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     

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.     

Supersymmetric instantons and topological quantum field theory

We present an action which generates the supersymmetric self-dual equations corresponding to euclidean super Yang-Mills theory in four dimensions. By adding additional constraint fields with new local symmetries, the classical equations of this system are the usual super self-dual equations when a gauge is chosen for the constraint fields. This construction is a supersymmetric generalization of the Labastida-Pernici action which corresponds to a gauge unfixed version of Witten's topological quantum field theory. We discuss some topological prospects for this model, and the role of supersymmetric instantons in Donaldson theory.     

Minkowski space Yang-Mills fields from Euclidean self-dual fields

It is examined, if it is possible, to obtain solutions of the SU(2) Yang-Mills field equations in Minkowski space from Euclidean self-dual Yang-Mills fields by method proposed by Bernreuther. It is shown that the conditions, which are imposed on the Euclidean self-dual fields by this method, make every Euclidean self-dual field and the corresponding Minkowski space field obtained from it, equivalent to a pure gauge field, F _ab 0.     

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.     

Non-local continuity equations for self-dual SU(N) Yang-Mills fields

Using a manifestly gauge invariant formulation, an infinite set of classical non-local continuity equations is constructed for the self-dual SU(N) Yang-Mills fields.     

Self-dual Yang-Mills fields in an Einstein universe

We find exact solutions of the self-consistent Einstein-Yang-Mills system of equations. These solutions are self-dual Yang-Mills fields inI ¹×S³ space-time.     

Infinite number of local conservation laws for the self-dual SU(2) Yang-Mills system within 't Hooft's ansatz

We derive infinite sets of local continuity equations for the four-dimensional classical self-dual SU(2) Yang-Mills fields subjected to 't Hooft's ansatz. In striking analogy to the two-dimensional CP(n) non-linear sigma model where local conservation laws obtain either from complex Cauchy-Riemann analyticity or from a matrix Riccati equation, our local sets derive from quaternionic Fueter analyticity or a Riccati equation associated with the geometric prolongation structure implied by the Belavin-Zakharov linear spectral problem for the self-dual Yang-Mills system. Our analysis underlines the close connection between local and non-local conservation laws and suggests that infinite sets of local continuity equations should be present in the general self-(antiself-)dual gauge field case.     

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.     

Local stability of deformations of self-dual Yang-Mills fields on S4

We prove that local deformations of irreducible self-dual Yang-Mills fields on S⁴ in the space of smooth soluions not assumed to be self-dual are indeed self-dual.     

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     

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.     

Bäcklund transformations and exact solutions of the stationary axially symmetric Einstein equations

The Bäcklund transformations of the self-dual Yang-Mills fields are used to derive exact solutions of the stationary axially symmetric Einstein equations. Three solutions which belong to an infinite series of such solutions are explicitly calculated, the first of which is the Lewis solution. Also the first of another infinite series of solutions is calculated, which for some value of a parameter becomes the Papapetrou solution.     

Stability and isolation phenomena for Yang-Mills fields

In this article a series of results concerning Yang-Mills fields over the euclidean sphere and other locally homogeneous spaces are proved using differential geometric methods. One of our main results is to prove that any weakly stable Yang-Mills field overS ⁴ with groupG=SU₂, SU₃ orU ₂ is either self-dual or anti-self-dual. The analogous statement for SO₄-bundles is also proved. The other main series of results concerns gap-phenomena for Yang-Mills fields. As a consequence of the non-linearity of the Yang-Mills equations, we can give explicitC ⁰-neighbourhoods of the minimal Yang-Mills fields which contain no other Yang-Mills fields. In this part of the study the nature of the groupG does not matter, neither is the dimension of the base manifold constrained to be four.Laboratoire Associé au C.N.R.S. No. 169Research partially supported by Volkswagen Grant and NSF Grant MCS-77-23579     

Self-duality,helicity and coherent states in non-Abelian gauge theories

Solutions of the non-linear classical Yang-Mills field equations obtained by iteration from self-dual solutions of the linearized equations are shown to be themselves self-dual. In Minkowski space, such solutions are necessarily complex. We give a quantum theory interpretation of these solutions which relates them to the matrix element of the field operators between the vacuum state and a coherent state of spin-one quanta of definite helicity.     

Connections on vector bundles over super Riemann surfaces

We show that the globally inequivalent off-shell N=1 super Yang-Mills theories in two dimensions classify the superholomorphic structures on vector bundles over super Riemann surfaces. More precisely, there is a one-to-one correspondence between superholomorphic structures on vector bundles over super Riemann surfaces and unitary connections satisfying certain curvature constraints. These curvature constraints are the canonical constraints used in superspace formulations of super Yang-Mills theories, but arise in our considerations as integrability requirements for the local existence of solutions to certain differential equations. Finally, we discuss the relationship of this work with some aspects of Witten's twistor-like transform.     

On the Yang R-gauge for SU(3)

The self-dual SU(3) Yang-Mills fields parametrised in an R-gauge are solved with a particular two-function Ansatz.     

N=2 super Yang-Mills theory in projective superspace

We constructN=2 Yang-Mills theory in projective superspace by exploiting the analogy to Ward's twistor construction of self-dual Yang-Mills fields.Work supported in part by NSF grant No. PHY 85-07627     

On the Weyl-Wigner-Moyal description of SU (∞) Nahm equations

We show how the reduced self-dual Yang-Mills theory described by the Nahm equations can be carried over to the Weyl-Wigner-Moyal formalism employed recently in self-dual gravity. Evidence of the existence of a correspondence between BPS magnetic monopoles and space-time hyper-Kähler metrics is given.