Ciocan-Fontanine–Kim wall-crossing conjecture for stable quasimaps

Let $Y$ be a complete intersection in projective space, and fix $g,n\geq 0$. Denote by $[\overline{\mathcal{M}}^{\epsilon}_{g,n}(Y,\beta)]^{\mathrm{vir}}$ and $[\overline{\mathcal{M}}^{\infty}_{g,n}(Y,\beta)]^{\mathrm{vir}}$ the virtual classes of the moduli spaces of $\epsilon$-stable and $\infty$-stable quasimaps, respectively. Let $\mu^{\epsilon}_{\beta}(z)$ be the relevant coefficients of the $I$-function, let $b_{\vec{\beta}}$ convert marked points to basepoints, and let $c$ be the contraction morphism from $\infty$-stable to $\epsilon$-stable quasimaps.

Ciocan-Fontanine–Kim wall-crossing conjecture.

$$\sum_{\beta}q^{\beta}[\overline{\mathcal{M}}^{\epsilon}_{g,n}(Y,\beta)]^{\mathrm{vir}}=\sum_{\beta_0,\beta_1,\ldots,\beta_k}\frac{q^{\beta_0}}{k!}b_{\vec{\beta}*}c_*\left(\prod_{i=1}^kq^{\beta_i}\operatorname{ev}_{n+i}^*(\mu^{\epsilon}_{\beta_i}(-\psi_{n+i}))\cap[\overline{\mathcal{M}}^{\infty}_{g,n+k}(Y,\beta_0)]^{\mathrm{vir}}\right).$$

This conjecture describes the change in quasimap theory as the stability parameter crosses walls and is intended to relate stable-quasimap and Gromov–Witten theories through the $\epsilon$-wall-crossing. It was formulated in all genus; the supplied source gives no resolution status for this precise statement.