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).

This post imported from StackExchange Physics at 2020-12-07 19:33 (UTC), posted by SE-user Ramiro Hum-Sah