We prove a conjecture of Hutchings and Lee relating the Seiberg–Witten invariants of a closed
3–manifold
with
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
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
. 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
in the
case that
is a mapping torus.
Keywords
Seiberg–Witten invariant, torsion, topological quantum
field theory
