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The contraction of invariant vector and spinor frames makes $H_3$ invariant under the Super-Poincaré group. Odd-coordinate differentials commute in the graded exterior algebra. The four-dimensional Fierz rearrangement identity $(C{\gamma }^m{)}_{(\alpha \beta }(C{\gamma }_m{)}_{\gamma \delta )}=0$ consequently gives $d{H}_3=i(d\bar{\theta }{\gamma }^{m}d\theta )(d\bar{\theta }{\gamma }_{m}d\theta )=0$. Its three-form degree is distinct from the four-form needed in the four-dimensional supermembrane Wess-Zumino coupling.

Anticommuting-variable dependence alone is not the defining condition: a superfield must carry a specified Super-Poincaré group transformation law. Any appropriately graded function on superspace becomes an unconstrained scalar superfield under coordinate pullback, but an arbitrary collection of coefficients in a Grassmann variable polynomial is not automatically a prescribed supermultiplet. For example, a chiral superfield has $\Phi (y,\theta )=A(y)+\sqrt{2}\theta \psi (y)+{\theta }^{2}F(y)$, with $y^{\mu }=x^{\mu }+i\theta {\sigma }^{\mu }\bar{\theta }$ and ${\bar{D}}_{\dot{\alpha }}\Phi =0$; a vector superfield obeys $V=V^{†}$ and contains a gauge vector, gaugino and auxiliary field in Wess-Zumino gauge. A polynomial with constant component fields can be a special chiral superfield; lack of explicit spacetime dependence alone does not disqualify it. This is superfield covariance is not Grassmann dependence alone.

The supersymmetry algebra closes on spacetime translations. Its supercharges are Weyl spinors, andThus the composition of two odd supersymmetry transformations moves the spacetime argument. The generators also change spin and connect bosons with fermions. They extend the Poincaré group to the Super-Poincaré group, rather than forming an internal symmetry that commutes with spacetime transformations.

Use the supertranslation-invariant superspace coframeIt is a basis because the change from $(dX,d\theta )$ is triangular with identity diagonal blocks. Under a rigid supertranslation, $\delta \theta =ϵ$, $\delta X^m=-i\overline{ϵ}{\gamma }^m\theta$ and $dϵ=0$. Therefore $\delta {\Pi }^m=-i\overline{ϵ}{\gamma }^{m}d\theta +i\overline{ϵ}{\gamma }^{m}d\theta =0$, and $\delta d\theta =0$.

Writing wedge products explicitly, the three-form isBoth factors are supertranslation invariant. Under a Lorentz transformation, ${\Pi }^m$ and the bilinear in $d\theta$ transform as vectors, so their contraction is invariant. Thus $H_3$ is invariant under the Super-Poincaré group.

The Maurer-Cartan equation is $d{\Pi }^m=id\bar{\theta }{\gamma }^{m}d\theta$. Since the other factor is closed,The required four-dimensional Fierz rearrangement identity, which may be stated without proof, isDifferentials of odd Grassmann variables commute in the graded exterior algebra. Thus the coefficient of the product of four $d\theta$ factors is the fully symmetric spinor-index part appearing in this identity. It vanishes, provingThis is the closed three-form on four-dimensional superspace; it should not be confused with the different closed four-form used for a membrane coupling in Question 10.

A superfield is a field on superspace with a specified Super-Poincaré group transformation law. Being a polynomial in anticommuting coordinates alone does not specify this law or make coefficients a supermultiplet. Conversely, a suitably graded function on full superspace, acted on by coordinate pullback, is an unconstrained scalar superfield. A polynomial with spacetime-independent coefficients can be a special constant-component chiral superfield; dependence on spacetime is not itself the defining criterion. Chirality and reality impose additional conditions.

The Super-Poincaré group combines Lorentz transformations and supertranslations. Its even part acts by spin-group transformations and ordinary spacetime translations; odd parameters generate the supercharges. Their graded commutator closes on translations through the Super-Poincaré algebra. A superspace tensor built from invariant coframes and Lorentz-invariant contractions is invariant under the full group.