Generalized dilatation operator method for non-relativistic holography

TL;DR

This paper introduces a comprehensive algorithm for constructing the holographic dictionary in non-relativistic Lifshitz backgrounds, accommodating arbitrary dynamical and hyperscaling violation exponents, and addressing unique non-relativistic asymptotic expansions.

Contribution

It develops a covariant algorithm for Lifshitz holography that handles arbitrary exponents and introduces a novel eigenfunction expansion based on non-relativistic grading.

Findings

Established a general asymptotic solution construction method.

Demonstrated the necessity of a two-operator eigenfunction expansion.

Provided a covariant framework for non-relativistic holographic dictionary derivation.

Abstract

We present a general algorithm for constructing the holographic dictionary for Lifshitz and hyperscaling violating Lifshitz backgrounds for any value of the dynamical exponent $z$ and any value of the hyperscaling violation parameter $\theta$ compatible with the null energy condition. The objective of the algorithm is the construction of the general asymptotic solution of the radial Hamilton-Jacobi equation subject to the desired boundary conditions, from which the full dictionary can be subsequently derived. Contrary to the relativistic case, we find that a fully covariant construction of the asymptotic solution for running non-relativistic theories necessitates an expansion in the eigenfunctions of two commuting operators instead of one. This provides a covariant but non-relativistic grading of the expansion, according to the number of time derivatives.