Asymptotic generators of fermionic charges and boundary conditions preserving supersymmetry

TL;DR

This paper develops a covariant phase space approach to define Hamiltonian generators for bosonic and fermionic symmetries in supergravity, establishing criteria for boundary conditions that preserve supersymmetry and analyzing their implications in AdS theories.

Contribution

It provides a general prescription for boundary conditions that preserve supersymmetry and defines conserved generators in supergravity theories, including novel boundary superpotentials.

Findings

Unique boundary conditions for supercharges in minimal AdS supergravity.

Multiple supersymmetry-preserving boundary conditions in extended supergravity.

Energy positivity holds under all supersymmetry-preserving boundary conditions.

Abstract

We use a covariant phase space formalism to give a general prescription for defining Hamiltonian generators of bosonic and fermionic symmetries in diffeomorphism invariant theories, such as supergravities. A simple and general criterion is derived for a choice of boundary condition to lead to conserved generators of the symmetries on the phase space. In particular, this provides a criterion for the preservation of supersymmetries. For bosonic symmetries corresponding to diffeomorphisms, our prescription coincides with the method of Wald et al. We then illustrate these methods in the case of certain supergravity theories in $d=4$. In minimal AdS supergravity, the boundary conditions such that the supercharges exist as Hamiltonian generators of supersymmetry transformations are unique within the usual framework in which the boundary metric is fixed. In extended ${\mathcal N}=4$ AdS supergravity, or more generally in the presence of chiral matter superfields, we find that there exist many boundary conditions preserving ${\mathcal N}=1$ supersymmetry for which corresponding generators exist. These choices are shown to correspond to a choice of certain arbitrary boundary ``superpotentials,'' for suitably defined ``boundary superfields.'' We also derive corresponding formulae for the conserved bosonic charges, such as energy, in those theories, and we argue that energy is always positive, for any supersymmetry-preserving boundary conditions. We finally comment on the relevance and interpretation of our results within the AdS-CFT correspondence.