each chamber. When $\ke\in(1,\infty)$ one recovers the Kontsevich moduli spaces of stable maps to ${W/\!\!/{{\bf G}}}$; we write $\ke=\infty$ for these stability conditions. The asymptotic stability condition $\ke=0+$ corresponds to being in the first chamber $(0,\frac{1}{\beta(L_\theta)}]$ for [*all*]{} $\beta$.

For a long list of examples to which quasimap theory applies, see §2.8 of [@CK0].

$\ke$-quasimap invariants, $\ke$-CohFT {#projective}
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Let $\QQ[ {\mathrm {Eff}}(W,{{\bf G}},\theta)]$ be the semigroup ring. We write $q^\beta$ for the element corresponding to $\beta\in {\mathrm {Eff}}(W,{{\bf G}},\theta)$. The Novikov ring associated to the triple $(W,{{\bf G}},\theta)$ is the $\mathfrak{m}$-adic completion $$\Lambda:=\widehat{\QQ[ {\mathrm {Eff}}(W,{{\bf G}},\theta)]}$$ with respect to the maximal ideal $\mathfrak{m}$ generated by $\{ q^\beta\; |\; \beta\neq 0\}$. Throughout the paper $H^*({W/\!\!/{{\bf G}}},\QQ)$ and $H^*({W/\!\!/{{\bf G}}},\Lambda)$ will denote the [*even*]{} cohomology with the indicated coefficients.

In this subsection we assume that ${W/\!\!/{{\bf G}}}$ is projective; the extension to quasi-projective targets is discussed in the next subsection. We have the intersection pairing $${\langle}\gamma,\delta{\rangle}:=\int_{{W/\!\!/{{\bf G}}}}\gamma\delta$$ on cohomology. We extend it $\Lambda$-linearly to $H^*({W/\!\!/{{\bf G}}},\Lambda)$.

Once and for all, fix homogeneous bases $\{\gamma_1,\dots,\gamma_s\}$ and $\{\gamma^1,\dots ,\gamma^s\}$ of $H^*({W/\!\!/{{\bf G}}},\QQ)$, dual with respect to the pairing ${\langle}\; , \;{\rangle}$. The cohomology class of the diagonal $\Delta\subset {W/\!\!/{{\bf G}}}\times{W/\!\!/{{\bf G}}}$ is then 