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Logarithmic twoPoint Correlation Functions from a z = 2 Lifshitz Model
"... The Einstein–Proca action is known to have asymptotically locally Lifshitz spacetimes as classical solutions. For dynamical exponent z = 2, twopoint correlation functions for fluctuations around such a geometry are derived analytically. It is found that the retarded correlators are stable in the se ..."
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The Einstein–Proca action is known to have asymptotically locally Lifshitz spacetimes as classical solutions. For dynamical exponent z = 2, twopoint correlation functions for fluctuations around such a geometry are derived analytically. It is found that the retarded correlators are stable in the sense that all quasinormal modes are situated in the lower halfplane of complex frequencies. Correlators in the longitudinal channel exhibit features that are reminiscent of a structure usually obtained in field theories that are logarithmic, i.e. contain an indecomposable highest weight representation. This suggests the model at hand as a candidate for a gravity dual of a logarithmic field theory with anisotropic scaling symmetry.
Boundary StressEnergy Tensor and NewtonCartan Geometry
 in Lifshitz Holography, JHEP 1401 (2014) 057. arXiv:1311.6471, doi:10.1007/JHEP01(2014)057
"... For a specific action supporting z = 2 Lifshitz geometries we identify the Lifshitz UV completion by solving for the most general solution near the Lifshitz boundary. We identify all the sources as leading components of bulk fields which requires a vielbein formalism. This includes two linear combin ..."
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For a specific action supporting z = 2 Lifshitz geometries we identify the Lifshitz UV completion by solving for the most general solution near the Lifshitz boundary. We identify all the sources as leading components of bulk fields which requires a vielbein formalism. This includes two linear combinations of the bulk gauge field and timelike vielbein where one asymptotes to the boundary timelike vielbein and the other to the boundary gauge field. The geometry induced from the bulk onto the boundary is a novel extension of Newton–Cartan geometry that we call torsional Newton–Cartan (TNC) geometry. There is a constraint on the sources but its pairing with a Ward identity allows one to reduce the variation of the onshell action to unconstrained sources. We compute all the vevs along with their Ward identities and derive conditions for the boundary theory to admit conserved currents obtained by contracting the boundary stressenergy tensor with a TNC analogue of a conformal Killing vector. We also obtain the anisotropic Weyl anomaly that takes the form of a Hořava–Lifshitz action defined on a TNC geometry. The Fefferman–Graham expansion contains a free function that does not appear in the variation of the onshell action. We show that this is related to an irrelevant deformation that selects between two different UV completions. ar X iv
UvADARE (Digital Academic Repository) On the Marginally Relevant Operator in z = 2 Lifshitz Holography
"... Abstract: We study holographic renormalization and RG flow in a stronglycoupled Lifshitztype theory in 2+1 dimensions with dynamical exponent z = 2. The bottomup gravity dual we use is 3+1 dimensional Einstein gravity coupled to a massive vector field. This model contains a marginally relevant o ..."
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Abstract: We study holographic renormalization and RG flow in a stronglycoupled Lifshitztype theory in 2+1 dimensions with dynamical exponent z = 2. The bottomup gravity dual we use is 3+1 dimensional Einstein gravity coupled to a massive vector field. This model contains a marginally relevant operator around the Lifshitz fixed point. We show how holographic renormalization works in the presence of this marginally relevant operator without the need to introduce explicitly cutoffdependent counterterms. A simple closedform expression is found for the renormalized onshell action. We also discuss how asymptotically Lifshitz geometries flow to AdS in the interior due to the marginally relevant operator. We study the behavior of the renormalized entanglement entropy and confirm that it decreases monotonically along the LifshitztoAdS RG flow.
From geometry towards Holography For Schrödinger Spacetimes
, 2012
"... We study various geometrical aspects of Schrödinger spacetimes. For z = 2 we obtain a global coordinate system and we explain the relations among its geodesic completeness, the choice of global time, and the harmonic trapping of nonrelativistic CFTs. We analyse the causal structure by probing the ..."
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We study various geometrical aspects of Schrödinger spacetimes. For z = 2 we obtain a global coordinate system and we explain the relations among its geodesic completeness, the choice of global time, and the harmonic trapping of nonrelativistic CFTs. We analyse the causal structure by probing the spacetime with point particles as well as with scalar fields. We show that, even though there is no global time function in the technical sense (Schrödinger spacetime being nondistinguishing), the time coordinate of the global Schrödinger coordinate system is, in a precise way, the closest one can get to having such a time function. In spite of this and the corresponding strongly Galilean and almost pathological causal structure of this spacetime, it is nevertheless possible to define a Hilbert space of normalisable scalar modes with a welldefined timeevolution. We also discuss how the Galilean causal structure is reflected and encoded in the scalar Wightman functions and the bulktobulk propagator. Furthermore, we study the complete class of 5dimensional asymptotically Schrödinger spacetimes that can be obtained as the TsT transform of an asymptotically AdS spacetime. We use a Fefferman–Graham type expansion to analyse the onshell action for
GKDimension of Birationally . . .
, 2007
"... Let k be an algebraically closed field, let K/k be a finitely generated field extension of transcendence degree 2 with automorphism σ ∈ Autk(K), and let A ⊆ Q = K[t; σ] be an Ngraded subalgebra ..."
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Let k be an algebraically closed field, let K/k be a finitely generated field extension of transcendence degree 2 with automorphism σ ∈ Autk(K), and let A ⊆ Q = K[t; σ] be an Ngraded subalgebra
Lifshitz holography: The whole shebang
"... 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 [1], that are ..."
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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 [1], 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 HamiltonJacobi equation is obtained recursively in the form of a graded expansion in eigenfunctions of two commuting operators [2], which are the appropriate generalization of the dilatation operator for non scale invariant and Lorentz violating boundary conditions. The FeffermanGraham expansions, the sources and 1point 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 HamiltonJacobi 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.