Construct quantum constraints for arbitrary rank

Construct, for general rank $M\geq 2$, a complete system of q-difference operators annihilating the rank-$M$ partition function $Z_{U(M)}[y_1,\ldots,y_M]$ whose classical limit defines the corresponding Lagrangian variety in $(\mathbb{C}^*)^{2M}$, and solve this system.

Background

For rank two, the paper constructs three quantum constraints whose classical limits define a reducible Lagrangian variety. At rank three, the authors verify only the analogue of one constraint. They explicitly state that the remaining independent constraints are unknown, and earlier in the section they state that the general-rank construction cannot yet be solved.

References

Unfortunately we do not know how to solve this problem for the general rank $M$.

— Lagrangian varieties from $q$-matrix models  (2609.01454 - Mishnyakov et al., 1 Sep 2026) in Section Higher rank quantum varieties, subsection Comment on higher rank

At present, however, we do not know how to derive the remaining independent quantum constraints.

— Lagrangian varieties from $q$-matrix models  (2609.01454 - Mishnyakov et al., 1 Sep 2026) in Section Higher rank quantum varieties, subsection Comment on higher rank

In general, it is not known whether superintegrability persists, and the $q$-Virasoro constraints need not provide a well-defined recursive prescription for determining all correlators. Consequently, for a generic $q$-deformed matrix model we do not know how to construct systematically an operator $\hat{A}{U(1)}$ annihilating $Z{U(1)}[y]$.

— Lagrangian varieties from $q$-matrix models  (2609.01454 - Mishnyakov et al., 1 Sep 2026) in Discussion, Section discussion