Entanglement entropy of Wilson loops: Holography and matrix models

A half-BPS circular Wilson loop in $\mathcal{N}=4$ $SU(N)$ supersymmetric Yang-Mills theory in an arbitrary representation is described by a Gaussian matrix model with a particular insertion. The additional entanglement entropy of a spherical region in the presence of such a loop was recently computed by Lewkowycz and Maldacena using exact matrix model results. In this note we utilize the supergravity solutions that are dual to such Wilson loops in a representation with order $N^2$ boxes to calculate this entropy holographically. Employing the matrix model results of Gomis, Matsuura, Okuda and Trancanelli we express this holographic entanglement entropy in a form that can be compared with the calculation of Lewkowycz and Maldacena. We find complete agreement between the matrix model and holographic calculations.Comment: 17 pages, 1 figur

Lifshitz entanglement entropy from holographic cMERA

We study entanglement entropy in free Lifshitz scalar field theories holographically by employing the metrics proposed by Nozaki, Ryu and Takayanagi in \cite{Nozaki:2012zj} obtained from a continuous multi-scale entanglement renormalisation ansatz (cMERA). In these geometries we compute the minimal surface areas governing the entanglement entropy as functions of the dynamical exponent $z$ and we exhibit a transition from an area law to a volume law analytically in the limit of large $z$. We move on to explore the effects of a massive deformation, obtaining results for any $z$ in arbitrary dimension. We then trigger a renormalisation group flow between a Lifshitz theory and a conformal theory and observe a monotonic decrease in entanglement entropy along this flow. We focus on strip regions but also consider a disc in the undeformed theory.Comment: 17 pages, v2: references added and improved discussions, v3: published versio

Holographic entanglement entropy of surface defects

We calculate the holographic entanglement entropy in type IIB supergravity solutions that are dual to half-BPS disorder-type surface defects in ${\cal N}=4$ Super Yang-Mills theory. The entanglement entropy is calculated for a ball-shaped region bisected by a surface defect. Using the bubbling supergravity solutions we also compute the expectation value of the defect operator. Combining our result with the previously-calculated one-point function of the stress tensor in the presence of the defect, we adapt the calculation of Lewkowycz and Maldacena to obtain a second expression for the entanglement entropy. Our two expressions agree up to an additional term, whose possible origin and significance is discussedComment: 41 pages. pdflatex, 3 figures. v2: typos corrected, reference corrected, some comments on CFT interpretation added. v3: references added, some clarification

Drude in D major

We study holographic momentum relaxation in the limit of a large number of spacetime dimensions D. For an axion model we find that momentum conservation is restored as D becomes large. To compensate we scale the strength of the sources with D so that momentum is relaxed even at infinite D. We analytically obtain the quasi-normal modes which control electric and heat transport, and give their frequencies in a 1/D expansion. We also obtain the AC thermal conductivity as an expansion in 1/D, which at leading order takes Drude form. To order 1/D our analytical result provides a reasonable approximation to the AC conductivity even at D=4, establishing large D as a practical method in this context. As a further application, we discuss the signature of the transition from coherent to incoherent behaviour known to exist in the system for finite D.Comment: 19 pages, 2 figure

Entanglement entropy of Wilson surfaces from bubbling geometries in M-theory

We consider solutions of eleven-dimensional supergravity constructed in [1,2] that are half-BPS, locally asymptotic to $AdS_7\times S^4$ and are the holographic dual of heavy Wilson surfaces in the six-dimensional $(2,0)$ theory. Using these bubbling solutions we calculate the holographic entanglement entropy for a spherical entangling surface in the presence of a planar Wilson surface. In addition, we calculate the holographic stress tensor and, by evaluating the on-shell supergravity action, the expectation value of the Wilson surface operator.Comment: 42 pages, 4 figures, v2: minor modification

Superconducting instabilities of R-charged black branes

We explore superconducting instabilities of black branes in SO(6) gauged supergravity at finite temperature and finite R-charge densities. We compute the critical temperatures for homogeneous neutral and superconducting instabilities in a truncation of 20 scalars and 15 gauge fields as a function of the chemical potentials conjugate to the three U(1) charges in SO(6). We find that despite the imbalance provided by multiple chemical potentials there is always at least one superconducting black brane branch, emerging at a temperature where the normal phase is locally thermodynamically stable. We emphasise that the three-equal charge solution, Reissner-Nordstrom, is subdominant to a thermodynamically unstable black brane at sufficiently low temperatures --- a feature which is hidden in an equal charge truncation.Comment: 23 pages, 6 figure

A soliton menagerie in AdS

We explore the behaviour of charged scalar solitons in asymptotically global AdS4 spacetimes. This is motivated in part by attempting to identify under what circumstances such objects can become large relative to the AdS length scale. We demonstrate that such solitons generically do get large and in fact in the planar limit smoothly connect up with the zero temperature limit of planar scalar hair black holes. In particular, for given Lagrangian parameters we encounter multiple branches of solitons: some which are perturbatively connected to the AdS vacuum and surprisingly, some which are not. We explore the phase space of solutions by tuning the charge of the scalar field and changing scalar boundary conditions at AdS asymptopia, finding intriguing critical behaviour as a function of these parameters. We demonstrate these features not only for phenomenologically motivated gravitational Abelian-Higgs models, but also for models that can be consistently embedded into eleven dimensional supergravity.Comment: 62 pages, 21 figures. v2: added refs and comments and updated appendice

Collective Excitations of Holographic Quantum Liquids in a Magnetic Field

We use holography to study N=4 supersymmetric SU(Nc) Yang-Mills theory in the large-Nc and large-coupling limits coupled to a number Nf << Nc of (n+1)-dimensional massless supersymmetric hypermultiplets in the Nc representation of SU(Nc), with n=2,3. We introduce a temperature T, a baryon number chemical potential mu, and a baryon number magnetic field B, and work in a regime with mu >> T,\sqrt{B}. We study the collective excitations of these holographic quantum liquids by computing the poles in the retarded Green's function of the baryon number charge density operator and the associated peaks in the spectral function. We focus on the evolution of the collective excitations as we increase the frequency relative to T, i.e. the hydrodynamic/collisionless crossover. We find that for all B, at low frequencies the tallest peak in the spectral function is associated with hydrodynamic charge diffusion. At high frequencies the tallest peak is associated with a sound mode similar to the zero sound mode in the collisionless regime of a Landau Fermi liquid. The sound mode has a gap proportional to B, and as a result for intermediate frequencies and for B sufficiently large compared to T the spectral function is strongly suppressed. We find that the hydrodynamic/collisionless crossover occurs at a frequency that is approximately B-independent.Comment: 45 pages, 8 png and 47 pdf images in 22 figure

From Geometry to Numerics: interdisciplinary aspects in mathematical and numerical relativity

This article reviews some aspects in the current relationship between mathematical and numerical General Relativity. Focus is placed on the description of isolated systems, with a particular emphasis on recent developments in the study of black holes. Ideas concerning asymptotic flatness, the initial value problem, the constraint equations, evolution formalisms, geometric inequalities and quasi-local black hole horizons are discussed on the light of the interaction between numerical and mathematical relativists.Comment: Topical review commissioned by Classical and Quantum Gravity. Discussion inspired by the workshop "From Geometry to Numerics" (Paris, 20-24 November, 2006), part of the "General Relativity Trimester" at the Institut Henri Poincare (Fall 2006). Comments and references added. Typos corrected. Submitted to Classical and Quantum Gravit