$\mathcal N=2$ AdS$_4$ supergravity, holography and Ward identities

TL;DR

This paper develops a detailed holographic framework for four-dimensional $ ext{AdS}_4$ $ ext{N}=2$ supergravity, including fermionic contributions, boundary symmetries, and Ward identities, extending previous results and clarifying boundary conditions.

Contribution

It generalizes existing holographic supergravity results by incorporating fermionic fields, boundary superconformal currents, and detailed gauge-fixing conditions, providing a comprehensive analysis of boundary Ward identities.

Findings

Superconformal currents satisfy Ward identities with bulk equations of motion.

Super-Schouten tensor includes gravitino bilinears, generalizing the bosonic case.

Vanishing supertorsion constrains boundary sources and super-Schouten tensor structure.

Abstract

We develop in detail the holographic framework for an $\mathcal{N}=2$ pure AdS supergravity model in four dimensions, including all the contributions from the fermionic fields and adopting the Fefferman-Graham parametrization. We work in the first order formalism, where the full superconformal structure can be kept manifest in principle, even if only a part of it is realized as a symmetry on the boundary, while the remainder has a non-linear realization. Our study generalizes the results presented in antecedent literature and includes a general discussion of the gauge-fixing conditions on the bulk fields which yield the asymptotic symmetries at the boundary. We construct the corresponding superconformal currents and show that they satisfy the related Ward identities when the bulk equations of motion are imposed. Consistency of the holographic setup requires the super-AdS curvatures to vanish at the boundary. This determines, in particular, the expression of the super-Schouten tensor of the boundary theory, which generalizes the purely bosonic Schouten tensor of standard gravity by including gravitini bilinears. The same applies to the superpartner of the super-Schouten tensor, the conformino. Furthermore, the vanishing of the supertorsion poses general constraints on the sources of the three-dimensional boundary conformal field theory and requires that the super-Schouten tensor is endowed with an antisymmetric part proportional to a gravitino-squared term.