The holographic supersymmetric Casimir energy

TL;DR

This work identifies a mismatch between standard holographic renormalization and supersymmetry on AlAdS$_5$ backgrounds, showing that SUSY Ward identities are not preserved by the usual counterterms. It introduces a finite boundary term $\Delta S_{ m new}$ constructed from boundary data and the graviphoton, yielding a SUSY-renormalized action $S_{ m susy}$ that reproduces the field-theory supersymmetric Casimir energy $E_{ m susy}$ and the BPS relation between charges. The authors prove that for $M_4=S^1_\beta\times S^3$ the on-shell action satisfies $S_{ m susy}=\beta E_{ m susy}$ and define a SUSY-corrected charge $Q_{ m susy}$ together with $H_{ m susy}$ and $J_{ m susy}$ that obey $\beta H_{ m susy}=S_{ m susy}$ and $J_{ m susy}=0$, confirming the BPS saturation. These results provide a consistent holographic framework for SUSY observables in 4d SCFTs on curved backgrounds and suggest generalizations to other dimensions and topologies.

Abstract

We consider a general class of asymptotically locally AdS_5 solutions of minimal gauged supergravity, that are dual to superconformal field theories on curved backgrounds S^1 x M_3 preserving two supercharges. We demonstrate that standard holographic renormalization corresponds to a scheme that breaks supersymmetry. We propose new boundary terms that restore supersymmetry, and show that for smooth solutions with topology S^1 x R^4 the improved on-shell action reproduces both the supersymmetric Casimir energy and the field theory BPS relation between charges.