The Riemann-Hilbert Problem and Fuchsian Differential Equations on the Riemann Sphere

1995 ◽  
pp. 1159-1168
Author(s):  
A. A. Bolibruch
Keyword(s):  
Differential Equations ◽  
Hilbert Problem ◽  
Riemann Sphere ◽  
Fuchsian Differential Equations ◽  
Riemann Hilbert Problem

MONODROMY ARISING FROM THE MARKOFF THEORY

2006 ◽  
Vol 20 (11n13) ◽  
pp. 1819-1832
Author(s):  
SERGE PERRINE
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Free Group ◽  
Monodromy Group ◽  
Hilbert Problem ◽  
Monodromy Representation ◽  
Poincare Group ◽  
Complete Family ◽  
Computation Theory ◽  
Riemann Hilbert Problem

In a former work, recalling what the Markoff theory is, we summarized some existing links with the group GL(2, ℤ) of 2 × 2 matrices. We also quoted the relation with conformal punctured toruses. The monodromy representation of the Poincaré group of such a torus was considered. Here we explicit the corresponding solution of the associated Riemann-Hilbert problem, and the resulting Fuchs differential equation. We precisely describe how the calculus runs. The main result is the description of a complete family of Fuchs differential equations with, as the monodromy group, the free group with two generators. We also identify a link with some eigenvalues of a Laplacian. The introduction explains the links that we see with information and computation theory (classical or quantum).

scholarly journals A Riemann–Hilbert approach to the Akhiezer polynomials

2007 ◽  
Vol 366 (1867) ◽  
pp. 973-1003 ◽  
Author(s):  
Yang Chen ◽  
Alexander R Its
Keyword(s):  
Riemann Surface ◽  
Meromorphic Function ◽  
Matrix Factorization ◽  
Hilbert Scheme ◽  
Hilbert Problem ◽  
Hankel Determinant ◽  
Recurrence Coefficients ◽  
Hyperelliptic Riemann Surface ◽  
Fuchsian Differential Equations ◽  
Riemann Hilbert Problem

In this paper, we study those polynomials, orthogonal with respect to a particular weight, over the union of disjoint intervals, first introduced by N. I. Akhiezer, via a reformulation as a matrix factorization or Riemann–Hilbert problem. This approach complements the method proposed in a previous paper, which involves the construction of a certain meromorphic function on a hyperelliptic Riemann surface. The method described here is based on the general Riemann–Hilbert scheme of the theory of integrable systems and will enable us to derive, in a very straightforward way, the relevant system of Fuchsian differential equations for the polynomials and the associated system of the Schlesinger deformation equations for certain quantities involving the corresponding recurrence coefficients. Both of these equations were obtained earlier by A. Magnus. In our approach, however, we are able to go beyond Magnus' results by actually solving the equations in terms of the Riemanni Θ -functions. We also show that the related Hankel determinant can be interpreted as the relevant τ -function.

scholarly journals On the Modular Operator of Mutli-component Regions in Chiral CFT

2021 ◽  
Author(s):  
Stefan Hollands
Keyword(s):  
Field Theory ◽  
Integral Equation ◽  
Hilbert Problem ◽  
Matrix Elements ◽  
New Approach ◽  
Conformal Field ◽  
The Matrix ◽  
Kms Condition ◽  
Modular Operator ◽  
Riemann Hilbert Problem

AbstractWe introduce a new approach to find the Tomita–Takesaki modular flow for multi-component regions in general chiral conformal field theory. Our method is based on locality and analyticity of primary fields as well as the so-called Kubo–Martin–Schwinger (KMS) condition. These features can be used to transform the problem to a Riemann–Hilbert problem on a covering of the complex plane cut along the regions, which is equivalent to an integral equation for the matrix elements of the modular Hamiltonian. Examples are considered.

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