Integration method for four-dimensional self-duality equations of Yang-Mills fields

Institute of Mathematics and Mechanics, Azerbaijan Academy of Sciences. Presented at the International Workshop Squeezing, Groups, and Quantum Mechanics, Baku, Azerbaijan, September 16–22, 1991.     

Solution and ellipticity properties of the self-duality equations of Corriganet al. in eight dimensions

We study the two sets of self-dual Yang-Mills equations in eight dimensions proposed in 1983 by E. Corriganet at. and show that one of these sets forms an elliptic system under the Coulomb gauge condition, and the other (overdetermined) set can have solutions that depend at most onN arbitrary constants, whereN is the dimension of the gauge group, hence the global solutions of both systems are finite dimensional. We describe a subvarietyP ₈ of the skew-symmetric 8 x 8 matrices by an eigenvalue criterion and we show that the solutions of the elliptic equations of Corriganet al. are among the maximal linear submanifolds ofP ₈. We propose an eighth-order action for which the elliptic set is a maximum.     

Solutions of gravitational field equations

In this paper we use iterative methods to generate series solutions of the gravitational field equations in a cosmological model with heat flow.     

Solutions of coupled KdV-type equations

We consider a family of coupled (Ito type) KdV equations in 1+1 dimensions and use Hlavatý's technique to obtain a class of explicit wave-type solutions.     

Kosterlitz-thouless transition and self-duality in three dimensions

A class of self-dual globally symmetric Z_N models in three dimensions is presented. The limit N → ∞ is a type of anisotropic U(1) model (XY model) dual to a gas of integer point charges, interacting via a logarithmic potential in three dimensions. The latter is, at low temperature, nothing but a sine-Gordon theory with an anisotropic, logarithmic propagator. It therefore has a low-temperature Kosterlitz-Thouless phase and KT phase transition to a massive phase.The relation of the U(1) model to the thermodynamics of a helical magnet along the ferromagnetic-helical phase boundary in zero applied field (or to the smectic A to amectic C phase boundary in a liquid crystal) is indicated.     

Chirality,self-duality,and supergravity counterterms

The restrictions imposed by chirality invariance on higher-loop counterterms in supergravity are obtained. It is shown that their dependence on gravitational curvature and on spin-$\text{3}\text{2}$ field strength is such that they vanish when these quantities are self-or anti-self-dual. Implications regarding quantum corrections in the instanton sector are discussed.     

Actions for non-abelian twisted self-duality

The dynamics of abelian vector and antisymmetric tensor gauge fields can be described in terms of twisted self-duality equations. These first-order equations relate the p-form fields to their dual forms by demanding that their respective field strengths are dual to each other. It is well known that such equations can be integrated to a local action that carries on equal footing the p-forms together with their duals and is manifestly duality invariant. Space-time covariance is no longer manifest but still present with a non-standard realization of space-time diffeomorphisms on the gauge fields. In this paper, we give a non-abelian generalization of this first-order action by gauging part of its global symmetries. The resulting field equations are non-abelian versions of the twisted self-duality equations. A key element in the construction is the introduction of proper couplings to higher-rank tensor fields. We discuss possible applications (to Yang-Mills and supergravity theories) and comment on the relation to previous no-go theorems.     

Solutions to generalized double sine-Gordon equations

A non-linear differential equation, which encompasses the sine-Gordon and double sine-Gordon equations as special cases, is presented and a solution given in terms of Weierstrass' $\text{P}$ functions. The equation is time independent, but a Lorentz transformation may in the usual way produce translational solutions. The special case of the double sine-Gordon equation is considered in some detail.     

Solutions of Hartree-Fock equations for Coulomb systems

This paper deals with the existence of multiple solutions of Hartree-Fock equations for Coulomb systems and related equations such as the Thomas-Fermi-Dirac-Von Weizäcker equation.Partially supported by CEA-DAM     

Solutions of the exact time-dependent microbending equations

Bandwidth improvement factor and losses of two-dimensional fibers with power-law index profiles, submitted to microbends with power-law curvature spectra, are given here in a graphical form. The domain of validity of the analytical solutions is discussed.     

Hyper-Kähler induction and self-duality on Riemann surfaces

Universal hyper-Kähler spaces are constructed from Lie groups acting on flat Kähler manifolds. These spaces are used to describe the moduli space of solutions of Hitchin's equation — self-duality equations on a Riemann surface — as the contangent bundle of the moduli space of flat connections on a Riemann surface.     

The Kirkwood-Salsburg equations: Solutions and spectral properties

It is shown that the Kirkwood-Salsburg equations for a classical lattice gas are equivalent to the Dobrushin-Lanford-Ruelle equilibrium equations. The term “Kirkwood-Salsburg equations” is used here in a restricted sense, and thus the known result for a larger system of equations is improved (see Table 1). Some information on the spectrum of the Kirkwood-Salsburg operator is found in connection with zeros of partition functions. An example is given to show that the Kirkwood-Salsburg equations can have other solutions than states in the space of uniformly bounded correlation functions.     

Massive self-duality solution associated with invariant one-forms

A massive self-duality solution associated with invariant 1-forms is presented. At the zero mass limit the massive self-dual theory of the SO(3) gauge group on 4 dimensions cannot be reduced to that of massless self-duality.In such a case the self-dual connection turns to the flat connection and one cannot obtain a massless theory in such an approach.     

Solutions of the Maxwell equations and photon wave functions

Properties of six-component electromagnetic field solutions of a matrix form of the Maxwell equations, analogous to the four-component solutions of the Dirac equation, are described. It is shown that the six-component equation, including sources, is invariant under Lorentz transformations. Complete sets of eigenfunctions of the Hamiltonian for the electromagnetic fields, which may be interpreted as photon wave functions, are given both for plane waves and for angular-momentum eigenstates. Rotationally invariant projection operators are used to identify transverse or longitudinal electric and magnetic fields. For plane waves, the velocity transformed transverse wave functions are also transverse, and the velocity transformed longitudinal wave functions include both longitudinal and transverse components. A suitable sum over these eigenfunctions provides a Green function for the matrix Maxwell equation, which can be expressed in the same covariant form as the Green function for the Dirac equation. Radiation from a dipole source and from a Dirac atomic transition current are calculated to illustrate applications of the Maxwell Green function.     

Solutions of the Einstein equations with zero coupling

A study is made of the algebraic properties and geometric structure of solutions of the Einstein equations the metric tensors of which differ by the product of two identical isotropic vectors. Proof is offered for a theorem which states that when the congruence of isotropic lines with a tangent vector field used for the coupling is geodesic, both spaces are algebraically special in the Petrov-Penrose sense. A noncoordinate transformation of this type can be used to find new exact solutions from known solutions.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, No. 10, pp. 32–37, October, 1971.The author thanks Professor V. I. Rodichev for interest in the study and for valuable discussions.     

Solutions of Klein-Gordon and Dirac equations on quantum Minkowski spaces

Covariant differential calculi and exterior algebras on quantum homogeneous spaces endowed with the action of inhomogeneous quantum groups are classified. In the case of quantum Minkowski spaces they have the same dimensions as in the classical case. Formal solutions of the corresponding Klein-Gordon and Dirac equations are found. The Fock space construction is sketched.This research was supported in part by NSF grant DMS-9508597 and in part by Polish KBN grant No. 2 P301 02007.     

Solutions of nonlinear equations simulating pair production and pair annihilation

Exact solutions of nonlinear evolution equations are constructed which correspond to a swelling lump which then divides itself into two (or more) solitary waves moving in opposite directions or, conversely, two (or more) solitary waves coming together and annihilating each other with an exponential decay of the amplitude for large times. A particle motion can be associated to the phenomenon.