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3

TL;DR: It is the wedge product $\wedge$ and the exterior derivative/differential $d$ (which squares to zero $d^2=0$) that give rise to Grassmann-odd elements and supersymmetry. More concretely, Ref. 1 first writes down a (non-relativistic) SUSY algebra ${\cal A}$ $$\tag{10} \{Q_1,Q_1\}_+~=~2H~=~\{Q_2,Q_2\}_+, \qquad \{Q_1,Q_2\}_+~=~0,$$ spanned by two ...

2

A bosonic symmetry that acts differently on the different components of a supermultiplet is an R-symmetry. Such a symmetry does not commute with the supercharges. Since the commutator between an R-symmetry and a supercharge gives something Grassmann odd, it has to given another supercharge. Schematically $$[R,Q] = Q$$ or in terms of variations acting on ...

1

A priori, it is hard to know without having any experience with dimensional reductions. One has to get a feeling for how certain quantities change under the process, e.g. how components of the higher-dimensional gauge fields may turn into adjoint scalars, how spinors behave. An interesting thing to note is that symmetries of the lower-dimensional theory ...

1

Yes, you are correct. $\mathcal{N}=2$ supersymmetry in 3d can be obtained by dimensionally reducing 4d $\mathcal{N}=1$. The 4d chiral superfield contains a complex scalar, a Weyl fermion, and a complex auxiliary field. Reducing this to 3d we get a compex scalar, a Dirac fermion, and the auxiliary. You can also impose reality conditions to rewrite the 4d ...

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