Holographic Renormalization for Asymptotically Lifshitz Spacetimes

TL;DR

This work develops a holographic renormalization framework for asymptotically Lifshitz spacetimes with a massive vector in four dimensions, focusing on the z=2 case. By identifying AL boundary conditions and constructing two actions—a Minimal Action and an Extended Action—with local boundary counterterms, the authors ensure a well-posed variational principle and finite on-shell action. They compare boundary stress tensors from Brown–York and Hollands–Ishibashi–Marolf formalisms, showing HIM charges are finite and robust under AL boundary data, while BY charges require the extended action for finiteness and are related to HIM charges by a chemical-potential term ΘΔΨ. The Lifshitz topological black hole serves as a concrete application, illustrating energy, entropy, and action relations, and the results illuminate how these holographic constructions connect to dual condensed-matter models and thermodynamics, with future work aimed at general z and broader AL boundary conditions.

Abstract

A variational formulation is given for a theory of gravity coupled to a massive vector in four dimensions, with Asymptotically Lifshitz boundary conditions on the fields. For theories with critical exponent z=2 we obtain a well-defined variational principle by explicitly constructing two actions with local boundary counterterms. As part of our analysis we obtain solutions of these theories on a neighborhood of spatial infinity, study the asymptotic symmetries, and consider different definitions of the boundary stress tensor and associated charges. A constraint on the boundary data for the fields figures prominently in one of our formulations, and in that case the only suitable definition of the boundary stress tensor is due to Hollands, Ishibashi, and Marolf. Their definition naturally emerges from our requirement of finiteness of the action under Hamilton-Jacobi variations of the fields. A second, more general variational principle also allows the Brown-York definition of a boundary stress tensor.