Квантовая алгебраическая геометрия

We describe how to construct “quantum” Riemann surfaces that are analogues of hyperelliptic Riemann surfaces. These quantum surfaces correspond to solutions of Shrödinger equation. We construct resolvelnts and quantum analogues of holomorphic and meromorphic differentials, integrals over A- and B-cycles, Abelian (bi-differentials of second and third kind, correlation functions, and symplectic invariants associated with solutions of the loop equation. As an example, I will prove that the standard Riemann bilinear identities are satisfied in the quantum surface case as well.