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We study the higher genus equivariant Gromov-Witten theory of the Hilbertscheme of n points of the plane. Since the equivariant quantum cohomology issemisimple, the higher genus theory is determined by an R-matrix via theGivental-Teleman classification of Cohomological Field Theories (CohFTs). Weuniquely specify the required R-matrix by explicit data in degree 0. As aconsequence, we lift the basic triangle of equivalences relating theequivariant quantum cohomology of the Hilbert scheme and theGromov-Witten/Donaldson-Thomas correspondence for 3-fold theories of localcurves to a triangle of equivalences in all higher genera. The proof uses thepreviously determined analytic continuation of the fundamental solution of theQDE of the Hilbert scheme. The GW/DT edge of the triangle in higher genusconcerns new CohFTs defined by varying the 3-fold local curve in the modulispace of stable curves.  The equivariant orbifold Gromov-Witten theory of the symmetric product of theplane is also shown to be equivalent to the theories of the triangle in allgenera. The result establishes a complete case of the crepant resolutionconjecture.

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