Equivariant Nahm Transforms and Minimal Yang--Mills Connections

This dissertation examines two different subjects within the study of instantons: the construction of Nahm transforms for instantons invariant under certain group actions; and a generalization of the proof that Yang--Mills minimizers are instantons.

The first Nahm transform examined is the ADHM construction for $S^1$-invariant instantons on $S^4$, which correspond to singular monopoles on $\RR^3$. In this case, there is a decomposition of the ADHM data in terms of $S^1$-subrepresentations of $\ker \Dir$. The moduli spaces of $S^1$-invariant $SU(2)$-instantons are given up to charge 3, and examples of ADHM data for instantons of charge $4$ are also provided.

The second Nahm transform considered is for instantons on a certain flat quotient of $\RR^4$ with nonabelian fundamental group. Equivalently, one can consider these to be $\ZZ_2$-invariant instantons on $T^4$, and the Nahm transform yields instantons invariant under a crystallographic action.

In our study of minimal Yang--Mills connections, we extend results of Bourguignon--Lawson--Simons and Stern, who showed that connections that minimize $\|F_\nabla\|^2$ on homogeneous manifolds must be instantons or have instanton subbundles. We extend the previous arguments by considering variations constructed using conformal vector fields, and also allow these vector fields to be incomplete. We prove a minimality result over a half-cylinder.