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

Bergére, Michel ; Borot, Gaëtan ; Eynard, Bertrand

In: Annales Henri Poincaré, 2015, vol. 16, no. 12, p. 2713-2782

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    Titel
    • Rational Differential Systems, Loop Equations, and Application to the q th Reductions of KP
    Autor
    • Bergére, Michel. Institut de Physique Théorique, CEA Saclay, Orme des Merisiers, 91191, Gif-sur-Yvette Cedex, France
    • Borot, Gaëtan. Section de Mathématiques, Université de Genève, Geneve, Switzerland
    • Eynard, Bertrand. Institut de Physique Théorique, CEA Saclay, Orme des Merisiers, 91191, Gif-sur-Yvette Cedex, France
    Dokumententyp
    • Postprint
    Sprache
    • Englisch
    Veröffentlicht in
    • Annales Henri Poincaré, 2015, vol. 16, no. 12, p. 2713-2782. Springer Basel
    Andere elektronische Ausgabe
    Klassifikation
    • Mathematik
    OAI-PMH-ID
    • oai:doc.rero.ch:333095
    Summary
    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 $${\hbar}$$ ħ 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.