Surface counterterms and boundary stress-energy tensors for asymptotically non-anti-de Sitter spaces

TL;DR

This work generalizes holographic renormalization to spacetimes that are not asymptotically AdS by using surface counterterms tied to the dilaton potential via an effective length scale $l_{\rm eff}$. By adding a counterterm $S_{\rm ct}$ with coefficient $c_0$, the authors obtain finite boundary stress-energy tensors and Euclidean actions for domain-wall black holes, 3D dilaton black holes, and topological dilaton black holes, and they verify consistency with direct D$p$-brane calculations where applicable. The study reveals new thermodynamic features, such as Hawking-Page-like transitions in hyperbolic dilaton backgrounds and topology-dependent stability, illustrating how dilaton dynamics modify holographic boundary data and black hole thermodynamics. Overall, the results extend the scope of holographic renormalization to a broader class of non-AdS geometries, enabling well-defined bulk-boundary thermodynamics and potential QFT interpretations on domain walls or curved boundaries.

Abstract

For spaces which are not asymptotically anti-de Sitter where the asymptotic behavior is deformed by replacing the cosmological constant by a dilaton scalar potential, we show that it is possible to have well-defined boundary stress-energy tensors and finite Euclidean actions by adding appropriate surface counterterms. We illustrate the method by the examples of domain-wall black holes in gauged supergravities, three-dimensional dilaton black holes and topological dilaton black holes in four dimensions. We calculate the boundary stress-energy tensor and Euclidean action of these black configurations and discuss their thermodynamics. We find new features of topological black hole thermodynamics.