Simple d=4 supergravity with a boundary

To construct rigidly or locally supersymmetric bulk-plus-boundary actions, one needs an extension of the usual tensor calculus. Its key ingredients are the extended (F-, D-, etc.) density formulas and the rule for the decomposition of bulk multiplets into (co-dimension one) boundary multiplets. Working out these ingredients for d = 4 N = 1 Poincare supergravity, we discover the special role played by R-symmetry (absent in the d = 3 N = 1 case we studied previously). The U(1) A R-symmetry has to be gauged which leads us to extend the old-minimal set of auxiliary fields S, P, A(mu) by a U(1) A compensator a. Our results include the "F + A" density formula, the "Q + L + A" formula for the induced supersymmetry transformations (closing into the standard d = 3 N = 1 algebra) and demonstration that the compensator a is the first component of the extrinsic curvature multiplet. We rely on the superconformal approach which allows us to perform, in parallel, the same analysis for new-minimal supergravity.