Given a free Majorana field theory in $D$ spacetime dimension. Say $D=2$ to $10$.

> Question: Which $D$ spacetime dimension,do we have a free Majorana field theory with Majorana spinor in the real representation to be supersymmetry? And what is the $\mathcal{N}$?

1. in Lorentz signature $(D-1,1)$

2. in Euclidean signature $(D ,0)$

I think for Lorentz signature $(3,1)$, we have a Majorana field theory to be supersymmetry with $\mathcal{N}=1$ if coupling to SU(2) gauge theory with adjoint rep.

> Do we have other examples?

Useful Table to look at: https://en.wikipedia.org/wiki/Spinor#Summary_in_low_dimensions