Yes, but some parts are reasonably concrete.

%'e discuss topological invariants

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.

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,

We construct all 5(7(2) Yang-Mills instantons on S 4 that admit a certain symmetry ("quadrupole symmetry"). This is accomplished by an equivariant version of the "ADHM monad" classification of instantons. This work is part of an attempt to better understand the structure of non-self-dual Yang-Mills connections with the same symmetry.