Quantum Kirwan Morphism and Gromov-witten Invariants of Quotients I

This is the first in a sequence of papers in which we construct a quantum version of the Kirwan map from the equivariant quantum cohomology QHG(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. In this part we introduce the moduli spaces used in the construction of the quantum Kirwan morphism.