Topological expansion of the Bethe ansatz, and quantum algebraic geometry

In this article, we solve the loop equations of the \beta-random matrix model, in a way similar to what was found for the case of hermitian matrices \beta=1. For \beta=1, the solution was expressed in terms of algebraic geometry properties of an algebraic spectral curve of equation y^2=U(x). For arbitrary \beta, the spectral curve is no longer algebraic, it is a Schroedinger equation ((\hbar\partial)^2-U(x)).\psi(x)=0 where \hbar\propto (\sqrt\beta-1/\sqrt\beta). In this article, we find a solution of loop equations, which takes the same form as the topological recursion found for \beta=1. This allows to define natural generalizations of all algebraic geometry properties, like the notions of genus, cycles, forms of 1st, 2nd and 3rd kind, Riemann bilinear identities, and spectral invariants F_g, for a quantum spectral curve, i.e. a D-module of the form y^2-U(x), where [y,x]=\hbar. Also, our method allows to enumerate non-oriented discrete surfaces.