= Solution
The <Noether gauging procedure> begins with a rigid symmetry parameter $\epsilon$. Replacing it by $\epsilon(x)$ leaves in the action variation a term proportional to $j^\mu\partial_\mu\epsilon$. Introduce a <gauge field> whose variation contains $\partial_\mu\epsilon$ and couple it to the <Noether current> $j^\mu$ to cancel that term. The new coupling produces further variations, so both the action and transformation laws are corrected successively in the coupling constant until local invariance is obtained. <Covariant derivatives> organize this iterative completion.
For the <Wess–Zumino model>, the parameter is a spinor and the current is the <supercurrent>. Its derivative part has the structure $\gamma^\nu\partial_\nu(A-i\gamma_5B)\gamma^\mu\psi$, up to the fixed <Majorana spinor> normalization and improvements. The <gauge field> is the <spin>-$3/2$ <gravitino> $\Psi_\mu$, with leading transformation $\delta\Psi_\mu=\kappa^{-1}D_\mu\epsilon+\cdots$. A coupling of $\bar\Psi_\mu$ to that current cancels the derivative of the local parameter. Since the commutator of two local <supersymmetries> includes a spacetime transformation, the <vierbein> and gravity must participate as well. This gives the <Wess-Zumino chiral multiplet coupled to supergravity>.
The terms displayed in the final expression have the following distinct roles:
* The <scalar field> <kinetic term> acquires the invariant volume density $e$ and the inverse <metric tensor>. For a <Lorentz scalar> the spacetime <covariant derivative> is just its ordinary derivative; the metric contraction is what changes from the rigid theory.
* The <Majorana spinor> <kinetic term> uses curved <gamma matrices> $\gamma^\mu=e_a{}^\mu\gamma^a$ and the <spinor covariant derivative>. The factor one half avoids double counting the real Majorana degrees of freedom. The <spin connection> makes the term locally <Lorentz covariant>.
* The curvature term is the <Einstein-Hilbert action>, with gravitational coupling $\kappa$. Its variation cancels the curvature terms generated when <covariant derivatives> are commuted in the <gravitino> variation. Its sign follows the source's curvature and metric conventions.
* The <gravitino> <kinetic term> is the curved-space <Rarita-Schwinger field> action. It supplies the dynamical <gauge field> needed to localize <supersymmetry>, just as a gauge-field kinetic term accompanies an ordinary gauging.
* The term of order $\kappa$ is the <gravitino>–<supercurrent> coupling. Its scalar factor must be differentiated: in standard notation it contains $\not\!D(A-i\gamma_5B)$. The glyph rendered as $\phi$ before the parentheses in the source formula must be read as a slashed derivative, or corrected to one; the undifferentiated scalar product would not be the free <Wess–Zumino model> <supercurrent>.
* The order-$\kappa^2$ term bilinear in <gravitini> couples their chiral current to the <scalar field> phase <Noether current> $J_\mu=A D_\mu B-(D_\mu A)B$.
* The remaining order-$\kappa^2$ term couples the matter <axial current> $\bar\psi\gamma_5\gamma^\mu\psi$ to the same $J_\mu$.
The last two interactions are part of the higher-order local <supersymmetry> completion. They can be organized through the <composite Kähler connection> in the fermion derivatives, or derived by eliminating the corresponding supergravity auxiliary connection. Matter fermions and <gravitini> carry the appropriate chiral Kähler weights, fixing the relative current couplings. They are not new adjustable <Yukawa interactions>. In reading this organization, one must not include the composite connection in $D_\mu$ and then count its separately displayed current terms a second time.
Finally the <spin connection> itself acquires <gravitino-induced torsion> and matter spin-current contributions. Eliminating this algebraic connection, or solving for <auxiliary fields>, can make further fermion contact terms implicit in the chosen <covariant derivatives> and component convention. The key mechanism is \b[current coupling at first order, followed by the covariant and higher-order terms required by local <supersymmetry>], rather than merely changing ordinary derivatives in the original matter Lagrangian.
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