Seiberg–Witten invariant

In mathematics, Seiberg–Witten invariants are invariants of compact smooth 4-manifolds introduced by Witten (1994), using the Seiberg–Witten theory studied by Seiberg and Witten (1994a, 1994b) during their investigations of Seiberg–Witten gauge theory.

Seiberg–Witten invariants are similar to Donaldson invariants and can be used to prove similar (but sometimes slightly stronger) results about smooth 4-manifolds. They are technically much easier to work with than Donaldson invariants; for example, the moduli spaces of solutions of the Seiberg–Witten equations tend to be compact, so one avoids the hard problems involved in compactifying the moduli spaces in Donaldson theory.

For detailed descriptions of Seiberg–Witten invariants see (Donaldson 1996), (Moore 2001), (Morgan 1996), (Nicolaescu 2000), (Scorpan 2005, Chapter 10). For the relation to symplectic manifolds and Gromov–Witten invariants see (Taubes 2000). For the early history see (Jackson 1995).

The Seiberg-Witten equations depend on the choice of a complex spin structure, Spin^c, on a 4-manifold M. In 4 dimensions the group Spin^c is

and there is a homomorphism from it to SO(4). A Spin^c-structure on M is a lift of the natural SO(4) structure on the tangent bundle (given by the Riemannian metric and orientation) to the group Spin^c. Every smooth compact 4-manifold M has Spin^c-structures (though most do not have spin structures).

Fix a smooth compact 4-manifold M, choose a spin^c-structure s on M, and write W⁺, W⁻ for the associated spinor bundles, and L for the determinant line bundle. Write φ for a self-dual spinor field (a section of W⁺) and A for a U(1) connection on L. The Seiberg–Witten equations for (φ,A) are

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