Seiberg–Witten invariants

In mathematics, and especially gauge theory, Seiberg–Witten invariants are invariants of compact smooth oriented 4-manifolds introduced by Edward Witten (1994), using the Seiberg–Witten theory studied by Nathan 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 Seiberg–Witten moduli space of solutions of the Seiberg–Witten equations up to gauge tends to be compact, so one avoids the hard problems involved in compactifying the Yang–Mills moduli space of solutions of the Yang–Mills equations up to gauge in Donaldson theory. Allegedly Eric Weinstein discovered the Seiberg–Witten equations prior to Edward Witten in 1987.

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).

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