Rational differential systems, loop equations, and application to the q-th reductions of KP

Abstract : To any solution of a linear system of differential equations, we associate a kernel, correlators satisfying a set of loop equations, and in presence of isomonodromic parameters, a Tau function. We then study their semiclassical expansion (WKB type expansion in powers of the weight hbar per derivative) of these quantities. When this expansion is of topological type (TT), the coefficients of expansions are computed by the topological recursion with initial data given by the semiclassical spectral curve of the linear system. This provides an efficient algorithm to compute them at least when the semiclassical spectral curve is of genus 0. TT is a non trivial property, and it is an open problem to find a criterion which guarantees it is satisfied. We prove TT and illustrate our construction for the linear systems associated to the q-th reductions of KP - which contain the (p,q) models as a specialization.
Document type :
Preprints, Working Papers, ...
Complete list of metadatas

https://hal-cea.archives-ouvertes.fr/cea-01117895
Contributor : Emmanuelle de Laborderie <>
Submitted on : Wednesday, February 18, 2015 - 10:28:01 AM
Last modification on : Friday, May 10, 2019 - 4:38:15 PM

Links full text

Identifiers

  • HAL Id : cea-01117895, version 1
  • ARXIV : 1312.4237

Citation

Michel Bergère, Gaëtan Borot, Bertrand Eynard. Rational differential systems, loop equations, and application to the q-th reductions of KP. 2015. ⟨cea-01117895⟩

Share

Metrics

Record views

63