Local BRST cohomology in minimal D=4, N=1 supergravity

TL;DR

This work systematizes the local BRST cohomology for D=4, N=1 supergravity in both old and new minimal formulations, including Yang–Mills couplings. By solving s ω4 + d ω3 = 0 through a covariant tilde{s} cohomology and a carefully chosen off-shell and on-shell tensor basis, the paper derives the most general gauge-invariant actions, all on-shell counterterms, candidate gauge anomalies, and nontrivial deformations. It shows old minimal supergravity admits a broad set of actions expressible via superspace integrals, while new minimal contains a few additional exceptional terms (including the supersymmetrized Einstein–Hilbert term and FI terms) that cannot be obtained in the same way; a duality-like deformation maps new minimal to old minimal with gauged R-symmetry. The results demonstrate the absence of supersymmetry anomalies, reveal a highly stable gauge structure under deformations, and provide a comprehensive classification that extends to matter multiplets and more general actions via spectral sequences. The analysis hinges on a QDS structure of the linearized supersymmetry algebra, enabling a tractable cohomological computation and precise identification of invariant functionals, anomalies, and dynamical conservation laws.

Abstract

The local BRST cohomology is computed in old and new minimal supergravity, including the coupling to Yang-Mills gauge multiplets. This covers the determination of all gauge invariant local actions for these models, the classification of all the possible counterterms that are invariant on-shell, of all candidate gauge anomalies, and of the possible nontrivial (continuous) deformations of the standard actions and gauge transformations. Among others it is proved that in old minimal supergravity the most general gauge invariant action can indeed be constructed from well-known superspace integrals, whereas in new minimal supergravity there are only a few additional (but important) contributions which cannot be constructed in this way without further ado. Furthermore the results indicate that supersymmetry itself is not anomalous in minimal supergravity and show that the gauge transformations are extremely stable under consistent deformations of the models. There is however an unusual deformation converting new into old minimal supergravity with local R-invariance which is reminiscent of a duality transformation.