Self-dual Solutions Research Articles

Yang-Mills self-dual solutions and the conformal group contraction

We show that it is possible to use the idea of group contraction together with the self-dual solutions for a Yang-Mills theory, to obtain the number of parameters which characterize such classical configurations. To be precise, quantum fluctuations about a Yang-Mills pseudo-particle solution are studied from the point of view of the spontaneous breakdown of conformal symmetry. It is argued that the linearization procedure by which one passes from the non-linear Yang-Mills equation to the linear equations for the small oscillation modes leads to a group contraction of the conformal group SO(5, 1) and it is this which allows the number of parameters of the classical self-dual (or anti-self-dual) solutions to be counted. This argument is applied to the gauge group SU( n), where n ⩾ 2.

Algebraic construction of static axially symmetric self-dual gauge fields for an arbitrary group

A linear scattering problem associated with the static axially symmetric self-duality equation is introduced in a natural way using a symmetry argument. In conjunction with previously found symmetry (Bäcklund) transformations it provides an algebraic construction method of the self-dual solutions. This algebraic procedure which can be used to construct multi-monopoles is presented in some detail.

Self-dual Kerr–Schild metrics and null Maxwell fields

A construction for self-dual solutions of Einstein’s equations with metric in Kerr–Schild form from null electromagnetic fields in Minkowski space is described. Two examples are given.

Transition matrices of self-dual solutions to SU(N) gauge theory

We extract self-dual potentials from SU(N) transition matrices of the Atiyah—Ward type. The simplest nontrivial example is studied in detail and we find a topologically nontrivial, regular, self-dual solution. Different gauges can be found for which the corresponding gauge potentials are respectively time-independent or real.

General classical solutions of a supersymmetric non-linear coupled boson-fermion model in two dimensions

The supersymmetric generalization SCP n−1 of the bosonic CP n−1 model in two-dimensional euclidean space-time describes a coupled boson-fermion model with a quartic fermion self-coupling. The self-dual classical solutions are given in terms of holomorphic supercurves in the space of scalar superfields and the general classical solutions can be constructed explicitly by transformations on these supercurves. The corresponding boson and fermion solutions interpreted as classical c-number fields are displayed and we discuss various features such as action and topological charge.

One description of the self-dual solutions of the Einstein equation with a cosmological term

One description of the self-dual solutions of the Einstein equation with a cosmological term

Asymptotic High-Frequency Modes of Homogeneous Waveguide Structures with Impedance Boundaries

Homogeneous waveguides with both isotropic and anisotropic impedance boundaries are considered and asymptotic high-frequency mode properties are systematically derived. Among the new results are orthogonality properties of the asymptotic HF fields, existence of self-dual solutions, construction of stationary functionals, and an explicit formula for the calculation of the asymptotic attenuation coefficient for the general waveguide.

Ann monopole solution with 4n — 1 degrees of freedom

An exact static monopole solution, possessingn units of magnetic charge and (4n-1) degrees of freedom, is constructed, generalising the recent work of Ward on two monopole solutions. The equations solved are those of anSU(2) gauge theory with adjoint representation Higgs field in the (BPS) limit of vanishing Higgs potential. The number of degrees of freedom is maximal for self-dual solutions. The construction is described in a deductive way, within the framework of the Atiyah-Ward formalism for self-dual gauge fields.

Ansätze for self-dual Yang-Mills fields

A sequenceA1,A2, ... of ansatze for generating self-dual solutions of the Yang-Mills equations is presented. For eachn,An produces a solution depending on two arbitrary functions of three variables. As an application, we see thatA2 generates a static Yang-Mills-Higgs 2-monopole solution.

Green's functions associated with the edth operators

An integral representation for the inverses of the differential operators called edth is given in the two cases of particular interest. This operator plays a large role in the study ofH spaces and of self-dual solutions to the Yang-Mills field equations in the asymptotic domain.

A derivation of the Prasad-Sommerfield solution

A certain form for the self-dual solutions of the Yang-Mills-Higgs system is tested. (When restricted to spherical symmetry this ansatz is the most general form in the Wigner-Eckart sense.) It is found that the only consistent solutions of this form are spherically symmetric, and that the only finite energy solution then is the Prasad-Sommerfield solution (1975).

Non selfdual complex solutions with finite, complex euclidean actions for SU(2) Yang-Mills fields

A pair of complex conjugate solutions are presented for classical SU(2) gauge fields in four-dimensional flat euclidean space. The solutions are non selfdual and have finite complex actions.

Summable chains of instantons: Green's functions and the Prasad-Sommerfield limit

We construct, for each homotopy class, a type of instanton configuration which exhibits many special, simple properties. The basic reason behind such properties is indicated by deriving our configuration in the 't Hooft gauge, starting from a class of particularly simple, static, and self-dual solutions in de Sitter space. Apart from this we consider, in this paper, mostly results for flat Euclidean space. We show that the static solutions are equivalent to multiply charged instantons at the origin in Witten's sense. Green's functions for this class of instanton background are studied. The known flat-space results of Brown et al. are shown to be reducible, for our case, to totally explicit and relatively compact forms. The sums over different indices arising in their formalism are performed. The inversion of a matrix, necessary for the isospin-1 massless scalar field, is carried out explicitly, for our configuration, for arbitrary index of the background instanton field. Green's functions for the Prasad-Sommerfield case are obtained as limits of our results directly in summed-up forms. Green's functions are studied also in de Sitter space. Special features due to periodic time are pointed out.

Interpretation and further properties of general classical CPn−1 solutions

We present arguments suggesting that non-(anti) self-dual classical solutions to the equations of motion of the euclidean CP n −1 model can be interpreted as unstable non-interacting mixtures of instantons and anti-instantons. Fermionic modes in the background of these solutions are discussed. We determine the modes explicitly for the case of an embedded O(3) solution and point out that they give rise to a non-trivial illustration of the Atiyah-Singer index theorem.

Some twisted self-dual solutions for the Yang-Mills equations on a hypertorus

TheSU(N) Yang-Mills equations are considered in a four-dimensional Euclidean box with periodic boundary conditions (hypertorus). Gauge-invariant twists can be introduced in these boundary conditions, to be labeled with integersn μν (= −n μν ), defined moduloN. The Pontryagin number in this space is often fractional. Whenever this number is zero there are solutions to the equationsG μν =0 HereG μν is the covariant curl. When this number is not zero we find a set of solutions to the equations $$G_{\mu \nu } = \tilde G_{\mu \nu } $$ , provided that the periodsa μ of the box satisfy certain relations.

Exact, Fractionally Charged Self-Dual Solution

A finite-action self-dual solution of SU(2) gauge theory with topological charge 3/2 is given.

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 A 4 a plays the role of the Higgs field. In similar fashion, we make A 3 a A 4 a serve as the two required isovector fields in an ansatz independent of x 3 and x 4. 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 −2 π/ 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.

On pseudoparticle solutions in Yang's theory of gravity

Within the framework of differential geometry, Yang's parallel-displacement gauge theory is considered with respect to “pure” gravitational fields. In afour-dimensional Riemannian manifold it is shown that thedouble self-dual solutions obey Einstein's vacuum equations with the cosmological term, whereas the doubleanti-self-dual configurations satisfy the Rainich conditions of Wheeler'sgeometrodynamics. Conformal methods reveal that the gravitational analog of the “instanton” or pseudoparticle solution of Yang-Mills theory was already known to Riemann.

STATIC SPHERICALLY SYMMETRIC SELF-DUAL SOLUTIONS OF THE YANG-MILLS FIELD EQUATION AND 'THOOFT-POLYAKOV MONOPOLES DISTRIBUTED CONTINUOUSLY ON A SPHERE

In this paper, all synchrono spherosymmetrical statical self-dual solutions of Yang-Mills guage field have been obtained. From these solutions, a result on the continuous surface distribution of the magnetic charge is derived.

Constant Yang-Mills fields

We discuss self-dual solutions with constant potentials for Euclidean and Minkowskian Yang-Mills theories. We show that the action and energy of these solutions must vanish and give two explicit solutions with nonvanishing field-strength tensors. The solutions are not expected to be stable.