Yang-Mills theory for semidirect products ${\rm G}\ltimes\mathfrak{g}^*$ and its instantons

TL;DR

This paper develops a method to construct and analyze instantons in Yang-Mills theory with a semidirect product gauge algebra, providing explicit solutions and moduli space structures for specific cases like SU(2)\ltimes\mathbb{R}^3.

Contribution

It introduces a novel approach to construct self-dual instantons in Yang-Mills theories with semidirect product gauge groups, extending the understanding of such solutions.

Findings

Explicit instanton solutions for SU(2)\ltimes\mathbb{R}^3

Construction of zero modes and moduli space metrics

Demonstration of embedded G-instantons within the semidirect product framework

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 ${\rm G}_{\mathfrak{h}}\ltimes{\mathfrak{h}^*}$, a noncompact group given by the coadjoint action on $\mathfrak{h}^*$ of the Lie group ${\rm G}_{\mathfrak{h}}$ 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 ${\rm G}_{\mathfrak{h}}\ltimes{\mathfrak{h}^*}$ instanton has an embedded ${\rm G}_{\mathfrak{h}}$ 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 $SU(2)\ltimes{\bf R}^3$. Explicit expressions for the selfdual connection, the zero modes and the metric and complex structures of the moduli space are given.