Lifshitz holography: The whole shebang

TL;DR

This paper develops a comprehensive method for constructing the holographic dictionary for Lifshitz and hyperscaling violating backgrounds, enabling detailed analysis of dual field theories with non-relativistic scaling and anisotropic features.

Contribution

It introduces a recursive algorithm for solving the radial Hamilton-Jacobi equation in general Lifshitz holography models, including hyperscaling violation, and derives all holographic data and counterterms.

Findings

Derived Fefferman-Graham expansions for Lifshitz backgrounds

Identified Ward identities and holographic renormalization counterterms

Presented exact solutions with specific dynamical and hyperscaling exponents

Abstract

We provide a general algorithm for constructing the holographic dictionary for any asymptotically locally Lifshitz background, with or without hyperscaling violation, and for any values of the dynamical exponents $z$ and $\theta$, as well as the vector hyperscaling violating exponent, that are compatible with the null energy condition. The analysis is carried out for a very general bottom up model of gravity coupled to a massive vector field and a dilaton with arbitrary scalar couplings. The solution of the radial Hamilton-Jacobi equation is obtained recursively in the form of a graded expansion in eigenfunctions of two commuting operators, which are the appropriate generalization of the dilatation operator for non scale invariant and Lorentz violating boundary conditions. The Fefferman-Graham expansions, the sources and 1-point functions of the dual operators, the Ward identities, as well as the local counterterms required for holographic renormalization all follow from this asymptotic solution of the radial Hamilton-Jacobi equation. We also find a family of exact backgrounds with $z>1$ and $\theta>0$ corresponding to a marginal deformation shifting the vector hyperscaling violating parameter and we present an example where the conformal anomaly contains the only $z=2$ conformal invariant in $d=2$ with four spatial derivatives.