Quantum Kirwan for Quantum K-theory

Abstract. For G a complex reductive group and X a smooth projective or convex quasi-projective polarized G-variety we construct a formal map in quantum K-theory κ^G_X: QK⁰_G (X) → QK⁰(X//G) from the equivariant quantum K-theory QK⁰_G to the quantum K-theory of the geometric invariant theory quotient X//G, assuming the quotient X//G is a smooth Deligne-Mumford stack with projective coarse moduli space. As an example, we give a presentation of the (possibly bulk-shifted) quantum K-theory of any smooth proper toric Deligne-Mumford stack with projective coarse moduli space, generalizing the presentation for quantum K-theory of projective spaces due to Buch-Mihalcea [7] and (implicitly) of Givental-Tonita [18]. We also provide a wall-crossing formula for the K-theoretic gauged potential τ^G_X: QK⁰_G → Λ^G_X under variation of geometric invariant theory quotient, a proof of the invariance of τ^G_X under (strong) crepant transformation assumptions, and a proof of the abelian non-abelian correspondence relating τ^G_X and τ^T_X for T ⊂ G a maximal torus..