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 Incoming links: Supersymmetric gauge theory

A four-dimensional

N = 4

1, 4, 6, 4, 1

1

- 1

With canonical normalization, the renormalizable supersymmetric gauge theory has

L = \int d^{4} θ Φ^{†} e^{2 V} Φ + [\frac{1}{16 g ^{2}} \int d^{2} θ Tr (W^{α} W_{α}) + h.c.] .

The first term is the gauge-covariant Kähler potential and the second is the Supersymmetric Yang-Mills action. A holomorphic superpotential would also be integrated as

\int d^{2} θ W (Φ) + h.c.

S U (3)

admits no nonconstant renormalizable gauge-invariant superpotential: the apparent cubic

ϵ_{ijk} Φ^{i} Φ^{j} Φ^{k}

vanishes.

In four-dimensional

N = 2

supersymmetric gauge theory, Seiberg–Witten theory determines exact low-energy quantities from the geometry of the vacuum parameter space. For rank one, the central charge in supersymmetry is proportional to

n_{e} a + n_{m} a_{D}

, with integral electric and magnetic charges and

a_{D}

the dual scalar period. The BPS bound in supersymmetry fixes the mass of a short state by this charge combination. Its vanishing identifies loci where a magnetic monopole or dyon can become massless, even when the original electric description is strongly coupled.

F_{mn} = - i g [X_{m}, X_{n}]

gives the nonnegative scalar potential

V = \frac{g ^{2}}{4} \sum_{m, n} ∥[X_{m}, X_{n}] ∥^{2}