Non-self-dual Yang-Mills connections with quadrupole symmetry

Abstract

We prove the existence of non-self-dual Yang-Mills connections on SU(2) bundles over the four-sphere, specifically on all bundles with second Chern number not equal ±1. We study connections equivariant under an SU(2) symmetry group to reduce the effective dimensionality from four to one, and then use variational techniques. The existence of non-self-dual SU(2) YM connections on the trivial bundle (second Chern number equals zero) has already been established by Sibner, Sibner, and Uhlenbeck via different methods.

Solitons in Field Theory and Nonlinear Analysis

“…Fairly recently, Sibner, Sibner, and Uhlenbeck [286] proved that there exists among the trivial topological dass C2 = 0 a family of non-self-dual critical points for the 8U(2) Yang-Mills theory on 8 4 . Moreover, Sadun and Segert [270] obtained the same condusion for C2 i-±l. Parker [242], Bor [43], and Bor and Montgomery [44] have established similar results, again on 8 4 • It will be desirable to solve the problem in all 4m dimensions.…”

“…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.…”

“…We have seen that for the problem over a full plane, there exist topological and non-topological N-vortex solutions realizing any givcn winding number N, and, for the problem over a finite periodic domain n, the vortex number N is confined by the size of n and for each prescribed distribution of vortex locations, there are at least two gauge-distinct solutions realizing such vortex locations whenever existence is ensured subcritically through the Chern-Simons coupling parameter, It is weH known that for thc classical Abelian Higgs model, also known in its static limit as the Ginzburg-Landau theory, with the Lagrangian action dcnsity (5.7.2) instead of (5.2.1), the answer to the same problem is affirmative [157,304], although such an equivalence result docs not hold for general Yang-Mills equations [43,44,59,60,242,270,286,305].…”

Hyperbolic vortices and some non-self-dual classical solutions of SU(3) gauge theory

“…We note that the solution for n = 4 is almost as low as only three times the n = 1 action. The lowest action for a non-self-dual solution in SU (2) found by Sadun and Segert [8] is roughly 5.4 times the instanton action. In SU (4) gauge theory, it is clear that we can find a non-self-dual solution with action twice that of the instanton: we pick two commuting SU (2) subalgebras and consider the potential which is composed of an instanton in one SU (2) and an anti-instanton in the other SU (2).…”