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

Abstract

We solve the loop equations of the $\beta$-ensemble model analogously to the solution found for the Hermitian matrices $\beta =1$. For β=1$,thesolutionwasexpressedusingthealgebraicspectralcurveofequation$y^2=U(x)$.Forarbitrary$β$,thespectralcurveconvertsintoaSchr\ddot{o}dingerequation$((\hbar\partial)^2-U(x))ψ(x)=0$with$\hbar\propto (\sqrtβ-1/\sqrtβ)/N$.ThispaperissimilartothesisterpaperI,inparticular,allthemainingredientsspecificforthealgebraicsolutionoftheproblemremainthesame,butherewepresentthesecondapproachtofindingasolutionofloopequationsusingsectorwisedefinitionofresolvents.Beingtechnicallymoreinvolved,itallowsdefiningconsistentlytheB-cyclestructureoftheobtainedquantumalgebraiccurve(aD-moduleoftheform$y^2-U(x)$,where$[y,x]=\hbar$)andtoconstructexplicitlythecorrelationfunctionsandthecorrespondingsymplecticinvariants$F_h$,orthetermsofthefreeenergy,in1/N^2$-expansion at arbitrary $\hslash$. The set of "flat" coordinates comprises the potential times $t_k$ and the occupation numbers \widetildeε_α$.WedefineandinvestigatethepropertiesoftheA-andB-cycles,formsof1st,2ndand3rdkind,andtheRiemannbilinearidentities.Weusetheseidentitiestofindexplicitlythesingularpartof$\mathcal F_0$thatdependsexclusivelyon$\widetildeε_α$.

58 pages, 7 figures

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