of $\theta$-[*effective*]{} classes (see §\[e-qmaps\] for the definition). The genus $g$ descendant potential of ${W/\!\!/{{\bf G}}}$ is defined by $$F^\ke_g({\bf t}):=\sum_{\beta \in {\mathrm {Eff}}(W,{{\bf G}},\theta)}\sum_{m\geq 0} \frac{q^\beta}{m!}{\langle}{\bf t}(\psi_1),{\bf t}(\psi_2),\dots, {\bf t}(\psi_m) {\rangle}^\ke_{g,m,\beta}.$$ As usual, the unstable terms in the sum corresponding to $(g,m,\beta)$ for which the moduli spaces do not exist are omitted.

The wall-crossing formula relates the Gromov-Witten potential $F_g^\infty$ to the potential $F^\ke_g$ for [*semi-positive*]{} triples $(W ,{{\bf G}}, \theta)$ (for these triples, the corresponding quotients ${W/\!\!/{{\bf G}}}$ have nef anti-canonical class). To state it, recall first from [@CK0] that in genus zero quasimap theory there is a $J$-function $J^\ke(q,t,z)$ for each stability parameter $\ke\geq 0+$. It depends on the Novikov variables, a general even cohomology element $t\in H^*({W/\!\!/{{\bf G}}},\QQ)$, and a formal variable $z$. For ${{\varepsilon }}>1$, it is the usual Givental (big) $J$-function of Gromov-Witten theory. The [*small*]{} $J^\ke$-function is defined as the restriction at $t=0$: $$J^\ke_{sm}(q,z):=J^\ke(q,0,z).$$ In the semi-positive case, $J^\ke$ takes values in $H^*({W/\!\!/{{\bf G}}},\Lambda)[[1/z]]$ and we will need the first two terms in the $1/z$-expansion of small $J^\ke$, $$J^\ke_{sm}(q,z)=J^\ke_0(q){{\mathbbm 1}}+ J^\ke_1(q)\frac{1}{z}+O\left(\frac{1}{z^2}\right),$$ where $\one$ is the unit in cohomology. For the asymptotic stability $\ke=0+$ we use the special notation $I_{sm}=J^{0+}_{sm}$, and call this the 