A connection between the Einstein and Yang-Mills equations

It is our purpose here to show an unusual relationship between the Einstein equations and the Yang-Mills equations. We give a correspondence between solutions of the self-dual Einstein vacuum equations and the self-dual Yang-Mills equations with a special choice of gauge group. The extension of the argument to the full Yang-Mills equations yields Einstein's unifield equations. We try to incorporate the full Einstein vacuum equations, but the approach is incomplete. We first consider Yang-Mills theory for an arbitrary Lie-algebra with the condition that the connection 1-form and curvature are constant on Minkowski space. This leads to a set of algebraic equations on the connection components. We then specialize the Lie-algebra to be the (infinite dimensional) Lie-algebra of a group of diffeomorphisms of some manifold. The algebraic equations then become differential equations for four vector fields on the manifold on which the diffeomorphisms act. In the self-dual case, if we choose the connection components from the Lie-algebra of the volume preserving 4-dimensional diffeomorphism group, the resulting equations are the same as those obtained by Ashtekar, Jacobsen and Smolin, in their remarkable simplification of the self-dual Einstein vacuum equations. (An alternative derivation of the same equations begins with the self-dual Yang-Mills connection now depending only on the time, then choosing the Lie algebra as that of the volume preserving 3-dimensional diffeomorphisms.) When the reduced full Yang-Mills equations are used in the same context, we get Einstein's equations for his unified theory based on absolute parallelism. To incorporate the full Einsteinvacuum equations we use as the Lie group the semi-direct product of the diffeomorphism group of a 4-dimensional manifold with the group of frame rotations of anSO(1, 3) bundle over the 4-manifold. This last approach, however, yields equations more general than the vacuum equations.Andrew Mellon Postdoctoral fellow and Fulbright ScholarSupported in part by NSF grant no. PHY 80023     

Clifford algebra structure and Einstein's equations

We point out that a Clifford algebra representation for the Riemannian curvature leads to an equation for gravity similar to the Yang-Mills equation, in a gauge model for gravity with the Lorentz group. Einstein's equation of general relativity emerges as a natural solution in this approach.     

Algebraic interpretation of the Yang-Mills field

Einstein's principle of general relativity is a dynamical-group approach in that all dynamics is implied by the invariance and no force is introduced (as an external, symmetry-breaking factor). In this spirit we take a Poincaré-invariant free wave equation and, deforming the Poincaré group to the de Sitter group, obtain interaction. This illustrates our algebraic approach to gauge invariance, whereby the (generalized) Maxwell tensor of the Yang-Mills field appears as structure constants of the homogeneous algebra obtained as a deformation of an inhomogeneous one, with interaction appearing via the same tensor, which plays a role corresponding to the curvature tensor in Einstein's general relativity.     

Superholomorphic structures,moment maps,and super self-dual Yang-Mills connections over super Riemann surfaces

We consider the space of superconnections with certain curvature constraints over super Riemann surfaces. We define a moment map over that space to the dual of the super Lie algebra of gauge transformations. The zero set of this moment map corresponds to the super self-dual Yang-Mills equations in two dimensions. This result generalizes the recently proposed scheme for the nonsupersymmetric case. The superfield equations also arise from super self-dual Yang-Mills equations in four dimensions by dimensional reduction.     

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 Infinitesimal Symmetries of the Self-Dual Yang-Mills Equations

Abstract

Infinite-dimensional algebra of all infinitesimal transformations of solutions of the self-dual Yang-Mills equations is described. It contains as subalgebras the infinite-dimensional algebras of hidden symmetries related to gauge and conformal transformations.     

Reduction of 4-dim self-dual super Yang-Mills equations onto super Riemann surfaces

Recently, a self-dual super Yang-Mills equation over a super Reimann surface was obtained as the zero set of a moment map on the space of superconnections to the dual of the super Lie algebra of gauge transformations. We present a new formulation of the 4-dim Euclidean self-dual super Yang-Mills equations in terms of constraints on the supercurvature. By dimensional reduction, we obtain the same set of superconformal field equations which define self-dual connections on a super Rieman surface.     

Gravitation and unified covariant formalism of classical gauge fields in equiaffine space

In this paper, we construct a unified covariant formalism for the classical gauge fields in an equiaffine space. The gauge transformation groups are the Lie groups, induced according to the third Lie theorem by the structure constants. As a result of the gauge transformations, one set of geometric objects is replaced by another. It is confirmed that the differential conservation laws in the equiaffine spaces are a result of the equations of the gauge fields. The particular case when the gauge transformation group is a four-parameter group and is abelian is distinguished. This group corresponds to gauge fields that are induced by an energy-momentum tensor and, which, as a result, are called gravitational fields. As a particular case of the equations of the given gravitational fields, we obtain Einstein's equations with the help of a Lagrangian, which is quadratic with respect to the gravitational field intensities. In concluding, we note the possibility of describing gauge fields, corresponding to nongravitational interactions of vector mesons with nonzero rest mass, without invoking the scalar Higgs mesons. This possibility appears both as a result of the generalization of the Yang-Mills covariant derivative and as a result of including gravitational interactions in the general gauge field formalism.Translated from Izvestiya Vysshykh Uchebnykh Zavedenii, Fizika, No. 12, 47–51, December, 1981.     

Comments on Yang-Mills thermodynamics: The Hagedorn spectrum and the gluon gas pictures for a generic gauge algebra

We discuss the dependence of pure Yang-Mills equation of state on the choice of gauge algebra. In the confined phase, we generalize to an arbitrary simple gauge algebra Meyer?s proposal of modeling the Yang-Mills matter by an ideal glueball gas in which the high-lying glueball spectrum is approximated by a Hagedorn spectrum of closed-bosonic-string type. Such a formalism is undefined above the Hagedorn temperature, corresponding to the phase transition toward a deconfined state of matter in which gluons are the relevant degrees of freedom. Under the assumption that the renormalization scale of the running coupling is gauge-algebra independent, we discuss about how the behavior of thermodynamical quantities such as the trace anomaly should depend on the gauge algebra in both the confined and deconfined phase. The obtained results compare favorably with recent and accurate lattice data in the su(3) case and support the idea that the more the gauge algebra has generators, the more the phase transition is of first-order type.     

Equations of motion of interacting massless fields of all spins as a free differential algebra

It is argued that the equations of motion of interacting massless fields of all spins s=0,1,…,∞ can naturally be formulated in terms of a free differential algebra (FDA) constructed from one-forms and zero-forms that belong both to the adjoint representation of the infinite-dimensional superalgebra of higher spins and auxiliary fields proposed previously. This FDA is found explicitly in the first non-trivial order in the zero-forms. Various properties of the proposed FDA are discussed including the ways for incorporating internal (Yang-Mills) gauge symmetries via associative algebras.     

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.     

Lagrangian of the N = 1 supersymmetric Yang-Mills theory in the formalism of unified field

A method is proposed to use matrix nonlinear equations in order to construct the Lagrangian of the N = 1 supersymmetric Yang-Mills theory and the algebra of generators of transformations, closed off-shell in the 1.5-order formalism. Contrary to the superfield approach, there is no need to impose a Wess—Zumino type gauge fixing.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 16–21, September 1989.     

Weyl group, <Emphasis Type="Italic">CP</Emphasis> and the kink-like field configurations in the effective <Emphasis Type="Italic">SU</Emphasis>(3) gauge theory

Effective Lagrangian for Yang-Mills gauge fields invariant under the standard space-time and local gauge SU(3) transformations is considered. It is demonstrated that a set of twelve degenerated minima exists as soon as a nonzero gluon condensate is postulated. The minima are connected to each other by the parity transformations and Weyl group transformations associated with the color su(3) algebra. The presence of degenerated discrete minima in the effective potential leads to the solutions of the effective Euclidean equations of motion in the form of the kink-like gauge field configurations interpolating between different minima. Spectrum of charged scalar field in the kink background is discussed.     

BRS gauge and ghost field supersymmetry in gravity and supergravity

Becchi-Rouet-Stora transformations are obtained for the following systems: (i) Pure Einstein gravity in first order form with vierbein and spin connection as independent fields. (ii) First order Einstein gravity coupled to Yang-Mills fields. (iii) Pure supergravity. For the first two systems the results are as in Yang-Mills theory. But for conventional supergravity the BRS transformations leave the effective action invariant only if the classical equations of motion are satisfied. New transformations of the gauge fields of supergravity have been proposed under which the supersymmetry algebra closes. The corresponding BRS transformations do leave the effective action invariant without the need to use the classical equation of motion; moreover, as in Yang-Mills theories, they are nilpotent and have unit Jacobian.     

Self-dual Yang-Mills fields in d=7,8, octonions and ward equations

The Yang-Mills theories in d=7 and d=8 with the arbitrary gauge group G are considered. Generalized self-duality-type relations for gauge fields are reduced to systems of nonlinear differential equations on functions of one variable (Ward equations). Ward equations may be reduced to equations which follow from Yang-Baxter equations. This permits us to obtain new classes of explicit solutions of the Yang-Mills equations in d=7 and d=8.     

Uniformly valid amplitude expansions for non-Abelian gauge theories

The imposition of a particular gauge condition in a non-Abelian gauge theory may lead to large-r nonuniformities in an amplitude expansion when used in the construction of an approximate solution of the field equations of the theory. We show that for both the Yang-Mills theory and general relativity it is always possible to find a class of gauge conditions that do not suffer from this defect and at the same time lead to solvable equations in each order of the approximation employed. We further show how one can construct such gauge conditions by a method similar to Lighthill's method of strained coordinates. An example of such a construction is given for the Yang-Mills theory.     

Symmetry and bifurcations of momentum mappings

The zero set of a momentum mapping is shown to have a singularity at each point with symmetry. The zero set is diffeomorphic to the product of a manifold and the zero set of a homogeneous quadratic function. The proof uses the Kuranishi theory of deformations. Among the applications, it is shown that the set of all solutions of the Yang-Mills equations on a Lorentz manifold has a singularity at any solution with symmetry, in the sense of a pure gauge symmetry. Similarly, the set of solutions of Einstein's equations has a singularity at any solution that has spacelike Killing fields, provided the spacetime has a compact Cauchy surface.This work was partially supported by the National Science Foundation. The second author was supported by a Killam Visiting fellowship at the University of Calgary during the completion of the paper     

Witten's gauge field equations and an infinite-dimensional Grassmann manifold

Witten's gauge fields are interpreted as motions on an infinite-dimensional Grassmann manifold. Unlike the case of self-dual Yang-Mills equations in Takasaki's work, the initial data must satisfy a system of differential equations since Witten's equations comprise a pair of spectral parameters. Solutions corresponding to (anti-) self-dual Yang-Mills fields are characterized in the space of initial data and in application, some Yang-Mills fields which are not self-dual, anti-self-dual nor abelian can be constructed.     

A new symmetry group of real self-dual Yang-Mills theory

A new infinite parameter symmetry group is found for real self-dual Yang-Mills theory in four euclidean dimensions. Whereas the gauge potentials transform under a group including local gauge transformations and Kac-Moody-like transformations, the gauge invariant object tr P exp(∮A·dξ) is seen to carry a representation of the Kac-Moody symmetry. Four-dimensional Polyakov loop-space currents restricted to the self-dual sector are constructed from this algebra.