2009/11/09 by Leonid Chekhov, Chekhov, L., Bertrand Eynard +3 · 2 citations
Mathematics · #05A15 #05C30 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.0911.1664
openalex publication_date 2009/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article, we solve the loop equations of the β-random matrix model, in a way similar to what was found for the case of hermitian matrices β=1. For β=1, the solution was expressed in terms of algebraic geometry properties of an algebraic spectral curve of equation y2=U(x). For arbitrary β, the spectral curve is no longer algebraic, it is a Schroedinger equation ((ℏ∂)2-U(x)).ψ(x)=0 where ℏ∝ (√β-1/√β). In this article, we find a solution of loop equations, which takes the same form as the topological recursion found for β=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 Fg, for a quantum spectral curve, i.e. a D-module of the form y2-U(x), where [y,x]=ℏ. Also, our method allows to enumerate non-oriented discrete surfaces.