Quaternionic monopoles

Andrei Teleman (1) , Christian Okonek (1)

1 Institut für Mathematik [Zürich]

We present the simplest non-abelian version of Seiberg-Witten theory: Quaternionic monopoles. These monopoles are associated with Spin^h(4)-structures on 4-manifolds and form finite-dimensional moduli spaces. On a Kahler surface the quaternionic monopole equations decouple and lead to the projective vortex equation for holomorphic pairs. This vortex equation comes from a moment map and gives rise to a new complex-geometric stability concept. The moduli spaces of quaternionic monopoles on Kahler surfaces have two closed subspaces, both naturally isomorphic with moduli spaces of canonically stable holomorphic pairs. These components intersect along Donaldsons instanton space and can be compactified with Seiberg-Witten moduli spaces. This should provide a link between the two corresponding theories.

Mots clés: Seiberg-Witten theory; non-abelian monopoles; quaternionic monopoles

Domaines: Géométrie différentielle [math.DG]; Topologie géométrique [math.GT]; Géométrie algébrique [math.AG]

Andrei Teleman, Christian Okonek. Quaternionic monopoles. Communications in Mathematical Physics, 1996, 180 (2), pp.363-388. ⟨10.1007/BF02099718⟩. ⟨hal-00881772⟩