Seiberg-Witten theory

Donaldson's instanton invariants were spectacular but brutally hard to compute, because the moduli spaces were noncompact and the equations nonabelian. In 1994 Seiberg and Witten, working in physics, wrote down a different set of equations on four-manifolds that capture much of the same information but are dramatically more tractable: the moduli spaces are compact and the gauge group is abelian. Within months these equations re-proved and extended nearly everything Donaldson theory had done, and they became the standard tool of four-dimensional differential topology.

The setup needs a spin-c structure on a smooth closed oriented 4-manifold X (always available, unlike spin). The unknowns are a pair: a U(1) connection A on the determinant line bundle, and a positive spinor phi (a section of the positive spin-c spinor bundle). The Seiberg-Witten equations are F_A^+ = sigma(phi) and D_A phi = 0 — the self-dual part of the curvature equals a quadratic expression sigma(phi) in the spinor, and the spinor is in the kernel of the twisted Dirac operator D_A. The moduli space M is the set of solutions modulo gauge. Two features make the theory work where Donaldson's struggled: (a) the Weitzenbock/Lichnerowicz formula plus the scalar curvature gives an a priori bound on phi, so M is COMPACT — no Uhlenbeck bubbling to fight; (b) the equation is abelian (gauge group of maps to U(1)), so the analysis is far gentler. For generic metric M is a smooth compact manifold, and the Seiberg-Witten invariant SW(s) is a count of points (or an integral of a cohomology class) over M, depending on the spin-c structure s.

The consequences are sweeping: SW invariants detect exotic smooth structures, prove the failure of the smooth h-cobordism theorem, give the Thom conjecture (the genus-minimizing property of complex curves in CP^2, proved by Kronheimer-Mrowka), and impose strong constraints (e.g. only finitely many basic classes; Kahler and symplectic manifolds have nonzero SW invariants by Taubes' theorem SW = Gromov). Honest caveats. First, SW invariants are conjecturally (Witten's conjecture) equivalent to Donaldson invariants but the two are not literally the same object; do not claim they are identical. Second, the whole theory is special to dimension four — there is no useful analogue in higher dimensions, which is part of why four dimensions remains uniquely mysterious. Third, an important subtlety: when b_2^+(X) = 1 the invariant depends on a chamber (a sign of the metric/perturbation), so the 'invariant' must be stated with that wall-crossing dependence, not as a single number.