Abstract

We present the simplest non-abelian version of Seiberg-Witten theory: quaternionic monopoles. On Kähler surfaces the quaternionic monopole equations decouple and lead to a projective vortex equation. This vortex equation comes from a moment map and gives rise to a new stability concept for holomorphic pairs. The moduli spaces of quaternionic monopoles on Kähler surfaces decompose into two closed subspaces, both naturally isomorphic with moduli spaces of canonically stable pairs. These components intersect along Donaldson's instanton space and can be compactified with spaces associated with (abelian) Seiberg-Witten monopoles [E. Witten, Math. Res. Lett. 1 (1994), no. 6, 769--796]