$W/_{\mathrm {aff}}{{\bf G}}$ is proper. This assumption is automatic if ${W/\!\!/{{\bf G}}}$ is projective and holds for the natural torus actions on all interesting examples of non-compact targets, such as quasiprojective toric varieties, total spaces of bundles over projective quotients, and Nakajima quiver varieties. It follows that the ${{\bf {T}}}$-fixed loci in ${W/\!\!/{{\bf G}}}$ and in all the moduli spaces of stable quasimaps are also proper. In this situation we get a ${{\bf {T}}}$-equivariant version of the CohFT, see e.g. §6.3 in [@CKM] and all results in §\[projective\] (which may be viewed as corresponding to the zero-dimensional torus ${{\bf {T}}}=\{1\}$) are valid in this setting.

Precisely, let $$\QQ[\lambda_1,\dots,\lambda_n]=H^*_{{{\bf {T}}}}({{\mathrm{Spec}}}(\CC),\QQ)$$ and $$K:=\QQ(\lambda_1,\dots,\lambda_n)=H^*_{{{\bf {T}}},\mathrm{loc}}({{\mathrm{Spec}}}(\CC),\QQ)$$ be the equivariant cohomology, respectively the localized equivariant cohomology of a point. The cohomology ring of ${W/\!\!/{{\bf G}}}$ is replaced by the localized equivariant cohomology $$H^*_{{{\bf {T}}},\mathrm{loc}}({W/\!\!/{{\bf G}}},\QQ)=H^*_{{{\bf {T}}}}({W/\!\!/{{\bf G}}},\QQ)\otimes_{\QQ[\lambda_1,\dots,\lambda_n]} K,$$ while the Novikov ring is now $\Lambda=K[[q]]$.

The pairing is defined by the localization formula $${\langle}\delta,\gamma{\rangle}=\int_{{W/\!\!/{{\bf G}}}^{{{\bf {T}}}}} \frac{i^*(\delta\gamma)}{\mathrm{e}(N)},$$ where $i:{W/\!\!/{{\bf G}}}^{{{\bf {T}}}}\hookrightarrow {W/\!\!/{{\bf G}}}$ is the inclusion of the fixed point locus and $\mathrm{e}(N)$ is the ${{\bf {T}}}$-equivariant Euler class of the normal bundle. Similarly, the ${{\bf {T}}}$-equivariant $\ke$-quasimap invariants are defined by the virtual localization 