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.

49 pages

References in corpus (4)

• Think globally, compute locally
• Large N expansion of convergent matrix integrals, holomorphic anomalies, and background independence
• The partition function of the two-matrix model as an isomonodromic tau-function
• Mixed correlation function and spectral curve for the 2-matrix model

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• Laguerre Ensemble: Correlators, Hurwitz Numbers and Hodge Integrals
• Painlev'e 2 equation with arbitrary monodromy parameter, topological recursion and determinantal formulas
• Whittaker vectors for $\mathcal{W}$-algebras from topological recursion
• WKB solutions of difference equations and reconstruction by the topological recursion
• The geometry of Casimir W-algebras
• The factorial growth of topological recursion