Hamilton-Jacobi Renormalization for Lifshitz Spacetime

TL;DR

This work develops a Hamilton–Jacobi holographic renormalization framework for asymptotically Lifshitz spacetimes by demanding that all divergences be removable with local counterterms. The authors derive boundary conditions from finiteness, identify a Lifshitz scaling anomaly, and construct a local counterterm action within the Einstein–Proca setup (with optional scalar) that renders the on-shell action finite to high orders. They provide nontrivial consistency checks via perturbative calculations of scalar, vector, and tensor sectors, and discuss boundary-data interpretations in terms of sources and vevs, including special cases such as $z=2$. The findings support the view that Lifshitz spacetimes can correspond to well-behaved UV-complete field theories and lay groundwork for further explorations of higher-derivative counterterms and non-relativistic holography.

Abstract

Just like AdS spacetimes, Lifshitz spacetimes require counterterms in order to make the on-shell value of the bulk action finite. We study these counterterms using the Hamilton-Jacobi method. Rather than imposing boundary conditions from the start, we will derive suitable boundary conditions by requiring that divergences can be canceled using only local counterterms. We will demonstrate in examples that this procedure indeed leads to a finite bulk action while at the same time it determines the asymptotic behavior of the fields. This puts more substance to the belief that Lifshitz spacetimes are dual to well-behaved field theories. As a byproduct, we will find the analogue of the conformal anomaly for Lifshitz spacetimes.