Quantum Kirwan morphism and Gromov-Witten invariants of quotients III

This is the third 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. We also give a formula for a solution to the quantum differential equation in terms of a localized gauged potential. These results overlap with those of Givental, Lian-Liu-Yau, Coates-Corti-Iritani-Tseng and Ciocan-Fontanine-Kim.