Yang–Mills theory for semidirect products

Abstract

Yang–Mills theory with a symmetry algebra that is the semidirect product $$\mathfrak {h}\ltimes \mathfrak {h}^*$$ defined by the coadjoint action of a Lie algebra $$\mathfrak {h}$$ on its dual $$\mathfrak {h}^*$$ is studied. The gauge group is the semidirect product $$\mathrm{G}_{\mathfrak {h}}\ltimes {\mathfrak {h}^*}$$, a noncompact group given by the coadjoint action on $$\mathfrak {h}^*$$ of the Lie group $$\mathrm{G}_{\mathfrak {h}}$$ of $$\mathfrak {h}$$. For $$\mathfrak {h}$$ simple, a method to construct the self–antiself dual instantons of the theory and their gauge nonequivalent deformations is presented. Every $$\mathrm{G}_{\mathfrak {h}}\ltimes {\mathfrak {h}^*}$$ instanton has an embedded $$\mathrm{G}_{\mathfrak {h}}$$ instanton with the same instanton charge, in terms of which the construction is realized. As an example, $$\mathfrak {h}=\mathfrak {s}\mathfrak {u}(2)$$ and instanton charge one is considered. The gauge group is in this case $$SU(2)\ltimes \mathbf{R}^3$$. Explicit expressions for the selfdual connection, the zero modes and the metric and complex structures of the moduli space are given