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

摘要

In this paper we show that the dimensionally reduced Seiberg-Witten equationslead to a Higgs field and study the resulting moduli spaces. The moduli spacearising out of a subset of the equations, shown to be non-empty for a compactRiemann surface of genus g >= 1, gives rise to a family of moduli spacescarrying a hyperkahler structure. For the full set of equations thecorresponding moduli space does not have the aforementioned hyperkahlerstructure 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''.