Topological expansion of the $\beta$-ensemble model and quantum algebraic geometry in the sectorwise approach

We construct the solution of the loop equations of the $\beta$-ensemble model in a form analogous to the solution in the case of the Hermitian matrices $\beta=1$. The solution for $\beta=1$ is expressed in terms of the algebraic spectral curve given by $y^2=U(x)$. The spectral curve for arbitrary $\beta$ converts into the Schrödinger equation $\bigl((\hbar\partial)^2-U(x)\bigr)\psi(x)=0$, where $\hbar\propto \bigl(\sqrt\beta-1/\sqrt\beta\,\bigr)/N$. The basic ingredients of the method based on the algebraic solution retain their meaning, but we use an alternative approach to construct a solution of the loop equations in which the resolvents are given separately in each sector. Although this approach turns out to be more involved technically, it allows consistently defining the $\mathcal B$-cycle structure for constructing the quantum algebraic curve (a D-module of the form $y^2-U(x)$, where $[y,x]=\hbar$) and explicitly writing the correlation functions and the corresponding symplectic invariants $\mathcal F_h$ or the terms of the free energy in an $1/N^2$-expansion at arbitrary $\hbar$. The set of “flat”; coordinates includes the potential times $t_k$ and the occupation numbers $\widetilde{\epsilon}_\alpha$. We define and investigate the properties of the $\mathcal A$- and $\mathcal B$-cycles, forms of the first, second, and third kinds, and the Riemann bilinear identities. These identities allow finding the singular part of $\mathcal F_0$, which depends only on $\widetilde{\epsilon}_\alpha$.