The ADHM construction and non-local symmetries of the self-dual Yang-Mills equations

Abstract

We consider the action on instanton moduli spaces of the non-local symmetries of the self-dual Yang-Mills equations on R 4 discovered by Chau and coauthors. Beginning with the ADHM construction, we show that a sub-algebra of the symmetry algebra generates the tangent space to the instanton moduli space at each point. We explicitly find the subgroup of the symmetry group that preserves the one-instanton moduli space. This action simply corresponds to a scaling of the moduli space.

Reducible Connections and Non-local Symmetries of the Self-dual Yang–Mills Equations

“…Such formulae bear a strong resemblance to those arising in the theory of harmonic maps of finite type (see, e.g., [14,Chapter 24]). This result is distinct from the instanton case discussed in [13], which bears more of a resemblance to the theory of harmonic maps of finite uniton number [25].…”

“…It has been investigated further in, for example, [17,18,22]. The first direct derivation of the infinitesimal result (2.11) from the generator (2.9), to my knowledge, appears in [13].…”

“…In this paper and its companion [13] we investigate the non-local symmetry algebra of the selfdual Yang-Mills equations on R 4 discussed in [7,8,9] and corresponding group actions on spaces of solutions of the self-dual Yang-Mills equations on open subsets of R 4 . In [13] we studied instanton moduli spaces, as explicitly described by the ADHM construction [1,2], and group actions that preserved the L 2 nature of the curvature of the connection. In the current work, we investigate reducible connections defined on a simply-connected, open subset of R 4 .…”

“…In the case of instanton solutions on R 4 , it was argued in [13] that the symmetry group that acts on the moduli space has orbits of high codimension in the moduli space. (In other words, the orbits are quite small.)…”

“…As in the companion paper [13], we study only the non-local symmetries of the self-dual Yang-Mills equations constructed in [7,8,9], and not symmetries that require the existence of a nontrivial conformal group on our manifold. We also specialise to the case of SU 2 structure group, although the generalisation to any classical Lie group is straightforward.…”