Topological expansion of the Bethe ansatz, and quantum algebraic geometry

Details Topological expansion of the Bethe ansatz, and quantum algebraic geometry Topological expansion of the Bethe ansatz, and quantum algebraic geometry

2009-11-09

arxiv.org/abs/0911.1664v2

Physics

Mathematical Physics

latex, 70 pages, 10 figures. Misprints corrected, references added

Scientific paper

In this article, we solve the loop equations of the \beta-random matrix model, in a way similar to what was found for the case of hermitian matrices \beta=1. For \beta=1, the solution was expressed in terms of algebraic geometry properties of an algebraic spectral curve of equation y^2=U(x). For arbitrary \beta, the spectral curve is no longer algebraic, it is a Schroedinger equation ((\hbar\partial)^2-U(x)).\psi(x)=0 where \hbar\propto (\sqrt\beta-1/\sqrt\beta). In this article, we find a solution of loop equations, which takes the same form as the topological recursion found for \beta=1. This allows to define natural generalizations of all algebraic geometry properties, like the notions of genus, cycles, forms of 1st, 2nd and 3rd kind, Riemann bilinear identities, and spectral invariants F_g, for a quantum spectral curve, i.e. a D-module of the form y^2-U(x), where [y,x]=\hbar. Also, our method allows to enumerate non-oriented discrete surfaces.

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