Chiral field-strength superfield ($W_{\alpha }$)

For a non-Abelian vector superfield, the chiral field-strength superfield isIt obeys ${\bar{D}}_{\dot{\alpha }}{W}_{\alpha }=0$ and transforms covariantly under a supergauge transformation.

For a real Abelian vector superfield, $W_{\alpha }=-{\bar{D}}^2{D}_{\alpha }V/4$ is chiral. The supercovariant derivative algebra with left derivatives gives $D^{\alpha }{\bar{D}}^2{D}_{\alpha }={\bar{D}}_{\dot{\alpha }}{D}^2{\bar{D}}^{\dot{\alpha }}$, proving the displayed reality condition. Its component equations contain the ordinary gauge-theory Bianchi identity, not the Maxwell equations of motion. For a non-Abelian gauge group, replace these by gauge-covariant derivatives and compare both field strengths in the same gauge frame; the identity follows from the graded Jacobi identity of the superconnection.

For an Abelian vector superfield, the chiral field-strength superfield is $W_{\alpha }=-{\bar{D}}^2{D}_{\alpha }V/4$. Three barred supersymmetric covariant derivatives vanish, proving ${\bar{D}}_{\dot{\beta }}{W}_{\alpha }=0$. This is a chiral spinor superfield with a gaugino as its lowest component. Phase and component signs depend on the stated Wess-Zumino gauge convention; the reality of $V$ alone does not fix those phase conventions.

The four-dimensional supersymmetric Yang-Mills action is the chiral superspace integral of $\mathrm{Tr}(W^{\alpha }{W}_{\alpha })$ plus its Hermitian conjugate. Its component fields are a Yang-Mills gauge field, a gaugino and an auxiliary field.