nLab R-symmetry

Definition

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 which pairs spinors to vectors 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).

References

• 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:

• Wikipedia, R-Symmetry

Discussion in heterotic M-theory:

• Anthony Ashmore, Sebastian Dumitru, Burt Ovrut, Explicit Soft Supersymmetry Breaking in the Heterotic M-Theory B−L MSSM (arXiv:2012.11029)