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

TL;DR

This paper extends the solution of beta-ensemble loop equations to arbitrary beta using a sectorwise approach, constructing quantum algebraic curves and explicit correlation functions with B-cycle structure and symplectic invariants.

Contribution

It introduces a sectorwise method for solving beta-ensemble loop equations, enabling the explicit construction of quantum algebraic curves and correlation functions for arbitrary beta.

Findings

Solution of loop equations for arbitrary beta using sectorwise resolvents

Construction of quantum algebraic curves with B-cycle structure

Explicit calculation of correlation functions and symplectic invariants

Abstract

We solve the loop equations of the $\beta$-ensemble model analogously to the solution found for the Hermitian matrices $\beta=1$. For \beta=1$, the solution was expressed using the algebraic spectral curve of equation $y^2=U(x)$. For arbitrary $\beta$, the spectral curve converts into a Schr\"odinger equation $((\hbar\partial)^2-U(x))\psi(x)=0$ with $\hbar\propto (\sqrt\beta-1/\sqrt\beta)/N$. 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 \widetilde{\epsilon}_\alpha$. 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 $\widetilde{\epsilon}_\alpha$.