Quantum Kirwan morphism and Gromov-Witten invariants of quotients II

This is the second in a sequence of papers in which we construct a quantum version of the Kirwan map from the equivariant quantum cohomology of a smooth polarized complex projective variety with the action of a connected complex reductive group to the orbifold quantum cohomology of its geometric invariant theory quotient, and prove that it intertwines the genus zero gauged Gromov-Witten potential with the genus zero Gromov-Witten graph potential. In this part we construct virtual fundamental classes on the moduli spaces used in the construction of the quantum Kirwan map and the gauged Gromov-Witten potential.