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Witten's Equations

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

$\displaystyle D_A\phi$ $\textstyle =$ $\displaystyle 0$  
$\displaystyle F_+^A$ $\textstyle =$ $\displaystyle i\sigma(\phi,\phi).$  

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


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.

© 1996-9 Eric W. Weisstein