Lifshitz holography: the whole shebang

Abstract

Journal of High Energy Physics 2015.1 (2015): 052 reproduced by permission of Scuola Internazionale Superiore di Studi Avanzati (SISSA)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 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 [3], 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 derivativesWC is supported by the SITP, Stanford and the Arab Fund for Economic and Social Development. The work of IP is funded by the Consejo Superior de Investigaciones Científicas and the European Social Fund under the contract JAEDOC068. This work has also been supported by the ESF Holograv Programme, the Spanish Ministry of Economy and Competitiveness under grant FPA2012-32828, Consolider-CPAN (CSD2007-00042), the Spanish MINECO’s “Centro de Excelencia Severo Ochoa” Programme under grant SEV-2012-0249, as well as by the grant HEPHACOS-S2009/ESP1473 from the C.A. de Madri

AdS2 holographic dictionary

“…The fact that Sct, which in this case is an algebraic function of the canonical variables, vanishes on-shell indicates that constant dilaton solutions satisfy a second class constraint. As it was shown in a different context in [71], the boundary counterterms are ambiguous up to second class constraints, but this ambiguity is lifted to linear order in the second class constraints by requiring that the canonical variables themselves are appropriately renormalized. More generically, the renormalization of n-point functions determines the counterterms up to order n in the second class constraints.…”

Quantum critical response: from conformal perturbation theory to holography

“…More generally, we expect that the pressure is related to the O(r d+z ) contributions to f (r), though we note that the holographic renormalization for T µν can require some care, especially in Lifshitz theories [48]. With this particular choice for f (r), we find that for a rather general family of x-independent matter backgrounds that support a Lifshitz geometry, the equations of motion for the metric perturbation h xy , defined in (4.37), are independent of the details of the bulk matter:…”

Disformal transformation in Newton–Cartan geometry

“…These covariant temporal and spatial projectors have been introduced in [52,53] in the context of Lifshitz holography. Their applications can be found in e.g.…”