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6

You need to be a bit careful about the counting of supercharges. In four dimensions, the smallest spinor representation is four-dimensional (over the real numbers), so $\mathcal{N}=1$ supersymmetry has four supercharges. When there are more supercharges, there are simply more states combined into a single 'multiplet'. For example, in $\mathcal{N}=1$ ...

4

If the integral $I:=\int d\theta$ on the algebra ${\cal A}$ of superfunctions $f(\theta)=\theta a + b$ should be 1) a (graded) linear operation, 2) translation invariant, i.e., $\int d\theta ~f(\theta+\theta') =\int d\theta~f(\theta)$, 3) and if the output $\int d\theta~ f(\theta)$ should not depend on the integration variable $\theta$, then it is ...

2

So I asked Jeff Harvey about this, and this is what he told me. "I suppose one could figure it out by first looking at whether there is an embedding of the bosonic subalgebra and if that works going on to look at the fermionic generators and checking if one can piece the two parts of the embedding together."

1

Here I will just make a couple of general remarks. 1) Graded algebras usually refer to $\mathbb{Z}$- or $\mathbb{N}$-graded algebras, while superalgebras are $\mathbb{Z}_2$-graded algebras. 2) Grassmann numbers are oddly graded supernumbers. Please click on the links for further information, important properties and references. References: 1) Bryce ...

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