Non-self-dual Yang-Mills connections with nonzero Chern number

Abstract

We prove the existence of non-self-dual Yang-Mills connections on SU(2) bundles over the standard four-sphere, specifically on all bundles with second Chern number not equal to ±1. A YangMills (YM) connection A is a critical point of the YM actionwhere F is the curvature of the connection A and * is the Hodge dual. The YM equations D * F = 0, where D denotes the covariant exterior derivative, are the variational equations of this functional, and constitute a system of second-order PDE's in A. Absolute minima of the YM action, in addition to satisfying the YM equations, also satisfy a first-order system of PDE's, the (anti)self-duality equations *F = ±F. We call a connection non-self-dual (NSD) if it is neither self-dual ( *F = F ) nor antiself-dual (*F = -F), i.e., if it is not a minimum of the YM action.(Anti) self-dual connections on *S 4 have been well-understood for some time. The first nontrivial example, the BPST instanton [BPST], was found in 1975, and three years later all self-dual solutions on S 4 were classified [ADHM], not only for SU(2) but for all classical groups. The study of self-dual SU(2) connections on other four-manifolds led to spectacular progress in topology, including the discovery of fake R 4 (see [FU] for an overview). The study of NSD YM connections has proceeded much more slowly. While some examples of NSD YM connections on fourmanifolds are known [I, Mai, Ma2, Ur, P], until recently NSD YM connections on SU(2) bundles over standard S 4 proved elusive,

Domain wall equations, Hessian of superpotential, and Bogomol'nyi bounds

“…In the Abelian Higgs vortex model all finite-energy solitons in the BPS coupling are BPS [28,47] but in the Yang-Mills-Higgs monopole model [48] and Yang-Mills instanton model [14,35,40,41,46] there exist non-BPS solitons which are of course of energies above the BPS bounds. In this paper we have carried out a systematic investigation on the puzzle for a general domain wall model governing a multiple real-component scalar field u = (u 1 , .…”

Stationary points of the Yang‐Mills action

“…The instanton number of the solution corresponding to the pair (n+, n-) is k = (n?n:)/8, as explained in [16], [3] . Every integer k except f l can be obtained from this formula, so we have solutions with arbitrary instanton number (except k = f l ) .…”

Existence of a new instanton in constrained Yang-Mills-Higgs theory

“…for integers m ≥ 2 and ∆A YM < 0, which, most likely, depends on m also. Later, several solutions were constructed explicitely by Sadun and Segert [11]. The existence proof of Sibner et al [10] goes by a mini-max procedure over non-contractible loops, where an important ingredient is the inequality contained in (28), which is analogous to ours (12).…”