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Witten's Equations

For a connection A and a positive spinor phi in Gamma(V_+), Witten's equations (also called the Seiberg-Witten invariants) are given by

D_Aphi=0
(1)
F_+^A=isigma(phi,phi).
(2)

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

int_X(|F_+^A-isigma(phi,phi)|^2+|D_Aphi|^2).
(3)

See also

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

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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.

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Witten's Equations

Cite this as:

Weisstein, Eric W. "Witten's Equations." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WittensEquations.html

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