Annales Henri Poincaré

, Volume 16, Issue 12, pp 2713–2782 | Cite as

Rational Differential Systems, Loop Equations, and Application to the qth Reductions of KP

  • Michel Bergére
  • Gaëtan BorotEmail author
  • Bertrand Eynard


To any solution of a linear system of differential equations, we associate a matrix kernel, correlators satisfying a set of loop equations, and in the 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 qth reductions of KP—which contain the (p, q) models as a specialization.


Matrix Model Meromorphic Function Spectral Curve Double Point Matrix Kernel 
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Copyright information

© Springer Basel 2015

Authors and Affiliations

  • Michel Bergére
    • 1
  • Gaëtan Borot
    • 2
    • 3
    • 4
    Email author
  • Bertrand Eynard
    • 1
    • 5
  1. 1.Institut de Physique ThéoriqueCEA SaclayGif-sur-Yvette CedexFrance
  2. 2.Section de MathématiquesUniversité de GenèveGeneveSwitzerland
  3. 3.Department of MathematicsMITCambridgeUSA
  4. 4.MPI für MathematikBonnGermany
  5. 5.Centre de Recherches MathématiquesMontréalCanada

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