The mirror-triviality characterization of the physical geometric window

Let $Fl$ be a partial flag manifold, and let $I_{Fl}^\ell$ denote its level-$\ell$ quasimap quantum $K$-theoretic $I$-function. Let $J_{Fl}^\ell$ denote the small stable-map $J$-function of level $\ell$.

Mirror-triviality conjecture. A level $\ell$ is in the physical geometric window if and only if

$$I_{Fl}^\ell=J_{Fl}^\ell.$$

The two functions are generating functions for Gromov–Witten-type invariants, with the $I$-function calculated on a quasimap space and the $J$-function on the moduli space of stable maps to $Fl$. The conjecture is verified directly for projective spaces and Grassmannians; the general flag-manifold case remains open.