Seiberg-Witten theory as a complex version of Abelian Higgs model

The adiabatic limit procedure associates with every solution of Abelian Higgs model in （2 ^- 1） dimensions a geodesic in the moduli space of static solutions. We show that the same procedure for Seiberg- Witten equations on 4-dimensional symplectic manifolds introduced by Taubes may be considered as a complex （2＋2）-dimensional version of the （2＋ 1）-dimensional picture. More precisely, the adiabatic limit procedure in the 4-dimensional case associates with a solution of Seiberg-Witten equations a pseudoholomorphic divisor which may be treated as a complex version of a geodesic in （2＋l）-dimensional case.

The adiabatic limit procedure associates with every solution of Abelian Higgs model in(2 + 1)dimensions a geodesic in the moduli space of static solutions. We show that the same procedure for SeibergWitten equations on 4-dimensional symplectic manifolds introduced by Taubes may be considered as a complex(2+2)-dimensional version of the(2 + 1)-dimensional picture. More precisely, the adiabatic limit procedure in the 4-dimensional case associates with a solution of Seiberg-Witten equations a pseudoholomorphic divisor which may be treated as a complex version of a geodesic in(2+1)-dimensional case.