The goal of this paper is to give a new proof of a theorem of Meng
and Taubes that identifies the Seiberg–Witten invariants of
3–manifolds with Milnor torsion. The point of view here will be
that of topological quantum field theory. In particular, we relate the
Seiberg-Witten equations on a 3–manifold with the Abelian vortex
equations on a Riemann surface. These techniques also give a new proof of
the surgery formula for the Casson invariant, interpreted as an invariant
of a homology S2×S1.
Keywords
Seiberg–Witten invariant, Casson
invariant, Alexander polynomial, Milnor torsion, topological
quantum field theory, moduli space, vortex equation
