Simple d=4 supergravity with a boundary

TL;DR

This paper develops a framework for constructing supersymmetric actions with boundaries in four-dimensional N=1 supergravity, highlighting the role of R-symmetry and introducing a new compensator field.

Contribution

It extends tensor calculus for supergravity with boundaries, incorporating R-symmetry gauging and new density formulas, and applies the superconformal approach to different supergravity formulations.

Findings

Extended density formulas for boundary actions.

Identification of the extrinsic curvature multiplet.

Demonstration of supersymmetry closure with the new compensator.

Abstract

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 Poincar\'e 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.