Holographic renormalization and supersymmetry

TL;DR

The work addresses whether holographic renormalization can be made supersymmetric in AdS/CFT. It shows that in four dimensions the standard scheme already respects SUSY Ward identities, while in five dimensions it does not, necessitating new finite boundary terms to restore SUSY and reproduce field-theory results such as the supersymmetric Casimir energy. By constructing general AlAdS5 solutions and testing on multiple boundary geometries, the authors demonstrate that the corrected renormalization prescription yields holographic charges and on-shell actions in agreement with localization-based field theory predictions, including BPS relations. The findings have significant implications for how supersymmetric observables are computed holographically and motivate deeper investigations into the origin of the non-standard boundary terms and their applicability to broader classes of theories and dimensions.

Abstract

Holographic renormalization is a systematic procedure for regulating divergences in observables in asymptotically locally AdS spacetimes. For dual boundary field theories which are supersymmetric it is natural to ask whether this defines a supersymmetric renormalization scheme. Recent results in localization have brought this question into sharp focus: rigid supersymmetry on a curved boundary requires specific geometric structures, and general arguments imply that BPS observables, such as the partition function, are invariant under certain deformations of these structures. One can then ask if the dual holographic observables are similarly invariant. We study this question in minimal N = 2 gauged supergravity in four and five dimensions. In four dimensions we show that holographic renormalization precisely reproduces the expected field theory results. In five dimensions we find that no choice of standard holographic counterterms is compatible with supersymmetry, which leads us to introduce novel finite boundary terms. For a class of solutions satisfying certain topological assumptions we provide some independent tests of these new boundary terms, in particular showing that they reproduce the expected VEVs of conserved charges.