Differential Equations on the Complex Sphere and the Riemann-Hilbert Problem

2003 ◽  
pp. 157-185
Author(s):  
Marius van der Put ◽  
Michael F. Singer
Keyword(s):  
Differential Equations ◽  
Hilbert Problem ◽  
Complex Sphere ◽  
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).

Solvability of singular integro-differential equations via Riemann–Hilbert problem

2018 ◽  
Vol 265 (11) ◽  
pp. 5455-5471 ◽  
Author(s):  
Pingrun Li ◽  
Guangbin Ren
Keyword(s):  
Differential Equations ◽  
Hilbert Problem ◽  
Riemann Hilbert Problem

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.

scholarly journals Isomonodromy Aspects of the tt* Equations of Cecotti and Vafa II: Riemann–Hilbert Problem

2015 ◽  
Vol 336 (1) ◽  
pp. 337-380 ◽  
Author(s):  
Martin A. Guest ◽  
Alexander R. Its ◽  
Chang-Shou Lin
Keyword(s):  
Hilbert Problem ◽  
Riemann Hilbert Problem
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