20 - The problem of gauging supersymmetry

Summary

The fundamental symmetries exhibited by the laws of nature are local rather than global, i.e. they are gauged. This is certainly true, say, of SU(3)_color and [SU(2) × U(1)]_electroweak. If the laws of nature turn out to exhibit some form of supersymmetry, it is then natural to ask whether supersymmetry can also be gauged. As we saw, all (N = 1 and N > 1) supersymmetries in four dimensions contain the Poincaré (or de-Sitter) algebra as a subalgebra. Gauging supersymmetry thus implies gauging the Poincaré or de-Sitter algebras. But such a gauge theory of the Poincaré or de-Sitter algebras is the Einstein theory of gravity (Kibble 1961). When gauging supersymmetry, gravity then must be included. This gives us an idea about the gauge fields to be expected when supersymmetry is gauged. Gravity is described by a massless spin two boson. Then at the very least, for local N = 1 supersymmetry we expect this spin two graviton to acquire a supersymmetric partner. Purely on representation–theoretic grounds (see chapter 5) this partner must be a massless fermion of spin three-halves or five-halves. The supersymmetry charges Q_α span a (spin one-half) Majorana spinor. The corresponding gauge field having one vector index beyond that of the charges, will describe spin three-halves, not five-halves. So the N = 1 supergravity multiplet contains one massless spin two boson, the graviton, and one Majorana spin three-halves fermion (Volkov & Akulov 1973) the gravitino.