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To any solution of a linear system of differential equations, we associate akernel, correlators satisfying a set of loop equations, and in presence ofisomonodromic parameters, a Tau function. We then study their semiclassicalexpansion (WKB type expansion in powers of the weight hbar per derivative) ofthese quantities. When this expansion is of topological type (TT), thecoefficients of expansions are computed by the topological recursion withinitial data given by the semiclassical spectral curve of the linear system.This provides an efficient algorithm to compute them at least when thesemiclassical spectral curve is of genus 0. TT is a non trivial property, andit is an open problem to find a criterion which guarantees it is satisfied. Weprove TT and illustrate our construction for the linear systems associated tothe q-th reductions of KP - which contain the (p,q) models as a specialization.

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