Existence of the quantum Kirwan map

Let $(M,\omega)$ be a symplectic manifold with a Hamiltonian action of a compact connected Lie group $G$, moment map $\mu:M\to\mathfrak g^*$, and smooth symplectic quotient

$$\bar M=\mu^{-1}(0)/G.$$

Assume that (H) holds, meaning that $G$ acts freely on $\mu^{-1}(0)$ and $\mu$ is proper. Let $(M,\omega)$ be equivariantly convex at infinity and aspherical, and suppose that $(\bar M,\bar\omega)$ is semi-positive. With $\Lambda_\omega^\mu$ the equivariant Novikov ring, $H_G^*(M)$ equivariant cohomology, $\kappa_G$ the Kirwan map, and $\operatorname{QH}^*(\bar M,\bar\omega)$ the small quantum cohomology, there exists a $\Lambda_\omega^\mu$-algebra homomorphism

$$\varphi:H_G^*(M)\otimes\Lambda_\omega^\mu\to\operatorname{QH}^*(\bar M,\bar\omega)$$

of the form

$$\varphi=\kappa_G\otimes\operatorname{id}+\sum_{0\neq B}\varphi^B\otimes e^B\cdot.$$

This is the quantum Kirwan map predicted by Salamon's conjecture. The theorem supplies its existence under the stated convexity, asphericity, freeness, properness, and semi-positivity assumptions; the source statement itself does not claim a stronger general result.