Supersymmetry $\mathcal{N}$ constrained by spacetime dimensions $D$ Given some spacetime dimensions $D$, are there only certain allowed supersymmetry charge nunbers $\mathcal{N}$?
What are the relations of $\mathcal{N}$ and $D$ for the following cases:

*

*When the theory is conformal.


*When the theory does not have to be conformal.


*When the theory is Lorentz invariant.


*When the theory does not have to be Lorentz invariant.
Possible other situations are worthwhile to comment on relating $\mathcal{N}$ and $D$?
 A: All the questions are basically answered in the classic paper "Supersymmetries and their representations". See also the wonderful talk: What's new with Q?.
1.- When the theory is conformal:
In $D=2$ $N=(1,0)$ (heterotic and type I strings), $N=(1,1)$ (type $IIB$ string), $N=(2,0)$ (type $IIA$ string), $N=(2,2)$ ( N=2 strings), $N=(2,1)$ ($N=2$ Heterotic strings) and $N=4$ strings are alloweed.
For the remaining I change the notation to enumerate the number of possible supercharges. In $D=3$ $N=2,4,6,8,10,12,16$ are alloweed. $D=4$ has $N=4,8,12,16$. $D=5$ $N=8$ is the only option and for $D=6$ the options are $N=8$ and 16 supercharges.
2.- No satisfactory answer can exist (to my poor knowledge). See https://arxiv.org/abs/hep-th/9409111 and https://arxiv.org/abs/hep-th/9506101 for interesting subtleties in $D=3$.
To answer 3) and 4): Supersymmetry is the "square root of the Poincaré group". Supersymmetry enforces Poincaré invariance. And basically all the possibilities are the number of supercharges of all string theories and the eleven dimensional supergravity. You can check the precise answers in The String Landscape, the Swampland, and the Missing Corner (page 5).
