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Hyperscaling violation : a unified frame for effective holographic theories
 JHEP 1211 (2012) 061, [arXiv:1210.0540
"... We investigate systematic classifications of low energy and lower dimensional effective holographic theories with Lifshitz and Schrödinger scaling symmetries only using metrics in terms of hyperscaling violation (θ) and dynamical (z) exponents. Their consistent parameter spaces are constrained by ..."
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We investigate systematic classifications of low energy and lower dimensional effective holographic theories with Lifshitz and Schrödinger scaling symmetries only using metrics in terms of hyperscaling violation (θ) and dynamical (z) exponents. Their consistent parameter spaces are constrained by null energy and positive specific heat conditions, whose validity is explicitly checked against a previously known result. From dimensional reductions of many microscopic string solutions, we observe the classifications are tied with the number of scales in the original microscopic theories. Conformal theories do not generate a nontrivial θ for a simple sphere reduction. Theories with Lifshitz scaling with one scale are completely fixed by θ and z, and have a universal emblackening factor at finite temperature. Dimensional reduction of intersecting M2M5 requires, we call, spatial anisotropic exponents (]), along with z = 1, θ = 0, because of another scale. Theories with Schrödinger scaling show similar simple classifications at zero temperature, while require more care due to an additional parameter being a thermodynamic variable at finite temperature. ar X iv
Universality and exactness of Schrodinger geometries in string and Mtheory
"... We propose an organizing principle for classifying and constructing Schrödingerinvariant solutions within string theory and Mtheory, based on the idea that such solutions represent nonlinear completions of linearized vector and graviton KaluzaKlein excitations of AdS compactifications. A crucial ..."
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We propose an organizing principle for classifying and constructing Schrödingerinvariant solutions within string theory and Mtheory, based on the idea that such solutions represent nonlinear completions of linearized vector and graviton KaluzaKlein excitations of AdS compactifications. A crucial simplification, derived from the symmetry of AdS, is that the nonlinearities appear only quadratically. Accordingly, every AdS vacuum admits infinite families of Schrödinger deformations parameterized by the dynamical exponent z. We exhibit the ease of finding these solutions by presenting three new constructions: two from M5 branes, both wrapped and extended, and one from the D1D5 (and Sdual F1NS5) system. From the boundary perspective, perturbing a CFT by a null vector operator can lead to nonzero βfunctions for spin2 operators; however, symmetry restricts them to be at most quadratic in couplings. This point of view also allows us to easily prove nonrenormalization theorems: for any Sch(z) solution of twoderivative supergravity constructed in the above manner, z is uncorrected to all orders in higher derivative corrections if the deforming KK mode lies in a short multiplet of an AdS supergroup. Furthermore, we find infinite classes of 1/4 BPS solutions with 4,5 and 7dimensional Schrödinger symmetry that are exact. ar
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