Isomonodromy deformations at an irregular singularity with coalescing eigenvalues

Abstract

We consider an n × n linear system of ODEs with an irregular singularity of Poincaré rank 1 at z = ∞, holomorphically depending on parameter t within a polydisc in C n centred at t = 0. The eigenvalues of the leading matrix at z = ∞ coalesce along a locus ∆ contained in the polydisc, passing through t = 0. Namely, z = ∞ is a resonant irregular singularity for t ∈ ∆. We analyse the case when the leading matrix remains diagonalisable at ∆. We discuss the existence of fundamental matrix solutions, their asymptotics, Stokes phenomenon and monodromy data as t varies in the polydisc, and their limits for t tending to points of ∆. When the deformation is isomonodromic away from ∆, it is well known that a fundamental matrix solution has singularities at ∆. When the system also has a Fuchsian singularity at z = 0, we show under minimal vanishing conditions on the residue matrix at z = 0 that isomonodromic deformations can be extended to the whole polydisc, including ∆, in such a way that the fundamental 1

On residual categories for Grassmannians

“…In a series of papers[5][6][7], Cotti, Dubrovin, and Guzzetti study semisimple Frobenius structures with coalescing eigenvalues. It would be very interesting to clarify the relation between Conjecture 1.12 and the Stokes phenomenon of the Dubrovin's connection in quantum cohomology studied in loc.…”

Notes on Non-Generic Isomonodromy Deformations

“…These notes partly touch the topics of my talk in Ann Arbor at the conference in memory of Andrei Kapaev, August 2017. They are a reworking of some of the main results of [11], concerning non-generic isomonodromy deformations of the differential system (2.1) below. The approach here is different from [11], since I will start from the point of view of Pfaffian systems.…”

“…They are a reworking of some of the main results of [11], concerning non-generic isomonodromy deformations of the differential system (2.1) below. The approach here is different from [11], since I will start from the point of view of Pfaffian systems. This allows to introduce the main theorem (Theorem 4.9 below) in a relatively simple way (provided we give for granted another result of [11] summarised in Theorem 4.7 below).…”

“…The approach here is different from [11], since I will start from the point of view of Pfaffian systems. This allows to introduce the main theorem (Theorem 4.9 below) in a relatively simple way (provided we give for granted another result of [11] summarised in Theorem 4.7 below). The approach here is also an opportunity to review the difference between weak and strong isomonodromy deformations.…”

“…In Section 3, we define and characterise "strong" isomonodromy deformations of the differential system (2.1) in terms of solutions of the Pfaffian system and in terms of essential monodromy data. We recover the total differential system used in the last part of [11], a special case of which (for A(u) skew-symmetric) has been well known in the theory of Frobenius manifolds [16,18]. In Section 4, we explain some of the main results of [11], particularly Theorem 4.9, which extend strong isomonodromy deformations to the non-generic case when the matrix Λ = diag(u 1 , .…”