Four-dimensional supersymmetry counting convention

Let $M_4={\mathbb{R}}^4$ with coordinates $x^0,x^1,x^2,x^3$, Lorentzian metric $\eta =-(d{x}^0{)}^2+{\sum }_{j=1}^3(d{x}^j{)}^2$, orientation $d{x}^0\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$, and time orientation ${\mathrm{\partial }}_{x^0}$. Its orthonormal frame transformations preserving both orientations form $S{O}^{+}(1,3)$; write ${\mathrm{Spin}}^{+}(1,3)$ for its connected double cover.

For this signature, let ${\mathrm{Cl}}_{1,3}^{\mathbb{C}}$ be the complex algebra generated by ${\gamma }_0,\dots ,{\gamma }_3$ subject to ${\gamma }_{\mu }{\gamma }_{\nu }+{\gamma }_{\nu }{\gamma }_{\mu }=2{\eta }_{\mu \nu }I$. A complex Dirac module $S_4^{\mathbb{C}}$ is a four-dimensional irreducible module for this algebra, with the induced ${\mathrm{Spin}}^{+}(1,3)$ action. Supply a ${\mathrm{Spin}}^{+}(1,3)$-equivariant antilinear involution $J:S_4^{\mathbb{C}}\to S_4^{\mathbb{C}}$, meaning $J(\lambda s)=\bar{\lambda }J(s)$ and $J^2=I$. Define the real Majorana module $S_4^{\mathrm{M}}:=\{s\in S_4^{\mathbb{C}}:J(s)=s\}$. It has real dimension $4$: for every $s\in S_4^{\mathbb{C}}$, both $(s+J(s))\mathrm{/}2$ and $(s-J(s))\mathrm{/}(2i)$ lie in $S_4^{\mathrm{M}}$ and $s$ is the first plus $i$ times the second. A four-dimensional Majorana supersymmetry parameter is a vector in $S_4^{\mathrm{M}}$; over $M_4$ its spacetime-dependent version is a section of the associated bundle for a supplied spin structure.

A four-dimensional supersymmetry count $N\in {\mathbb{Z}}_{\ge 0}$ is the number of independent copies of $S_4^{\mathrm{M}}$ in a specified parameter space $\mathcal{Q}\cong (S_4^{\mathrm{M}}{)}^{\oplus N}$ as real ${\mathrm{Spin}}^{+}(1,3)$-modules. Equivalently, ${\dim }_{\mathbb{R}}\mathcal{Q}=4N$ for this specified direct-sum module. In this convention the theory is called $N$-extended; in particular, $32$ real dimensions of such parameters correspond to $N=8$. This is a naming convention for a stated supersymmetry-parameter space; it does not assert that such a space exists in a given compactification.