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

L. O. Chekhov; B. Eynard; O. Marchal

doi:10.1007/s11232-011-0012-3

paper

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

arXiv:1009.6007 · doi:10.1007/s11232-011-0012-3

Abstract

We solve the loop equations of the $β$ -ensemble model analogously to the solution found for the Hermitian matrices $β = 1$ . For β=1 $, t h eso l u t i o n w a se x p r esse d u s in g t h e a l g e b r ai cs p ec t r a l c u r v eo f e q u a t i o n$ y^2=U(x) $. F or a r bi t r a r y$ β $, t h es p ec t r a l c u r v eco n v er t s in t o a S c h r o ¨ d in g er e q u a t i o n$ ((\hbar\partial)^2-U(x))ψ(x)=0 $w i t h$ \hbar\propto (\sqrtβ-1/\sqrtβ)/N $. T hi s p a p er i ss imi l a r t o t h es i s t er p a p er I, in p a r t i c u l a r, a l l t h e mainin g r e d i e n t ss p ec i f i c f or t h e a l g e b r ai cso l u t i o n o f t h e p r o b l e m r e main t h es am e, b u t h er e w e p r ese n tt h eseco n d a pp r o a c h t o f in d in g a so l u t i o n o f l oo p e q u a t i o n s u s in g sec t or w i se d e f ini t i o n o f r eso l v e n t s . B e in g t ec hni c a l l y m or e in v o l v e d, i t a l l o w s d e f inin g co n s i s t e n tl y t h e B - cy c l es t r u c t u r eo f t h eo b t ain e d q u an t u ma l g e b r ai cc u r v e (a D - m o d u l eo f t h e f or m$ y^2-U(x) $, w h er e$ [y,x]=\hbar $) an d t oco n s t r u c t e x pl i c i tl y t h ecor r e l a t i o n f u n c t i o n s an d t h ecor r es p o n d in g sy m pl ec t i c in v a r ian t s$ F_h $, or t h e t er m so f t h e f r eee n er g y, in 1/ N^{2}$ -expansion at arbitrary $ℏ$ . The set of "flat" coordinates comprises the potential times $t_{k}$ and the occupation numbers \widetildeε_α $. W e d e f in e an d in v es t i g a t e t h e p r o p er t i eso f t h e A - an d B - cy c l es, f or m so f 1 s t, 2 n d an d 3 r d k in d, an d t h e R i e mannbi l in e a r i d e n t i t i es . W e u se t h ese i d e n t i t i es t o f in d e x pl i c i tl y t h es in g u l a r p a r t o f$ \mathcal F_0 $t ha t d e p e n d se x c l u s i v e l y o n$ \widetildeε_α$.

58 pages, 7 figures

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

Abstract

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