Solutions to Yang—Mills equations that are not self-dual

Abstract

The Yang-Mills functional for connections on principle SU(2) bundles over S4 is studied. Critical points of the functional satisfy a system of second-order partial differential equations, the Yang-Mills equations. If, in particular, the critical point is a minimum, it satisfies a first-order system, the self-dual or anti-self-dual equations. Here, we exhibit an infinite number of finite-action nonminimal unstable critical points. They are obtained by constructing a topologically nontrivial loop of connections to which mi-max theory is applied. The construction exploits the fundamental relationship between certain invariant instantons on S4 and magnetic monopoles on H3. This result settles a question in gauge field theory that has been open for many years.The existence of non-self-dual solutions to the Yang-Mills equations on 54 has been an open problem for more than a decade. Evidence for nonexistence rests primarily on the analogy with the problem of harmonic maps from S2 to S2, where it is not difficult to show that all solutions to the harmonic map equation are conformal, anticonformal, or trivial. The general philosophy has been that the SU(2) Yang-Mills equations on S4 = R4 U oo are similar, with the quaternions replacing the complex numbers. Partial success has been achieved by Bourguignon and Lawson (1), Bourguignon et al. (2), and Taubes (3), who have shown that non-self-dual solutions cannot take on local minima.In this paper, we show that there exist an infinite number of nontrivial non-self-dual solutions to Yang-Mills equations on 54 by exploiting the properties of solutions with U(1) symmetries. We produce solutions in the trivial bundle which are left invariant by a U(1) action for every integer m 2 2. The m = 1 symmetry describes the basic BPST instanton, and one might hope that the most elementary non-self-dual solution would describe an instanton-anti-instanton balanced pairing with a U(1) symmetry corresponding to m = 1. This we have not been able to find. At best we have a procedure for generating a solution that should correspond to an m instanton and m anti-instanton pair for m > 2.Our ideas are based on the fundamental relationship between m-equivariant gauge fields on S4 and monopoles on hyperbolic 3-space H-3, as described by Atiyah (4) [see also Braam (5)]. Taubes (6) has shown the existence of nonself-dual monopoles on Euclidean R3, and our arguments parallel his but on Hl. By translating our instanton arguments into monopole language, our methods produce hyperbolic monopoles for all real m > 1. These monopoles correspond to nonsingular Yang-Mills connections on 54 exactly when m is an integer. The details of this will appear later.The analytic techniques in this paper are entirely due to Taubes, who has developed an extensive variational theory for both instantons (3) and Euclidean monopoles (6-9). His analytic arguments carry over exactly to our situation, and we depend on his methods and papers rather than reproducing them here. Section 1. The Basic ResultWe follow the descri...

Solitons in Field Theory and Nonlinear Analysis

“…The absence of degree one solutions of (2.2.23), (2.2.24) already indieates that (2.1.13) is never attained at deg( tP) = ± 1. Thus if one could prove that the energy (2.2.13) has a minimizer, or a critieal point, among all degree one field configurations, it would mean that the system allows non-self-dual solutions as SU (2) instantons [43,242,270,286]. At this moment the question in either direction is open.…”

“…We first illustrate this idea with a simple example, the Maxwell equations. Then it is easy to examine the validity of the following Bianchi identity With the Hodge dual * like that defined by (3.1.14), [286] (see also [242,270]) shows that there are solutions of (3.1.8) which do not satisfy either self-dual or anti-selfdual equations, (3.1.19). The solutions of (3.1.19) are called self-dual or anti-self-dual solutions or instantons.…”

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