Publication : t10/154

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 Schrodinger equation ((hbarpartial)^2-U(x))psi(x)=0 with hbarpropto (sqrtbeta-1/sqrtbeta)/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.

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