Symplectic and Hyperkahler structures in a dimensional reduction of the Seiberg-Witten equations with a Higgs field

Details Symplectic and Hyperkahler structures in a dimensional reduction of the Seiberg-Witten equations with a Higgs field Symplectic and Hyperkahler structures in a dimensional reduction of the Seiberg-Witten equations with a Higgs field

2001-12-20

arxiv.org/abs/math/0112219v2

Rept.Math.Phys. 50 (2002) 277-290

Mathematics

Differential Geometry

12 pages, AMS-LaTeX. To appear in Reports on Mathematical Physics

Scientific paper

10.1016/S0034-4877(02)80058-1

In this paper we show that the dimensionally reduced Seiberg-Witten equations lead to a Higgs field and study the resulting moduli spaces. The moduli space arising out of a subset of the equations, shown to be non-empty for a compact Riemann surface of genus g >= 1, gives rise to a family of moduli spaces carrying a hyperkahler structure. For the full set of equations the corresponding moduli space does not have the aforementioned hyperkahler structure but has a natural symplectic structure. For the case of the torus, g=1, we show that the full set of equations has a solution, different from the ``vortex solutions''.

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