Witten's Equations

Also called the Seiberg-Witten Invariants. For a connection $A$ and a Positive Spinor $\phi\in\Gamma(V_+)$,

$$D_A\phi = 0$$

$$F_+^A = i\sigma(\phi,\phi).$$

The solutions are called monopoles and are the minima of the functional

$$\int_X (\vert F_+^A-i\sigma(\phi,\phi)\vert^2+\vert D_A\phi\vert^2).$$

See also Lichnerowicz Formula, Lichnerowicz-Weitzenbock Formula, Seiberg-Witten Equations

References

Cipra, B. ``A Tale of Two Theories.'' What's Happening in the Mathematical Sciences, 1995-1996, Vol. 3. Providence, RI: Amer. Math. Soc., pp. 14-25, 1996.

Donaldson, S. K. ``The Seiberg-Witten Equations and 4-Manifold Topology.'' Bull. Amer. Math. Soc. 33, 45-70, 1996.

Kotschick, D. ``Gauge Theory is Dead!--Long Live Gauge Theory!'' Not. Amer. Math. Soc. 42, 335-338, 1995.

Seiberg, N. and Witten, E. ``Monopoles, Duality, and Chiral Symmetry Breaking in $N=2$ Supersymmetric QCD.'' Nucl. Phys. B 431, 581-640, 1994.

Witten, E. ``Monopoles and 4-Manifolds.'' Math. Res. Let. 1, 769-796, 1994.