Holographic renormalization for irrelevant operators and multi-trace counterterms

We investigate the structure of holographic renormalization in the presence of sources for irrelevant operators. By working perturbatively in the sources we avoid issues related to the non-renormalizability of the dual field theory. We find new classes of divergences which appear to be non-local on the gravity side. However in all cases a systematic renormalization procedure exists involving either standard local counterterms or new counterterms which may be interpreted as multi-trace counterterms in the field theory. The multi-trace counterterms reflect a more intricate relation between sources and the asymptotics of bulk fields.Comment: 42 page

A holographic model for the fractional quantum Hall effect

Experimental data for fractional quantum Hall systems can to a large extent be explained by assuming the existence of a modular symmetry group commuting with the renormalization group flow and hence mapping different phases of two-dimensional electron gases into each other. Based on this insight, we construct a phenomenological holographic model which captures many features of the fractional quantum Hall effect. Using an SL(2,Z)-invariant Einstein-Maxwell-axio-dilaton theory capturing the important modular transformation properties of quantum Hall physics, we find dyonic diatonic black hole solutions which are gapped and have a Hall conductivity equal to the filling fraction, as expected for quantum Hall states. We also provide several technical results on the general behavior of the gauge field fluctuations around these dyonic dilatonic black hole solutions: We specify a sufficient criterion for IR normalizability of the fluctuations, demonstrate the preservation of the gap under the SL(2,Z) action, and prove that the singularity of the fluctuation problem in the presence of a magnetic field is an accessory singularity. We finish with a preliminary investigation of the possible IR scaling solutions of our model and some speculations on how they could be important for the observed universality of quantum Hall transitions.Comment: 86 pages, 16 figures; v.2 references added, typos fixed, improved discussion of ref. [39]; v.3 more references added and typos fixed, several statements clarified, v.4 version accepted for publication in JHE

Quasinormal modes and holographic correlators in a crunching AdS geometry

We calculate frequency space holographic correlators in an asymptotically AdS crunching background, dual to a relevant deformation of the M2-brane CFT placed in de Sitter spacetime. For massless bulk scalars, exploiting the connection to a solvable supersymmetric quantum mechanical problem, we obtain the exact frequency space correlator for the dual operator in the deformed CFT. Controlling the shape of the crunching surface in the Penrose diagram by smoothly dialling the deformation from zero to infinity, we observe that in the large deformation limit the Penrose diagram becomes a `square', and the exact holographic correlators display striking similarities to their counterparts in the BTZ black hole and its higher dimensional generalisations. We numerically determine quasinormal poles for relevant and irrelevant operators, and find an intricate pattern of these in the complex frequency plane. In the case of relevant operators, the deformation parameter has an infinite sequence of critical values, each one characterised by a pair of poles colliding and moving away from the imaginary frequency axis with increasing deformation. In the limit of infinite deformation all scalar operators have identical quasinormal spectra. We compare and contrast our strongly coupled de Sitter QFT results with strongly coupled thermal correlators from AdS black holes

Probing crunching AdS cosmologies

Holographic gravity duals of deformations of CFTs formulated on de Sitter spacetime contain FRW geometries behind a horizon, with cosmological big crunch singularities. Using a specific analytically tractable solution within a particular single scalar truncation of N=8 supergravity on AdS_4, we first probe such crunching cosmologies with spacelike radial geodesics that compute spatially antipodal correlators of large dimension boundary operators. At late times, the geodesics lie on the FRW slice of maximal expansion behind the horizon. The late time two-point functions factorise, and when transformed to the Einstein static universe, they exhibit a temporal non-analyticity determined by the maximal value of the scale factor a_max. Radial geodesics connecting antipodal points necessarily have de Sitter energy E < a_max, while geodesics with E > a_max terminate at the crunch, the two categories of geodesics being separated by the maximal expansion slice.The spacelike crunch singularity is curved ``outward'' in the Penrose diagram for the deformed AdS backgrounds, and thus geodesic limits of the antipodal correlators do not directly probe the crunch. Beyond the geodesic limit, we point out that the scalar wave equation, analytically continued into the FRW patch, has a potential which is singular at the crunch along with complex WKB turning points in the vicinity of the FRW crunch. We then argue that the frequency space Green's function has a branch point determined by a_max which corresponds to the lowest quasinormal frequency

Holography for chiral scale-invariant models

Deformation of any d-dimensional conformal field theory by a constant null source for a vector operator of dimension (d + z -1) is exactly marginal with respect to anisotropic scale invariance, of dynamical exponent z. The holographic duals to such deformations are AdS plane waves, with z=2 being the Schrodinger geometry. In this paper we explore holography for such chiral scale-invariant models. The special case of z=0 can be realized with gravity coupled to a scalar, and is of particular interest since it is related to a Lifshitz theory with dynamical exponent two upon dimensional reduction. We show however that the corresponding reduction of the dual field theory is along a null circle, and thus the Lifshitz theory arises upon discrete light cone quantization of an anisotropic scale invariant field theory.Comment: 62 pages; v2, published version, minor improvements and references adde

Holographic Hall conductivities from dyonic backgrounds

We develop a general framework for computing the holographic 2-point functions and the corresponding conductivities in asymptotically locally AdS backgrounds with an electric charge density, a constant magentic field, and possibly non-trivial scalar profiles, for a broad class of Einstein-Maxwell-Axion-Dilaton theories, including certain Chern-Simons terms. Holographic renormalization is carried out for any theory in this class and the computation of the renormalized AC conductivities at zero spatial momentum is reduced to solving a single decoupled first order Riccati equation. Moreover, we develop a first order fake supergravity formulalism for dyonic renormalization group flows in four dimensions, allowing us to construct analytically infinite families of such backgrounds by specifying a superpotential at will. These RG flows interpolate between AdS$_4$ in the UV and a hyperscaling violating Lifshitz geometry in the IR with exponents $1<z<3$ and $\theta=z+1$. For $1<z<2$ the spectrum of fluctuations is gapped and discrete. Our hope and intention is that this analysis can serve as a manual for computing the holographic 1- and 2-point functions and the corresponding transport coefficients in any dyonic background, both in the context of AdS/CMT and AdS/QCD

The a-theorem and conformal symmetry breaking in holographic RG flows

We study holographic models describing an RG flow between two fixed points driven by a relevant scalar operator. We show how to introduce a spurion field to restore Weyl invariance and compute the anomalous contribution to the generating functional in even dimensional theories. We find that the coefficient of the anomalous term is proportional to the difference of the conformal anomalies of the UV and IR fixed points, as expected from anomaly matching arguments in field theory. For any even dimensions the coefficient is positive as implied by the holographic a-theorem. For flows corresponding to spontaneous breaking of conformal invariance, we also compute the two-point functions of the energy-momentum tensor and the scalar operator and identify the dilaton mode. Surprisingly we find that in the simplest models with just one scalar field there is no dilaton pole in the two-point function of the scalar operator but a stronger singularity. We discuss the possible implications.Comment: 50 pages. v2: minor changes, added references, extended discussion. v3: we have clarified some of the calculations and assumptions, results unchanged. v4: published version in JHE

Irrelevant deformations and the holographic Callan-Symanzik equation

We discuss the systematics of obtaining the Callan-Symanzik equation within the framework of the gauge/gravity dualities. We present a completely general formula which in particular takes into account the new holographic renormalization results of arXiv:1102.2239. Non-trivial beta functions are obtained from new logarithmic terms in the radial expansion of the fields. The appearance of multi-trace counterterms is also discussed in detail and we show that mixing between single- and multi-trace operators leads to very specific non-linearities in the Callan-Symanzik equation. Additionally, we compute the conformal anomaly for a scalar three-point function in a CFT.Comment: 40 page