research of I.C.-F. was partially supported by the NSA grant H98230-11-1-0125 and the NSF grant DMS-1305004. The research of B.K. was partially supported by NRF-2007-0093859. In addition, I.C-F. thanks KIAS for financial support, excellent working conditions, and an inspiring research environment. The authors thank the referee for helpful suggestions and Hyenho Lho for pointing out an error in an earlier version.

Quasimap CohFT
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To fix notation, in this section we recall briefly (after [@CKM]) the Cohomological Field Theory defined by the moduli spaces of $\ke$-stable quasimaps.

Quotients
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Consider a triple $(W,{{\bf G}}, \theta)$ with $W$ an affine complex algebraic variety, ${{\bf G}}$ a reductive complex algebraic group acting on $W$, and $\theta\in\chi({{\bf G}})$ a character of ${{\bf G}}$. The ${{\bf G}}$-equivariant line bundle $L_\theta=W\times \CC_\theta$, exhibits $\theta$ as a linearization of the action on the trivial line bundle. Hence $(W,{{\bf G}}, \theta)$ determines a GIT quotient ${W/\!\!/{{\bf G}}}$. It has a projective morphism to the [*affine quotient*]{} $W/_{\mathrm {aff}}{{\bf G}}={{\mathrm{Spec}}}(A(W)^{{\bf G}})$. The line bundle $L_\theta$ descends to a relative polarization on ${W/\!\!/{{\bf G}}}$, denoted by ${{\mathcal{O}}}(\theta)$, which may be taken (without loss of generality) to be relatively very ample over $W/_{\mathrm {aff}}{{\bf G}}$.

We assume that the semi-stable 