nLab number of supersymmetries

Idea

In discussion of supersymmetry: the number of generators of odd degree, suitably conceived.

Definition

A supersymmetry super Lie algebra or super Lie group is determined by the underlying bosonic algebra/group (body) and a real spin representation $\mathbf{N}$.

One says that the corresponding number of supersymmetries is either the dimension

$$N \;\coloneqq\; dim_{\mathbb{R}}\big( \mathbf{N} \big)$$

of the real spin representation $\mathbf{N}$, and as such denoted by a roman “$N$”,

or, alternatively, the multiplicity

$$\mathcal{N} \;\coloneqq\; (n_i)_{i \in I}$$

of the irreducible real spin representations $\mathbf{N}^{irr}_{i}$ in a direct sum decomposition

$$\mathbf{N} \;\simeq\; \underset{i \in I}{\oplus} n_i \mathbf{N}^{irr}_i$$

of this real spin representation $\mathbf{N}$, and as such denoted by a list of calligraphic “$\mathcal{N}$”s.

Typically there is either a single irrep or precisely two, in which case these multiplicities are either a single natural number

$$\mathcal{N} \;\in\; \mathbb{N}$$

or a pair of them

$$\mathcal{N} \;=\; (\mathcal{N}_+, \mathcal{N}_-) \,,$$

respectively.

References

See the references at supersymmetry, for instance

• Urs Schreiber, geometry of physics – supersymmetry