Yang-Mills Instantons and Dyons on Group Manifolds

TL;DR

This paper explores solutions to Euclidean and Minkowski SU(N) Yang-Mills theories on group manifolds, introducing BPS equations, instantons, dyons, and algebraic reductions, with applications to flux compactifications.

Contribution

It introduces BPS-type equations for Yang-Mills on group manifolds and derives explicit instanton and dyon solutions, including algebraic reductions to Nahm and Toda equations.

Findings

Constructed explicit instanton solutions on GxR.

Derived finite-energy dyon solutions in Minkowski signature.

Reduced matrix equations to Nahm and Toda systems for specific groups.

Abstract

We consider Euclidean SU(N) Yang-Mills theory on the space GxR, where G is a compact semisimple Lie group, and introduce first-order BPS-type equations which imply the full Yang-Mills equations. For gauge fields invariant under the adjoint G-action these BPS equations reduce to first-order matrix equations, to which we give instanton solutions. In the case of G=SU(2)=S^3, our matrix equations are recast as Nahm equations, and a further algebraic reduction to the Toda chain equations is presented and solved for the SU(3) example. Finally, we change the metric on GxR to Minkowski and construct finite-energy dyon-type Yang-Mills solutions. The special case of G=SU(2)xSU(2) may be used in heterotic flux compactifications.