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 QH G ( X ) of a smooth polarized complex projective variety X with the action of a connected complex reductive group G to the orbifold quantum cohomology QH ( X // G ) of its geometric invariant theory quotient X // G , and prove that it intertwines the genus zero gauged Gromov–Witten potential of X with the genus zero Gromov–Witten graph potential of X // G . We also give a formula for a solution to the quantum differential equation on X // G in terms of a localized gauged potential for X . These results overlap with those of Givental [ 14 ], Lian–Liu–Yau [ 21 ], Iritani [ 20 ], Coates–Corti–Iritani–Tseng [ 11 ], and Ciocan–Fontanine–Kim [ 7 ], [ 8 ].