Unified gravitational and Yang-Mills fields

We unify the gravitational and Yang-Mills fields by extending the diffeomorphisms in (N=4+n)-dimensional space-time to a larger group, called the conservation group. This is the largest group of coordinate transformations under which conservation laws are covariant statements. We present two theories that are invariant under the conservation group. Both theories have field equations that imply the validity of Einstein's equations for general relativity with the stress-energy tensor of a non-Abelian Yang-Mills field (with massive quanta) and associated currents. Both provide a geometrical foundation for string theory and admit solutions that describe the direct product of a compactn-dimensional space and flat four-dimensional space-time. One of the theories requires that the cosmological constant shall vanish. The conservation group symmetry is so large that there is reason to believe the theories are finite or renormalizable.     

Yang-Mills Detour Complexes and Conformal Geometry

Working over a pseudo-Riemannian manifold, for each vector bundle with connection we construct a sequence of three differential operators which is a complex (termed a Yang-Mills detour complex) if and only if the connection satisfies the full Yang-Mills equations. A special case is a complex controlling the deformation theory of Yang-Mills connections. In the case of Riemannian signature the complex is elliptic. If the connection respects a metric on the bundle then the complex is formally self-adjoint. In dimension 4 the complex is conformally invariant and generalises, to the full Yang-Mills setting, the composition of (two operator) Yang-Mills complexes for (anti-)self-dual Yang-Mills connections. Via a prolonged system and tractor connection a diagram of differential operators is constructed which, when commutative, generates differential complexes of natural operators from the Yang-Mills detour complex. In dimension 4 this construction is conformally invariant and is used to yield two new sequences of conformal operators which are complexes if and only if the Bach tensor vanishes everywhere. In Riemannian signature these complexes are elliptic. In one case the first operator is the twistor operator and in the other sequence it is the operator for Einstein scales. The sequences are detour sequences associated to certain Bernstein-Gelfand-Gelfand sequences.     

Curved Space-Times by Crystallization of Liquid Fiber Bundles

Motivated by the search for a Hamiltonian formulation of Einstein equations of gravity which depends in a minimal way on choices of coordinates, nor on a choice of gauge, we develop a multisymplectic formulation on the total space of the principal bundle of orthonormal frames on the 4-dimensional space-time. This leads quite naturally to a new theory which takes place on 10-dimensional manifolds. The fields are pairs of \(((\alpha ,\omega ),\varpi )\), where \((\alpha ,\omega )\) is a 1-form with coefficients in the Lie algebra of the Poincaré group and \(\varpi \) is an 8-form with coefficients in the dual of this Lie algebra. The dynamical equations derive from a simple variational principle and imply that the 10-dimensional manifold looks locally like the total space of a fiber bundle over a 4-dimensional base manifold. Moreover this base manifold inherits a metric and a connection which are solutions of a system of Einstein–Cartan equations.     

Higher-dimensional vacuum solutions of Einstein's field equations

Some general solutions of the (general)D-dimensional vacuum Einstein field equations are obtained. The four-dimensional properties of matter are studied by investigating whether the higher-dimensional vacuum field equations reduce (formally) to Einstein's four-dimensional theory with matter. It is found that the solutions obtained give rise to an induced four-dimensional cosmological perfect fluid with a (physically reasonable) linear equation of state.     

Gravity theory on Poisson manifold with R‐flux

A novel gravity theory based on Poisson Generalized Geometry is investigated. A gravity theory on a Poisson manifold equipped with a Riemannian metric is constructed from a contravariant version of the Levi‐Civita connection, which is based on the Lie algebroid of a Poisson manifold. Then, we show that in Poisson Generalized Geometry the R‐fluxes are consistently coupled with such a gravity. An R‐flux appears as a torsion of the corresponding connection in a similar way as an H‐flux which appears as a torsion of the connection formulated in the standard Generalized Geometry. We give an analogue of the Einstein‐Hilbert action coupled with an R‐flux, and show that it is invariant under both β‐diffeomorphisms and β‐gauge transformations.     

Application of Lie Algebroid Structures to Unification of Einstein and Yang-Mills Field Equations

Yang-Mills field equations describe new forces in the context of Lie groups and principle bundles. It is of interest to know if the new forces and gravitation can be described in the context of algebroids. This work was intended as an attempt to answer last question. The basic idea is to construct Einstein field equation in an algebroid bundle associated to space-time manifold. This equation contains Einstein and Yang-Mills field equations simultaneously. Also this equation yields a new equation that can have interesting experimental results.     

A symplectic structure on the set of Einstein metrics

A symplectic structure i.e. a symplectic form on the set of all solutions of the Einstein equations on a given 4-dimensional manifold is defined. A degeneracy distribution of is investigated and its connection with an action of the diffeomorphism group is established. A multiphase formulation of General Relativity is presented. A superphase space for General Relativity is proposed.     

A geometric notion of complete integrability

A system of partial differential equations which can be described as a harmonic mapping of riemannian manifolds is called completely integrable when the corresponding n-dimensional manifold of fields admits 2n?1 independent Killing vector fields. It is conjectured that, for systems of two independent variables, complete integrability in the present sense implies the existence of a Lax pair for the system, for which the theory of the inverse scattering method is applicable. The stationary axisymmetric Einstein and Einstein-Maxwell equations, the SU(n) self-dual Yang-Mills fields in 1+1 dimensions, and the two-dimensional non-linear σ-models are shown to satisfy the conjecture; the conjecture is also proved for any system of n = 2 and n = 3 partial differential equations for n unknown scalar fields.     

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.     

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.     

Geometry of the submanifolds of SEXn. II. The generalized fundamental equations for the hypersubmanifold of SEXn

A connection which is both Einstein and semisymmetric is called an SE connection, and a generalizedn-dimensional Riemannian manifold on which the differential geometric structure is imposed byg through an SE connection is called ann-dimensional SE manifold and denoted by SEX_n. This paper is a direct continuation of earlier work. In this paper, we derive the generalized fundamental equations for the hypersubmanifold of SEX_n, including generalized Gauss formulas, generalized Weingarten equations, and generalized Gauss-Codazzi equations.     

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.     

The O(3,1) Yang-Mills equations and the Einstein equations

The vacuum Einstein equations (with cosmological constant) written in a slightly unconventional manner, can be decomposed into three parts: the first two parts are the ordinary self dual Yang-Mills equations and the anti-self dual Yang-Mills equations for anO(3,1) gauge group, on an unspecified background space-time, the third part are equations that solder or relate these two Y-M fields and connections to the curvature and connection of that unknown space-time. It is the purpose of this note to take this point of view seriously and concentrate on the first two parts in their own right. We apply to them generalizations of solution construction techniques which have arisen from the study of self dual Yang-Mills equations on Minkowski space. At the end we discuss how to solder or bootstrap these results to the determination of the space-time itself.     

The twistor theory of equations of KdV type: I

This article is the first of two concerned with the development of the theory of equations of KdV type from the point of view of twistor theory and the self-dual Yang-Mills equations. A hierarchy on the self-dual Yang-Mills equations is introduced and it is shown that a certain reduction of this hierarchy is equivalent to then-generalized KdV-hierarchy. It also emerges that each flow of then-KdV hierarchy is a reduction of the self-dual Yang-Mills equations with gauge group SL _n . It is further shown that solutions of the self-dual Yang-Mills hierarchy and their reductions arise via a generalized Ward transform from holomorphic vector bundles over a twistor space. Explicit examples of such bundles are given and the Ward transform is implemented to yield a large class of explicit solutions of then-KdV equations. It is also shown that the construction of Segal and Wilson of solutions of then-KdV equations from loop groups is contained in our approach as an ansatz for the construction of a class of holomorphic bundles on twistor space.A summary of the results of the second part of this work appears in the Introduction.Most of this work was done while Darby Fellow of Mathematics at Lincoln College, Oxford     

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.     

Self-dual solutions to euclidean gravity

Recent work on Euclidean self-dual gravitational fields is reviewed. We discuss various solutions to the Einstein equations and treat asymptotically locally Euclidean self-dual metrics in detail. These latter solutions have vanishing classical action and nontrivial topological invariants, and so may play a role in quantum gravity resembling that of the Yang-Mills instantons.     

The Hidden Symmetries and Their Algebraic Structure of the Static Axially Symmetric SDYM Fields

A new explicit transformation about the static axially symmetric self-dual Yang-Mills(SDYM) fields is presented.The theory has proved that the new transformation is a symmetric one.For the two kinds of the Lie algebraic generators of the Lie group SL (N.R)/SO(N),the corresponding transformations are given.By making use of the Yang-Baxter equality and their square brackets we have obtained the Loop and comformal algebraic structures of the symmetric transformations for the basic fields.All the results obtained in this paper can be directly generalized to the other models.     

Deformation quantization of Heisenberg manifolds

ForM a smooth manifold equipped with a Poisson bracket, we formulate aC*-algebra framework for deformation quantization, including the possibility of invariance under a Lie group of diffeomorphisms preserving the Poisson bracket. We then show that the much-studied non-commutative tori give examples of such deformation quantizations, invariant under the usual action of ordinary tori. Going beyond this, the main results of the paper provide a construction of invariant deformation quantizations for those Poisson brackets on Heisenberg manifolds which are invariant under the action of the Heisenberg Lie group, and for various generalizations suggested by this class of examples. Interesting examples are obtained of simpleC*-algebras on which the Heisenberg group acts ergodically.This work was supported in part by National Science Foundation grant DMS 8601900