Holographic renormalization and supersymmetry

TL;DR

This paper investigates how holographic renormalization can be made compatible with supersymmetry in AdS/CFT, introducing new boundary terms in five dimensions to preserve supersymmetric invariance and matching field theory results.

Contribution

It demonstrates that standard holographic renormalization aligns with supersymmetric field theory in four dimensions, but requires novel boundary terms in five dimensions to maintain supersymmetry.

Findings

Holographic renormalization reproduces field theory results in 4D.

Standard counterterms in 5D are incompatible with supersymmetry.

New finite boundary terms in 5D restore supersymmetric invariance.

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.