THE YANG–MILLS EQUATIONS AND THE INTERSECTION OF QUARTICS IN PROJECTIVE 4-SPACE ℂℙ4

2006 ◽  
Vol 03 (02) ◽  
pp. 201-208 ◽  
Author(s):  
A. LESFARI ◽  
A. ELACHAB
Keyword(s):  
Algebraic Geometry ◽  
Differential Equations ◽  
Gauge Group ◽  
Algebraic Surfaces ◽  
Abelian Surface ◽  
System Of Differential Equations ◽  
Yang Mills ◽  
Completely Integrable ◽  
Affine Part

In this paper, we discuss an interesting interaction between complex algebraic geometry and dynamics: the integrability of the Yang–Mills system for a field with gauge group SU(2) and the intersection of quartics in projective 4-space ℂℙ4. Using Enriques classification of algebraic surfaces and dynamics, we show that these two quartics intesect in the affine part of an abelian surface and it follows that the system of differential equations is algebraically completely integrable.

CYCLIC COVERINGS OF ABELIAN VARIETIES AND THE GENERALIZED YANG–MILLS SYSTEM FOR A FIELD WITH GAUGE GROUP SU(2)

2008 ◽  
Vol 05 (06) ◽  
pp. 947-961
Author(s):  
A. LESFARI
Keyword(s):  
Dynamical System ◽  
Gauge Group ◽  
Integrable System ◽  
Double Cover ◽  
Abelian Surface ◽  
Invariant Surface ◽  
Yang Mills ◽  
Completely Integrable ◽  
Genus 2 ◽  
Constants Of Motion

In this paper, we consider a dynamical system related to the Yang–Mills system for a field with gauge group SU(2). We solve this system in terms of genus two hyperelliptic functions. The corresponding invariant surface defined by the two constants of motion can be completed as a cyclic double cover of an abelian surface (the jacobian of a genus 2 curve) and we show that this system is algebraic completely integrable in the generalized sense. Also we show that this system is part of an algebraic completely integrable system in five unknowns having three constants of motion.

scholarly journals Symmetries of the self-dual Yang-Mills equations

1994 ◽  
Vol 49 (1) ◽  
pp. 151-158
Author(s):  
Rod Halburd
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Gauge Group ◽  
Finite Dimension ◽  
The Self ◽  
Partial Differential ◽  
Yang Mills ◽  
Master System ◽  
Group Of Symmetries ◽  
Conformal Symmetries

It has been conjectured by R. S. Ward that the self-dual Yang-Mills Equations (SDYMEs) form a “master system” in the sense that most known integrable ordinary and partial differential equations are obtainable as reductions. We systematically construct the group of symmetries of the SDYMEs on R4 with semisimple gauge group of finite dimension and show that this yields only the well known gauge and conformal symmetries.

ALGEBRAIC SOLUTIONS OF YANG-MILLS-HIGGS EQUATIONS

1995 ◽  
Vol 10 (11) ◽  
pp. 925-930
Author(s):  
Y. BRIHAYE ◽  
STEFAN GILLER ◽  
PIOTR KOSINSKI
Keyword(s):  
Numerical Analysis ◽  
Irreducible Representation ◽  
Differential Equations ◽  
Gauge Group ◽  
Higgs Field ◽  
Regular Solutions ◽  
Yang Mills ◽  
Spatially Homogeneous ◽  
Classical Fields ◽  
Algebraic Solutions

We study the SU(2) Yang-Mills-Higgs equations with the Higgs field in an arbitrary irreducible representation of the gauge group. We propose an ansatz for the classical fields which, solving the classical equations, leads to systems of coupled differential equations. Several regular solutions of these systems can be constructed explicitly; they are spatially homogeneous and periodic in time. The connection between our solutions and a recent numerical analysis of sphaleron (monopole)’s evolution is discussed.

scholarly journals Systèmes dynamiques algébriquement complètement intégrables et géométrie

2015 ◽  
Vol 53 (1) ◽  
pp. 109-136 ◽  
Author(s):  
A. Lesfari
Keyword(s):  
Integrable Systems ◽  
Abelian Integrals ◽  
Completely Integrable Systems ◽  
Yang Mills ◽  
Completely Integrable ◽  
Algebraic Tori ◽  
Straight Line ◽  
Integrable Dynamical Systems ◽  
Basic Ideas ◽  
Affine Part

Résumé In this paper I present the basic ideas and properties of the complex algebraic completely integrable dynamical systems. These are integrable systems whose trajectories are straight line motions on complex algebraic tori (abelian varieties). We make, via the Kowalewski-Painlevé analysis, a detailed study of the level manifolds of the system. These manifolds are described explicitly as being affine part of complex algebraic tori and the flow can be solved by quadrature, that is to say their solutions can be expressed in terms of abelian integrals. The Adler-van Moerbeke method’s which will be used is primarily analytical but heavily inspired by algebraic geometrical methods. We will also discuss several examples of algebraic completely integrable systems : Kowalewski’s top, geodesic flow on SO(4), Hénon-Heiles system, Garnier potential, two coupled nonlinear Schrödinger equations and Yang-Mills system.

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