A topological existence proof for \(SO(n)\)-anti-self-dual connections (Q5957282)

Instantons are minimizers of the Yang-Mills energy for connections on a principal \(SU(2)\) bundle over a 4-manifold. They solve the anti-self-duality (ASD) equations and provide the basis for the construction of Donaldson invariants (of the 4-manifold). More precisely, it is the space of all such instantons, i.e. the instanton moduli space, which is used to define the invariants. Non-emptiness results for these moduli spaces are thus of great significance. A key existence theorem for such instantons was proved by \textit{C. H. Taubes} [J. Differ. Geom. 19, 517-560 (1984; Zbl 0552.53011)]. NEWLINENEWLINENEWLINEThe Yang-Mills energy and its minimizing anti-self-duality equations are equally well defined on principal \(G\) bundles over 4-manifolds, where \(G\) is any compact semisimple Lie group. Thus, at least in principle, higher rank analogues of the Donaldson invariants can be contemplated. These would, of course, require that the instanton moduli spaces be non-empty. The main result in this paper gives a necessary condition for this to be so in the case \(G=SO(n)\). The author proves: NEWLINENEWLINENEWLINETheorem. Let \(X\) be a closed, oriented, smooth Riemannian 4-manifold. Let \(P\) be a principal \(SO(n)\)-bundle with \(w_2(P)=0\). Then \(P\) admits an irreducible ASD connection if NEWLINE\[NEWLINE-\frac{1}{4}p_1(\operatorname {ad} P)\geq \text{rank}(SO(n))b^{2+}\biggl[\frac{n}{4}\biggr],NEWLINE\]NEWLINE where \(p_1\) is the first Pontryagin class, \(b^{2+}\) is the dimension of the space of self-dual harmonic 2-forms on \(X\), and \([a]\) denotes the least integer greater than or equal to \(a\). NEWLINENEWLINENEWLINEThe proof uses techniques adapted from Taubes, i.e. it uses glueing techniques to construct approximate solutions and then looks for perturbations which make the solutions exact. The existence of such perturbations is reduced to the existence of zeros in a suitably defined finite dimensional obstruction map. In a departure from Taubes's original work [op. cit.] (though following suggestions in that work), this finite dimensional problem is solved by topological obstruction theory methods.