The quasimap–Gromov–Witten S-operator comparison conjecture

Let $(W,{\bf G},\theta)$ be a GIT presentation and let $(\theta,\varepsilon)$ be arbitrary, including all asymptotic cases. Write $\mathds{S}_{\bf t}^{\theta,\varepsilon}$ and $P^{\theta,\varepsilon}$ for the quasimap $S$-operator and $P$-series, and let $\kappa({\bf t})$ be the corresponding divisor-class shift. Define

$$\tau({\bf t})=\kappa({\bf t})+\sum_{\beta\ne0}q^\beta\sum_i\gamma_i\left\langle\!\left\langle\gamma^i,\mathbbm{1}\right\rangle\!\right\rangle_{0,2,\beta}^{\theta,\varepsilon}.$$

Quasimap–Gromov–Witten comparison conjecture. One has

$$\mathds{S}_{\bf t}^{\theta,\varepsilon}(\mathbbm{1})=\mathds{S}_{\tau({\bf t})}^{(\infty,1)}(\mathbbm{1}),$$

and, for ${\bf t}=\sum_i t_i\tilde{\gamma}_i$, there are unique

$$P^{(\infty,1),\theta,\varepsilon}({\bf t},z)=\mathbbm{1}+O(q)\in H^*_{{\bf T},\mathrm{loc}}(W/\!/{\bf G})[z][[q,t_j]],$$ $$\tau^{(\infty,1),\theta,\varepsilon}({\bf t})=\kappa({\bf t})+O(q)\in H^*_{{\bf T},\mathrm{loc}}(W/\!/{\bf G})[[q,t_j]],$$

so that

$$\mathds{S}_{\bf t}^{\theta,\varepsilon}(z)\bigl(P^{\theta,\varepsilon}({\bf t},z)\bigr)=\mathds{S}_{\tau^{(\infty,1),\theta,\varepsilon}({\bf t})}^{(\infty,1)}(z)\bigl(P^{(\infty,1),\theta,\varepsilon}(\tau^{(\infty,1),\theta,\varepsilon}({\bf t}),z)\bigr).$$

This predicts a comparison between quasimap theories for arbitrary stability conditions and the stable-map Gromov–Witten theory. The supplied text gives no resolution status.