  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.
