Stokes Matrices Research Articles

Explicit evaluation of the Stokes matrices for certain quantum confluent hypergeometric equations

Explicit evaluation of the Stokes matrices for certain quantum confluent hypergeometric equations

Correction: Stokes matrices and exceptional isomorphisms

Correction: Stokes matrices and exceptional isomorphisms

Stokes matrices and exceptional isomorphisms

AbstractWe prove that the integral Stokes matrices of size 4 with given Coxeter invariant of nonzero discriminant decompose into finitely many braid group orbits. We also establish exceptional isomorphisms between moduli spaces of points on algebraic unit 3-spheres and certain moduli of local systems on positive genus surfaces.

High-resolution 3D imaging in light-field microscopy through Stokes matrices and data fusion.

The trade-off between the lateral and vertical resolution has long posed challenges to the efficient and widespread application of Fourier light-field microscopy, a highly scalable 3D imaging tool. Although existing methods for resolution enhancement can improve the measurement result to a certain extent, they come with limitations in terms of accuracy and applicable specimen types. To address these problems, this paper proposed a resolution enhancement scheme utilizing data fusion of polarization Stokes vectors and light-field information for Fourier light-field microscopy system. By introducing the surface normal vector information obtained from polarization measurement and integrating it with the light-field 3D point cloud data, 3D reconstruction results accuracy is highly improved in axial direction. Experimental results with a Fourier light-field 3D imaging microscope demonstrated a substantial enhancement of vertical resolution with a depth resolution to depth of field ratio of 0.19%. This represented approximately 44 times the improvement compared to the theoretical ratio before data fusion, enabling the system to access more detailed information with finer measurement accuracy for test samples. This work not only provides a feasible solution for breaking the limitations imposed by traditional light-field microscope hardware configurations but also offers superior 3D measurement approach in a more cost-effective and practical manner.

On some Lie-theoretic solutions of the tt*-Toda equations with integer Stokes data

We describe some solutions of the tt*-Toda equations with integer Stokes data, based on a Lie-theoretic approach. In particular, we find all such solutions in a certain 1-parameter family. For general solutions, we compare the results of Guest–Its–Lin (based on solving the equations) with the original classification of Cecotti–Vafa (based on braid group orbits of Stokes matrices for hypothetical solutions), and find that these agree in low dimensions.

Moduli Spaces for the Fifth Painlevé Equation

Isomonodromy for the fifth Painlevé equation ${\rm P}_5$ is studied in detail in the context of certain moduli spaces for connections, monodromy, the Riemann-Hilbert morphism, and Okamoto-Painlevé spaces. This involves explicit formulas for Stokes matrices and parabolic structures. The rank 4 Lax pair for ${\rm P}_5$, introduced by Noumi-Yamada et al., is shown to be induced by a natural fine moduli space of connections of rank 4. As a by-product one obtains a polynomial Hamiltonian for ${\rm P}_5$, equivalent to the one of Okamoto.

Nonintegrability of the Painlevé IV equation in the Liouville–Arnold sense and Stokes phenomena

AbstractIn this paper we study the integrability of the Hamiltonian system associated with the fourth Painlevé equation. We prove that one two‐parametric family of this Hamiltonian system is not integrable in the sense of the Liouville–Arnold theorem. Computing explicitly the Stokes matrices and the formal invariants of the second variational equations, we deduce that the connected component of the unit element of the corresponding differential Galois group is not Abelian. Thus the Morales–Ramis–Simó theory leads to a nonintegrable result. Moreover, combining the new result with our previous one we establish that for all values of the parameters for which the equation has a particular rational solution the corresponding Hamiltonian system is not integrable by meromorphic first integrals which are rational in t.

The sixth Painlevé equation as isomonodromy deformation of an irregular system: monodromy data, coalescing eigenvalues, locally holomorphic transcendents and Frobenius manifolds

The sixth Painlevé equation PVI is both the isomonodromy deformation condition of a 2-dimensional isomonodromic Fuchsian system and of a 3-dimensional irregular system. Only the former has been used in the literature to solve the nonlinear connection problem for PVI, through the computation of invariant quantities . We prove a new simple formula expressing the invariants p jk in terms of the Stokes matrices of the irregular system, making the irregular system a concrete alternative for the nonlinear connection problem. We classify the transcendents such that the Stokes matrices and the p jk can be computed in terms of special functions, providing a full non-trivial class of 3-dim. examples such that the theory of non-generic isomonodromy deformations of Cotti et al (2019 Duke Math. J. 168 967–1108) applies. A sub-class of these transcendents realises the local structure of all the 3-dim Dubrovin–Frobenius manifolds with semisimple coalescence points of the type studied in Cotti et al (2020 SIGMA 16 105). We compute all the monodromy data for these manifolds (Stokes matrix, Levelt exponents and central connection matrix).

Peacock patterns and resurgence in complex Chern–Simons theory

The partition function of complex Chern–Simons theory on a 3-manifold with torus boundary reduces to a finite-dimensional state-integral which is a holomorphic function of a complexified Planck’s constant tau in the complex cut plane and an entire function of a complex parameter u. This gives rise to a vector of factorially divergent perturbative formal power series whose Stokes rays form a peacock-like pattern in the complex plane. We conjecture that these perturbative series are resurgent, their trans-series involve two non-perturbative variables, their Stokes automorphism satisfies a unique factorization property and that it is given explicitly in terms of a fundamental matrix solution to a (dual) linear q-difference equation. We further conjecture that a distinguished entry of the Stokes automorphism matrix is the 3D-index of Dimofte–Gaiotto–Gukov. We provide proofs of our statements regarding the q-difference equations and their properties of their fundamental solutions and illustrate our conjectures regarding the Stokes matrices with numerical calculations for the two simplest hyperbolic textbf{4}_1 and textbf{5}_{2} knots.

Stokes Phenomenon and Reflection Equations

In this paper, we study the Stokes phenomenon of the cyclotomic Knizhnik–Zamolodchikov equation associated to the symmetric pair \((\mathfrak {gl}_n,\mathfrak {so}_n)\), and prove that its two types of Stokes matrices satisfy the Yang–Baxter and reflection equations respectively. We briefly discuss its isomonodromy deformation, and its relations with cyclotomic associators, twists and quantum symmetric pairs.

Stokes matrices for Airy equations

We compute Stokes matrices for generalised Airy equations and prove that they are regular unipotent (up to multiplication with the formal monodromy). This class of differential equations was defined by Katz and includes the classical Airy equation. In addition, it includes differential equations which are not rigid. Our approach is based on the topological computation of Stokes matrices of the enhanced Fourier-Sato transform of a perverse sheaf due to D'Agnolo, Hien, Morando and Sabbah.

A short proof of a theorem of Cotti, Dubrovin and Guzzetti

We give a short proof of a theorem of G. Cotti, B. Dubrovin and D. Guzzetti (arXiv:1706.04808 and arXiv:2101.03397) asserting the vanishing of some entries of the Stokes matrices at coalescing points of an isomonodromic deformation.

Correction to : Stokes Matrices of a Reducible Double Confluent Heun Equation via Monodromy Matrices of a Reducible General Huen Equation with Symmetric Finite Singularities

A Correction to this paper has been published: https://doi.org/10.1007/s10883-021-09575-w

Stokes phenomena in 3d mathcal{N} = 2 SQED2 and mathbbm{CP} 1 models

We propose a novel approach of uncovering Stokes phenomenon exhibited by the holomorphic blocks of mathbbm{CP} 1 model by considering it as a specific decoupling limit of SQED2 model. This approach involves using a ℤ3 symmetry that leaves the supersymmetric parameter space of SQED2 model invariant to transform a pair of SQED2 holomorphic blocks to get two new pairs of blocks. The original pair obtained by solving the line operator identities of the SQED2 model and the two new transformed pairs turn out to be related by Stokes-like matrices. These three pairs of holomorphic blocks can be reduced to the known triplet of mathbbm{CP} 1 blocks in a particular decoupling limit where two of the chiral multiplets in the SQED2 model are made infinitely massive. This reduction then correctly reproduces the Stokes regions and matrices of the mathbbm{CP} 1 blocks. Along the way, we find six pairs of SQED2 holomorphic blocks in total, which lead to six Stokes-like regions covering uniquely the full parameter space of the SQED2 model.

Stokes Matrices of a Reducible Double Confluent Heun Equation via Monodromy Matrices of a Reducible General Huen Equation with Symmetric Finite Singularities

Stokes Matrices of a Reducible Double Confluent Heun Equation via Monodromy Matrices of a Reducible General Huen Equation with Symmetric Finite Singularities

Isomonodromic Laplace transform with coalescing eigenvalues and confluence of Fuchsian singularities

We consider a Pfaffian system expressing isomonodromy of an irregular system of Okubo type, depending on complex deformation parameters u=(u_1,ldots ,u_n), which are eigenvalues of the leading matrix at the irregular singularity. At the same time, we consider a Pfaffian system of non-normalized Schlesinger-type expressing isomonodromy of a Fuchsian system, whose poles are the deformation parameters u_1,ldots ,u_n. The parameters vary in a polydisc containing a coalescence locus for the eigenvalues of the leading matrix of the irregular system, corresponding to confluence of the Fuchsian singularities. We construct isomonodromic selected and singular vector solutions of the Fuchsian Pfaffian system together with their isomonodromic connection coefficients, so extending a result of Balser et al. (I SIAM J Math Anal 12(5): 691–721, 1981) and Guzzetti (Funkcial Ekvac 59(3): 383–433, 2016) to the isomonodromic case, including confluence of singularities. Then, we introduce an isomonodromic Laplace transform of the selected and singular vector solutions, allowing to obtain isomonodromic fundamental solutions for the irregular system, and their Stokes matrices expressed in terms of connection coefficients. These facts, in addition to extending (Balser et al. in I SIAM J Math Anal 12(5): 691–721, 1981; Guzzetti in Funkcial Ekvac 59(3): 383–433, 2016) to the isomonodromic case (with coalescences/confluences), allow to prove by means of Laplace transform the main result of Cotti et al. (Duke Math J arXiv:1706.04808, 2017), namely the analytic theory of non-generic isomonodromic deformations of the irregular system with coalescing eigenvalues.

Stokes Matrices for Confluent Hypergeometric Equations

Abstract We apply the method of [7] to compute the Stokes matrices of non-resonant confluent hypergeometric differential equations. We discuss the ambiguity of the presentation of the Stokes matrices regarding different choices. The results rely on an explicit description of the perverse sheaf associated to the non-confluent regular singular hypergeometric system arising via Fourier–Laplace transform. We give assumptions on the parameter such that the Stokes matrices have rational or real values. Under some more restrictive conditions, the Stokes matrices had been computed by Duval–Mitschi before. We compare our results with their formulae in the unramified case.

Confluent conformal blocks of the second kind

We construct confluent conformal blocks of the second kind of the Virasoro algebra. We also construct the Stokes transformations which map such blocks in one Stokes sector to another. In the BPZ limit, we verify explicitly that the constructed blocks and the associated Stokes transformations reduce to solutions of the confluent BPZ equation and its Stokes matrices, respectively. Both the confluent conformal blocks and the Stokes transformations are constructed by taking suitable confluent limits of the crossing transformations of the four-point Virasoro conformal blocks.

Topological computation of Stokes matrices of some weighted projective lines

By mirror symmetry, the quantum connection of a weighted projective line is closely related to the localized Fourier–Laplace transform of some Gauß–Manin system. Following an article of D’Agnolo, Hien, Morando, and Sabbah, we compute the Stokes matrices for the latter at infty for the cases {mathbb {P}}(1,3) and {mathbb {P}}(2,2) by purely topological methods. We compare them to the Gram matrix of the Euler–Poincaré pairing on D^b(mathrm{Coh}({mathbb {P}}(1,3))) and D^b(mathrm{Coh}({mathbb {P}}(2,2))), respectively. This article is based on the doctoral thesis of the author.

Confluence of Singularities in Hypergeometric Systems

A system in a Birkhoff normal form with an irregular singularity of Poincare rank 1 at the origin and a regular singularity at infinity is through the Borel-Laplace transform dual to a system in an Okubo form. Schafke has showed that the Birkhoff system can also be obtained from the Okubo system by a simple limiting procedure. The Okubo system comes naturally with two kinds of mixed solution bases, both of which converge under the limit procedure to the canonical solutions of the limit Birkhoff system on sectors near the irregular singularity at the origin. One can then define Stokes matrices of the Okubo system as connection matrices between different branches of the mixed solution bases and use them to relate the monodromy matrices of the Okubo system to the usual Stokes matrices of the limit system at the irregular singularity. This is illustrated on the example of confluence in the generalized hypergeometric equation.