Local-system conjecture for derived categories of GLSM phases

Let $X_{\beta}$ be the phase associated with a VGIT parameter $\beta \in L^{\vee}_{\mathbb{R}}$, and let FIPS denote the Fayet–Iliopoulos parameter space of the corresponding gauged linear sigma model. The derived category of the phase is $D^b(X_{\beta})$. GLSM local-system conjecture. There is an action of $\pi_1(\operatorname{FIPS})$ on $D^b(X_{\beta})$ for any phase $X_{\beta}$. This predicts the mathematically testable existence of the physicists' local system of categories over the FIPS, despite the fact that D-brane transport and the GLSM itself are not yet rigorous in general.