This paper shows that certain observables in three solvable q‑deformed matrix models can be read as “quantizations” of geometric objects. The authors study the Chern–Simons, q‑Laguerre, and q‑Gaussian matrix models. They focus on one‑point and two‑point functions of inverse characteristic polynomials and interpret them as quantum versions of Lagrangian subvarieties living in (C^*)^2 and (C^*)^4, respectively. In plain terms, an observable that depends on one or two variables is matched to a curve or a surface in a four‑dimensional space of pairs of multiplicative coordinates, and the quantum rules turn these geometric equations into q‑difference operators acting on the partition function.

To get these results the authors use explicit formulas that follow from superintegrability, which gives closed expressions for averages of certain symmetric polynomials (Schur polynomials). They then reorganize the matrix‑model Ward identities into q‑difference equations. At “rank one” the partition function satisfies a single q‑difference equation whose classical limit is an algebraic curve. At “rank two” they find a system of three q‑difference equations; the common classical limit of their symbols defines a two‑dimensional Lagrangian variety in (C^*)^4.

A new and concrete finding is that these higher‑rank varieties are reducible. Each rank‑two solution contains the obvious product component made from two rank‑one curves, but there are additional components. Those extra pieces can be described as graphs of special maps called anti‑symplectic birational involutions. In the Chern–Simons example the matrix‑model construction reproduces known geometric features (the mirror curve of the resolved conifold at rank one and augmentation‑type varieties at rank two), which serves as a consistency check. The similar but previously unexplored appearances of this structure in the q‑Laguerre and q‑Gaussian models are new.