# Supersymmetric Noether theorem and supercurrents — invariance requirements

Consider $\mathcal{N}=1,d=4$ SUSY with $n$ chiral superfields $\Phi^i,$ Kaehler potential $K,$ superpotential $W$ and action ($\overline{\Phi}_i$ is complex conjugate of $\Phi^i$) $$S= \int d^4x \left[ \int d^2\theta d^2\overline{\theta} K(\Phi,\overline{\Phi})+ \int d^2\theta W(\Phi) + \int d^2\overline{\theta} \overline{W}(\overline{\Phi})\right].$$

As in Weinberg (QFT, volume 3, page 89), requiring invariance of $K$ and $W$ under \begin{align}\delta\Phi^i &= i\epsilon Q^{i}_{~j}\Phi^j \\ \delta\overline{\Phi}_i &=- i\epsilon \overline{\Phi}_jQ_{~i}^{j}\end{align} with $\epsilon$ small positive constant real parameter and $Q$ hermitian matrix, one obtains some conditions on $K$ and $W,$ namely (denoting $K_i=\frac{\partial K}{\partial\Phi^i}$ and $K^i=\frac{\partial K}{\partial\overline{\Phi}_i},$ and the same for $W$) $$K_i Q^i_{~j} \Phi^j = \overline{\Phi}_j Q^j_{~i} K^i$$ $$W_i Q^i_{~j}\Phi^j=0=\overline{\Phi}_j Q^j_{~i} \overline{W}^i$$ and can then compute the Noether current associated to such transformations, promote $\epsilon$ to a full chiral superfield $\epsilon\Lambda$, etc.

The obtained invariance implies $\delta_{\epsilon}S=0$ for the above transformations.

QUESTION: is the reverse also true? namely, by requiring $\delta_{\epsilon}S=0$ (instead of the apparently stronger invariance condition on $K$ and $W$), does one obtain the same constraints as above on $K$ and $W?$

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$$\delta_\epsilon K(\Phi,\bar{\Phi})=\frac{\partial K(\Phi,\bar{\Phi})}{\partial\Phi^i}\delta_\epsilon \Phi^i+\frac{\partial K(\Phi,\bar{\Phi})}{\partial\bar{\Phi}_i}\delta_\epsilon\bar{\Phi}_i=i\epsilon K_i {Q^i}_j\Phi^j-i\epsilon K^i\bar{\Psi}_j{Q^j}_i,$$
$$\delta_\epsilon W(\Phi)=\frac{\partial W(\Phi)}{\partial \Phi^i}\delta_\epsilon \Phi^i=i\epsilon W_i {Q^i}_j\Phi^j$$
$$\delta_\epsilon \overline{W}(\bar{\Phi})=\frac{\partial \overline{W}(\bar{\Phi})}{\partial \bar{\Phi}_i}\delta_\epsilon \bar{\Phi}_i=-i\epsilon \overline{W}^i\bar{\Psi}_j{Q^j}_i,$$