Supersymmetric vacuum

A vacuum of a globally supersymmetric gauge theory preserves supersymmetry exactly when every auxiliary F-field and D-field can vanish simultaneously. Its vacuum energy is then zero because the scalar potential is a sum of nonnegative squares.

F-flatness is the system $\partial W/\partial {\varphi }_i=0$ obtained by setting every chiral-multiplet auxiliary field to zero.

D-flatness sets every vector-multiplet auxiliary field to zero. Quotienting its solution set by the compact gauge group is equivalent, under standard stability conditions, to quotienting the F-flat variety by the complexified gauge group.

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.

For canonical chiral superfields, nonzero $\lambda$, 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 $\lambda =0$, equal nonzero charged magnitudes instead permit a gauge-Higgsed supersymmetric branch. A neutral modulus alone does not imply internal gauge breaking.

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.

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.