Holographic RG Flows Research Articles

Holographic RG flows on curved manifolds and quantum phase transitions

Holographic RG flows dual to QFTs on maximally symmetric curved manifolds (dSd, AdSd, and Sd) are considered in the framework of Einstein-dilaton gravity in d + 1 dimensions. A general dilaton potential is used and the flows are driven by a scalar relevant operator. The general properties of such flows are analyzed and the UV and IR asymptotics computed. New RG flows can appear at finite curvature which do not have a zero curvature counterpart. The so-called ‘bouncing’ flows, where the β-function has a branch cut at which it changes sign, are found to persist at finite curvature. Novel quantum first-order phase transitions are found, triggered by a variation in the d-dimensional curvature in theories allowing multiple ground states.

Membrane paradigm and RG flows for anomalous holographic theories

Holographic RG flows can be better understood with the help of radially conserved charges. It was shown by various authors that the bulk gauge and diffeomorphism symmetries lead to the conservation of the zero mode of the holographic U(1) current and, if the spacetime is stationary, to that of the holographic heat current. In describing dual theories with ’t Hooft anomalies the bulk gauge invariance is broken by Chern-Simons terms. We show that conservation laws can still be derived and used to characterize the anomalous transport in terms of membrane currents at the horizon. We devote particular attention to systems with gravitational anomalies. These are known to be problematic due to their higher derivative content. We show that this feature alters the construction of the membrane currents in a way which is deeply tied with the anomalous gravitational transport.

Holographic RG flow at zero and finite temperatures

We consider a 5d holographic model with a dilaton potential representing a sum of exponential functions. We construct Poincaré invatiant and black brane solutions with AdS and non-AdS boundaries. Under the holographic duality these solutions can be interpreted as RG flows. We discuss the dependence of the running coupling on the energy through the constructed solutions.

Holographic self-tuning of the cosmological constant

We propose a brane-world setup based on gauge/gravity duality in which the four-dimensional cosmological constant is set to zero by a dynamical self-adjustment mechanism. The bulk contains Einstein gravity and a scalar field. We study holographic RG flow solutions, with the standard model brane separating an infinite volume UV region and an IR region of finite volume. For generic values of the brane vacuum energy, regular solutions exist such that the four-dimensional brane is flat. Its position in the bulk is determined dynamically by the junction conditions. Analysis of linear fluctuations shows that a regime of 4-dimensional gravity is possible at large distances, due to the presence of an induced gravity term. The graviton acquires an effective mass, and a five-dimensional regime may exist at large and/or small scales. We show that, for a broad choice of potentials, flat-brane solutions are manifestly stable and free of ghosts. We compute the scalar contribution to the force between brane-localized sources and show that, in certain models, the vDVZ discontinuity is absent and the effective interaction at short distances is mediated by two transverse graviton helicities.

Supersymmetric RG flows and Janus from type II orbifold compactification

We study holographic RG flow solutions within four-dimensional N=4 gauged supergravity obtained from type IIA and IIB string theories compactified on T^6/mathbb {Z}_2times mathbb {Z}_2 orbifold with gauge, geometric and non-geometric fluxes. In type IIB non-geometric compactifications, the resulting gauged supergravity has ISO(3)times ISO(3) gauge group and admits an N=4mathrm{AdS}_4 vacuum dual to an N=4 superconformal field theory (SCFT) in three dimensions. We study various supersymmetric RG flows from this N=4 SCFT to N=4 and N=1 non-conformal field theories in the IR. The flows preserving N=4 supersymmetry are driven by relevant operators of dimensions Delta =1,2 or alternatively by one of these relevant operators, dual to the dilaton, and irrelevant operators of dimensions Delta =4 while the N=1 flows in addition involve marginal deformations. Most of the flows can be obtained analytically. We also give examples of supersymmetric Janus solutions preserving N=4 and N=1 supersymmetries. These solutions should describe two-dimensional conformal defects within the dual N=4 SCFT. Geometric compactifications of type IIA theory give rise to N=4 gauged supergravity with ISO(3)ltimes U(1)^6 gauge group. In this case, the resulting gauged supergravity admits an N=1mathrm{AdS}_4 vacuum. We also numerically study possible N=1 RG flows to non-conformal field theories in this case.

Exploring the bulk inAdS/CFT: A covariant approach

I propose a general, covariant way of defining when one region is "deeper in the bulk" than another. This definition is formulated outside of an event horizon (or in the absence thereof) in generic geometries; it may be applied to both points and surfaces, and may be used to compare the depth of bulk points or surfaces relative to a particular boundary subregion or relative to the entire boundary. Using the recently proposed "lightcone cut" formalism, the comparative depth between two bulk points can be determined from the singularity structure of Lorentzian correlators in the dual field theory. I prove that, by this definition, causal wedges of progressively larger regions probe monotonically deeper in the bulk. The definition furthermore matches expectations in pure AdS and in static AdS black holes with isotropic spatial slices, where a well-defined holographic coordinate exists. In terms of holographic RG flow, this new definition of bulk depth makes contact with coarse-graining over both large distances and long time scales.

Second-order hydrodynamics and universality in non-conformal holographic fluids

We study second-order hydrodynamic transport in strongly coupled non-conformal field theories with holographic gravity duals in asymptotically anti-de Sitter space. We first derive new Kubo formulae for five second-order transport coefficients in non-conformal fluids in $(3+1)$ dimensions. We then apply them to holographic RG flows induced by scalar operators of dimension $\Delta=3$. For general background solutions of the dual bulk geometry, we find explicit expressions for the five transport coefficients at infinite coupling and show that a specific combination, $\tilde{H}=2\eta\tau_\pi-2(\kappa-\kappa^*)-\lambda_2$, always vanishes. We prove analytically that the Haack-Yarom identity $H=2\eta\tau_\pi-4\lambda_1-\lambda_2=0$, which is known to be true for conformal holographic fluids, also holds when taking into account leading non-conformal corrections. The numerical results we obtain for two specific families of RG flows suggest that $H$ vanishes regardless of conformal symmetry. Our work provides further evidence that the Haack-Yarom identity $H=0$ may be universally satisfied by strongly coupled fluids.

Anomaly matching condition in two-dimensional systems

Based on Son-Yamamoto relation obtained for transverse part of triangle axial anomaly in ${\rm QCD}_4$, we derive its analog in two-dimensional system. It connects the transverse part of mixed vector-axial current two-point function with diagonal vector and axial current two-point functions. Being fully non-perturbative, this relation may be regarded as anomaly matching for conductivities or certain transport coefficients depending on the system. We consider the holographic RG flows in holographic Yang-Mills-Chern-Simons theory via the Hamilton-Jacobi equation with respect to the radial coordinate. Within this holographic model it is found that the RG flows for the following relations are diagonal: Son-Yamamoto relation and the left-right polarization operator. Thus the Son-Yamamoto relation holds at wide range of energy scales.

Holography for N $$ \mathcal{N} $$ = 1∗ on S 4

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 mass-deformation parameters. Important aspects of the analysis involve characterizing the ambiguities in the partition function of non-conformal $\mathcal{N}=1$ supersymmetric theories on $S^4$ as well as the action of S-duality on the $\mathcal{N}=1^*$ theory.

Renormalized entanglement entropy

We develop a renormalization method for holographic entanglement entropy based on area renormalization of entangling surfaces. The renormalized entanglement entropy is derived for entangling surfaces in asymptotically locally anti-de Sitter spacetimes in general dimensions and for entangling surfaces in four dimensional holographic renormalization group flows. The renormalized entanglement entropy for disk regions in $AdS_4$ spacetimes agrees precisely with the holographically renormalized action for $AdS_4$ with spherical slicing and hence with the F quantity, in accordance with the Casini-Huerta-Myers map. We present a generic class of holographic RG flows associated with deformations by operators of dimension $3/2 < \Delta < 5/2$ for which the F quantity increases along the RG flow, hence violating the strong version of the F theorem. We conclude by explaining how the renormalized entanglement entropy can be derived directly from the renormalized partition function using the replica trick i.e. our renormalization method for the entanglement entropy is inherited directly from that of the partition function. We show explicitly how the entanglement entropy counterterms can be derived from the standard holographic renormalization counterterms for asymptotically locally anti-de Sitter spacetimes.

Asymptotically AdS spacetimes with a timelike Kasner singularity

Exact solutions to Einstein's equations for holographic models are presented and studied. The IR geometry has a timelike cousin of the Kasner singularity, which is the less generic case of the BKL (Belinski-Khalatnikov-Lifshitz) singularity, and the UV is asymptotically AdS. This solution describes a holographic RG flow between them. The solution's appearance is an interpolation between the planar AdS black hole and the AdS soliton. The causality constraint is always satisfied. The entanglement entropy and Wilson loops are discussed. The boundary condition for the current-current correlation function and the Laplacian in the IR is examined. There is no infalling wave in the IR, but instead, there is a normalizable solution in the IR. In a special case, a hyperscaling-violating geometry is obtained after a dimensional reduction.

Gaugings of four-dimensionalN=3supergravity andAdS4/CFT3holography

We study matter-coupled $N=3$ gauged supergravity in four dimensions with various semisimple gauge groups. When coupled to $n$ vector multiplets, the gauged supergravity contains $3+n$ vector fields and $3n$ complex scalars parametrized by $SU(3,n)/SU(3)\times SU(n)\times U(1)$ coset manifold. Semisimple gauge groups take the form of $G_0\times H\subset SO(3,n)\subset SU(3,n)$ with $H$ being a compact subgroup of $SO(n+3-\textrm{dim}(G_0))$. The $G_0$ groups considered in this paper are of the form $SO(3)$, $SO(3,1)$, $SO(2,2)$, $SL(3,\mathbb{R})$ and $SO(2,1)\times SO(2,2)$. We find that $SO(3)\times SO(3)$, $SO(3,1)$ and $SL(3,\mathbb{R})$ gauge groups admit a maximally supersymmetric $AdS_4$ critical point. The $SO(2,1)\times SO(2,2)$ gauge group admits a supersymmetric Minkowski vacuum while the remaining gauge groups admit both half-supersymmetric domain wall vacua and $AdS_4$ vacua with completely broken supersymmetry. For the $SO(3)\times SO(3)$ gauge group, there exists another supersymmetric $N=3$ $AdS_4$ critical point with $SO(3)_{\textrm{diag}}$ symmetry. We explicitly give a detailed study of various holographic RG flows between $AdS_4$ critical points, flows to non-conformal theories and supersymmetric domain walls in each gauge group. The results provide gravity duals of $N=3$ Chern-Simons-Matter theories in three dimensions.

The holographic Weyl semi-metal

We present a holographic model of a Weyl semi-metal. We show the evidences that upon varying a mass parameter the model undergoes a sharp crossover at small temperature from a topologically non-trivial state to a trivial one. The order parameter is the anomalous Hall effect (AHE) and we find that it is very strongly suppressed above a critical value of the mass parameter. This can be taken as a hint for an underlying topological quantum phase transition. We give an interpretation of the results in terms of a holographic RG flow and compare to a weakly coupled field theoretical model. Since there are no fermionic quasiparticle excitations in the strongly coupled holographic model the presence of an anomalous Hall effect cannot be bound to notions of topology in momentum spaces.

Flowing to higher dimensions: a new strongly-coupled phase on M2 branes

We describe a one-parameter family of new holographic RG flows that start from $AdS_4 \times S^7$ and go to $\widehat{AdS_5} \times {\cal B}_6$, where ${\cal B}_6$ is conformal to a K\"ahler manifold and $\widehat{AdS_5}$ is Poincar\'e $AdS_5$ with one spatial direction compactified and fibered over ${\cal B}_6$. The new solutions "flow up dimensions," going from the $(2+1)$-dimensional conformal field theory on M2 branes in the UV to a $(3+1)$-dimensional field theory on intersecting M5 branes in the infra-red. The M2 branes completely polarize into M5 branes along the flow and the Poincar\'e sections of the $\widehat{AdS_5}$ are the $(3+1)$-dimensional common intersection of the M5 branes. The emergence of the extra dimension in the infra-red suggests a new strongly-coupled phase of the M2 brane and ABJM theories in which charged solitons are becoming massless. The flow solution is first analyzed by finding a four-dimensional $N \! = \! 2$ supersymmetric flow in $N \! = \! 8$ gauged supergravity. This is then generalized to a one parameter family of non-supersymmetric flows. The infra-red limit of the solutions appears to be quite singular in four dimensions but the uplift to eleven-dimensional supergravity is remarkable and regular (up to orbifolding). Our construction is a non-trivial application of the recently derived uplift formulae for fluxes, going well beyond the earlier constructions of stationary points solutions. The eleven-dimensional supersymmetry is also analyzed and shows how, for the supersymmetric flow, the M2-brane supersymmetry in the UV is polarized entirely into M5-brane supersymmetry in the infra-red.

Formula omitted] superpotentials

Motivated by their applications to holographic RG flows and hairy black holes in Einstein-scalar systems, we present a collection of superpotentials driving the dynamics of N=2 and N=1 four-dimensional supergravities. These theories arise as consistent truncations of the electric/magnetic families of CSO(p,q,r)c maximal supergravities, with p+q+r=8, discovered by Dall'Agata et al. The N=2 and N=1 truncations describe SU(3) and Z2×SO(3) invariant sectors, respectively, and contain AdS4 solutions preserving N=1,2,3,4 supersymmetry within the full theories, as well as various gauge symmetries. Realisations in terms of non-geometric type IIB as well as geometric massive type IIA backgrounds are also discussed. The aim of this note is to provide easy to handle superpotentials that facilitate the study of gravitational and gauge aspects of the CSO(p,q,r)c maximal supergravities avoiding the technicalities required in their construction.

Universal consistent truncation for 6d/7d gauge/gravity duals

Recently, AdS7 solutions of IIA supergravity have been classified; there are infinitely many of them, whose expression is known analytically, and with internal space of S 3 topology. Their field theory duals are six-dimensional (1,0) SCFT’s. In this paper we show that for each of these AdS7 solutions there exists a consistent truncation from massive IIA supergravity to minimal gauged supergravity in seven dimensions. This theory has an SU(2) gauge group, and a single scalar, whose value is related to a certain distortion of the internal S 3. This explains the universality observed in recent work on AdS5 and AdS4 solutions dual to compactifications of the (1, 0) SCFT6’s. Thanks to previous work on the minimal gauged supergravity, the truncation also implies the existence of holographic RG-flows connecting those solutions to the AdS7 vacuum, as well as new classes of IIA AdS3 solutions.

Consistent type IIB reductions to maximal 5D supergravity

We use exceptional field theory as a tool to work out the full non-linear reduction ansaetze for the AdS$_5\times S^5$ compactification of IIB supergravity and its non-compact counterparts in which the sphere $S^5$ is replaced by the inhomogeneous hyperboloidal space $H^{p,q}$. The resulting theories are the maximal 5D supergravities with gauge groups SO(p,q). They are consistent truncations in the sense that every solution of 5D supergravity lifts to a solution of IIB supergravity. In particular, every stationary point and every holographic RG flow of the scalar potentials for the compact and non-compact 5D gaugings directly lift to solutions of IIB supergravity.

Holographic RG flow in a new SO(3) × SO(3) sector of ω-deformed SO(8) gauged N = 8 $$ \mathcal{N}=8 $$ supergravity

We consider a certain $$ \mathcal{N}=1 $$ supersymmetric, SO(3) × SO(3) invariant, subsector of the ω-deformed family of SO(8)-gauged $$ \mathcal{N}=8 $$ four-dimensional supergravities. The theory contains two scalar fields and two pseudoscalar fields. We look for stationary points of the scalar potential, corresponding to AdS vacua in the theory. One of these, which breaks all supersymmetries but is nonetheless stable, is new. It exists only when ω ≠ 0. We construct supersymmetric domain wall solutions in the truncated theory, and we give a detailed analysis of their holographic dual interpretations using the AdS/CFT correspondence. Domain walls where the pseudoscalars vanish were studied previously, but those with non-vanishing pseudoscalars, which we analyse numerically, are new. The pseudoscalars are associated with supersymmetric mass deformations in the CFT duals. When ω is zero, the solutions can be lifted to M-theory, where they approach the Coulomb-branch flows of dielectric M5-branes wrapped on S3 in the deep IR.

Holographic RG flows with nematic IR phases

We construct zero-temperature geometries that interpolate between a Lifshitz fixed point in the UV and an IR phase that breaks spatial rotations but preserves translations. We work with a simple holographic model describing two massive gauge fields coupled to gravity and a neutral scalar. Our construction can be used to describe RG flows in non-relativistic, strongly coupled quantum systems with nematic order in the IR. In particular, when the dynamical critical exponent of the UV fixed point is z=2 and the IR scaling exponents are chosen appropriately, our model realizes holographically the scaling properties of the bosonic modes of the quadratic band crossing model.

Erratum to: Holographic RG flows in six dimensional F(4) gauged supergravity

Erratum to: Holographic RG flows in six dimensional F(4) gauged supergravity