Torsion, TQFT, and Seiberg–Witten invariants of $3$–manifolds

Abstract

We prove a conjecture of Hutchings and Lee relating the Seiberg–Witten invariants of a closed 3–manifold $X$ with $b_1\ge 1$ to an invariant that “counts” gradient flow lines—including closed orbits—of a circle-valued Morse function on the manifold. The proof is based on a method described by Donaldson for computing the Seiberg–Witten invariants of 3–manifolds by making use of a “topological quantum field theory,” which makes the calculation completely explicit. We also realize a version of the Seiberg–Witten invariant of $X$ as the intersection number of a pair of totally real submanifolds of a product of vortex moduli spaces on a Riemann surface constructed from geometric data on $X$. The analogy with recent work of Ozsváth and Szabó suggests a generalization of a conjecture of Salamon, who has proposed a model for the Seiberg–Witten–Floer homology of $X$ in the case that $X$ is a mapping torus.