A double-trace deformation is the simplest perturbation of a conformal field theory that has a gravity dual. In this paper we review the existing results for the case in which the deformation is composed from a scalar operator, and extend them to the case of a spinor operator. In particular we check the validity of the c-conjecture along the RG flow induced by the deformation, using both Cardy's c-function and the recent proposal by Myers and Sinha of a c-function from entanglement entropy.
We generalize the $T\overline{T}$ deformation of CFT$_2$ to higher-dimensional large-$N$ CFTs, and show that in holographic theories, the resulting effective field theory matches semiclassical gravity in AdS with a finite radial cutoff. We also derive the deformation dual to arbitrary bulk matter theories. Generally, the deformations involve background fields as well as CFT operators. By keeping track of these background fields along the flow, we demonstrate how to match correlation functions on the two sides in some simple examples, as well as other observables.
A few supergravity solutions representing configurations of NS5-branes admit exact conformal field theory (CFT) description. Deformations of these solutions should be described by exactly marginal operators of the corresponding theories. We briefly review the essentials of these constructions and present, as a new case, the operators responsible for turning on angular momentum.
The simplest gravity duals for quantum critical theories with z=2 `Lifshitz' scale invariance admit a marginally relevant deformation. Generic black holes in the bulk describe the field theory with a dynamically generated momentum scale Lambda as well as finite temperature T. We describe the thermodynamics of these black holes in the quantum critical regime where T >> Lambda^2. The deformation changes the asymptotics of the spacetime mildly and leads to intricate UV sensitivities of the theory which we control perturbatively in Lambda^2/T.
We investigate deformations of Gauss-Bonnet-Lifshitz holography in $(n+1)$ dimensional spacetime. Marginally relevant operators are dynamically generated by a momentum scale $Λ\sim 0$ and correspond to slightly deformed Gauss-Bonnet-Lifshitz spacetimes via a holographic picture. To admit (non-trivial) sub-leading orders of the asymptotic solution for the marginal mode, we find that the value of the dynamical critical exponent $z$ is restricted by $z= n-1-2(n-2) \tildeα$, where $\tildeα$ is the (rescaled) Gauss-Bonnet coupling constant. The generic black hole solution, which is characterized by…
In recent years, there have been two independent but related developments in the study of irrelevant deformations in two dimensional quantum field theories (QFTs). The first development is the deformation of a two dimensional QFT by the determinant of the energy momentum stress tensor, commonly referred to as $T{\bar T}$ deformation. The second development is in two dimensional holographic field theories which are dual to string theory in asymptotically Anti-de Sitter (AdS) spacetimes. In this latter development, the deformation is commonly referred to as single-trace $T{\bar T}$ deformation. …
Recently, a 2d coset model with N=3 superconformal symmetry was proposed to be holographic dual to a higher spin supergravity on AdS_3 and the relation to superstring theory was discussed. However, away from the tensionless limit, there is no higher spin symmetry and the higher spin states are massive. In this paper, we examine the deformations of the coset model which preserve N=3 superconformal symmetry, but break generic higher spin symmetry. We focus on double-trace type deformations which are dual to changes of boundary conditions for the bulk matter fields. In the bulk theory, the symmet…
The holographic representation of the entanglement entropy of four dimensional conformal field theories is studied. By generalizing the replica trick the anomalous terms in the entanglement entropy are evaluated. The same terms in the holographic representation are calculated by a method which does not require the solution of the equations of motion or a cut off. The two calculations disagree for rather generic geometries. The reasons for the disagreement are analyzed.
Using six-dimensional Euclidean $F(4)$ gauged supergravity we construct a holographic renormalization group flow for a CFT on $S^5$. Numerical solutions to the BPS equations are obtained and the free energy of the theory on $S^5$ is determined holographically by calculation of the renormalized on-shell supergravity action. In the process, we deal with subtle issues such as holographic renormalization and addition of finite counterterms. We then propose a candidate field theory dual to these solutions. This tentative dual is a supersymmetry-preserving deformation of the strongly-coupled non-Lag…
We consider the Moyal deformation of self-dual gravity. In the conformal primary basis, holomorphic collinear limits of the amplitudes of this theory show that it enjoys a perturbatively exact symmetry algebra $LW_\wedge$ that generalises $Lw_\wedge$, the loop algebra of the wedge algebra of $w_{1+\infty}$, which appears in self-dual gravity.
In this article we study large central charge partition function and entanglement entropy of $T\bar{T}$ deformed two dimensional conformal field theory, following the approach to $T\bar{T}$ deformation as integrated infinitesimal double trace deformation used by Guica et al.. For sphere partition function and entanglement entropy of half great circle with antipodal points being the entangling surface, we obtain different results compared to previous works, with more reasonable CFT limits and qualitatively different behaviour as the deformation parameter $μ$ goes to infinity, which contradicts …
We review recent progress in the study of S-folds in light of the gauge/gravity duality and the AdS swampland conjecture. S-folds correspond to non-geometric backgrounds of type IIB supergravity of the form $\,\textrm{AdS}_4 \, \times \, \textrm{S}^1 \, \times \, \mathcal{M}\,$ that involve a non-trivial $\,\textrm{SL}(2,\,\mathbb{Z})\,$ (S-duality) monodromy for the type IIB fields when moving around the $\,\textrm{S}^1$. We present four such solutions with $\,\mathcal{M}=\textrm{S}^{5}\,$ that preserve $\,\mathcal{N}=4,2,1,0\,$ supersymmetries. Via the AdS/CFT correspondence, these solutions…
We investigate deformations of Lifshitz holography in $(n+1)$ dimensional spacetime. After discussing the situation for general Lifshitz scaling symmetry parameter $z$, we consider $z=n-1$ and the associated marginally relevant operators. These operators are dynamically generated by a momentum scale $Λ\sim 0$ and correspond to slightly deformed Lifshitz spacetimes via a holographic picture. We obtain renormalization group flow at finite temperature from UV Lifshitz to IR AdS, and evaluate how physical quantities such as the free energy density and the energy density depend on $\log(Λ^z/T)$ in …
Gukov, Martinec, Moore and Strominger found that the D1-D5-D5' system with the D5-D5' angle at 45 degrees admits a deformation "rho" preserving supersymmetry. Under this deformation, the D5-branes and D5'-branes reconnect along a single special Lagrangian manifold. We construct the near-horizon limit of this brane setup (for which no supergravity solution is currently known), imposing the requisite symmetries perturbatively in the deformation rho. Reducing to the three-dimensional effective gauged supergravity, we compute the scalar potential and verify the presence of a deformation with the e…
I review recent work on the holographic relation between higher-spin theories in Anti-de Sitter spaces and conformal field theories. I present the main results of studies concerning the higher-spin holographic dual of the three-dimensional O(N) vector model. I discuss the special role played by certain double-trace deformations in Conformal Field Theories that have higher-spin holographic duals. Using the canonical formulation I show that duality transformations in a U(1) gauge theory on AdS4 induce boundary double-trace deformations. I argue that a similar effect takes place in the holography…
In this article, we consider a Renormalization Group flow of the Thermofield-Double state in a UV-complete description of Holography, by introducing a relevant deformation to the ${\cal N}=4$ super Yang-Mills theory at strong coupling. This RG-flow is known to have a non-trivial, interacting ${\cal N}=1$ fixed point at the IR. Geometrically, an RG-flow of the TFD-state naturally continues radially and inside the black hole event horizon and yields a Kasner-structure near singularity, as has been observed in recent works. We show that for a generic value of the deformation, the putative IR fixe…
We study thermodynamic aspects of a tractable toy model of holography for extremal Kerr black holes proposed in [arXiv:1806.10127]. On the gravity side, the theory can be described by the worldsheet action of string theory on a warped AdS$_3$ background supported by NS-NS flux. Once we turn on temperature, the deformed background is described by a black string solution of type IIB supergravity that features a locally warped AdS$_3$ factor. The dual field theory is conjectured to be a single-trace version of a $J\bar{T}$-deformed CFT at finite temperature. As evidence for the correspondence we …
In this paper, based on the principles of Gauge/gavity duality, we explore the field theory description of certain special class of strongly coupled hyperscaling violating QFTs in the presence of scalar deformations near the \textit{effective} dynamical scale ($ r_F $) of the theory. In the language of the AdS/CFT duality, the scalar deformations of the above type could be thought of as being sourced due to some massless scalar excitation in the bulk which explicitly break the $ SO(2) $ rotational invariance along the spatial directions of the brane. As a consequence of these deformations, it …
We find new supersymmetric backgrounds of ${\cal N} = 8$ gauged supergravity in four Euclidean dimensions that are dual to deformations of ABJM theory on $S^3$. The deformations encode the most general choice of $U(1)_R$ symmetry used to define the theory on $S^3$. We work within an ${\cal N} = 2$ truncation of the ${\cal N} = 8$ supergravity theory obtained via a group theory argument. We find perfect agreement between the $S^3$ free energy computed from our supergravity backgrounds and the previous field theory computations of the same quantity based on supersymmetric localization and matrix…
We consider the coupling of a free scalar to a single-trace operator of a large N CFT in d dimensions. This is equivalent to a double-trace deformation coupling two primary operators of the CFT, in the limit when one of the two saturates the unitarity bound. At leading order, the RG-flow has a non-trivial fixed point where multiplets recombine. We show this phenomenon in field theory, and provide the holographic dual description. Free scalars correspond to singleton representations of the AdS algebra. The double-trace interaction is mapped to a boundary condition mixing the singleton with the …
In the vacuum state of a CFT, the entanglement entropy of singular surfaces contains a logarithmic universal term which is only due to the singularity of the entangling surface. We consider the relevant perturbation of a three dimensional CFT for singular entangling surface. We observe that in addition to the universal term due to the entangling surface, there is a new logarithmic term which corresponds to a relevant perturbation of the conformal field theory with a coefficient depending on the scaling dimension of the relevant operator. We also find a new power law divergence in the holograph…
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 study the entanglement entropy (EE) and the Rényi entropy (RE) of multiple intervals in two-dimensional $T\overline{T}$-deformed conformal field theory (CFT) at finite temperature by field theoretic and holographic methods. First, by the replica method with the twist operators, we construct the general formula of the RE and EE up to the first order of a deformation parameter. By using our general formula, we show that the EE of multiple intervals for a holographic CFT is just a summation of the single interval case even with the small deformation. This is a non-trivial consequence from the …
We extend Bousso's notion of a lightsheet - a surface where entropy can be defined in a way so that the entropy bound is satisfied - to more general surfaces. Intuitively these surfaces may be regarded as deformations of the Bousso choice; in general, these deformations will be timelike and so we refer to them as `timesheets'. We show that a timesheet corresponds to a section of a certain twistor bundle over a given spacelike two-surface B. We further argue that increasing the entropy flux through a given region corresponds to increasing the volume of certain regions in twistor space. We furth…
We consider Einstein gravity in AdS in the presence of a deformed conformal boundary metric, in the limit of large spacetime dimension. At leading order we find a new set of effective near-horizon equations. These can be understood as covariant generalisations of the undeformed equations with new source terms due to the curvature. We show that these equations are given by the conservation of the exact second-order Landau-frame hydrodynamic stress tensor. No derivative expansions are invoked in this identification. We use the new equations to study CFTs with 2d lattice deformations, computing t…
We construct the five-dimensional supergravity dual of the $\mathcal{N}=1^*$ mass deformation of the $\mathcal{N} =4$ supersymmetric Yang-Mills theory on $S^4$ and use it to calculate the universal contribution to the corresponding $S^4$ free energy at large 't Hooft coupling in the planar limit. The holographic RG flow solutions are smooth and preserve four supercharges. As a novel feature compared to the holographic duals of $\mathcal{N} = 1^*$ on $\mathbb{R}^4$, in our backgrounds the five-dimensional dilaton has a non-trivial profile, and the gaugino condensate is fixed in terms of the mas…
Asymptotically AdS wormhole solutions are considered in the context of holography. Correlation functions of local operators on distinct boundaries are studied. It is found that such correlators are finite at short distances. Correlation functions of non-local operators (Wilson loops) on distinct boundaries are also studied, with similar conclusions. Deformations of the theory with multi-trace operators on distinct boundaries are considered and studied. As a consequence of these results, the dual theory is expected to factorize in the UV, and the two sectors to be coupled by a soft non-local in…
School of Science, Huzhou Normal University（湖州师范学院理学院）；Mandelstam Institute for Theoretical Physics, School of Physics, University of the Witwatersrand（威特沃特斯兰德大学物理学院曼德尔斯坦理论物理研究所）；Department of Physics, Brown University（布朗大学物理系）；Brown Center for Theoretical Physics and Innovation, Brown University（布朗大学理论与创新物理中心）；Department of Physics, College of Science, Kyung Hee University（庆熙大学理学院物理系）；Research Institute for Basic Sciences, Kyung Hee University（庆熙大学基础科学研究院）；International Research Center for Quantum Matter, Kyung Hee University（庆熙大学量子物质国际研究中心）
We rigorously define the Liouville action functional for finitely generated, purely loxodromic quasi-Fuchsian group using homology and cohomology double complexes naturally associated with the group action. We prove that the classical action - the critical point of the Liouville action functional, considered as a function on the quasi-Fuchsian deformation space, is an antiderivative of a 1-form given by the difference of Fuchsian and quasi-Fuchsian projective connections. This result can be considered as global quasi-Fuchsian reciprocity which implies McMullen's quasi-Fuchsian reciprocity. We …
We study phase transitions driven by fermionic double-trace deformations in gauge-gravity duality. Both the strength of the double trace deformation and the infrared conformal dimension/self-energy scaling of the quasiparticle can be used to decrease the critical temperature to zero, leading to a line of quantum critical points. The self-energy scaling is controlled indirectly through an applied magnetic field and the quantum phase transition naturally involves the condensation of a fermion bilinear which models the spin density wave in an antiferromagnetic state. The nature of the quantum cri…
We discuss further aspects of the higher spin dS/CFT correspondence. Using a recent result of Dunne and Kirsten, it is shown how to numerically compute the partition function of the free Sp(N) model for a large class of SO(3) preserving deformations of the flat/round metric on R^3/S^3 and the source of the spin-zero single-trace operator dual to the bulk scalar. We interpret this partition function as a Hartle-Hawking wavefunctional. It has a local maximum about the pure de Sitter vacuum. Restricting to SO(3) preserving deformations, other local maxima (which exceed the one near the de Sitter …
Charged and symmetry-resolved Rényi entropies are entanglement measures quantifying the degree of entanglement within different charge sectors of a theory with a conserved global charge. We use holography to determine the dependence of charged Rényi entropies on small shape deformations away from a spherical or planar entangling surface in general dimensions. This dependence is completely characterized by a single coefficient appearing in the two point function of the displacement operator associated with the Rényi defect. We extract this coefficient using its relation to the one point functio…
We evaluate the partition function of the free and interacting O(N) vector model on a two-parameter family of squashed three spheres in the presence of a scalar deformation. We also find everywhere regular solutions of Einstein gravity coupled to a scalar field in AdS and in dS with the same double squashed boundary geometry. Remarkably, the thermodynamic properties of the AdS solutions qualitatively agree with the behavior predicted by the free O(N) model with a real mass deformation. The dS bulk solutions specify the semiclassical `no-boundary' measure over anisotropic deformations of inflat…
We consider the entanglement entropies in dS$_d$ sliced (A)dS$_{d+1}$ in the presence of a hard radial cutoff for $2\le d\le 6$. By considering a one parameter family of analytical solutions, parametrized by their turning point in the bulk $r^\star$, we are able to compute the entanglement entropy for generic intervals on the cutoff slice. It has been proposed that the field theory dual of this scenario is a strongly coupled CFT, deformed by a certain irrelevant deformation -- the so-called $T\bar T$ deformation. Surprisingly, we find that we may write the entanglement entropies formally in th…
Recently proposed double trace deformations of large $N$ holographic CFTs in four dimensions define a one parameter family of quantum field theories, which are interpreted in the bulk dual as living on successive finite radius hypersurfaces. The transformation of variables that turns the equation defining the deformation of a four dimensional large $N$ CFT by such operators into the expression for the radial ADM Hamiltonian in the bulk is found. This prescription clarifies the role of various functions of background fields that appear in the flow equation defining the deformed holographic CFT,…
A pedagogical introduction to invariant theory and finite-$N$ holography
Huzhou Normal University（湖州师范学院）；Mandelstam Institute for Theoretical Physics, School of Physics, University of the Witwatersrand（威特沃特斯兰德大学物理学院曼德尔斯坦理论物理研究所）；Research Institute of Basic Sciences, Seoul National University（首尔大学基础科学研究院）；Department of Mathematics, University of Ghana（加纳大学数学系）
We show how the $T^2$ deformation of large $N$, holographic conformal field theories arise from coupling them to the quadratic mass term of ghost-free massive gravity. We then show that in a certain approximation, the equations arising from the variation of the background frame field are those of the Randall--Sundrum II scenario.
