Stokes Matrices Research Articles

Unfolding of the unramified irregular singular generalized isomonodromic deformation

We introduce an unfolded moduli space of connections, which is an algebraic relative moduli space of connections on complex smooth projective curves, whose generic fiber is a moduli space of regular singular connections and whose special fiber is a moduli space of unramified irregular singular connections. On the moduli space of unramified irregular singular connections, there is a subbundle of the tangent bundle defining the generalized isomonodromic deformation produced by the Jimbo-Miwa-Ueno theory. On an analytic open subset of the unfolded moduli space of connections, we construct a non-canonical lift of this subbundle, which we call an unfolding of the unramified irregular singular generalized isomonodromic deformation. Our construction of an unfolding of the unramified irregular singular generalized isomonodromic deformation is not compatible with the asymptotic property in the unfolding theory established by Hurtubise, Lambert and Rousseau which gives unfolded Stokes matrices for an unfolded linear differential equation in a general framework.

Stokes Phenomenon and Yang–Baxter Equations

We describe the monodromy of dynamical Knizhnik-Zamolodchikov equations via Stokes phenomenon. It defines a family of braid groups representations by certain Stokes matrices. In particular, these Stokes matrices satisfy the Yang-Baxter equation.

Isomonodromy deformations at an irregular singularity with coalescing eigenvalues

We consider an $n\times n$ linear system of ODEs with an irregular singularity of Poincar\'e rank 1 at $z=\infty$, holomorphically depending on parameter $t$ within a polydisc in $\mathbb{C}^n$ centred at $t=0$. The eigenvalues of the leading matrix at $z=\infty$ coalesce along a locus $\Delta$ contained in the polydisc, passing through $t=0$. Namely, $z=\infty$ is a resonant irregular singularity for $t\in \Delta$. We analyse the case when the leading matrix remains diagonalisable at $\Delta$. 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 $\Delta$. When the deformation is isomonodromic away from $\Delta$, it is well known that a fundamental matrix solution has singularities at $\Delta$. 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 $\Delta$, in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisc. These data can be computed just by considering the system at fixed $t=0$. Conversely, if the $t$-dependent system is isomonodromic in a small domain contained in the polydisc not intersecting $\Delta$, if the entries of the Stokes matrices with indices corresponding to coalescing eigenvalues vanish, then we show that $\Delta$ is not a branching locus for the fundamental matrix solutions. The importance of these results for the analytic theory of Frobenius Manifolds is explained. An application to Painlev\'e equations is discussed.

Results on the extension of isomonodromy deformations to the case of a resonant irregular singularity

We explain some results of [G. Cotti, B. A. Dubrovin and D. Guzzetti, Isomonodromy deformations at an irregular singularity with coalescing eigenvalues, preprint (2017); arXiv:1706.04808 .], discussed in our talk [G. Cotti, Monodromy of semisimple Frobenius coalescent structures, in Int. Workshop Asymptotic and Computational Aspects of Complex Differential Equations, CRM, Pisa, February 13–17, (2017).] in Pisa, February 2017. Consider an [Formula: see text] linear system of ODEs with an irregular singularity of Poincaré rank 1 at [Formula: see text] and Fuchsian singularity at [Formula: see text], holomorphically depending on parameter [Formula: see text] within a polydisk in [Formula: see text] centered at [Formula: see text]. The eigenvalues of the leading matrix at [Formula: see text], which is diagonal, coalesce along a coalescence locus [Formula: see text] contained in the polydisk. Under minimal vanishing conditions on the residue matrix at [Formula: see text], we show in [G. Cotti, B. A. Dubrovin and D. Guzzetti, Isomonodromy deformations at an irregular singularity with coalescing eigenvalues, preprint (2017); arXiv:1706.04808 .] that isomonodromic deformations can be extended to the whole polydisk, including [Formula: see text], in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisk. These data can be computed just by considering the system at point of [Formula: see text], where it simplifies. Conversely, if the [Formula: see text]-dependent system is isomonodromic in a small domain contained in the polydisk not intersecting [Formula: see text], and if suitable entries of the Stokes matrices vanish, then [Formula: see text] is not a branching locus for the fundamental matrix solutions. The results have applications to Frobenius manifolds and Painlevé equations.

Zero Level Perturbation of a Certain Third-Order Linear Solvable ODE with an Irregular Singularity at the Origin of Poincaré Rank 1

We study an irregular singularity of Poincare rank 1 at the origin of a certain third-order linear solvable homogeneous ODE. We perturb the equation by introducing a small parameter $\varepsilon \in ({\mathbb R}_{+},0)$ (e < 1), which causes the splitting of the irregular singularity into two finite Fuchsian singularities. We show that when the solutions of the perturbed equation contain logarithmic terms, the Stokes matrices of the initial equation are limits of the part of the monodromy matrices around the finite resonant Fuchsian singularities of the perturbed equation.

On the Stokes Matrices of the $tt^*$-Toda Equation

We derive a formula for the signature of the symmetrized Stokes matrix $\mathcal{S}+\mathcal{S}^\mathrm{T}$ for the $tt^*$-Toda equation, reminiscent of a formula of Beukers and Heckmann for the generalized hypergeometric equation. The condition $\mathcal{S}+\mathcal{S}^\mathrm{T} > 0$ is prominent in the work of Cecotti and Vafa on the $tt^*$ equation; using our formula, we show that the Stokes matrices $\mathcal{S}$ satisfying this condition are parameterized by the points of an open convex polytope.

Moduli space for generic unfolded differential linear systems

In this paper, we identify the moduli space for germs of generic unfoldings of nonresonant linear differential systems with an irregular singularity of Poincaré rank k at the origin, under analytic equivalence. The modulus of a given family was determined in [10]: it comprises a formal part depending analytically on the parameters, and an analytic part given by unfoldings of the Stokes matrices. These unfoldings are given on “Douady–Sentenac” (DS) domains in the parameter space covering the generic values of the parameters corresponding to Fuchsian singular points. Here we identify exactly which moduli can be realized. A necessary condition on the analytic part, called compatibility condition, is saying that the unfoldings define the same monodromy group (up to conjugacy) for the different presentations of the modulus on the intersections of DS domains. With the additional requirement that the corresponding cocycle is trivial and good limit behavior at some boundary points of the DS domains, this condition becomes sufficient. In particular we show that any modulus can be realized by a k-parameter family of systems of rational linear differential equations over CP1 with k+1, k+2 or k+3 singular points (with multiplicities). Under the generic condition of irreducibility, there are precisely k+2 singular points which are Fuchsian as soon as simple. This in turn implies that any unfolding of an irregular singularity of Poincaré rank k is analytically equivalent to a rational system of the form y′=A(x)pϵ(x)⋅y, with A(x) polynomial of degree at most k and pϵ(x) is the generic unfolding of the polynomial xk+1.

Intrinsic character of Stokes matrices

Two germs of linear analytic differential systems xk+1Y′=A(x)Y with a non-resonant irregular singularity are analytically equivalent if and only if they have the same eigenvalues and equivalent collections of Stokes matrices. The Stokes matrices are the transition matrices between sectors on which the system is analytically equivalent to its formal normal form. Each sector contains exactly one separating ray for each pair of eigenvalues. A rotation in S allows supposing that R+ lies in the intersection of two sectors. Reordering of the coordinates of Y allows ordering the real parts of the eigenvalues, thus yielding triangular Stokes matrices. However, the choice of the rotation in x is not canonical. In this paper we establish how the collection of Stokes matrices depends on this rotation, and hence on a chosen order of the projection of the eigenvalues on a line through the origin.

On Stokes Matrices in terms of Connection Coefficients

The classical problem of computing a complete system of Stokes multipliers of a linear system of ODEs of rank one in terms of some connection coefficients of an associated hypergeometric system of ODEs, is solved with no genericness assumptions on the residue matrix at zero, by an extension of the method of [3].

The Stokes Phenomenon and Some Applications

Multisummation provides a transparent description of Stokes matrices which is reviewed here together with some applications. Examples of moduli spaces for Stokes matrices are computed and discussed. A moduli space for a third Painlev\'e equation is made explicit. It is shown that the monodromy identity, relating the topological monodromy and Stokes matrices, is useful for some quantum differential equations and for confluent generalized hypergeometric equations.

Confluence of meromorphic solutions of q-difference equations

Dans cet article, nous considérons un q-analogue du processus de sommation de Borel-Laplace, avec q>1 paramètre réel. En particulier, nous prouvons que la sommation de Borel-Laplace d’une série formelle solution d’une équation différentielle linéaire peut être approchée, dans un secteur convenable, par une solution méromorphe d’une certaine famille d’équations aux q-différences linéaire. Nous faisons les calculs pour les séries hypergéométriques. En s’inspirant de Sauloy, nous prouvons comment une base de solutions d’une equation différentielle linéaire peut être approchée, sur un secteur convenable, par une base de solutions d’une famille correspondante d’équations aux q-différences. Cela nous mène à l’approximation des matrices de Stokes et de monodromies de l’équation différentielle, par des matrices dont les entrées sont invariantes par multiplication par q.

Stokes matrices for the quantum differential equations of some Fano varieties

The classical Stokes matrices for the quantum differential equation of $$\mathbb {P}^n$$ are computed using multisummation and the ‘monodromy identity’. Thus, we recover the results of D. Guzzetti that confirm Dubrovin’s conjecture for projective spaces. The same method yields explicit formulas for the Stokes matrices of the quantum differential equations of smooth Fano hypersurfaces in $$\mathbb {P}^n$$ and for weighted projective spaces.

Stokes matrices for the quantum cohomologies of a class of orbifold projective lines

We prove the Dubrovin's conjecture for the Stokes matrices for the quantum cohomology of orbifold projective lines. The conjecture states that the Stokes matrix of the first structure connection of the Frobenius manifold constructed from the Gromov-Witten theory coincides with the Euler matrix of a full exceptional collection of the bounded derived category of the coherent sheaves. Our proof is based on the homological mirror symmetry, primitive forms of affine cusp polynomials, and the Picard-Lefschetz theory.

Linear differential equations and multiple zeta-values. III. Zeta(3)

We consider the hypergeometric equation (1 − t)∂t∂t∂g + x3g = 0, whose unique analytic solution φ1(t; x) = 1 + O(t) near t = 0 for t = 1 becomes a generating function for multiple zeta values φ1(1; x) = f3(x) = 1 − ζ(3)x3 + ζ(3, 3)x6 − …. We apply the so-called WKB method to study solutions of the hypergeometric equation for large x and we calculate corresponding Stokes matrices. We prove that the function f3(x) near x = ∞ is also expressed via WKB type functions which subject to some Stokes phenomenon. This implies that f3(x) satisfies a sixth order linear differential equation with irregular singularity at infinity.

Complete System of Analytic Invariants for Unfolded Differential Linear Systems with an Irregular Singularity of Poincaré Rank 1

In this paper, we give a complete system of analytic invariants for the unfoldings of nonresonant linear differential systems with an irregular singularity of Poincare rank 1 at the origin over a fixed neighborhood Dr . It consists of a formal part, given by polynomials, and an analytic part, given by an equivalence class of unfolded Stokes matrices. The unfolding parameter ǫ is taken in a sector S pointed at the origin of opening larger than 2π in the complex plane, thus covering a whole neighborhood of the origin. For each parameter value ǫ ∈ S, we cover Dr with two sectors and, over each sector, we construct a well chosen basis of solutions of the unfolded linear differential systems. This basis is used to find the unfolded Stokes matrices, which are analytic invariants linked to the monodromy of the chosen basis around the singular points. The unfolded Stokes matrices give a complete geometric interpretation to the well-known Stokes matrices at ǫ = 0: this includes the link (existing at least for the generic cases) between the divergence of the solutions at ǫ = 0 and the presence of logarithmic terms in the solutions for resonant values of the unfolding parameter. Finally, we give a realization theorem for a given complete system of analytic invariants satisfying a necessary and sufficient condition, thus identifying the set of modules.

Linear differential equations and multiple zeta values. II. A generalization of the WKB method

For linear differential equations x ( n ) + a 1 x ( n − 1 ) + ⋯ + a n x = 0 (and corresponding linear differential systems) with large complex parameter λ and meromorphic coefficients a j = a j ( t ; λ ) we prove existence of analogues of Stokes matrices for the asymptotic WKB solutions. These matrices may depend on the parameter, but under some natural assumptions such dependence does not take place. We also discuss a generalization of the Hukuhara–Levelt–Turritin theorem about formal reduction of a linear differential system near an irregular singular point t = 0 to a normal form with ramified change of time to the case of systems with large parameter. These results are applied to some hypergeometric equations related with generating functions for multiple zeta values.

ASYMPTOTIC AdS STRING SOLUTIONS FOR NULL POLYGONAL WILSON LOOPS IN R1,2

For the asymptotic string solution in AdS3 which is represented by the AdS3 Poincaré coordinates and yields the planar multi-gluon scattering amplitude at strong coupling in arXiv:0904.0663, we express it by the AdS4 Poincaré coordinates and demonstrate that the hexagonal and octagonal Wilson loops surrounding the string surfaces take closed contours consisting of null vectors in R1,2 owing to the relations of Stokes matrices. For the tetragonal Wilson loop we construct a string solution characterized by two parameters by solving the auxiliary linear problems and demanding a reality condition, and analyze the asymptotic behavior of the solution in R1,2. The freedoms of two parameters are related with some conformal SO(2,4) transformations.

Stokes phenomenon for the prolate spheroidal wave equation

The prolate spheroidal wave functions can be characterized as the eigenfunctions of a differential operator of order 2: ( t 2 − τ 2 ) φ ″ + 2 t φ ′ + σ 2 t 2 φ = μ φ . In this article we study the formal solutions of this equation in the neighborhood of the singularities (the regular ones ± τ, and the irregular one, at infinity) and perform some numerical experiments on the computation of Stokes matrices and monodromy, using formal/numerical algorithms we developed recently in the Maple package Desir. This leads to the following conjecture: the series appearing in the formal solutions at infinity, depending on the parameter μ, are in general divergent; they become convergent for some particular values of the parameter, corresponding exactly to the eigenvalues of the prolate operator. We give the proof of this result and its interpretation in terms of differential Galois groups.

Correlations between polycrystalline fabric and the polarization of transmitted light

Strong correlations have been found in the polarization of light transmitted through a polycrystalline material and the grain sizes and orientations of that material. Experiments and supporting simulations with irregularly shaped single quartz crystals show that linear polarization is lost more rapidly as grain sizes decrease and the angular spread of the crystal orientations increase. A quantitative method using Stokes matrices to predict such changes is described and experimentally verified using an apparatus to vary the orientation of irregular quartz crystals. Grain sizes are varied between 1mm and 4mm, and the angular spreads in the crystal orientation are varied between 9 degrees and 27 degrees . This technique has applications to identify changes in crystal structure of transparent uniaxial polycrystalline materials, especially in the nondestructive characterization of glacial ice.

Stokes matrices of hypergeometric integrals

Dans cet article, nous calculons les matrices de Stokes de l'equation differentielle ordinaire satisfaites par les integrales hypergeometriques, associees a un arrangement d'hyperplans en position generique. Cela generalise le calcul fait par J.-P. Ramis pour les fonctions hypergeometriques confluentes, qui correspondent a l'arrangement de deux points sur une droite. La demonstration est basee sur une description explicite d'une base de solutions canoniques comme integrales sur les cones de l'arrangement et les relations combinatoires entre les integrales sur cones et sur domaines.