space*]{} with coordinates $t_{ji}$. We will often need to specialize them to $t_1=t_2=\dots =0$ and will write $${\langle\!\langle}\delta_1\psi_1^{a_1},\dots,\delta_k\psi_k^{a_k} {\rangle\!\rangle}^\ke_{g,k}(t)={\langle\!\langle}\delta_1\psi_1^{a_1},\dots,\delta_k\psi_k^{a_k} {\rangle\!\rangle}^\ke_{g,k}\mid_{{\bf t}=t}$$ for the specializations, with $t:=t_0=\sum_i t_{0i}\gamma_i$.

Apriori the unstable terms are omitted from the sum , though in many cases certain conventions will be made to include them as well. When we do that, the appropriate conventions will be spelled out explicitly. In this notation, the genus $g$ potential (when $g\geq 1$) is the empty bracket $$F^\ke_g ({\bf t}(\psi))={\langle\!\langle}\;\; {\rangle\!\rangle}_{g,0}^{{\varepsilon }}({\bf t}(\psi)),$$ and then with $\delta_j=\gamma_{i_j}$ is the derivative $\frac{\partial}{\partial t_{a_1i_1}}\dots \frac{\partial}{\partial t_{a_ki_k}}F^\ke_g ({\bf t}(\psi))$.

### $\ke$-quasimap classes

Let $$f:{\mathrm{Q}_{g,k}^\ke({W/\!\!/{{\bf G}}},\beta)}\lra{\overline{M}_{g,k}}$$ be the composition of the forgetful morphism ${\mathrm{Q}_{g,k}^\ke({W/\!\!/{{\bf G}}},\beta)}\lra{\mathfrak{M}_{g,k}}$ with the stabilization morphism ${\mathfrak{M}_{g,k}}\lra{\overline{M}_{g,k}}$. If $2g-2+k>0$, define $\Lambda$-linear maps $$\label{Omega}
\begin{split}
&\Omega^\ke_{g,k} :H^*({W/\!\!/{{\bf G}}},\Lambda)^{\otimes k}{\longrightarrow}H^*({\overline{M}_{g,k}},\Lambda), \\& \Omega^\ke_{g,k}(\otimes_{j=1}^k\delta_j)=
\sum_\beta q^\beta \Omega^\ke_{g,k,\beta}(\otimes_{j=1}^k\delta_j),
\end{split}$$ by setting $$\label{Omega b}
 \Omega^\ke_{g,k,\beta}(\otimes_{j=1}^k\delta_j)=f_*( [{\mathrm{Q}_{g,k}^\ke({W/\!\!/{{\bf G}}},\beta)}]^{\mathrm{vir}}\cap\prod_{j=1}^k ev_j^*(\delta_j)).$$ The maps are clearly equivariant for the actions of the symmetric group $S_k$ on the source and target.

### $\ke$-CohFT

The boundary of ${\mathrm{Q}_{g,k}^\ke({W/\!\!/{{\bf G}}},\beta)}$ is the complement of the open stratum of quasimaps with irreducible and nonsingular domain curve. It has a recursive structure, with strata indexed by modular graphs and the virtual classes behave in a functorial 