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.
Highlights
The use of holographic techniques in order to gain insight into the strongly coupled dynamics of condensed matter systems has attracted considerable interest in the last few years
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 [1, 2], that are compatible with the null energy condition
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
Summary
The use of holographic techniques in order to gain insight into the strongly coupled dynamics of condensed matter systems has attracted considerable interest in the last few years. In this way there is no need for making an ansatz for the asymptotic solutions of the equations of motion — the asymptotic form is determined algorithmically by integrating order by order the flow equations This is useful in the case of non AdS boundary conditions where the form of the asymptotic expansions is a priori unknown and may even contain multiple scales [40]. This is achieved by covariantly expanding the solution of the HJ equation in simultaneous eigenfunctions of two commuting operators, which as we show are the appropriate generalization of the dilatation operator for anisotropic and non scale invariant boundary conditions.
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