Yang–Mills theory for semidirect products $$\mathrm{G}\ltimes \mathfrak {g}^*$$ G ⋉ g ∗ and its instantons

Abstract

Dedicated to Ramón F. Alvarez-Estrada on occasion of his 70th birthday Yang-Mills theory with a symmetry algebra that is the semidirect product h ⋉ h * defined by the coadjoint action of a Lie algebra h on its dual h * is studied. The gauge group is the semidirect product G h ⋉ h * , a noncompact group given by the coadjoint action on h * of the Lie group G h of h. For h simple, a method to construct the self-antiself dual instantons of the theory and their gauge nonequivalent deformations is presented. Every G h ⋉ h * instanton has an embedded G h instanton with the same instanton charge, in terms of which the construction is realized. As an example, h = su(2) and instanton charge one is considered. The gauge group is in this case SU(2) ⋉ R 3 . Explicit expressions for the selfdual connection, the zero modes and the metric and complex structures of the moduli space are given.

Perturbative quantization of Yang–Mills theory with classical double as gauge algebra

“…The pattern observed for the gauge invariant degrees of freedom in the quantum theory resembles very much that for the self-antiself dual instantons of the classical theory [16]. In the classical case, the number of collective coordinates of the G ⋉ instantons is twice that of the embedded G instantons, yet ω ab F a Tµν F bµν Z does not contribute to the instanton number.…”

“…So far no restriction has been placed on g. Assume now that it is semisimple. In this case, g ⋉ can be viewed as a limit of the direct product of g with itself [16]. To see this, take ω ab in eq.…”

“…This metric determines a Yang-Mills Lagrangian for the field that results from gauging the algebra. For g simple, the self-antiself dual instantons of the g ⋉ Yang-Mills theory in four-dimensional Euclidean space have been studied elsewhere [16]. Every g ⋉ instanton has an embeded g instanton with the same instanton number and twice the number of collective coordinates.…”