Noether superpotentials in supergravities

TL;DR

The paper tackles the problem that a naïve Noether-based superpotential in supergravity yields only half of the true supercharge. By applying a Lagrangian criterion due to Si, the authors derive the correct superpotential $U^{\mu\nu}_{\bar{\epsilon}} = - i \bar{\epsilon}^A \gamma^{\mu\nu\sigma} \psi^A_\sigma$ and show its consistency with the Hamiltonian (RT) framework, establishing equivalence between first- and second-order formalisms. They demonstrate that the Nester-Witten term $H_{\xi}$ embodies the gravitational contribution while electric and magnetic charges $Q$ and $P$ appear as central charges in the ${\cal N}_4=2$ SUSY algebra, including a magnetic central charge under appropriate boundary conditions. The Appendices provide the boundary data needed to realize these charges in the presence of monopoles, clarifying the structure of asymptotic charges in supergravity. This work solidifies the correct definition of conserved charges and their algebra in supergravity theories, with explicit implications for magnetic charges and boundary conditions.

Abstract

Straightforward application of the standard Noether method in supergravity theories yields an incorrect superpotential for local supersymmetry transformations, which gives only half of the correct supercharge. We show how to derive the correct superpotential through Lagrangian methods, by applying a criterion proposed recently by one of us. We verify the equivalence with the Hamiltonian formalism. It is also indicated why the first-order and second-order formalisms lead to the same superpotential. We rederive in particular the central extension by the magnetic charge of the ${\cal N}_4 =2$ algebra of SUGRA asymptotic charges.