small $I$-function of ${W/\!\!/{{\bf G}}}$. The series $I_0(q)\in\Lambda$ is invertible, of the form $1+O(q)$, while the series $I_1(q)$ is in $H^{\leq 2}({W/\!\!/{{\bf G}}},\Lambda)$, with vanishing constant term in $q$. For ${{\varepsilon }}>0$, the coefficients $J_0^\ke$ and $J_1^\ke$ are polynomial $q$-truncations of $I_0$ and $I_1$.

\[main conj intro\] If ${W/\!\!/{{\bf G}}}$ is a GIT quotient corresponding to a semi-positive triple $(W,{{\bf G}},\theta)$, then for every ${{\varepsilon }}\geq 0+$ we have $$\label{GW to epsilon}
(J_0^\ke)^{2g-2}F_g^\ke({\bf t}(\psi))=  F^\infty_g\left(\frac{{\bf t}(\psi)+J_1^\ke}{J^\ke_0}\right).$$ Further, for every $\ke_1\neq \ke_2$ $$\label{all epsilon}
(J_0^{\ke_1})^{2g-2} F_g^{\ke_1}(J_0^{\ke_1}{\bf t}(\psi)-J_1^{\ke_1})= (J_0^{\ke_2})^{2g-2} F^{\ke_2}_g\left(J^{\ke_2}_0{\bf t}(\psi)-J_1^{\ke_2}\right).$$

Since the transformation ${\bf t}(\psi)\mapsto ({\bf t}(\psi)+J_1^\ke)/J_0^\ke$ is invertible, the (apriori stronger) wall-crossing formula follows from .

Let $\{\gamma_1,\dots ,\gamma_s\}$ be a homogeneous basis of $H^{2*}({W/\!\!/{{\bf G}}},\QQ)$ and write $t_j=\sum_i t_{ji}\gamma_i$, so that $F^\ke_g$ are formal series in the $t_{ji}$ variables. To have the uniform statement in the conjecture for all stability parameters, some modifications to the potentials are needed in low genus, as follows:

-   When $g=0$ we must discard from both sides of equations and the parts of degree $\leq 1$ in the $t_{ji}$’s. There is, however, a natural way to include all terms of $t_{ji}$-degree $1$ in the genus zero potentials, so that becomes an equality up to 