Asymptotic behaviour of classical Yang-Mills fields in Minkowski space

We show how to define incoming and outgoing asymptotic fields for classical solutions of the Yang-Mills field equations without fixing the gauge. It is then seen that the Gribov ambiguities for putting the field in the Coulomb gauge reduce asymptotically to a field-independent, infinite parameter group of gauge transformations. This obscures the notion of color charge already at the classical level.     

Yang-Mills fields which are not self-dual

The purpose of this paper is to prove the existence of a new family of non-self-dual finite-energy solutions to the Yang-Mills equations on Euclidean four-space, withSU(2) as a gauge group. The approach is that of equivariant geometry: attention is restricted to a special class of fields, those that satisfy a certain kind of rotational symmetry, for which it is proved that (1) a solution to the Yang-Mills equations exists among them; and (2) no solution to the self-duality equations exists among them. The first assertion is proved by an application of the direct method of the calculus of variations (existence and regularity of minimizers), and the second assertion by studying the symmetry properties of the linearized self-duality equations. The same technique yields a new family of non-self-dual solutions on the complex projective plane.     

Permutability property for self-dual Yang-Mills fields

A permutability property for Bäcklund transformations of the self-dual SU(2) Yang-Mills fields is shown to exist. We give a superposition-type formula, whose iteration permits the simple algebraic construction of solutions of the self-dual Yang-Mills equations.This Letter has been authored under contract DE-AC02-76H00016 with the U.S. Department of Energy.     

Ansätze for self-dual Yang-Mills fields

A sequenceA ₁,A ₂, ... of ansätze for generating self-dual solutions of the Yang-Mills equations is presented. For eachn,A _n produces a solution depending on two arbitrary functions of three variables. As an application, we see thatA ₂ generates a static Yang-Mills-Higgs 2-monopole solution.     

Finite energy solutions for interacting Yang-Mills and Dirac fields on Minkowski space

The Lagrangian for a SU(2) Yang-Mills field interacting with a massless isospin 1/2 Dirac field is conformally invariant. Finite energy solutions are obtained by a conformal mapping of Minkowski space onto the compact manifoldS ¹×S³ with pseudo-Riemannian metric. They are symmetric with respect to the isometric group SO(2)×SO(4) ofS ¹×S³.     

Spinor analysis of Yang-Mills theory in the Minkowski space

Self-duality equations for Yang-Mills fields and the Dirac equation with an external (anti-) selfdual gauge field are studied in the Minkowski space by spinorial methods. For the Dirac equations, all (four) possible combinations of the fermion chirality and duality of the external fields are considered.     

Regularity of constraints in the Minkowski space Yang-Mills theory

For Yang-Mills theory in the Minkowski space it is proved that the constraint set is a smooth submanifold of the phase space consisting of square integrable Cauchy data.Partially supported by the NSERC Operating Grant No. A8091     

Some global solutions of the Yang-Mills equations in Minkowski space

A class of global solutions of the Yang-Mills equations whose Cauchy data depend on a pair of arbitrary functions is constructed. The asymptotic propagation of the energy in space-time is studied. The same results are valid if the Yang-Mills field is coupled to a scalar field.Research supported in part by NSF Grants MCS 77-01340 and MCS 79-01965     

Non-local continuity equations for self-dual SU(N) Yang-Mills fields

Using a manifestly gauge invariant formulation, an infinite set of classical non-local continuity equations is constructed for the self-dual SU(N) Yang-Mills fields.     

Local stability of deformations of self-dual Yang-Mills fields on S4

We prove that local deformations of irreducible self-dual Yang-Mills fields on S⁴ in the space of smooth soluions not assumed to be self-dual are indeed self-dual.     

Stability of Yang-Mills fields system in the homogeneous (anti)self-dual background field

Stability of Yang-Mills fields system in the background field is investigated based on the Toda criterion, Poincare sections, and the values of the maximal Lyapunov exponents. The existence of the region of regular motion at low densities of energy is demonstrated. Critical energy density of the order-chaos transition is analyzed for different values of the model parameter. The text was submitted by the authors in English.     

Frame-like actions for massless mixed-symmetry fields in Minkowski space

A frame-like action for arbitrary mixed-symmetry bosonic massless fields in Minkowski space is constructed. The action is given in a simple form and consists of two terms for a field of any spin. The fields and gauge parameters are certain tensor-valued differential forms. The formulation is based on the unfolded form of equations for mixed-symmetry fields.     

A universal phase space for particles in Yang-Mills fields

Given a principal G-bundle P over M and a Hamiltonian G-space Q, one may construct the reduced symplectic manifold (T*P x Q)₀. When a connection on P is chosen, this manifold becomes a bundle over T*M with fibre Q. It is shown that this bundle is precisely the phase space constructed by Sternberg for a classical particle in a Yang-Mills field.Research partially supported by NSF Grant MCS 74-23180.A01.     

Spinorial linear system for self-dual Yang-Mills equations

A conformally covariant linear system acting on a spinor (instead of a scalar) function is used to construct infinite series of nontrivially conserved currents for the restriction of self-dual Yang-Mills to the so-called A (1) set, or't Hooft ansatz.     

A string-like self-dual solution of Yang-Mills theory

We attempt to obtain Nielsen-Olesen strings in pure Yang-Mills theory without Higgs scalars. This is inspired by recent work interpreting the Prasad-Sommerfield monopole as a static self-dual Yang-Mills solution in which A₄^a plays the role of the Higgs field. In similar fashion, we make A₃^aA₄^a serve as the two required isovector fields in an ansatz independent of x₃ and x₄. The condition of self-duality results in a single Painlevé equation of the third kind (or equivalently, a radial sinh-Gordon equation in 2 + 0 dimensions), the solution of which determines A_μ^a. We make use of the extensive analysis of the former equation carried out by Wu, McCoy and collaborators in the context of the scaling limit of the two-dimensional Ising model. Their simplest solution yields a flux value of ?2π/e just as in the Nielsen-Olesen model and flux is quantized in multiples of this unit. The string tension (action per unit time per unit distance) diverges as r^?2 (In r)^?2 as r → 0 for the same Wu-McCoy solution.     

Non-commutative Euclidean and Minkowski structures

A noncommutative *-algebra that generalizes the canonical commutation relations and that is covariant under the quantum groups SO _q (3) or SO _q(1, 3) is introduced. The generating elements of this algebra are hermitean and can be identified with coordinates, momenta and angular momenta. In addition a unitary scaling operator is part of the algebra.     

Improved Hamiltonian for Minkowski Yang-Mills theory

I develop an improved Hamiltonian for classical, Minkowski Yang-Mills theory, which evolves infrared fields with tree level corrections from lattice spacing a beginning at O(a⁴). I use it to investigate the response of Chern-Simons number to a chemical potential, and to compute the maximal Lyapunov exponent. The Lyapunov exponent has a small a limit, and the Chern-Simons number response appears to be approaching one at the finest lattices considered. In both cases the limit is within 10% of the limit found using the unimproved (Kogut-Susskind) Hamiltonian. For the maximal Lyapunov exponent the limits differ between Hamiltonians by about 5%, significant at about 5σ, indicating that while a small a limit exists, its value depends on the specifics of the lattice cutoff. For Chern-Simons number the difference between Hamiltonians is within statistical errors of about 10%, which constitutes an upper bound on the lattice regulation dependence.     

Green functions in a super self-dual Yang-Mills background

In euclidean supersymmetric theories of chiral superfields and vector superfields coupled to a super-self-dual Yang-Mills background, we define Green functions for the Laplace-type differential operators which are obtained from the quadratic part of the action. These Green functions are expressed in terms of the Green function on the space of right chiral superfields, and an explicit expression for the right chiral Green function in the fundamental representation of an SU(n) gauge group is presented using the supersymmetric version of the ADHM formalism. The superfield kernels associated with the Laplace-type operators are used to obtain the one-loop quantum corrections to the super-self-dual Yang-Mills action, and also to provide a superfield version of the super-index theorems for the components of chiral superfields in a self-dual background.     

Affine Yang-Mills theory and affine self-dual Chern-Simons Solitons

A four-dimensional affine Yang-Mills theory, i.e. Yang—Mills gauge theory with values in an affine Kac-Moody algebra, is constructed which can give rise to a spontaneous breakdown of the affine symmetry. The affine self-dual Yang-Mills equation (which is a special kind of affine YM theory) in four dimensions is dimensionally reduced to the affine self-dual Chem-Simons equation in two dimensions. The latter is shown to have soliton solutions which satisfy the conformal affine Toda equations.K. C. Wong Research Award Winner; address after 28 October 1992: Department of Mathematics, University of Queensland, Brisbane, Queensland 4072, Australia.