Self-dual Yang-Mills fields in eight dimensions

Strongly self-dual Yang-Mills fields in even-dimensional spaces are characterised by a set of constraints on the eigenvalues of the Yang-Mills fieldsF . We derive a topological bound on R⁸, , wherep1 is the first Pontryagin class of the SO(n) Yang-Mills bundle, andk is a constant. Strongly self-dual Yang-Mills fields realise the lower bound.     

Lorentz-invariant solutions of the classical Yang-Mills equations

A gauge representation of the noncompact group SL(2,C) is defined. All the corresponding invariant (singular) solutions of the classical Yang-Mills equations in Minkowski space are found. It is shown that they are related to a family of real SU(2) x SU(2)-invariant Euclidean solutions containing the self-dual (one-instanton) configurations.     

Monopole vacuum in non-Abelian theories

It is shown that, in the theory of interacting Yang-Mills fields and a Higgs field, there is a topological degeneracy of Bogomol'nyi-Prasad-Sommerfield (BPS) monopoles and that there arises, in this case, a chromoelectric monopole characterized by a new topological variable that describes transitions between topological states of the monopole in Minkowski space (in just the same way as an instanton describes such transitions in Euclidean space). The limit of an infinitely large mass of the Higgs field at a finite density of the Bogomol'nyi-Prasad-Sommerfield monopole is considered as a model of the stable vacuum in pure Yang-Mills theory. It is shown that, in QCD, such a monopole vacuum may lead to a growing potential, a topological confinement, and an additional mass of the η₀ meson. The relationship between the result obtained here for the generating functional of perturbation theory and the Faddeev-Popov integral is discussed.     

Decay of classical Yang-Mills fields

The classical Yang-Mills equations in four-dimensional Minkowski space are invariant under the conformal group. The resulting conservation laws are explicitly exhibited in terms of the Cauchy data at a fixed time. In particular, it is shown that, for any finite-energy solution of the Yang-Mills equations, the local energy tends to zero ast.Research supported in part by NSF grants MCS 77-01340 and MCS 78-03567     

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     

Non-linear multi-plane wave solutions of self-dual Yang-Mills theory

New solutions of self-dual Yang-Mills (SDYM) equations are constructed in Minkowski space-time for the gauge groupSL(2, ). After proposing a Lorentz covariant formulation of Yang's equations, a set of Ansätze for exact non-linear multiplane wave solutions are proposed. The gauge fields are rational functions ofe ^x·ki (K _i ² =0, 1iN) for these Ansätze. At least, three families of multisoliton type solutions are derived explicitly. Their asymptotic behaviour shows that non-linear waves scatter non-trivially in Minkowski SDYM.On leave from LPTHE Université Paris VI, 4, Place Jussieu, Tour 16, ler étage, F-75230 Paris Cedex 05, France     

Instantons and algebraic geometry

Minimum action solutions for SU(2) Yang-Mills fields in Euclidean 4-space correspond, via the Penrose twistor transform, to algebraic bundles on the complex projective 3-space. These bundles in turn correspond to algebraic curves. The implication of these results for the Yang-Mills fields is described. In particular all solutions are rational and can be constructed from a series of AnsätzeA _l forl1.     

The Positive Action conjecture and asymptotically Euclidean metrics in quantum gravity

The Positive Action conjecture requires that the action of any asymptotically Euclidean 4-dimensional Riemannian metric be positive, vanishing if and only if the space is flat. Because any Ricci flat, asymptotically Euclidean metric has zero action and is local extremum of the action which is a local minimum at flat space, the conjecture requires that there are no Ricci flat asymptotically Euclidean metrics other than flat space, which would establish that flat space is the only local minimum. We prove this for metrics onR ⁴ and a large class of more complicated topologies and for self-dual metrics. We show that ifR 0 there are no bound states of the Dirac equation and discuss the relevance to possible baryon non-conserving processes mediated by gravitational instantons. We conclude that these are forbidden in the lowest stationary phase approximation. We give a detailed discussion of instantons invariant under anSU(2) orSO(3) isometry group. We find all regular solutions, none of which is asymptotically Euclidean and all of which possess a further Killing vector. In an appendix we construct an approximate self-dual metric onK3 — the only simply connected compact manifold which admits a self-dual metric.     

Yang-Mills fields which are not self-dual

The purpose of this paper is to prove the existence of a new family of non-self-dual finite-energy solutions to the Yang-Mills equations on Euclidean four-space, withSU(2) as a gauge group. The approach is that of equivariant geometry: attention is restricted to a special class of fields, those that satisfy a certain kind of rotational symmetry, for which it is proved that (1) a solution to the Yang-Mills equations exists among them; and (2) no solution to the self-duality equations exists among them. The first assertion is proved by an application of the direct method of the calculus of variations (existence and regularity of minimizers), and the second assertion by studying the symmetry properties of the linearized self-duality equations. The same technique yields a new family of non-self-dual solutions on the complex projective plane.     

Removable singularities in Yang-Mills fields

We show that a field satisfying the Yang-Mills equations in dimension 4 with a point singularity is gauge equivalent to a smooth field if the functional is finite. We obtain the result that every Yang-Mills field overR ⁴ with bounded functional (L ² norm) may be obtained from a field onS ⁴=R ⁴{}. Hodge (or Coulomb) gauges are constructed for general small fields in arbitrary dimensions including 4.     

Self dual solutions of the temperature SU(2) Yang-Mills theory

Continuing previous work we elaborate on the method of “heating” the self-dual axially symmetric fields of the SU(2) Yang-Mills theory to finite temperature. Heating consists of performing—in certain Ansatz functions which are two-dimensional (2D) conformally invariant—a 2D conformal transformation x = x₀ + i ∥x∥ → y(x), where the analytic function y(x) is periodic in the Euclidean time variable x₀. Solutions are preserved by this manipulation, which automatically changes zero-temperature fields into finite temperature ones. One can exploit this simple fact in various ways. The Harrington-Shepard caloron solution of the temperature Yang-Mills theory can be gotten from the T = 0 instanton by the transformation y(x) = (πT)^?1 tan πTx. One can generate a multicaloron solution from the T = 0 one instanton solution by a conformal transformation. Generally, self-dual axially symmetric Yang-Mills fields can be heated without spoiling self duality. The caloron and three other temperature solutions are studied in some detail. One of the new solutions is a generalized caloron with interesting properties. Our study reveals a remarkable property of the self-dual sector of the temperature Yang-Mills theory: it is full of Wu-Yang (color) monopoles at high temperature. At low temperature these monopoles disappear.     

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.     

Calculation of BRS cohomology with spectral sequences

A method for finding the general form of the BRS cohomology spaceH for the various gauge and supersymmetry theories is presented. The method is adapted for use in the space of integrated local polynomials of the gauge fields and ghosts with arbitrary numbers of fields and dervivatives. The technique uses the Hodge decomposition in a Fock space with a Euclidean inner product, and combines this with spectral sequences to generate simple and soluble equations whose solutions span a simple spaceE isomorphic to the complicated spaceH. The technique is illustrated for pedagogic purposes by the detailed calculation of the ghost charge zero and one sectors ofH for Yang-Mills theory with gauge groupSO (32) in ten dimensions. The method is appropriate for supersymmetric theories, gravity, supergravity and superstrings where higher order terms with many derivatives occur naturally in the effective action.Research supported in part by the Robert A. Welch Foundation and NSF Grants PHY 77-18762 and PHY 9009850     

Geometroelectrodynamics

We present an elementary particle model that can be thought of as a unification of certain topological ideas abstracted from the string model and the standard Yang-Mills theory. The basic dynamical entity of the model is a spacelike 3-surfaceX ³ in some metric spaceH and is interpreted as a particle. The dynamics of the model is based on two ideas. First the model is formally a Yang-Mills theory on the surfaceX ⁴ representing the orbit(s) of the particle(s) inH. Secondly the Yang-Mills structure onX ⁴ is constructed using only the natural geometric structures of the space H by a process which we call induction. It is found that some rather general requirements highly fix the choice of the space H. In fact the minimal model, for which the space H is the product of Minkowski space and the 2-sphere, is obtained by requiring that the symmetry group of the theory is the product of the Poincaré group and the color groupSO(3). The unique feature of the minimal model is that it affords a purely topological mechanism for quark confinement.     

On the Prasad-Sommerfield dyon (monopole) solution

We prove that the Prasad-Sommerfield dyon (monopole) solution for an SU(2) Yang-Mills field coupled with an SU(2) Higgs multiplet can be associated to a certain minimal immersion in S ³ (SU(2)) i.e. it has a differential-geometric content similar to that of self-dual solutions for the pure SU(2) Yang-Mills field. Implications of this result as well as possibilities to extend it to higher gauge groups are discussed.     

The semiclassical limit for gauge theory onS 2

It is shown that the Yang-Mills measureZ _h ^–1 e^–S()/h[D], whereh>0, describing gauge fields on the two-sphere converges to a probability measure on the moduli space of Yang-Mills connections onS ², ash0.This work was partially supported by NSF Grants DMS-8922941, and PHY-8912067     

Analytic structure of gauge fields

The analytic structure of gauge fields in the presence of fermions is studied in arbitrary symmetry. A Hamiltonian formalism is developed which relates Cauchy-Riemann equations to the symmetry. The formalism is applied to three problems in (2+1)-dimensional Euclidean space: (1) a free fermion, (2) a fermion interacting with a massless scalar field, and (3) a fermion interacting with a vector field. We find that the Hamiltonian for the free fermion is analytic and single-valued in a finite region of momentum space. With the addition of an auxiliary field, the Hamiltonian can be analytic in the entire momentum space. The scalar field then acquires spin-dependent coordinates by interaction with the fermion; the interactions break the Abelian symmetry of so that ₁ 1/(x₁ –-im ₁ ^–1 (x₁ –-im ₁ ^–1 ), wherem ₁ are spin-dependent and multivalued. There are four solutions for each chirality eigenvalue of the fermion. For spinless fermions gives the Jackiw-Nohl-Rebbi solution and is separable into Coulomb-like 1/x analytic functions on the first and fourth quadrants. For a vector field the results are similar except that the coordinates are not spindependent or multivalued; interactions break the initial symmetry andA (x )A ¹ (x ) and theA ¹ have a non-Abelian algebra. Thel indices represent directions fixed by spin matrices in a spin-dependent color space.     

Vector-spinor space and field equations

Generalizing the work of Einstein and Mayer, it is assumed that at each point of space-time there exists a vector-spinor space with N_v vector dimensions and N_s spinor dimensions, where N_v=2k and N_s=2 ^k, k3. This space is decomposed into a tangent space with4 vector and4 spinor dimensions and an internal space with N_v–4 vector and N_s–4 spinor dimension. A variational principle leads to field equations for geometric quantities which can be identified with physical fields such as the electromagnetic field, Yang-Mills gauge fields, and wave functions of bosons and fermions.     

Green's functions in Bianchi type-I spaces. Relation between Minkowski and Euclidean approaches

A theory is considered for a free scalar field with a conformal connection in a curved space-time with a Bianchi type-I metric. A representation is obtained for the Green's functionGin<0¦T(x)(x)¦0> _in in the form of an integral of a Schwinger-DeWitt kernel along a contour in a plane of complex-valued proper time. It is shown how a transition may be accomplished from Green's functions in space with the Euclidean signature to Green's functions in space with Minkowski signature and vice versa.Translated from Izvestiya Vyssnikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 20–27, June, 1988.