Seiberg-Witten invariant of a three-manifold

For a closed oriented smooth 3-manifold $Y$, the three-dimensional Seiberg–Witten equations involve a unitary connection and a spinor field associated with a Spin-c structure. A Dirac operator equation couples the spinor field to the unitary connection, while a curvature equation couples the unitary connection back to a quadratic expression in the spinor field. After perturbing and quotienting by gauge transformations, suitable signed counts, with corrections where required, define Seiberg-Witten invariant of a three-manifold.

The hypotheses and conventions matter. For $b_1(Y)>1$, the resulting function on Spin-c structures has a description by refined Reidemeister torsion; for $b_1(Y)=1$, chamber choices enter, and rational homology spheres require additional correction terms. For example, on $S^1\times S^2$ the first Betti number is one, so an unqualified count omitting the chamber data is incomplete. The invariants connect nonlinear differential equations with computable topological invariants.^[ref]

The dependence on perturbations, including wall crossing when $b_1=1$ and a correction term when $b_1=0$, is made explicit in the analytic construction.^[ref]