Fuchsian Systems Research Articles

Fredholm Determinant and Nekrasov Sum Representations of Isomonodromic Tau Functions

We derive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems with $n$ regular singular points on the Riemann sphere and generic monodromy in $\mathrm{GL}(N,\mathbb C)$. The corresponding operator acts in the direct sum of $N(n-3)$ copies of $L^2(S^1)$. Its kernel has a block integrable form and is expressed in terms of fundamental solutions of $n-2$ elementary 3-point Fuchsian systems whose monodromy is determined by monodromy of the relevant $n$-point system via a decomposition of the punctured sphere into pairs of pants. For $N=2$ these building blocks have hypergeometric representations, the kernel becomes completely explicit and has Cauchy type. In this case Fredholm determinant expansion yields multivariate series representation for the tau function of the Garnier system, obtained earlier via its identification with Fourier transform of Liouville conformal block (or a dual Nekrasov-Okounkov partition function). Further specialization to $n=4$ gives a series representation of the general solution to Painlev\'e VI equation.

Notes on Non-Generic Isomonodromy Deformations

Some of the main results of [Cotti G., Dubrovin B., Guzzetti D., Duke Math. J., to appear, arXiv:1706.04808], concerning non-generic isomonodromy deformations of a certain linear differential system with irregular singularity and coalescing eigenvalues, are reviewed from the point of view of Pfaffian systems, making a distinction between weak and strong isomonodromic deformations. Such distinction has a counterpart in the case of Fuchsian systems, which is well known as Schlesinger and non-Schlesinger deformations, reviewed in Appendix A.

On Some Applications of Sakai's Geometric Theory of Discrete Painlevé Equations

Although the theory of discrete Painlev\'e (dP) equations is rather young, more and more examples of such equations appear in interesting and important applications. Thus, it is essential to be able to recognize these equations, to be able to identify their type, and to see where they belong in the classification scheme. The definite classification scheme for dP equations was proposed by H. Sakai, who used geometric ideas to identify 22 different classes of these equations. However, in a major contrast with the theory of ordinary differential Painlev\'e equations, there are infinitely many non-equivalent discrete equations in each class. Thus, there is no general form for a dP equation in each class, although some nice canonical examples in each equation class are known. The main objective of this paper is to illustrate that, in addition to providing the classification scheme, the geometric ideas of Sakai give us a powerful tool to study dP equations. We consider a very complicated example of a dP equation that describes a simple Schlesinger transformation of a Fuchsian system and we show how this equation can be identified with a much simpler canonical example of the dP equation of the same type and moreover, we give an explicit change of coordinates transforming one equation into the other. Among our main tools are the birational representation of the affine Weyl symmetry group of the equation and the period map. Even though we focus on a concrete example, the techniques that we use are general and can be easily adapted to other examples.

Monodromy dependence and connection formulae for isomonodromic tau functions

We discuss an extension of the Jimbo-Miwa-Ueno differential 1-form to a form closed on the full space of extended monodromy data of systems of linear ordinary differential equations with rational coefficients. This extension is based on the results of M. Bertola generalizing a previous construction by B. Malgrange. We show how this 1-form can be used to solve a long-standing problem of evaluation of the connection formulae for the isomonodromic tau functions which would include an explicit computation of the relevant constant factors. We explain how this scheme works for Fuchsian systems and, in particular, calculate the connection constant for generic Painlev\'e VI tau function. The result proves the conjectural formula for this constant proposed in \cite{ILT13}. We also apply the method to non-Fuchsian systems and evaluate constant factors in the asymptotics of Painlev\'e II tau function.

On the dimension of the subspace of Liouvillian solutions of a Fuchsian system

The paper deals with the problem of determining the dimension of the subspace of Liouvillian solutions of a Fuchsian system of linear differential equations in the case where this question can be answered directly in terms of the matrix of coefficients of the system.

Fuchsian systems for Dotsenko–Fateev multipoint correlation functions and similar integrals of hypergeometric type

The Dotsenko-Fateev integral is an analytic function of one complex variable expressing the amplitude in the 4-point correlator of the 2D conformal field theory. Dotsenko-Fateev found ODE of third order with Fuchsian singularities satisfied by their integral. In the present paper, this work is extended to generalized Dotsenko-Fateev integrals, in particular those associated to arbitrary multipoint correlators, and Pfaff systems of PDE of Fuchsian type are constructed for them. The ubiquity of the Fuchsian systems is in that they permit to obtain local expansions of solutions in the neighborhoods of singularities of the system.

English

This paper explores some nonlinear systems of singular partial differential equations written in the form t A D t U = Λ(t, x)U + f t, x, ζU, t A D x U. Under an assumption on Λ, unique solvability theorems are provided in the space of functions that are holomorphic in x on an open set, differentiable with respect to t on a real interval ]0, r] and extending to a continuous function at t = 0. The studied systems contain Fuchsian systems.

Sur les processus linéaires de naissance et de mort sous-critiques dans un environnement aléatoire.

An explicit formula is found for the rate of extinction of subcritical linear birth-and-death processes in a random environment. The formula is illustrated by numerical computations of the eigenvalue with largest real part of the truncated matrix for the master equation. The generating function of the corresponding eigenvector satisfies a Fuchsian system of singular differential equations. A particular attention is set on the case of two environments, which leads to Riemann's differential equation.

A generalization of the Dotsenko-Fateev integral

The Dotsenko-Fateev integral, an analytic function of one complex variable arising in conformal field theory, is generalized in a natural way to an analytic function of two complex variables. A system of partial differential equations and a Pfaffian system of Fuchsian type are derived for this generalized Dotsenko- Fateev integral. The Fuchsian system permits to obtain local expansions of solutions in the neighborhoods of singularities of the system.

Computation of RS-Pullback Transformations for Algebraic Painlevé VI Solutions

Algebraic solutions of the sixth Painleve equation can be constructed with the help of RS-transformations of hypergeometric equations. Construction of these transformations includes specially ramified rational coverings of the Riemann sphere and the corresponding Schlesinger transformations (S-transformations). Some algebraic solutions can be constructed from rational coverings alone, without obtaining the corresponding pullbacked isomonodromy Fuchsian system, i.e., without the S part of the RS transformations. At the same time, one and the same covering can be used to pullback different hypergeometric equations, resulting in different algebraic Painleve VI solutions. In the case of high degree coverings, construction of the S parts of the RS-transformations may represent some computational difficulties. This paper presents computation of explicit RS pullback transformations and derivation of algebraic Painleve VI solutions from them. As an example, we present a computation of all seed solutions for pullbacks of hyperbolic hypergeometric equations. Bibliography: 26 titles.

Abelianization of Fuchsian systems on a $4$-punctured sphere and applications

In this paper we consider special linear Fuchsian systems of rank $2$ on a $4-$punctured sphere and the corresponding parabolic structures. Through an explicit abelianization procedure we obtain a $2-$to$-1$ correspondence between flat line bundle connections on a torus and these Fuchsian systems. This naturally equips the moduli space of flat $SL(2,\mathbb C)-$connections on a $4-$punctured sphere with a new set of Darboux coordinates. Furthermore, we apply our theory to give a complex analytic proof of Witten's formula for the symplectic volume of the moduli space of unitary flat connections on the $4-$punctured sphere.

Constructive Solutions to the Riemann–Hilbert Problem and Middle Convolution

In this paper, we present a general scheme to generate constructive solutions to the Riemann---Hilbert problem via middle convolution and illustrate this approach for a Fuchsian system with four singular points.

Isomonodromic τ-functions and W N conformal blocks

We study the solution of the Schlesinger system for the 4-point $\mathfrak{sl}_N$ isomonodromy problem and conjecture an expression for the isomonodromic $\tau$-function in terms of 2d conformal field theory beyond the known $N=2$ Painlev\'e VI case. We show that this relation can be used as an alternative definition of conformal blocks for the $W_N$ algebra and argue that the infinite number of arbitrary constants arising in the algebraic construction of $W_N$ conformal block can be expressed in terms of only a finite set of parameters of the monodromy data of rank $N$ Fuchsian system with three regular singular points. We check this definition explicitly for the known conformal blocks of the $W_3$ algebra and demonstrate its consistency with the conjectured form of the structure constants.

Integral Solutions to Schlesinger Equations

It is shown that Schlesinger equations for isomonodromic deformations of Fuchsian systems of order p on the Riemann spheres with upper triangular monodromy are reduced to multidimensional linear homogeneous (p = 2) and inhomogeneous (≥ 3) Pfaffian systems. For components of the solutions to the multidimensional linear Pfaffian systems (p = 2) we obtain integral representations of hypergeometric type and expressions in quadratures close to the hypergeometric Schlesinger equations describing deformations of upper triangular Fuchsian systems of order p = 3.

Middle convolution and non-Schlesinger deformations

Middle convolution is an operation for Fuchsian systems of differential equations which preserves Schlesinger’s deformation equations. In this paper we announce that Bolibruch’s non-Schlesinger deformations of Fuchsian systems are, in general, not preserved by middle convolution.

Isomonodromic Deformations and Painlevé Equations

The connection between isomonodromic deformation of the Fuchsian system of linear differential equations and the Painleve VI equation is considered. Namely, any Fuchsian system can be transformed in accordance with some dynamical system that is called the Schlesinger equations. Such deformation can be considered as a flow on the space of Fuchsian systems; it is the Schlesinger flow. It is shown that the space of all Fuchsian equations can be rationally projected on the symplectic space \(\mathbb {C}\times \mathbb {C}\) in such a way that the preimage of any point consists of the systems with the same monodromy. The Schlesinger flow can be projected on this symplectic space because the corresponding vector field has the following property: the projections of the values of the field at points coincide if the projections of the points coincide. The resulting flow on the extended phase space \(\mathbb {C}\times \mathbb {C}\times \mathbb {C}\ni (p,q,t)\) is called a Painleve VI Hamiltonian system. The nonlinear ODE Painleve VI is the Euler–Lagrange equation corresponding to the Hamiltonian equations under the Legendre transformation. The compatibility of the Schlesinger deformation with the initial Fuchsian system proves that the Painleve VI implies the constant monodromy; the converse does not hold because of the counterexample. The foregoing theory can be considered as an explanation to the presenting of the system of two compatible linear systems and a verification of the fact that the compatibility condition is Painleve VI. Other various linear systems with four poles, taking into account their orders, are also considered, and the corresponding pairs of the compatible linear systems are presented. It is verified directly that the compatibility conditions give all the other equations from Painleve’s list.

Non-Schlesinger Isomonodromic Deformations of Fuchsian Systems and Middle Convolution

The paper is devoted to non-Schlesinger isomonodromic deformations for resonant Fuchsian systems. There are very few explicit examples of such deformations in the literature. In this paper we construct a new example of the non-Schlesinger isomonodromic deformation for a resonant Fuchsian system of order 5 by using middle convolution for a resonant Fuchsian system of order 2. Moreover, it is known that middle convolution is an operation that preserves Schlesinger's deformation equations for non-resonant Fuchsian systems. In this paper we show that Bolibruch's non-Schlesinger deformations of resonant Fuchsian systems are, in general, not preserved by middle convolution.

Katz's middle convolution and Yokoyama's extending operation

We give a concrete relation between Katz's middle convolution and Yokoyama's extension and show the equivalence of both algorithms using these operations for the reduction of Fuchsian systems on the Riemann sphere.

Six-dimensional Painlevé systems and their particular solutions in terms of rigid systems

In this article, we propose a class of six-dimensional Painlevé systems given as the monodromy preserving deformations of the Fuchsian systems. They are expressed as polynomial Hamiltonian systems of sixth order. We also discuss their particular solutions in terms of the hypergeometric functions defined by fourth order rigid systems.

4-Dimensional Frobenius manifolds and Painleve’ VI

A Frobenius manifold has tri-Hamiltonian structure if it is even-dimensional and its spectrum is maximally degenerate. We study the case of the lowest nontrivial dimension $$n=4$$ and show that, under the assumption of semisimplicity, the corresponding isomonodromic Fuchsian system is described by the Painleve $$\hbox {VI}\mu $$ equation. Since the solutions of this equation are known to parametrize semisimple Frobenius manifolds of dimension $$n=3$$ , this leads to an explicit procedure mapping 3-dimensional Frobenius structures of 4-dimensional ones, and giving all tri-Hamiltonian structures in four dimensions. We illustrate the construction by computing two examples in the framework of Frobenius structures on Hurwitz spaces.