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 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 pp, qq models as a specialization.x k q, indexed by integer a 1 , . . . , a k P v1, dw, called k-points correlators, or shortly, correlators. and we show that the k-point correlators satisfy a set of linear equations (Theorem 2.1) and a set of quadratic equations (Theorem 2.2). We use the name loop equations to refer collectively to those set of equations. We also introduce a notion of "insertion operator" (Definition 2.5) allowing the derivation of k-linear loop equations for k ď d (the size of the differential system) from the master ones. These results are of purely algebraic nature and hold for any system (1-1). When L depends on a set of parameters t preserving the monodromy of the solutions, we can also associate to Ψpx, tq a Tau function T p tq, defined up to a constant prefactor.Secondly, in Section 3, we study the semiclassical expansion in powers of and describe in detail the monodromy of its coefficients (Section 3.2-3.4). We introduce in Definition 3.3 the notion of "expansion of topological type" -also referred to as the TT property -and show that the expansion can be computed by the topological recursion of [EO07] when the TT property holds. In practice, the main consequence of our theory is Theorem 3.1, and in presence of isomonodromic times, this also allows the computation of the expansion of ln T p tq (Corollary 4.2).Finally, in Section 5, we apply our theory to the linear system associated to the q-th reduction of KP, and illustrate it more specifically with examples of the pp, qq models (Section 6). As a motivation, those hierarchies are believed to describe the algebraic critical edge behavior that can be reached in the two hermitian matrix model, and universality classes of 2d quantum gravity coupled to conformal field theories [Moo90, DS90, GM90, dFGZJ94]. In any q-th reduction of KP, we show ( § 5.6-5.8) that the TT property holds, and that our Theorem 3.1 can be applied. CommentsThe earlier work [BE09a] described the construction of Section 2 for general 2ˆ2 rational systems, but implicitly assumed the TT property. It was illustrated for p2m`1, 2q systems in [BE09b], and entails a rigorous proof -modulo checking the TT property, wh...

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A renormalized Feynman amplitude expressed in the α-parameters is defined by introducing a subtraction operator acting directly upon the α-integrand. Different forms of this subtraction operator are discussed. We define the isotropic and nonisotropic normal products and we give a more general oversubtraction rule which ensures both the absolute convergence of the amplitude and the Bogolubov, Parasiuk and Hepp recurrence. The proof of absolute convergence of the amplitude is performed using Hepp’s sectors and equivalence classes of nests.

A new substraction formula is presented to renormalize Feynman amplitudes written in Schwinger's integral representation.The substractions are generated by an operator acting on the integrand, which only depends on the total number of internal lines but is completely independent of the structure of the graph.This formulation is also valid for non-renormalizable theories and is shown to reduce to Zimmermann's .R-operation for scalar theories. It satisfies in any case Bogoliubov's recursive formula and yields an explicit tool for actual computations of renormalized Feynman amplitudes with a minimal number of substractions.£(*) denotes the integral part of x.