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Lifshitz holography: The whole shebang

Wissam Chemissany, Ioannis Papadimitriou

TL;DR

The paper develops a comprehensive, algorithmic holographic dictionary for asymptotically Lifshitz and hvLf spacetimes by solving the radial Hamilton-Jacobi equation with a covariant, double expansion in derivative and Lifshitz-constraint eigenfunctions. It provides a systematic way to obtain Fefferman-Graham expansions, identify sources and 1-point functions, and derive Ward identities and counterterms without resorting to second-order equations of motion. The approach applies to generic z and θ obeying the NEC, and is demonstrated through Lifshitz and hvLf backgrounds, plus explicit Einstein-Proca and exponential-potential examples. The framework also yields exact marginal deformations and reveals a conformal anomaly for z=2 in d=2 that depends on the frame, illustrating the method’s power to generate non-relativistic conformal invariants and holographic data for a wide class of non-AdS geometries.

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 $θ$, 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 $θ>0$ 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.

Lifshitz holography: The whole shebang

TL;DR

The paper develops a comprehensive, algorithmic holographic dictionary for asymptotically Lifshitz and hvLf spacetimes by solving the radial Hamilton-Jacobi equation with a covariant, double expansion in derivative and Lifshitz-constraint eigenfunctions. It provides a systematic way to obtain Fefferman-Graham expansions, identify sources and 1-point functions, and derive Ward identities and counterterms without resorting to second-order equations of motion. The approach applies to generic z and θ obeying the NEC, and is demonstrated through Lifshitz and hvLf backgrounds, plus explicit Einstein-Proca and exponential-potential examples. The framework also yields exact marginal deformations and reveals a conformal anomaly for z=2 in d=2 that depends on the frame, illustrating the method’s power to generate non-relativistic conformal invariants and holographic data for a wide class of non-AdS geometries.

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 and , 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 and corresponding to a marginal deformation shifting the vector hyperscaling violating parameter and we present an example where the conformal anomaly contains the only conformal invariant in with four spatial derivatives.

Paper Structure

This paper contains 28 sections, 1 theorem, 361 equations, 12 tables.

Table of Contents

  1. Introduction
  2. The model and radial Hamiltonian formalism
  3. Holography for homogeneous anisotropic backgrounds
  4. Hamiltonian algorithm for the holographic dictionary
  5. Lif solutions
  6. hvLf solutions
  7. Weyl transforming hvLf solutions to Lif solutions
  8. Lif boundary conditions as a second class constraint
  9. Fefferman-Graham expansions and anisotropic RG flows from a superpotential
  10. Recursive solution of the HJ equation for asymptotically locally Lif backgrounds
  11. Locally Lif boundary conditions
  12. Graded expansion in eigenfunctions of the derivative and gradation operators
  13. Taylor expansion in the Lifshitz constraint
  14. Solving the recursion equations
  15. Solution at order $k=1$
  16. ...and 13 more sections

Key Result

Lemma B.1

For any $\varrho[\varphi]$ such that where we have i.e. $v_\varphi^i=0$.

Theorems & Definitions (1)

  • Lemma B.1