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The "simplest" link, is NOT to begin with a Superconformal gauge multiplet coupled to a chiral multiplet, but to couple the Weyl multiplet to a superconformal chiral multiplet. It is just a mathematical tool to make your life easier. As a matter of fact, you can take the superconfromal action and make a field redefinition to get the Poincare action, thus ...


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(4.1) is NOT superconformal, it is just conformal. It is invariant under the full conformal group. Conformal = Poincare + Dilatations + Special Conformal Symmetry (4.1) is obviously invariant under Poincare transformations. So, Toine provides you the conformal transformations of the scalar $\phi$ and the metric $g_{\mu\nu}$ in case you want to check that ...



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