Boundary CFT Research Articles

Quantum extremal islands made easy. Part I. Entanglement on the brane

Recent progress in our understanding of the black hole information paradox has lead to a new prescription for calculating entanglement entropies, which involves special subsystems in regions where gravity is dynamical, called quantum extremal islands. We present a simple holographic framework where the emergence of quantum extremal islands can be understood in terms of the standard Ryu-Takayanagi prescription, used for calculating entanglement entropies in the boundary theory. Our setup describes a d-dimensional boundary CFT coupled to a (d−1)-dimensional defect, which are dual to global AdSd+1 containing a codimension-one brane. Through the Randall-Sundrum mechanism, graviton modes become localized at the brane, and in a certain parameter regime, an effective description of the brane is given by Einstein gravity on an AdSd background coupled to two copies of the boundary CFT. Within this effective description, the standard RT formula implies the existence of quantum extremal islands in the gravitating region, whenever the RT surface crosses the brane. This indicates that islands are a universal feature of effective theories of gravity and need not be tied to the presence of black holes.

From microscopic black hole entropy toward Hawking radiationc

The AdS/CFT correspondence has recently provided a novel approach for counting the microstates of black holes impressively matching the macroscopic Bekenstein–Hawking entropy formula of rotating electrically charged asymptotically AdS black holes in four to seven dimensions. This approach is designed for supersymmetric extremal black holes, but can also be extended to nonsupersymmetric, near-extremal black holes. Besides the dual higher-dimensional boundary CFT, an effective 2D CFT emerges in a certain near-horizon limit accounting for both the extremal and the near-extremal black hole entropies. This effective 2D description is universal across dimensions and comes naturally equipped with an approach to quantitatively tackle aspects of Hawking radiation.

Spontaneously broken 3d Hietarinta/Maxwell Chern\u2013Simons theory and minimal massive gravity

We show that minimal massive 3d gravity (MMG) of (Bergshoeff et al. in Class Quantum Grav 31:145008, 2014), as well as the topological massive gravity, are particular cases of a more general ‘minimal massive gravity’ theory (with a single massive propagating mode) arising upon spontaneous breaking of a local symmetry in a Chern–Simons gravity based on a Hietarinta or Maxwell algebra. Similar to the MMG case, the requirements that the propagating massive mode is neither tachyon nor ghost and that the central charges of an asymptotic algebra associated with a boundary CFT are positive, impose restrictions on the range of the parameters of the theory.

On operator growth and emergent Poincar\xe9 symmetries

We consider operator growth for generic large-N gauge theories at finite temperature. Our analysis is performed in terms of Fourier modes, which do not mix with other operators as time evolves, and whose correlation functions are determined by their two-point functions alone, at leading order in the large-N limit. The algebra of these modes allows for a simple analysis of the operators with whom the initial operator mixes over time, and guarantees the existence of boundary CFT operators closing the bulk Poincaré algebra, describing the experience of infalling observers. We discuss several existing approaches to operator growth, such as number operators, proper energies, the many-body recursion method, quantum circuit complexity, and comment on its relation to classical chaos in black hole dynamics. The analysis evades the bulk vs boundary dichotomy and shows that all such approaches are the same at both sides of the holographic duality, a statement that simply rests on the equality between operator evolution itself. In the way, we show all these approaches have a natural formulation in terms of the Gelfand-Naimark-Segal (GNS) construction, which maps operator evolution to a more conventional quantum state evolution, and provides an extension of the notion of operator growth to QFT.

Large-d phase transitions in holographic mutual information

In the AdS/CFT correspondence, the entanglement entropy of subregions in the boundary CFT is conjectured to be dual to the area of a bulk extremal surface at leading order in GN in the holographic limit. Under this dictionary, distantly separated regions in the CFT vacuum state have zero mutual information at leading order, and only attain nonzero mutual information at this order when they lie close enough to develop significant classical and quantum correlations. Previously, the separation at which this phase transition occurs for equal-size ball-shaped regions centered at antipodal points on the boundary was known analytically only in 3 spacetime dimensions. Inspired by recent explorations of general relativity at large-d, we compute the separation at which the phase transition occurs analytically in the limit of infinitely many spacetime dimensions, and find that distant regions cannot develop large correlations without collectively occupying the entire volume of the boundary theory. We interpret this result as illustrating the spatial decoupling of holographic correlations in the large-d limit, and provide intuition for this phenomenon using results from quantum information theory. We also compute the phase transition separation numerically for a range of bulk spacetime dimensions from 4 to 21, where analytic results are intractable but numerical results provide insight into the dimension-dependence of holographic correlations. For bulk dimensions above 5, our exact numerical results are well approximated analytically by working to next-to-leading order in the large-d expansion.

The functional bootstrap for boundary CFT

We introduce a new approach to the study of the crossing equation for CFTs in the presence of a boundary. We argue that there is a basis for this equation related to the generalized free field solution. The dual basis is a set of linear functionals which act on the crossing equation to give a set of sum rules on the boundary CFT data: the functional bootstrap equations. We show these equations are essentially equivalent to a Polyakov-type approach to the bootstrap of BCFTs, and show how to fix the so-called contact term ambiguity in that context. Finally, the functional bootstrap equations diagonalize perturbation theory around generalized free fields, which we use to recover the Wilson-Fisher BCFT data in the ϵ-expansion to order ϵ2.

String theory on AdS3 × M7 in the tensionless limit

We review old and recent results on a special limit of string theory on [Formula: see text] with pure NS–NS fluxes: the limit in which the string length [Formula: see text] equals the [Formula: see text] radius [Formula: see text]. At this point of the moduli space, the theory exhibits special properties, which we discuss. Special attention is focused on features of correlation functions that are related to the noncompactness of the boundary CFT target space, and on how these features change when the point [Formula: see text] is approached. Also, we briefly review the recent proposals for exact realizations of AdS/CFT correspondence at this special point.

Deformed Heisenberg charges in three-dimensional gravity

We consider the bulk plus boundary phase space for three-dimensional gravity with negative cosmological constant for a particular choice of conformal boundary conditions: the conformal class of the induced metric at the boundary is kept fixed and the mean extrinsic curvature is constrained to be one. Such specific conformal boundary conditions define so-called Bryant surfaces, which can be classified completely in terms of holomorphic maps from Riemann surfaces into the spinor bundle. To study the observables and gauge symmetries of the resulting bulk plus boundary system, we will introduce an extended phase space, where these holomorphic maps are now part of the gravitational bulk plus boundary phase space. The physical phase space is obtained by introducing two sets of Kac-Moody currents, which are constrained to vanish. The constraints are second-class and the corresponding Dirac bracket yields an infinite-dimensional deformation of the Heisenberg algebra for the spinor-valued surface charges. Finally, we compute the Poisson algebra among the generators of conformal diffeomorphisms and demonstrate that there is no central charge. Although the central charge vanishes and the boundary CFT is likely non-unitary, we will argue that a version of the Cardy formula still applies in this context, such that the entropy of the BTZ black hole can be derived from the degeneracy of the eigenstates of quasi-local energy.

Entanglement wedge reconstruction using the Petz map

At the heart of recent progress in AdS/CFT is the question of subregion duality, or entanglement wedge reconstruction: which part(s) of the boundary CFT are dual to a given subregion of the bulk? This question can be answered by appealing to the quantum error correcting properties of holography, and it was recently shown that robust bulk (entanglement wedge) reconstruction can be achieved using a universal recovery channel known as the twirled Petz map. In short, one can use the twirled Petz map to recover bulk data from a subset of the boundary. However, this map involves an averaging procedure over bulk and boundary modular time, and hence it can be somewhat intractable to evaluate in practice. We show that a much simpler channel, the Petz map, is sufficient for entanglement wedge reconstruction for any code space of fixed finite dimension — no twirling is required. Moreover, the error in the reconstruction will always be non-perturbatively small. From a quantum information perspective, we prove a general theorem extending the use of the Petz map as a general-purpose recovery channel to subsystem and operator algebra quantum error correction.

Boundary gauge and gravitational anomalies from Ward identities

We consider the two-point functions of conserved bulk currents and energy-momentum tensor in a boundary CFT defined on ℝ1,2. Starting from the consistent forms of boundary gauge and gravitational anomalies we derive their respective contributions to the correlation functions in the form of anomalous Ward identities. Using the recently developed momentum space formalism we find an anomalous solution to each of these identities depending on a single undetermined form-factor. We study the solution in two different kinematic limits corresponding to small and large momentum pn, perpendicular to the boundary. We find that the anomalous term interpolates between a non-local form resembling the standard anomaly-induced term in a two-dimensional CFT at small pn and Chern-Simons contact terms at large pn. Using this we derive some consistency conditions regarding the dependence of these anomalies on the boundary conditions and discuss possible cancellation mechanisms. These ideas are then demonstrated on the explicit example of free, massless three-dimensional fermion. In particular we manage to obtain the respective anomalies via a diagrammatic momentum space computation and expose the well-known relation between bulk parity anomaly and boundary gauge anomalies.

Holographic complexity for defects distinguishes action from volume

We explore the two holographic complexity proposals for the case of a 2d boundary CFT with a conformal defect. We focus on a Randall-Sundrum type model of a thin AdS2 brane embedded in AdS3. We find that, using the “complexity=volume” proposal, the presence of the defect generates a logarithmic divergence in the complexity of the full boundary state with a coefficient which is related to the central charge and to the boundary entropy. For the “complexity=action” proposal we find that the logarithmically divergent term in the complexity is not influenced by the presence of the defect. This is the first case in which the results of the two holographic proposals differ so dramatically. We consider also the complexity of the reduced density matrix for subregions enclosing the defect. We explore two bosonic field theory models which include two defects on opposite sides of a periodic domain. We point out that for a compact boson, current free field theory definitions of the complexity would have to be generalized to account for the effect of zero-modes.

Holographic excited states in AdS black holes

We have recently presented a geometry dual to a Schwinger-Keldysh closed time contour, with two equal β/2 length Euclidean sections, which can be thought of as dual to the Thermo Field Dynamics formulation of the boundary CFT. In this work we study non-perturbative holographic excitations of the thermal vacuum by turning on asymptotic Euclidean sources. In the large-N approximation the states are found to be thermal coherent states and we manage to compute its eigenvalues. We pay special attention to the high temperature regime where the manifold is built from pieces of Euclidean and Lorentzian black hole geometries. In this case, the real time segments of the Schwinger-Keldysh contour get connected by an Einstein-Rosen wormhole through the bulk, which we identify as the exterior of a single maximally extended black hole. The Thermal-AdS case is also considered but, the Lorentzian regions become disconnected, its results mostly follows from the zero temperature case.

Exact quantum scale invariance of three-dimensional reduced QED theories

An effective quantum field theory description of graphene in the ultra-relativistic regime is given by reduced QED aka. pseudo QED aka. mixed-dimensional QED. It has been speculated in the literature that reduced QED constitutes an example of a specific class of hard-to-find theories: an interacting CFT in more than two dimensions. This speculation was based on two-loop perturbation theory. Here, we give a proof of this feature, namely the exact vanishing of the b-function, thereby showing that reduced QED can effectively be considered as an interacting (boundary) CFT, underpinning recent work in this area. The argument, valid for both two- and four-component spinors, also naturally extends to an exactly marginal deformation of reduced QED, thence resulting in a non-supersymmetric conformal manifold. The latter corresponds to boundary layer fermions between two different dielectric half-spaces.

The gravitational dynamics of kinematic space

We show that the dynamics of the kinematic space of a 2-dimensional CFT is gravitational and described by Jackiw-Teitelboim theory. We discuss the first law of this 2-dimensional dilaton gravity theory to support the relation between modular Hamiltonian and dilaton that underlies the kinematic space construction. It is further argued that Jackiw-Teitelboim gravity can be derived from a 2-dimensional version of Jacobson’s maximal vacuum entanglement hypothesis. Applied to the kinematic space context, this leads us to the statement that the kinematic space of a 2-dimensional boundary CFT can be obtained from coupling the boundary CFT to JT gravity through a maximal vacuum entanglement principle.

Analytic bootstrap for boundary CFT

We propose a method to analytically solve the bootstrap equation for two point functions in boundary CFT. We consider the analytic structure of the correlator in Lorentzian signature and in particular the discontinuity of bulk and boundary conformal blocks to extract CFT data. As an application, the correlator 〈ϕϕ〉 in ϕ4 theory at the Wilson-Fisher fixed point is computed to order ϵ2 in the ϵ expansion.

A Truncation of 11‐Dimensional Supergravity for Fubini‐Like Instantons in AdS4/CFT3

AbstractFrom a consistent Kaluza‐Klein truncation of 11‐dimensional supergravity over , with a general 4‐form ansatz, we arrive at a set of equations and solutions for the included fields. In particular, we have a bulk equation for a self‐interacting (pseudo)scalar with arbitrary mass. By computing the energy‐momentum tensors of the associated Einstein equations, to include the backreaction, and setting them to zero, we solve the resulting equations with the bulk one and get solutions corresponding to marginal and marginally relevant deformations of the boundary CFT3, which break the conformal symmetry. These bulk (pseudo)scalars are ‐singlet and the corresponding solutions break all supersymmetries and parity because of the associated (anti)M‐branes wrapping around specific and mixed internal and external directions. As a result and according to AdS4/CFT3 duality rules, we would realize the boundary counterpart in three‐dimensional Chern‐Simon field theories with matters in fundamental representations. In particular, we build a ‐invariant Fubini‐like instanton solution by setting a specific boundary Lagrangian. The resulting solution is used to describe the dynamics of thin‐wall bubbles that cause instability and big crunch singularities in the bulk because of the unboundedness of the boundary double‐hump potential from below. Relations to mass‐deformed ABJM model, vector models and scale invariance breaking are also discussed. Meanwhile, we evaluate corrections for the background actions because of the bulk and boundary instantons.

Bulk metric reconstruction from boundary entanglement

Most of the literature in the \emph{bulk reconstruction program} in holography focuses on recovering local bulk operators propagating on a quasilocal bulk geometry and the knowledge of the bulk geometry is always assumed or guessed. The fundamental problem of the bulk reconstruction program, which is \emph{recovering the bulk background geometry (metric)} from the boundary CFT state is still outstanding. In this work, we formulate a recipe to extract the bulk metric itself from the boundary state, specifically, the modular Hamiltonian information of spherical subregions in the boundary. Our recipe exploits the recent construction of Kabat and Lifschytz \cite{Kabat:2017mun} to first compute the bulk two point function of scalar fields directly in the CFT without knowledge of the bulk metric or the equations of motion, and then to take a large scaling dimension limit (WKB) to extract the geodesic distance between two close points in the bulk i.e. the metric. As a proof of principle, we consider three dimensional bulk and selected CFT states such as the vacuum and the thermofield double states. We show that they indeed reproduce the pure AdS and the regions outside the Rindler wedge and the BTZ black hole \emph{up to a rigid conformal factor}. Since our approach does not rely on symmetry properties of the CFT state, it can be applied to reconstruct asymptotically AdS geometries dual to arbitrary general CFT states provided the modular Hamiltonian is available. We discuss several obvious extensions to the case of higher spacetime dimensions as well as some future applications, in particular, for constructing metric beyond the causal wedge of a boundary region. In the process, we also extend the construction of \cite{Kabat:2017mun} to incorporate the first order perturbative locality for AdS scalars.

Holographic second laws of black hole thermodynamics

Recently, it has been shown that for out-of-equilibrium systems, there are additional constraints on thermodynamical evolution besides the ordinary second law. These form a new family of second laws of thermodynamics, which are equivalent to the monotonicity of quantum Rényi divergences. In black hole thermodynamics, the usual second law is manifest as the area increase theorem. Hence one may ask if these additional laws imply new restrictions for gravitational dynamics, such as for out-of-equilibrium black holes? Inspired by this question, we study these constraints within the AdS/CFT correspondence. First, we show that the Rényi divergence can be computed via a Euclidean path integral for a certain class of excited CFT states. Applying this construction to the boundary CFT, the Rényi divergence is evaluated as the renormalized action for a particular bulk solution of a minimally coupled gravity-scalar system. Further, within this framework, we show that there exist transitions which are allowed by the traditional second law, but forbidden by the additional thermodynamical constraints. We speculate on the implications of our findings.

Fermionic halos at finite temperature in AdS/CFT

We explore the gravitational backreaction of a system consisting in a very large number of elementary fermions at finite temperature, in asymptotically AdS space. We work in the hydrodynamic approximation, and solve the Tolman-Oppenheimer-Volkoff equations with a perfect fluid whose equation of state takes into account both the relativistic effects of the fermionic constituents, as well as its finite temperature effects. We find a novel dense core-diluted halo structure for the density profiles in the AdS bulk, similarly as recently reported in flat space, for the case of astrophysical dark matter halos in galaxies. We further study the critical equilibrium configurations above which the core undergoes gravitational collapse towards a massive black hole, and calculate the corresponding critical central temperatures, for two qualitatively different central regimes of the fermions: the diluted-Fermi case, and the degenerate case. As a probe for the dual CFT, we construct the holographic two-point correlator of a scalar operator with large conformal dimension in the worldline limit, and briefly discuss on the boundary CFT effects at the critical points.

Holographic mutual information of two disjoint spheres

We study quantum corrections to holographic mutual information for two disjoint spheres at a large separation by using the operator product expansion of the twist field. In the large separation limit, the holographic mutual information is vanishing at the semiclassical order, but receive quantum corrections from the fluctuations. We show that the leading contributions from the quantum fluctuations take universal forms as suggested from the boundary CFT. We find the universal behavior for the scalar, the vector, the tensor and the fermionic fields by treating these fields as free fields propagating in the fixed background and by using the 1/n prescription. In particular, for the fields with gauge symmetries, including the massless vector boson and massless graviton, we find that the gauge parts in the propagators play an indispensable role in reading the leading order corrections to the bulk mutual information.