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It is maybe simpler to consider all the generators as representations of $SL(2,C)$, so, using spinor indices, you will have : $M^{\alpha \dot \alpha \beta \dot \beta}, P^{\beta \dot \beta}, Q_\alpha, \bar Q^\dot\beta$ Indices are raised and lowered with the Levi-Civita symbols $\epsilon_{\alpha \beta}, \epsilon^{\alpha \beta},\epsilon_{\dot \alpha \dot ... 2 This answer is mostly rephrasing Trimok's correct answer in other words. The super-Poincare group is supposed to be an extension of the Poincare group, which contains the Lorentz group and translations. We will complexify the Lorentz group. The Lie group$G:=SL(2,\mathbb{C})\times SL(2,\mathbb{C})$is (isomorhic to the double cover of) the complexified ... 1 A partial answer: "I do not see a coupling between the MSSM fields and the messenger fields" There are gauge interactions between observable and messenger fields. Let us discuss with fig$(1)$page$11$of this reference. You have not direct interactions between observable sector gauginos ($\lambda$) and s-fermions ($\tilde f\$), and the superfield ...