superalgebra and (synthetic ) supergeometry
The connected component of the subgroup of outer automorphisms of the super Poincaré group (in some given dimension for some real spin representation (= “number of supersymmetries”)) which fixes the underlying Poincaré group is called its R-symmetry group (e.g. de Wit 02, p. 5-6).
So these are transformations acting on just the spinors, leaving the vectors alone, such that in particular the super-bracket that pairs two spinors to a vector remains invariant.
For the real spin representations in dimension 3,4 and 6 which are induced by the normed division algebra $\mathbb{A} = \mathbb{R}, \mathbb{C}, \mathbb{H}$, respectively (see at division algebra and supersymmetry) and for $N$ supersymmetries (i.e. the spin representation being the direct sum of $N$ copies of a real irrep) then the R-symmetry group at least includes the $N \times N$ $\mathbb{A}$-unitary matrices (Varadarajan 04, section 6.7).
Geometrically, when realizing supergravity theories by KK-compactification of 11-dimensional supergravity then this R-symmetry may be thought of as arising from residual isometries of the compactification space.
In gauged supergravity, in turn, part of the global R-symmetry is promoted to an actual gauge group.
For instance the $SO(8)$ R-symmetry and gauging of 4d supergravity (…say which version…) may be understood geometrically by compactification of 11d supergravity on an $S^7$ fiber (e.g. de Wit 02, section 5).
Daniel Freed, lectures 3 and 5 of Five lectures on supersymmetry, 1998
Bernard de Wit, section 2.5 of Supergravity, 2002 (pdf)
Veeravalli Varadarajan, section 6.7 of Supersymmetry for mathematicians: An introduction, Courant lecture notes in mathematics, American Mathematical Society, Providence, R.I (2004)
See also:
Discussion in heterotic M-theory:
Last revised on December 22, 2020 at 09:21:24. See the history of this page for a list of all contributions to it.