Yang-Mills theory for semidirect products $\mathrm{G}⋉{\mathfrak{g}}^{\ast }$ and its instantons

Yang-Mills theory with a symmetry algebra that is the semidirect product $\mathfrak{h}⋉{\mathfrak{h}}^{\ast }$ defined by the coadjoint action of a Lie algebra $\mathfrak{h}$ on its dual ${\mathfrak{h}}^{\ast }$ is studied. The gauge group is the semidirect product ${\mathrm{G}}_{\mathfrak{h}}⋉{\mathfrak{h}}^{\ast }$, a noncompact group given by the coadjoint action on ${\mathfrak{h}}^{\ast }$ 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 non-equivalent deformations is presented. Every ${\rm G}{\mathfrak{h}}\ltimes{\mathfrak{h}^*}$instantonhasanembedded${\rm G}{\mathfrak{h}}$ 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 $SU(2)⋉{\mathbf{R}}^3$. Explicit expressions for the selfdual connection, the zero modes and the metric and complex structures of the moduli space are given.