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A four-dimensional $\mathcal{N}=4$ supersymmetric gauge theory has one supersymmetric vector multiplet containing a gauge boson, four Weyl spinors, and six real scalar fields in the adjoint group representation. Its massless helicity multiplicities are $1,4,6,4,1$ from helicity $1$ to $-1$.

With the index convention $I(\mathbf{N})=1$ for an $SU(N)$ fundamental, an N=1 supersymmetric gauge theory has $b_0=\frac{3}{2}I(\mathrm{adj})-\frac{1}{2}{\sum }_{i}I(R_i)$. Positive $b_0$ gives asymptotic freedom, while negative $b_0$ makes the gauge interaction infrared free near the Gaussian fixed point.

With canonical normalization, the renormalizable supersymmetric gauge theory hasThe 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 $\int d^2\theta \mathcal{W}(\Phi )+h.c.$, but one commuting field in the fundamental representation of $SU(3)$ admits no nonconstant renormalizable gauge-invariant superpotential: the apparent cubic ${ϵ}_{ijk}{\Phi }^i{\Phi }^j{\Phi }^k$ vanishes.

Seiberg duality states that two different N=1 supersymmetric gauge theories flow to the same infrared quantum field theory. Gauge-invariant operators, global symmetries, 't Hooft anomalies and deformations match even though the ultraviolet gauge groups and elementary fields differ.

In four-dimensional $\mathcal{N}=2$ 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.

The supersymmetric conformal window is the range of flavor numbers for which an asymptotically free supersymmetric gauge theory flows to an interacting infrared fixed point. Its lower edge is often detected when a gauge-invariant chiral operator reaches the scalar unitarity bound.

A supersymmetric gauge theory combines vector superfields with charged matter superfields so that gauge invariance and supersymmetry are both manifest.

Supersymmetric quantum electrodynamics is an Abelian supersymmetric gauge theory with charged chiral multiplets. Its scalar potential combines F-term constraints with the U(1) D-term and any Fayet–Iliopoulos term.

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.

Minimal ten-dimensional supersymmetric gauge theory contains a gauge field and an adjoint Majorana-Weyl spinor. They have eight propagating bosonic and eight propagating fermionic states per gauge generator and sixteen real supercharges. A toroidal zero-mode dimensional reduction to four dimensions gives one gauge field, six real adjoint scalar fields, and four adjoint Weyl spinors: a four-dimensional N=4 super Yang-Mills theory. The internal field strength $F_{mn}=-ig[X_m,X_n]$ gives the nonnegative scalar potential $V=\frac{g^2}{4}{\sum }_{m,n}\Vert [X_m,X_n]{\Vert }^2$, where the norm uses a positive invariant inner product on the Lie algebra. Its minima are commuting tuples of adjoint scalar fields.