R-symmetry (changes) in nLab

Showing changes from revision #6 to #7: Added | Removed | Changed

Contents

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 underlyingnumber 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.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 𝔸=ℝ,ℂ,ℍ\mathbb{A} = \mathbb{R}, \mathbb{C}, \mathbb{H}, respectively (see at division algebra and supersymmetry) and for NN supersymmetries (i.e. the spin representation being the direct sum of NN copies of a real irrep) then the R-symmetry group at least includes the N×NN \times N 𝔸\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)SO(8) R-symmetry and gauging of 4d supergravity (…say which version…) may be understood geometrically by compactification of 11d supergravity on an S 7S^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)