R-symmetry

In theoretical physics, the R-symmetry is an internal symmetry present in some supersymmetric theories which interacts non-trivially with the supersymmetry; it is often described as a symmetry which "rotates the supercharges". More precisely, it is a symmetry which commutes with spacetime symmetries but not with the supersymmetries.

In four spacetime dimensions, the Haag–Łopuszański–Sohnius theorem determines which such symmetries are allowed for Poincaré-invariant theories. For minimal supersymmetry, the R-symmetry group is $\mathrm{U}(1)$. For theories with $𝒩$-extended supersymmetry, it is the non-abelian group $\mathrm{U}(𝒩)$; in the special case of $𝒩=4$, it may also be $\mathrm{SU}(4)$.

A theory which is supersymmetric need not be R-invariant, and the symmetry may be broken at the quantum level even where it is present classically, i.e. the symmetry may be anomalous. Both aspects are exhibited by the Wess-Zumino model, which is only invariant under R-symmetry when the superpotential is monomial, and the R-symmetry is always anomalous. A theory may only be invariant under a subgroup of the R-symmetry; for example, the minimal Supersymmetric Standard Model has only R-parity, i.e. invariance under the discrete ${\mathbb{Z}}_2$ subgroup of $\mathrm{U}(1)$.

History and terminology

In 1974, Wess and Zumino introduced their aforementioned model in which a ${\gamma }_5$ transformation appears in the commutator of two supersymmetry transformations. More realistic models with such a symmetry were introduced in 1975 by Salam and Strathdee and by Fayet, with the latter using the notation "$R$" for the associated tranformation. Parallel to this, Haag, Łopuszański, Sohnius and Dondi showed that in the $𝒩$-extended case, the internal symmetry group acting on the supercharges is enlarged to include $\mathrm{SU}(𝒩)$. The term "R-invariance" was used in early instances, but "R-symmetry" was in use by at least 1978.

Fayet later showed that in physically realistic models, in which the superpartners of known particles must be very massive, the $\mathrm{U}(1)$ R-symmetry must be broken to the ${\mathbb{Z}}_2$ subgroup (R-parity) to allow the gluinos and gravitino to acquire mass. The minimal supersymmetric standard model therefore does not possess R-symmetry, but only R-parity.

Relation to supersymmetry algebras

The R-symmetry group can be considered as a group of (external) automorphisms of the Poincaré superalgebra, and its Lie algebra is contained in more general supersymmetry algebras.

A model invariant under both supersymmetry and conformal transformations (such as the massless Wess-Zumino model) must also be R-invariant because the closure of the superconformal algebra requires R-symmetry terms in the odd-odd brackets.

Gauged R-symmetry

In rigid supersymmetric theories, any R-symmetry is global and cannot be gauged. However, in supergravity theories, in which the supersymmetry is local, R-symmetry can be gauged, though this has to be achieved by "mixing" it with another global symmetry group.

Other dimensions and signatures

The definition of the R-symmetry group depends on the reality structure of the spinor representation, hence it varies with dimension and signature. The R-symmetry group is always isomorphic to a classical Lie group or a product of two such groups.

In Lorentzian signature, we have the following. In dimensions 2, 6 and 10, where the reality condition is Weyl-compatible, there are two inequivalent real spinor modules distinguished by chirality, and the amount of supersymmetry is characterised by a pair of integers $(N_{+},N_{-})$ indicating the number of left- and right-handed sets of supercharges.

R-symmetry groups in Lorentzian signature

Spacetime dimension	Spinor type	R-symmetry group
2	Majorana-Weyl	$\mathrm{SO}({𝒩}_{+})\times \mathrm{SO}({𝒩}_{-})$
3	Majorana	$\mathrm{SO}(𝒩)$
4	Majorana	$\mathrm{U}(𝒩)$
5	Symplectic Majorana	$\mathrm{USp}(2𝒩)$
6	Symplectic Majorana-Weyl	$\mathrm{USp}(2{𝒩}_{+})\times \mathrm{USp}(2{𝒩}_{-})$
7	Symplectic Majorana	$\mathrm{USp}(2𝒩)$
8	Majorana	$\mathrm{U}(𝒩)$
9	Majorana	$\mathrm{SO}(𝒩)$
10	Majorana-Weyl	$\mathrm{SO}({𝒩}_{+})\times \mathrm{SO}({𝒩}_{-})$

Theories with extended supersymmetry generally have a number of scalar fields described by a sigma model, meaning that they can be considered as being coordinates on an abstract homogeneous Riemannian manifold. The R-symmetry can be identified as part of the isotropy of this "scalar manifold", which in turn is often viewed as the internal geometry in a compactification from a higher-dimensional theory (usually string theory or M-theory).

With more general signature or with non-standard choices of squaring map (or equivalently "twisted" reality conditions on spinors) there are more possibilities for the R-symmetry group. They are relevant to the study of time-like T-duality.