Quaternionic monopoles. (English) Zbl 0872.58015

Okonek, Christian; Teleman, Andrei

Commun. Math. Phys. 180, No. 2, 363-388 (1996).

Summary: We present the simplest non-abelian version of Seiberg-Witten theory: Quaternionic monopoles. These monopoles are associated with \(\text{Spin}^h(4)\)-structures on 4-manifolds and form finite-dimensional moduli spaces. On a Kähler 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 Kähler surfaces have two closed subspaces, both naturally isomorphic with moduli spaces of canonically stable holomorphic pairs. These components intersect along a Donaldson instanton space and can be compactified with Seiberg-Witten moduli spaces. This should provide a link between the two corresponding theories.

MSC:

58D27	Moduli problems for differential geometric structures
57R57	Applications of global analysis to structures on manifolds
58J20	Index theory and related fixed-point theorems on manifolds
32Q20	Kähler-Einstein manifolds

Keywords:

Seiberg-Witten theory; quaternionic monopole; vortex equation; moduli spaces

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