Abstract
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 ∝ \( {{\left( {{{\sqrt \beta - 1} \mathord{\left/ {\vphantom {{\sqrt \beta - 1} {\sqrt \beta }}} \right. \kern-\nulldelimiterspace} {\sqrt \beta }}} \right)} \mathord{\left/ {\vphantom {{\left( {{{\sqrt \beta - 1} \mathord{\left/ {\vphantom {{\sqrt \beta - 1} {\sqrt \beta }}} \right. \kern-\nulldelimiterspace} {\sqrt \beta }}} \right)} N}} \right. \kern-\nulldelimiterspace} N} \). 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 \( \tilde \varepsilon _\alpha \). 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 \( \mathcal{F}_0 \), which depends only on \( \tilde \varepsilon _\alpha \).
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This paper was written at the request of the Editorial Board.
Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 166, No. 2, pp. 163–215, February, 2011.
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Chekhov, L.O., Eynard, B. & Marchal, O. Topological expansion of the β-ensemble model and quantum algebraic geometry in the sectorwise approach. Theor Math Phys 166, 141–185 (2011). https://doi.org/10.1007/s11232-011-0012-3
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DOI: https://doi.org/10.1007/s11232-011-0012-3