Topological contents of 3D Seiberg-Witten theory

Abstract

Using physical arguments (Higgs mechanism, superconductivity, infrared regime, duality) and a geometric-topological construction (scalar curvature distribution compatible with surgery), we propose a topological interpretation of 3D SW theory in terms of the abelian Casson invariant. Further algebraic reasoning shows equivalence of that invariant to the Alexander ``polynomial''. Our starting point is a 3D version of the original SW theory. Observing that the scalar curvature $R$ plays the role of a mass-squared parameter for the monopole field we can use that observation to control the theory in low-energy limit.