the pair $(ev_\bullet, ev_\star)$) whose image gives a boundary divisor $D_0^\ke$ with virtual class $$\label{bdry3}
[D^\ke_0]^{\mathrm{vir}}=(h_0)_* \Delta^! [\mathrm{Q}^\ke_{g-1,[k]+\bullet+\star}({W/\!\!/{{\bf G}}},\beta) ]^{\mathrm{vir}}.$$

This leads to the usual splitting properties for the brackets and the classes . For example, if $[k]=S_1\coprod S_2$, $g_1+g_2=g$, with $|S_i|=k_i$, is a stable ordered partition, then $$\label{splitting 1} \rho^*\Omega_{g,k}^{{\varepsilon }}(\otimes_{j=1}^k\delta_j)=\sum_i \Omega_{g_1,k_1+1}^{{\varepsilon }}((\otimes_{j\in S_1}\delta_j)\otimes\gamma_i)
\Omega_{g_2,k_2+1}^{{\varepsilon }}((\otimes_{j\in S_2}\delta_j)\otimes\gamma^i),$$ where $$\rho:\overline{M}_{g_1,S_1\cup\bullet}\times \overline{M}_{g_2,S_2\cup\star}\lra{\overline{M}_{g,k}}$$ is the gluing map on stable curves. Analogously, $$\label{splitting 2}
\phi^* \Omega_{g,k}^{{\varepsilon }}(\otimes_{j=1}^k\delta_j)=\sum_i \Omega_{g-1,k+2}^{{\varepsilon }}((\otimes_{j=1}^k\delta_j)\otimes\gamma_i\otimes\gamma^i),$$ where $(g-1,k+2)$ is stable and $$\phi :\overline{M}_{g-1,k+2}\lra{\overline{M}_{g,k}}$$ is again the gluing map. In other words, we have the following

\[CohFT\] For each $\ke\geq 0+$ the maps give the structure of a Cohomological Field Theory (CohFT) over $\Lambda$ on $H^*({W/\!\!/{{\bf G}}},\Lambda)$ with the metric ${\langle}\; , \;{\rangle}$.

The primary invariants in the stable range $2g-2+k>0$ are given by the degree zero part of the CohFT, $$\sum_\beta q^\beta\langle \delta_1,\dots,\delta_k
\rangle^\ke_{g,k,\beta}=\int_{{\overline{M}_{g,k}}} \Omega_{g,k}^{{\varepsilon }}(\otimes_{j=1}^k\delta_j) .$$ The genus zero potential $F^\ke_0(t)$ satisfies the WDVV equation and determines the $\ke$-quasimap Frobenius manifold. The $\ke$-quantum product is given by $$\gamma_i\circ_{{\varepsilon }}\gamma_j=\sum_{l=1}^s \gamma_l{\langle\!\langle}\gamma_i,\gamma_j,\gamma^l{\rangle\!\rangle}^\ke_{0,3}(t).$$ The unit for this product is discussed in Remarks \[string etc\] and \[unit\] below.

Including $\psi$-classes on ${\overline{M}_{g,k}}$ in the integrals, the CohFT gives the “ancestor" invariants $$\label{ancestors}
\langle 