Higher dimensional holography

TL;DR

This work addresses the need for a higher-dimensional holographic renormalization framework beyond KK reductions by evaluating the on-shell Type IIB action on bubbling geometries that are dual to 1/2-BPS surface operators in ${\mathcal N}=4$ SYM. It introduces a 9d boundary counterterm and a defect-adapted Fefferman–Graham-like cut-off to regulate divergences, and shows that the renormalized action reproduces the Euler anomaly $b=3(\dim G-\dim\mathbb L)=3(N^2-\sum_l N_l^2)$ via a vacuum-subtracted log divergence, i.e. $S_{on-shell}-S^{\text{vacuum}}_{on-shell}=\log \Lambda\,(N^2-\sum_l N_l^2)$. This provides a concrete demonstration of higher-dimensional holography for asymptotically $AdS_{d+1}\times M^q$ spaces and suggests a broader framework for holographic renormalization in the presence of defects, with potential extensions to other bubbling geometries and defect observables.

Abstract

The holographic dictionary is well developed for gravity in asymptotically anti de Sitter $A(AdS_{d+1})$ spacetimes. However, this approach is limited, since many physically relevant configurations, such as bubbling geometries dual to heavy operators, do not arise as an uplift of a lower dimensional $A(AdS_{d+1})$ solution. Instead, they are intrinsic solutions of ten- or eleven-dimensional supergravity. In this work, we address this problem by evaluating the on-shell Type IIB supergravity action, with a suitable boundary counterterm, on the bubbling solutions describing half-BPS surface operators in $\mathcal{N}=4$ super Yang-Mills. This calculation exactly reproduces the surface operator Euler conformal anomaly, an observable known exactly in the field theory but previously inaccessible holographically. This example illustrates the need for a higher-dimensional approach to holographic renormalization applicable to general asymptotically $AdS_{d+1} \times M_q$ geometries.