$g$ to ${W/\!\!/{{\bf G}}}$, carrying virtual fundamental classes. They come equipped with evaluation maps and with tautological cotangent $\psi$-classes at the markings and therefore determine (for projective ${W/\!\!/{{\bf G}}}$) descendant $\ke$-quasimap invariants

$$\label{invariants}
\langle \delta_1\psi_1^{a_1},\dots,\delta_k\psi_k^{a_k}
\rangle^\ke_{g,k,\beta}.$$

As usual, if the target is only quasi-projective, but has a torus action with good properties, are well-defined as equivariant invariants. When $(g,k)\neq (0,0)$ and $\ke\in (1,\infty]$, are the Gromov-Witten invariants of ${W/\!\!/{{\bf G}}}$. For $(g,k)=(0,0)$, the same holds when $\ke\in (2,\infty]$.

As $\ke$ varies, we expect the invariants to be related via wall-crossing formulas. For the genus zero sector, with arbitrary number of primary insertions and one descendant insertion, such formulas are obtained in [@CK0], where we also show that they may be interpreted as a vast generalization of Givental’s toric mirror theorems. The genus zero wall-crossing formulas are conjectured to hold for general ${W/\!\!/{{\bf G}}}$ and are proved in [@CK0] for many GIT targets by localization methods. The present paper begins the exploration of wall-crossing at higher genus.

Conjectures
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Let $${\bf t}={\bf t}(\psi)=t_0+t_1\psi+t_2\psi^2+t_3\psi^3+\dots,$$ where $t_j\in H^*({W/\!\!/{{\bf G}}},\QQ)$ are general even cohomology classes. Let the [Novikov ring]{} $\Lambda=\QQ[[q]]$ be the $q$-adic completion of the semigroup ring on the semigroup ${\mathrm {Eff}}(W,{{\bf G}},\theta)$ 