Solutions of Yang–Mills equations on generalized Hopf bundles

Abstract

Trautman has constructed natural self-dual connections on the Hopf bundles over complex and quaternionic projective spaces CP n and HP n ; the associated connections are SU( n+1) and Sp( n+1) invariant. Trautman wondered if these connections could be generalized to the case of the corresponding projective spaces defined by indefinite metrics. In this note, we extend the work of Trautman in two different directions. We first define self-dual connections on the Hopf bundles over the projective spaces CP (p,q) and HP (p,q) which are U( p, q+1) and Sp( p, q+1) invariant. We also define self-dual connections over the Hopf bundles associated with the para-complex and para-quarternionic projective spaces C ̃ P (p,q) and H ̃ P (p,q) . Finally, the topology of these projective spaces is investigated.