$ 也被证明等同于所有属的三角形理论。结果建立了 crepant 分辨率猜想的完整案例 [Bryan 和 Graber，代数几何 - 西雅图 2005，第 1 部分, 纯数学研讨会论文集，80（美国数学会，普罗维登斯，罗德岛州，2009 年），23-42；科茨等。,几何。白杨。 13(2009), 2675–2744; 科茨和阮，安。研究所。傅立叶（格勒诺布尔） 63(2013), 431–478]。