  We prove a conjecture of Hutchings and Lee relating the Seiberg-Witten
invariants of a closed 3-manifold X with b_1 > 0 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 Ozsvath and Szabo 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.
