Higher genus Gromov-Witten theory of the Hilbert scheme of points of the plane and CohFTs associated to local curves

Rahul Pandharipande ETH-Zurich Hsian-Hua Tseng Ohio State University

Algebraic Geometry mathscidoc:1708.01001

We study the higher genus equivariant Gromov-Witten theory of the Hilbert scheme of n points of the plane. Since the equivariant quantum cohomology is semisimple, the higher genus theory is determined by an R-matrix via the Givental-Teleman classification of Cohomological Field Theories (CohFTs). We uniquely specify the required 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 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 previously determined analytic continuation of the fundamental solution of the QDE of the Hilbert scheme. 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 of the plane 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.
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@inproceedings{rahulhigher,
  title={Higher genus Gromov-Witten theory of the Hilbert scheme of points of the plane and CohFTs associated to local curves},
  author={Rahul Pandharipande, and Hsian-Hua Tseng},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170818104320405581812},
}
Rahul Pandharipande, and Hsian-Hua Tseng. Higher genus Gromov-Witten theory of the Hilbert scheme of points of the plane and CohFTs associated to local curves. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170818104320405581812.
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