Vacuum moduli space of a supersymmetric gauge theory

The vacuum moduli space is the space of simultaneous F-flat and D-flat configurations modulo gauge transformations. Gauge-invariant chiral operators often provide holomorphic coordinates, subject to algebraic constraints.

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Neutral flat direction with oppositely charged chiral fields ( $W = λ X Φ_{+} Φ_{-}, ϕ_{+} = ϕ_{-} = 0, X arbitrary$ )

 0  0

For canonical chiral superfields, nonzero

λ

, a gauged

U (1)

with charges

(0, + 1, - 1)

, and no Fayet–Iliopoulos term, F-flatness requires the charged product to vanish and D-flatness requires equal charged magnitudes. Together they force both charged scalar values to zero, leaving the neutral scalar arbitrary. The vacuum moduli space of a supersymmetric gauge theory has zero energy, unbroken gauge symmetry and unbroken supersymmetry. At

λ = 0

, equal nonzero charged magnitudes instead permit a gauge-Higgsed supersymmetric branch. A neutral modulus alone does not imply internal gauge breaking.

Higgs branch

 0  0

A Higgs branch is a component on which charged scalar expectation values break some or all of the gauge group. It is often obtained as a Kähler or hyperkähler quotient.

Coulomb branch

 0  0

A Coulomb branch is a component on which scalar expectation values in vector or neutral multiplets leave an Abelian gauge sector unbroken. Charged fields can become massless at special points where additional branches meet it.