number of supersymmetries

**superalgebra** and (synthetic ) **supergeometry**

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

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.

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.