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

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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 y²=U(x). For arbitrary β, the spectral curve converts into a Schrödinger equation ((ℏ∂)²-U(x))ψ(x)=0 with ℏ∝ (√β-1/√β)/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 y²-U(x), where [y,x]=ℏ) and to construct explicitly the correlation functions and the corresponding symplectic invariants Fₕ, or the terms of the free energy, in 1/N²-expansion at arbitrary ℏ. The set of "flat" coordinates comprises the potential times tₖ and the occupation numbers εα. 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 F₀ that depends exclusively on εα.