On the complete integrability of the static axially symmetric self-dual gauge field equations for an arbitrary group

We present an analysis of static axially symmetric gauge fields for an arbitrary gauge group G. Two ansätze are considered. The full ansatz involves a total of 4d(d = dim G), the reduced ansatz only 2d functions of (?, z). Imposing self-duality is shown to reduce the problem to a sigma model in the curved two-dimensional (?, z) space over the coset spaces G?/G for the full, and G¹/K for the reduced ansatz. G? is the complexification of G. ¹ is a particular non-compact form of G, and K the local form-preserving symmetry group of the reduced ansatz. We give explicitly the Lax-pair type representations (linear scattering problem) of the sigma model, indicating that the standard methods available for certain non-linear two-dimensional problems can be used to generate solutions. Our procedure has the advantage that only real fields over a real manifold enter the analysis.     

On the Atiyah-Drinfeld-Hitchin-Manin construction for self-dual gauge fields

The vector spacesA, B, C, in terms of which the general construction due to Atiyah, Drinfeld, Hitchin and Manin for self-dual gauge fields defined over some region of Euclidean space is phrased, are shown to be expressible in terms of the spaces spanned by the solutions of certain linear covariant differential equations depending on the gauge field. The corresponding linear maps betweenA andB, B andC are given with the properties required by ADHM and the results then necessary to verify the construction informally proved. The local problems associated with assuming the gauge field to obey the self-duality equations are separated from the global problems of assuring the required boundary conditions for a particular solution. With suitable global conditionsC is shown to be the dual ofA and a natural scalar product defined onB so as to reconstruct the gauge field in the standard form given by the construction. A discussion is given of the requirements entailed by the condition of a symmetry on the gauge field and the relation to the usual cohomological treatment is outlined in an appendix.On leave of absence from DAMTP, Silver Street, Cambridge, England     

Factorizations for self-dual gauge fields

For a particular class of patching matrices onP ₃(?), including those for the complex instanton bundles with structure group Sp(k,?) orO(2k,?), we show that the associated Riemann-Hilbert problemG(x, λ)=G?(x, λ)·G ₊ ^?1 (x, λ) can be generically solved in the factored formG _?=φ ₁ φ ₂.....φ _n . IfГ=Г _n is the potential generated in the usual way fromG _?, and we setψ _i =φ ₁.....,φ _i withψ _n =G _?, then eachψ _i also generates a selfdual gauge potentialΓ _i . The potentials are connected via the “dressing transformations” $$\Gamma _\iota = \phi _i^{ - 1} \cdot \Gamma _{\iota - 1} \cdot \phi _i + \phi _i ^{ - 1} D\phi _i$$ of Zakharov-Shabat. The factorization is not unique; it depends on the (arbitrary) ordering of the poles of the patching matrix.     

Spherically symmetric equations in gauge theories for an arbitrary semisimple compact lie group

An explicit construction of spherically symmetric equations (not only static and/or self-dual) in gauge theories for the minimal embedding of SU(2) in an arbitrary semisimple compact Lie group G is given. The final equations are written in a form containing only gauge invariant quantities in R₂. The whole group structure is concentrated in the only matrix, which is directly related to the Cartan matrix of G. In particular, the developed technique allows to generalize the Witten duality equation [1] and to obtain the spectrum of pointlike solutions in G.     

On a construction of self-dual gauge fields in seven dimensions

Abstract

We consider gauge fields associated with a semisimple Malcev algebra. We construct a gauge-invariant Lagrangian and found a solution of modified Yang-Mills equations in seven dimensions.     

On static,axially symmetric Einstein-Maxwell fields. Part I

The Weyl solution of the problem, obtained on the assumption thatg ₀₀ is a function of the electrostatic potential, is varied and the linearized field equations for the variation are discussed. The complete solution of the problem is determined for the special case of the Weyl solution that generalizes the Reissner-Nordstrom solution withm = ¦e¦.     

On static,axially symmetric Einstein-Maxwell fields. Part II

Starting from the Reissner-Nordström solution withm =e we consider a variation representing a second particle situated outside the horizon. A formal dipole term in the potential of the second particle ensures equilibrium without additional stresses between the particles. The complete solution for the variation is determined and discussed in detail.     

Semiclassical functional integrals for self-dual gauge fields

The semiclassical approximation to the functional integral for four-dimensional Euclidean gauge theories is discussed in detail for general stationary points of the action. It is shown how to take the limit from a compact space to flat space, and the zero modes corresponding to global gauge transformations are carefully discussed. The results are specialised to general self-dual multi-instanton gauge fields given by the general construction of Atiyah et al. It is shown how the normalization matrix of the zero modes can be determined and the complete expression for the functional measure is given for the two instanton case. This is shown to factorise for well-separated instantons. Some technical matters are discussed in an appendix and a resume of results for multi-instanton functional determinants is included.     

Bäcklund transformations for static axially-symmetric self-dual SU(N) gauge fields

Bäcklund transformations are derived for static axially-symmetric self-dual SU(N) gauge fields. In the case of SU(3) broken to U(1) × U(1), they produce field configurations with fractional topological charges.     

On symmetric gauge fields

The subgroups of the symmetry group of the gauge invariant Lagrangian are studied. For given subgroupG theG-invariant gauge fields are listed.     

Stationary axially symmetric Brans-Dicke-Maxwell fields

A method is presented which enables one to obtain solutions to the stationary axially symmetric Brans-Dicke fields coupled to source-free Maxwell fields from the solutions of the Einstein-Maxwell equations in Einstein's theory. The Brans-Dicke analog of the Kerr-Newman solution has been obtained as an example.     

The radiation gauge static potential for non-abelian gauge fields

The static potential between a fermion and an anti-fermion in a group singlet state is calculated, through two loops, in the radiation gauge first order formalism. The results of this calculation imply that the Coulomb propagator is not sufficient to determine the static potential: a new function of the coupling constant α_s(?t) is also required.     

A new construction technique for exact axially symmetric static electrovac solutions of the Einstein-Maxwell equations

Coordinates adapted to the full electrostatic symmetries of a charged axially symmetric static system are constructed and applied to the coupled Einstein-Maxwell equations. A new solution is presented and the framework for general solutions is developed. The Weyl solutions g₀₀(Φ) are readily extracted from the formalism.     

Non-existence of axially symmetric massive scalar fields

It has been observed that for the axially symmetric Einstein-Rosen metric, the stress-tensor of a scalar meson field associated with meson of rest mass cannot be the source term for generating gravitation. The above result also holds even when this meson field is coupled with an electromagnetic field.     

On the spherically symmetric gauge fields

The spherically symmetric gauge fields with a compact gauge group over 4-dimensional Minkowski space are determined completely. Expressions for the gauge potentials of these fields are obtained.     

Exact monopole solutions in gauge theories for an arbitrary semisimple compact group

We construct an exact n-parametric monopole and dyon solutions for an arbitrary compact gauge group G of rank n by using the symmetry between cylindrically symmetric instanton equations in Euclidean space R ₄ and monopole equations in Minkowski space R _3,1 (with Higgs scalar field in adjoint representation). The solutions are spherically symmetric with respect to the total momentum operator represents the minimal embedding of SU(2) in G. Explicit expressions for the monopole magnetic charge and mass matrices are obtained. The remarkable aspect of our results is the existence of discrete series of the monopole solutions, which are labelled by n quantum numbers and degenerated in the latter ones at a fixed monopole mass matrix.     

The linear system for self-dual gauge fields in a spacetime of signature 0

The overdetermined linear system for the self-dual Yang—Mills (SDYM) equations is examined in a flat four-dimensional space whose metric has signature 0. There are three different domains for the system, and correspondingly three (essentially) different solutions to the linear system for a given gauge field. If the gauge potential is real analytic, two of the solutions patch together to give a holomorphic function in an annular region of projective twistor space. Conversely, an arbitrary holomorphic GL(n, )-valued function in such a domain can be uniquely factored (on the real lines) to give a solution to SDYM with gauge group U(n). The set of all real analytic u(n)-valued gauge fields can thus be parametrized by the points of a certain double coset space.