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

Abstract

We solve the loop equations of the $β$-ensemble model analogously to the solution found for the Hermitian matrices $β=1$. For β=1$, the solution was expressed using the algebraic spectral curve of equation $y^2=U(x)$. For arbitrary $β$, the spectral curve converts into a Schrödinger equation $((\hbar\partial)^2-U(x))ψ(x)=0$ with $\hbar\propto (\sqrtβ-1/\sqrtβ)/N$. This paper is similar to the sister paper~I, in particular, all the main ingredients specific for the algebraic solution of the problem remain the same, but here we present the second approach to finding a solution of loop equations using sectorwise definition of resolvents. Being technically more involved, it allows defining consistently the B-cycle structure of the obtained quantum algebraic curve (a D-module of the form $y^2-U(x)$, where $[y,x]=\hbar$) and to construct explicitly the correlation functions and the corresponding symplectic invariants $F_h$, or the terms of the free energy, in 1/N^2$-expansion at arbitrary $\hbar$. The set of "flat" coordinates comprises the potential times $t_k$ and the occupation numbers \widetildeε_α$. We define and investigate the properties of the A- and B-cycles, forms of 1st, 2nd and 3rd kind, and the Riemann bilinear identities. We use these identities to find explicitly the singular part of $\mathcal F_0$ that depends exclusively on $\widetildeε_α$.