Off-shell supersymmetry of the rigid $N=1$ gauge-matter superspace action

For an even smooth matrix-valued superspace integrand $X$, expand $\delta X$ using [F1]. The odd Leibniz rule and constancy of the parameters give $\delta X=-{\mathrm{\partial }}_{{\theta }^{\alpha }}^L({ϵ}^{\alpha }X)+{\mathrm{\partial }}_{{\bar{\theta }}^{\dot{\alpha }}}^L({\overline{ϵ}}^{\dot{\alpha }}X)+{\mathrm{\partial }}_{\mu }(-i{ϵ}^{\alpha }{\sigma }_{\alpha \dot{\beta }}^{\mu }{\bar{\theta }}^{\dot{\beta }}X+i{\overline{ϵ}}^{\dot{\alpha }}{\theta }^{\beta }{\sigma }_{\beta \dot{\alpha }}^{\mu }X)$. The same identity holds entrywise for matrices because $Q$ acts by derivations.

In a local unitary frame choose complex-linear coordinates $z^A$ on $V_{\mathrm{m}\mathrm{a}\mathrm{t}}$ and write the chiral components as ${\mathrm{\Phi }}^A$. The postulate defines $W_G(\mathrm{\Phi })$ by the finite Taylor polynomial at the body $v(x)$ with nilpotent increments ${\nu }^A={\mathrm{\Phi }}^A-v^A(x)$. Each ${\mathrm{\Phi }}^A$ is chiral by [A1], so ${\bar{D}}_{\dot{\alpha }}{\mathrm{\Phi }}^A=0$ by [F3]. Holomorphicity ${\mathrm{\partial }}_{{\bar{z}}^A}{W}_G=0$ removes barred-coordinate terms from the multivariable chain rule. Applying ${\bar{D}}_{\dot{\alpha }}$ term-by-term to the finite Taylor polynomial and collecting the multi-index terms gives ${\bar{D}}_{\dot{\alpha }}{W}_G(\mathrm{\Phi })={\sum }_A({\mathrm{\partial }}_{z^A}{W}_G)(\mathrm{\Phi }){\bar{D}}_{\dot{\alpha }}{\mathrm{\Phi }}^A=0$. The vector field strength is chiral by [F2]. Therefore both $W_G(\mathrm{\Phi })$ and ${\kappa }_{\mathbb{C}}({\mathcal{W}}^{\alpha },{\mathcal{W}}_{\alpha })$ are chiral even superfields.

For $X={\sum }_a{\bar{\mathrm{\Phi }}}^a{e}^{2g{V}_{R_a}}{\mathrm{\Phi }}^a$, the finite Grassmann expansion has smooth coefficients, and the ordinary finite-dimensional matrix exponential is smooth in the body coefficients; hence step 1.1 applies. The odd-coordinate derivatives have no top Grassmann coefficient by [F4], so their D-term integrals vanish. The spacetime divergence integrates to zero under the compact-support or relative-action boundary condition in [A1]. The physical D bracket is the fixed multiple $-1\mathrm{/}4$ of that raw integral, so every matter D-term is invariant.

For either chiral integrand $Y$ in step 1.2, change to $y=x+i\theta \sigma \bar{\theta }$. By [F1] and [F3], $Q_{\alpha }={\mathrm{\partial }}_{{\theta }^{\alpha }}^L{\mathrm{\mid }}_y$ and ${\bar{Q}}_{\dot{\alpha }}=-{\mathrm{\partial }}_{{\bar{\theta }}^{\dot{\alpha }}}^L{\mathrm{\mid }}_y+2i{\theta }^{\beta }{\sigma }_{\beta \dot{\alpha }}^{\mu }{\mathrm{\partial }}_{y^{\mu }}$. Since $Y$ is independent of $\bar{\theta }$, $\delta Y={ϵ}^{\alpha }{\mathrm{\partial }}_{{\theta }^{\alpha }}^{L}Y+2i{\overline{ϵ}}^{\dot{\alpha }}{\theta }^{\beta }{\sigma }_{\beta \dot{\alpha }}^{\mu }{\mathrm{\partial }}_{\mu }Y$. The first term integrates to zero by Berezin coefficient extraction, and the second is a spacetime divergence with vanishing boundary contribution. The normalized physical F bracket multiplies this integral by $1\mathrm{/}2$, so it also vanishes; the Hermitian-conjugate extracted action obeys the conjugate identity.

The D-term and all chiral F-term variations in steps 2.1 and 2.2 integrate to zero. Their sum is the complete action in [A1], so $\delta S_{\mathrm{G}\mathrm{M}}=0$. The argument uses only the declared superspace variation and auxiliary fields; it invokes no equations of motion, gauge fixing, or quantum construction.