nLab
number of supersymmetries

Contents

Contents

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 N\mathbf{N}.

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

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

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

or, alternatively, the multiplicity

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

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

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

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

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

𝒩 \mathcal{N} \;\in\; \mathbb{N}

or a pair of them

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

respectively.

References

See the references at supersymmetry, for instance

Created on May 17, 2019 at 14:53:21. See the history of this page for a list of all contributions to it.