Topological expansion of the β-ensemble model and quantum algebraic geometry in the sectorwise approach
Résumé
We construct the solution of the loop equations of the β-ensemble model in a form analogous to the solution in the case of the Hermitian matrices β = 1. The solution for β = 1 is expressed in terms of the algebraic spectral curve given by y2 = U(x). The spectral curve for arbitrary β converts into the Schrödinger equation (ħ∂)2 − U(x) ψ(x) = 0, where ħ ∝ (β√−1/β√)/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 B-cycle structure for constructing the quantum algebraic curve (a D-module of the form y2 − U(x), where [y, x] = ħ) and explicitly writing the correlation functions and the corresponding symplectic invariants Fh or the terms of the free energy in an 1/N2-expansion at arbitrary ħ. The set of "flat" coordinates includes the potential times tk and the occupation numbers ε˜α . We define and investigate the properties of the A- and B-cycles, forms of the first, second, and third kinds, and the Riemann bilinear identities. These identities allow finding the singular part of F0 , which depends only on ε˜α.
Domaines
Physique mathématique [math-ph]
Origine : Fichiers produits par l'(les) auteur(s)
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