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The AdS/CFT correspondence : bulk to boundary map and applications Nogueira, Fernando Michell Falieri 2014

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The AdS / CFT Correspondence:bulk to boundary map and applicationsbyFernando Michell Falieri NogueiraB.Sc., Federal University of Minas Gerais, 2007M.Sc., Federal University of Minas Gerais, 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2014c© Fernando Michell Falieri Nogueira 2014AbstractThe holographic principle connects theories with gravity to lower dimen-sional theories without gravity. Notably, the AdS/CFT correspondence —the first concrete realization of the holographic principle — provides a oneto one map between string theory in Anti de Sitter space, and a stronglycoupled, large N , SU(N) super Yang-Mills gauge theory in one less dimen-sion.In this thesis, within the context of holographic field theories, I improveon the current understanding of the map between gravity (bulk) and gaugetheory (boundary) degrees of freedom. Furthermore, I explore some of theapplications of the AdS/CFT correspondence to the study of strongly cou-pled field theories.I study the map between bulk and boundary degrees of freedom mainlyby trying to determine what is the gravity dual of a subset of the boundaryfield theory. In the process of doing so I show how extremal surfaces, entan-glement entropy, hyperbolic black holes, and boson stars are fundamentaltools in this quest.Next, I explore a few examples of direct applications of the correspon-dence as a model building device. I discuss how AdS/CFT can be used toconstruct quasi realistic strongly coupled physical systems ranging from rel-ativistic fluids to plasmas and high temperature superconductors. Finally,I compare some of the results obtained in this thesis with known standardfield theory results.iiPrefaceChapter 1 is the sole work of the candidate.Chapter 2 is an edited version of the work by Bartlomiej Czech, JoannaL. Karczmarek, Fernando Nogueira, and Mark Van Raamsdonk. The Grav-ity Dual of a Density Matrix. Class.Quant.Grav., 29:155009, 2012.. It wasa collaboration between the candidate’s supervisor, co-supervisor, postdoc-toral fellow Bartlomiej Czech, and the candidate.Chapter 3 is an edited version of the work by Fernando Nogueira. Ex-tremal Surfaces in Asymptotically AdS Charged Boson Stars Backgrounds.Phys.Rev., D87(10):106006, 2013. And it is the sole work of the candidate.Chapter 4 is an edited version of the work by Bartlomiej Czech, JoannaL. Karczmarek, Fernando Nogueira, and Mark Van Raamsdonk. RindlerQuantum Gravity. Class.Quant.Grav., 29:235025, 2012.. It was a collabora-tion between the candidate’s supervisor, co-supervisor, postdoctoral fellowBartlomiej Czech, and the candidate.Chapter 5 is the sole work of the candidate.Chapter 6 is an edited version of the work by Fernando Nogueira andJared B. Stang. Density versus chemical potential in holographic field the-ories. Phys.Rev., D86:026001, 2012.. It was an even collaboration betweenfellow Ph.D. candidate Jared B. Stang and the candidate.Chapter 7 is an edited version of the work by Pallab Basu, FernandoNogueira, Moshe Rozali, Jared B. Stang, and Mark Van Raamsdonk. To-wards A Holographic Model of Color Superconductivity. New J. Phys.,13:055001, 2011. It was a collaboration between the candidate’s supervi-sor, professor Moshe Rozali, postdoctoral fellow Pallab Basu, fellow Ph.D.candidate Jared B. Stang, and the candidate.Chapter 8 is the sole work of the candidate.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xviDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Holography and AdS/CFT . . . . . . . . . . . . . . . . . . . 11.1.1 Two sides, one theory . . . . . . . . . . . . . . . . . . 21.1.2 The bulk to boundary map . . . . . . . . . . . . . . . 61.2 Phenomenological applications . . . . . . . . . . . . . . . . . 91.2.1 Holographic fluids and metric perturbations . . . . . . 101.2.2 Finite temperature in holographic field theories . . . . 101.2.3 Finite density in holographic field theories . . . . . . . 111.2.4 Holographic QCD . . . . . . . . . . . . . . . . . . . . 11I Nature of Spacetime2 Dual of a Density Matrix . . . . . . . . . . . . . . . . . . . . . 142.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Field theory considerations . . . . . . . . . . . . . . . . . . . 162.3 The gravity dual of ρA . . . . . . . . . . . . . . . . . . . . . . 182.4 Constraints on the region dual to ρA . . . . . . . . . . . . . . 192.5 Possibilities for R(A) . . . . . . . . . . . . . . . . . . . . . . . 20ivTable of Contents2.5.1 The causal wedge z(DA) . . . . . . . . . . . . . . . . . 202.5.2 The wedge of minimal-area extremal surfaces w(DA). 242.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Boson Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . 373.3 Mass, charge and scalar central density . . . . . . . . . . . . . 403.4 Overlapping extremal surfaces . . . . . . . . . . . . . . . . . . 433.5 Phase diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 513.6 Final comments . . . . . . . . . . . . . . . . . . . . . . . . . . 554 Rindler Quantum Gravity . . . . . . . . . . . . . . . . . . . . . 574.1 Introduction and summary . . . . . . . . . . . . . . . . . . . 574.2 A Rindler description of asymptotically global AdS spacetimes 614.2.1 Asymptotically AdS spacetimes as entangled states oftwo hyperbolic space CFTs . . . . . . . . . . . . . . . 614.2.2 The description of a single Rindler wedge . . . . . . . 644.3 The microstates of a Rindler wedge of AdS . . . . . . . . . . 664.4 Rindler space results . . . . . . . . . . . . . . . . . . . . . . . 714.5 Effects of disentangling on geometry . . . . . . . . . . . . . . 754.5.1 Review of the hyperbolic black holes . . . . . . . . . . 764.5.2 Geometrical effects of changing the temperature / en-tanglement . . . . . . . . . . . . . . . . . . . . . . . . 774.5.3 CFT on Sd interpretation of the Hd states at differenttemperatures . . . . . . . . . . . . . . . . . . . . . . . 804.6 Comments on generalization to cosmological spacetimes . . . 83II Applications of AdS/CFT5 Holographic Fluids and Metric Perturbations . . . . . . . . 865.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.2 Fluid / gravity correspondence . . . . . . . . . . . . . . . . . 875.2.1 Long wavelength limit . . . . . . . . . . . . . . . . . . 875.2.2 The Fefferman-Graham expansion . . . . . . . . . . . 885.2.3 Fluid / gravity correspondence . . . . . . . . . . . . . 895.3 Conservation equations and the stress energy tensor . . . . . 915.3.1 A simple example . . . . . . . . . . . . . . . . . . . . 915.3.2 Generalization . . . . . . . . . . . . . . . . . . . . . . 94vTable of Contents5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946 Density versus Chemical Potential in Holographic FieldTheories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.2 CFT thermodynamics . . . . . . . . . . . . . . . . . . . . . . 1016.3 General holographic field theories at finite density . . . . . . 1036.3.1 Finite density . . . . . . . . . . . . . . . . . . . . . . . 1036.3.2 Gauge field actions . . . . . . . . . . . . . . . . . . . . 1046.4 Holographic probes . . . . . . . . . . . . . . . . . . . . . . . . 1066.4.1 Probe branes and the Born-Infeld action . . . . . . . . 1076.4.2 Bottom-up models and the Einstein-Maxwell action . 1106.5 ρ− µ in backreacted systems . . . . . . . . . . . . . . . . . . 1166.5.1 Charged black holes . . . . . . . . . . . . . . . . . . . 1166.5.2 Hairy black holes . . . . . . . . . . . . . . . . . . . . . 1186.5.3 Backreacted soliton . . . . . . . . . . . . . . . . . . . . 1196.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1207 Color Super Conductivity . . . . . . . . . . . . . . . . . . . . . 1227.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1227.2 Basic setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297.3 Review: ψ = 0 solutions . . . . . . . . . . . . . . . . . . . . . 1337.3.1 AdS Soliton solution . . . . . . . . . . . . . . . . . . . 1347.3.2 Reissner-Nordstrom black hole solution . . . . . . . . 1347.4 Neutral scalar field: color superconductivity . . . . . . . . . . 1357.4.1 Numerical evaluation of solutions . . . . . . . . . . . . 1377.4.2 Critical temperature . . . . . . . . . . . . . . . . . . . 1397.4.3 Properties of the superconducting phase . . . . . . . . 1407.5 Charged scalar field: flavor superconductivity . . . . . . . . . 1417.5.1 Low-temperature horizon free solutions with scalar . . 1417.5.2 Hairy black hole solutions . . . . . . . . . . . . . . . . 1457.5.3 Phase diagrams . . . . . . . . . . . . . . . . . . . . . . 1477.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1498 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155viTable of ContentsAppendicesA Appendix to Chapter 4 . . . . . . . . . . . . . . . . . . . . . . 166A.1 Coordinate transformations . . . . . . . . . . . . . . . . . . . 166B Appendix to Chapter 7 . . . . . . . . . . . . . . . . . . . . . . 168B.1 Large charge limit . . . . . . . . . . . . . . . . . . . . . . . . 168B.1.1 Order of phase transitions in the probe limit . . . . . 171B.2 Critical µ for solutions with infinitesimal charged scalar . . . 172viiList of Tables6.1 The power α in the relationship ρ ∝ µα at large ρ for 3 + 1dimensional field theories dual to the given brane backgroundwith the indicated probe brane, with d − 1 shared spacelikedirections. For d = 5 the theory is considered to have a smallperiodic spacelike direction while for background Dp braneswith p > 3, the background is compactified to 3+ 1 dimensions. 986.2 The power α in the relationship ρ ∝ µα at large ρ for 3 +1 dimensional field theories dual to the given gravitationalbackground with the stated fields considered in either theprobe or backreacted limits. φ is the time component of thegauge field, ψ is a charged scalar field, and d is the numberof spacetime dimensions. For d = 5 the theory is consideredto have a small periodic spacelike direction. . . . . . . . . . . 100viiiList of Figures1.1 A number of extremal surfaces anchored at the boundary of atime slice of global AdS space. The boundary region A is thehemisphere bounded by the endpoints of one of the extremalsurfaces γA (blue lines), and the boundary of A, δA, is theboundary circle to which the surface attaches to. . . . . . . . 82.1 A spacelike slice Σ of a boundary manifold B (= S1× time)with a region A and its domain of dependence DA. The samedomain of dependence arises from any spacelike boundaryregion A˜ homologous to A with ∂A = ∂A˜. . . . . . . . . . . . 172.2 Causal wedge z(DA) associated with a domain of dependenceDA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 In pure global AdS, causal wedges of complementary hemi-spherical regions of the τ = 0 slice intersect along a codimension-two surface. In generic asymptotically AdS spacetimes, theyintersect only at the boundary. . . . . . . . . . . . . . . . . . 222.4 Different possible behaviors of extremal surfaces in sphericallysymmetric static spacetimes. Shaded region indicates w(DA)where A is the right hemisphere. The boundary of the shadedregion on the interior of the spacetime is the minimal areaextremal surface bounded by the equatorial Sd−1. . . . . . . . 292.5 Region w(DA) (shaded) where A is a boundary sphere of an-gular size greater than pi. No minimal surface with boundaryin A penetrates the unshaded middle region. . . . . . . . . . . 312.6 Spatial t = 0 slice of w(DA) (light shaded plus dark shaded)and z(DA) (dark shaded) for a planar AdS black hole. Thedashed curve is a spatial geodesic with endpoints in A¯. Knowl-edge of observables obtained from ρA¯ alone allow us to com-pute the length of this geodesic. . . . . . . . . . . . . . . . . . 32ixLIST OF FIGURES2.7 The region of spacetime reconstructible from density matricesρB and ρC (shaded, right hand side picture) is smaller thanthat reconstructible from ρB∪C (shaded, left hand side pic-ture). Reconstruction of R(B ∪C)− (R(B)∪R(C)) (interiorof dotted frame outside of the two shaded regions) requiresknowledge of entanglement between degrees of freedom in Band C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.8 The region of spacetime reconstructible from density matricesρAi lies arbitrarily close to the boundary (illustrated here ona spatial slice). The ability to reconstruct the bulk geometrydepends entirely on the knowledge of entanglement amongthe various boundary regions. . . . . . . . . . . . . . . . . . . 333.1 Plot of the star’s mass (orange) and charge (green) versus thecentral value of the scalar field ψ0 with m2 = 0 and q = 0.2in D = 4 dimensions. . . . . . . . . . . . . . . . . . . . . . . . 433.2 Plot of the star’s mass (orange) and charge (green) versus thecentral value of the scalar field ψ0 with m2 = 0 and q = 0.2in D = 3 dimensions. . . . . . . . . . . . . . . . . . . . . . . . 443.3 A plot of the Penrose diagram of a time slice of multiple ex-tremal surfaces on a four dimensional, asymptotically AdS,charged boson boson star background in global coordinatesfor m2 = 0, q = 0.1, and ψ(0) = 0.2. From top to bottomwe have θ = ±0.251pi,±0.355pi,±0.446pi,±0.5pi. In this par-ticular case the central density of the scalar field is belowthe threshold ψh, therefore there are no degenerate extremalsurfaces (see figure 3.6), in other words, we observe a solidw(DA). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47xLIST OF FIGURES3.4 Again, a plot of the Penrose diagram of a time slice of mul-tiple extremal surfaces on a four dimensional, asymptoticallyAdS, charged boson boson star background in global coordi-nates for m2 = 0, q = 0.1, and ψ(0) = 1.2. However, in thisexample, the scalar central density is above the threshold ψhand we observe the existence of degenerate extremal surfacesfor a range of boundary anchor points θ (see figure 3.6). Inparticular, for anchor points θ = ±pi/2 there are three solu-tions two of which (blue line) lie on top of each other, haveminimal area and do not penetrate the dashed small circle,while the third (red line) corresponds to a non minimal areaextremal surface. As discussed in this chapter, no minimalarea surface penetrates the deepest bulk points within thesmall dashed circle. . . . . . . . . . . . . . . . . . . . . . . . . 483.5 A similar plot as figures 3.3 and 3.4 highlighting the behaviourof all three extremal surfaces anchored at the same boundarypoints θ = ±0.49pi. Although two of the solutions (red lines)penetrate the dashed circle, these are not the minimal areaand therefore are not part of the w(DA) set. . . . . . . . . . . 493.6 A comparison of two different central densities for the plot ofrmin(θ(r)) in four dimensions, both cases with m2 = 0 andq = 0.1. It is clear how for anchor points roughly between0.17pi < θ < pi/2 there exist three distinct extremal surfaceswith different values of rmin, however, only one of them hasminimal area. See figures 3.3, 3.4 and 3.5 for specific examples. 513.7 Critical scalar field central density separating solid and hollowconfigurations for D = 3 dimensions with q = 0.1 in blue,q = 0.2 in light blue (dashed) and q = 0.3 in green (dotted).In all cases a central density value below the line correspondto solid solutions, while above it lies the hollow regime. . . . 523.8 Critical scalar field central density separating solid and hollowconfigurations for D = 4 dimensions with q = 0.1 in blue,q = 0.2 in light blue (dashed) and q = 0.3 in green (dotted).In all cases a central density value below the line correspondto solid solutions, while above it lies the hollow regime. . . . 53xiLIST OF FIGURES3.9 A phase diagram displaying both transitions found in D = 4dimensions (stable → unstable and solid → hollow) with q =0.1. The red line (below) is the stability threshold, a valueof ψ0 above it (the light blue region) renders a dynamicallyunstable configuration. The blue line (above), once again,represents the transition between solid and hollow configura-tions. It is clear from this figure how, in the range studied,only solid configurations are physically allowed. . . . . . . . . 543.10 A phase diagram comparing the central densities ψc and ψh asa function of m2 for different values of q. The dotted lines arefor q = 0.3, the dashed lines for q = 0.2, while the solid linesfor q = 0.1. The warm coloured lines (below) correspond totransition between stable and unstable configurations, whilethe cold coloured lines (above) transition between solid andhollow phases. As we lower the charge q both ψc and ψhincrease, however their difference remains roughly unchanged,highlighting how hollow solutions are unstable for all rangeof parameters investigated. . . . . . . . . . . . . . . . . . . . 554.1 A pair of accelerating observers in pure global AdS. Thespacetime region accessible to each is a wedge whose bound-ary geometry can be chosen as Hd × R. Each wedge has adual description as a thermal state of a CFT on this Hd ×Rboundary geometry. The full spacetime is described by anentangled state of the two Hd CFTs. . . . . . . . . . . . . . . 584.2 Quantum superposition of microstate geometries yielding pureAdS spacetime. Each choice of complementary Rindler wedgesleads to a different decomposition of AdS into a superpositionof disconnected spacetimes. . . . . . . . . . . . . . . . . . . . 594.3 Static observer in de Sitter space (left) and accelerated ob-server in AdS. Both have access to only a portion of the fullspacetime, bounded (on one side in the AdS case) by a hori-zon with a thermal character. . . . . . . . . . . . . . . . . . . 604.4 Conformal map from the boundary of a Poincare patch toMinkowski space. Region DL (solid) maps to one Rindlerwedge of Minkowski space, while the dotted region, DR, mapsto the other wedge. The Poincare patch boundary DP is theregion bounded by dashed lines. . . . . . . . . . . . . . . . . . 62xiiLIST OF FIGURES4.5 Wedges of pure AdS spacetime. Field theory observables inDR (the shaded part of the boundary cylinder) probe the bulkregion J+(DR)∩J−(DR) (the shaded region of the bulk). Anypoint in this region can receive a light signal (blue line) fromand send a light signal to DR. Physics outside this region canbe altered by changes on the boundary that do not affect thestate of the fields in DR. One trajectory along which suchchanges propagate is shown in red. . . . . . . . . . . . . . . . 654.6 Subregion D (shaded) of boundary D of a Rindler wedge ofAdS. The complement of D in D is shown dotted. (a) Re-gions D and D on the boundary of AdS. The correspondingbulk regions are also shown. W is a surface in the bulk whosearea computes the entanglement entropy of the fields in D.(b) D is mapped to portion of a Rindler wedge. (c) D ismapped to a finite portion of the infinite hyperbolic plane. . . 674.7 Position of cuts in the U-plane for formulas (4.10) and (4.11). 734.8 Numerical integration of regularized distance for d=1, 2, 3, 4(orange, blue, red, green, respectively). . . . . . . . . . . . . . 784.9 Numerical integration of regularized distance with u0 = 1 ford=1, 2, 3, 4 (orange, blue, red, green, respectively). . . . . . . 806.1 Charge density versus chemical potential for the probe gaugeand scalar fields, section 6.4.2, on a log-log scale. The thickdashed line is for the system with no scalar field for which, an-alytically, ρ ∝ µ. At a critical chemical potential, dependingon the mass of the scalar field, configurations with non-zeroscalar field become available. The thin dotted line is a modelpower law ρ ∝ µ3, as described in equation (6.57). Fromleft to right, the thick solid lines are for scalar field massesm2 = −15/4, −14/4, −13/4, and −3. A more negative scalarfield mass results in a denser field theory state at a givenchemical potential. . . . . . . . . . . . . . . . . . . . . . . . . 114xiiiLIST OF FIGURES6.2 Charge density versus chemical potential for the probe gaugeand scalar fields in the soliton background, section 6.4.2, andthe d = 5 black hole background, section 6.4.2. The thindashed line is the probe gauge field in the black hole back-ground for which, analytically, ρ ∝ µ. The thick solid linesare the soliton results (from left to right, the squared massof the scalar field is −22/4, −5, −18/4, and −4) while thethick dashed lines are the black hole results (again, from leftto right, m2 = −22/4, −5, −18/4, and −4). Each of the thicklines approaches the power law ρ ∝ µ4, equation (6.63). At agiven chemical potential, the soliton background gives a fieldtheory in a denser state. . . . . . . . . . . . . . . . . . . . . . 1177.1 Phase diagram of our model gauge theory with m2 = −6,R = 2/5. Region in dashed box is expanded in next figure. . . 1277.2 Phase diagram of our model gauge theory with m2 = −6,R = 2/5. Region in dashed box is expanded in next figure. . . 1287.3 Phase diagram of our model gauge theory with m2 = −6,R = 2/5. The dashed curve represents the phase boundaryin theory without a scalar field. . . . . . . . . . . . . . . . . . 1287.4 Phase diagram without scalar field, in units where R = 2/5. . 1367.5 Critical T/µ vs m2 of neutral scalar (filled circles). Massis above BF bound asymptotically but below BF bound innear-horizon region of zero-temperature background solutionin the range −6.25 ≤ m2 < −5. Unfilled circles representcritical values in the theory with alternate quantization ofthe scalar field, possible in the range −6.25 ≤ m2 < −5.25. . . 1407.6 Action vs chemical potential for soliton with scalar solutions,taking m2 = −6 and q = 2. . . . . . . . . . . . . . . . . . . . 1437.7 Action vs chemical potential for soliton with scalar solutions,taking m2 = −6 and q = 1.3. . . . . . . . . . . . . . . . . . . 1447.8 Action vs chemical potential for soliton with scalar solutions,taking m2 = −6 and q = 1.2. . . . . . . . . . . . . . . . . . . 1447.9 Phase diagram for m2 = −6 and q = 2. Clockwise from theorigin, the phases correspond to the AdS soliton (confined),RN black hole, black hole with scalar, and soliton with scalar. 1477.10 Phase diagram for m2 = −6 and q = 1.3. Clockwise from theorigin, the phases correspond to the AdS soliton (confined),RN black hole, black hole with scalar, and soliton with scalar. 148xivLIST OF FIGURES7.11 Small temperature region of phase diagram for m2 = −6 andq = 1.3. Dashed line represents a first order transition withinthe soliton with scalar phase. . . . . . . . . . . . . . . . . . . 1487.12 Phase diagram for large q, m2 = −6. . . . . . . . . . . . . . 149B.1 Critical values of µq vs m2 for scalar condensation in largeq limit. The top curve is the critical value for µ in blackhole phase (just above the transition temperature), while thebottom curve is the critical µ in low temperature phase. . . . 171B.2 Critical T/µ vs charge q for condensation of m2 = −6 scalarfield in Reissner-Nordstrom background. . . . . . . . . . . . . 173xvAcknowledgementsI would like to thank my advisors Mark Van Raamsdonk and Joanna Kar-czmarek for all the invaluable help given throughout the years, all the post-doctoral fellows that were part of the string theory group, in particularBartlomiej Czech and Pallab Basu, and all other members of the StringTheory group that in one way or another helped in this journey, in particu-lar Jared Stang, Charles Rabideau, Michael B. McDermott, Connor Behanand Philippe Sabella-Garnier.I would also like to thank all my office colleagues with whom I sharedmany laughs and many memorable moments, and Oliva Dela Cruz-Cordero,the graduate program coordinator for all the help, in particular with all theinternational student bureaucracy.And I would also like to thank my family and friends, specially Crystal,for being by my side all these years.xviDedicationI dedicate this thesis to my parents, without whom I could not have com-pleted it.xviiChapter 1Introduction1.1 Holography and AdS/CFTThe holographic principle is a remarkable idea relating theories with grav-ity to theories without gravity [129, 133]. The best understood example ofholography is the AdS/CFT correspondence which proposes a one to onerelation between a gravitational theory in anti-de Sitter space (bulk theory)and a strongly coupled, large N conformal field theory living on the confor-mal boundary of AdS [92, 140] (boundary theory). It advocates that eachstate of the boundary field theory should correspond to a configuration (notnecessarily classical) of the bulk gravitational theory. This correspondencehas been extensively used to study a variety of strongly coupled field theoriesand has led to the formulation of successful formalisms such as AdS/QCDand AdS/CMT.In this thesis I address a particular gap in our understanding of AdS/CFTthat could shine light on a myriad of long standing problems in quantumgravity and the nature of space-time. In addition, I explore some of thevast number of applications of AdS/CFT, in particular, holographic fluids,holographic field theories at finite temperature and chemical potential, andQCD like models.First, I will show how the correspondence can be used to answer somedeep questions about quantum gravity and the nature of spacetime. In par-ticular, I argue that entanglement entropy and its holographic formulationcan be seen as fundamental building blocks of holographic spacetimes. Thisdiscussion will pave the way to tackling one of the major questions sur-rounding the holographic principle: what is the exact map between bulkand boundary degrees of freedom.Understanding the connection between bulk and boundary degrees offreedom is one of the main barriers in the way of the gauge / gravity dual-ity fulfilling its potential of answering long standing problems in quantumgravity by recasting them in field-theoretic language. In order to improvethis situation, further investigation of the map between bulk and boundarydegrees of freedom is necessary. Optimistically, one should be able to ex-11.1. Holography and AdS/CFTpress any variation of the boundary theory state in terms of a well definedperturbation of the bulk geometry in a one to one fashion.Second, I will use the AdS/CFT correspondence directly to constructmodels for a variety of physical systems. In particular, I will show howrelativistic conformal fluids arise naturally from perturbations of the bulkmetric, and try to generalize this discussion hoping to elucidate how certainbulk features are connected to boundary observables. Moreover, I will usethe AdS/CFT correspondence in a bottom up approach to create a modelfor relativistic field theories at finite temperature and chemical potentialand study the main properties of these theories. Finally, as a special caseof holographic field theories, I will show how a richer model can be used todescribe the phenomenon of colour superconductivity and the various phasetransitions expected to take place.In the remainder of this chapter I will describe the basic tools that willbe used throughout this thesis. I start by discussing how the correspondencewas originally formulated and limits of its applicability. That is followed bya quick review on how to generalize holographic models to finite temperatureand chemical potential regimes, and the definitions of important tools suchas the holographic entanglement entropy.1.1.1 Two sides, one theoryThe most fundamental object in string theory — a string — can eitherbe open or closed. Closed strings are free to propagate throughout space,their internal degrees of freedom are subject to (anti)periodic boundaryconditions and give rise to tachyonic (negative squared mass), massless andmassive excitations. The endpoints of open strings, however, can be subjectto Dirichlet boundary conditions, in which case the strings require an objectto which their ends can attach to, these objects are called Dp-branes.Branes can extend in an arbitrary number p of spacial dimensions andprovide a (1+p) dimensional sub manifold to which open strings end pointscan attach to. Nevertheless, these objects are more than mere open stringshangers, branes also have mass, charge, interact with each other, and playa central role in the AdS/CFT correspondence.While today AdS/CFT — and, more generally, the holographic principle— is believed to apply to a myriad of different scenarios, the original argu-ment for the correspondence involved low energy excitations of D3-branesin type IIB string theory.Let us consider type IIB string theory on a ten dimensional flat Minkowskispace. Let us consider also a stack of N parallel D3-branes extending along21.1. Holography and AdS/CFTa (1 + 3) dimensional hyper plane embedded within this ten dimensionalspace.In the low energy limit, string theory in this background has only twokinds of excitations, massless closed strings, describing excitations of emptyspace, and massless open strings, describing excitations of the D-branes. Itis known that the low energy effective Lagrangian of these states is that oftype IIB supergravity for massless closed strings, and N = 4 U(N) superYang Mills for massless open strings.The full effective action of the massless modes is therefore written asS = Sbulk + Sbrane + Sint, (1.1)where Sbulk is the effective action of ten dimensional supergravity in additionto possible higher derivative corrections. Sbrane is the brane effective actiondefined on the (1+3) dimensional hyper plane covered by the branes, whichnot only includes N = 4 U(N) super Yang Mills, but also possible higherderivative corrections. And Sint is the effective action for the interactionbetween bulk and brane modes.The total effective action in equation (1.1) can be understood as theeffective description of the system once all the massive modes are integratedout. While in this general form the action is quite complicated — and, infact, non renormalizable — in the low energy limit it simplifies greatly. Inthis limit, all the higher derivative terms disappear, and the interaction sec-tor can also be neglected, leaving as a final product two decoupled systems,namely, free gravity in the bulk and a four dimensional gauge theory.The same stack of D-branes in flat Minkowski space can be studied froma different perspective. As mentioned before, D-branes are massive objects,when their number is large (N → ∞) they backreact with the backgroundmetric and are described as a black p-brane, a well known classical super-gravity solution. The metric for a black p-brane is given byds2 = f−1/2(−dt2 + dx21 + dx22 + dx23)+ f1/2(dr2 + r2dΩ25), (1.2)withf = 1 +R4r4, and R4 = 4pigsl4sN, (1.3)where gs is the string coupling, ls is the string length and N the number ofbranes1.1N is related to the total flux of the (3 + 1)-form potential F5 that will be omitted inthe present discussion.31.1. Holography and AdS/CFTFrom equation (1.2) we see that E, the energy of an object measured byan observer at a given position r and, E∞, the energy of the same objectmeasured by another observer at infinity, are related by the red shift factor,E∞ = f−1/4E. (1.4)Due to this red shift factor we see that from the point of view of anobserver at infinity there are two kinds of low energy excitations. Longwavelength, massless particles propagating in the bulk, and arbitrary energyexcitations near r = 0. Note that for large enough wavelengths, masslessparticles in the bulk are effectively blind to the existence of branes whosetypical size is of order R; conversely, the closer we bring higher energy exci-tations to r = 0, the deeper the potential well becomes, effectively trappingthem. In other words, these two types of low energy excitation decouple,leaving on the one hand free bulk supergravity, and on the other arbitraryexcitations in the near horizon (r → 0) limit.The near horizon limit of the black p-brane metric in equation 1.2 is justds2 =r2R2(−dt2 + dx21 + dx22 + dx23)+R2dr2r2+R2dΩ25, (1.5)which turns out to be the geometry of AdS5 × S5. Therefore in the nearhorizon limit we are left with string theory in AdS5 × S5.Notice that the above discussion was merely two different ways of look-ing at the same system. While the effective descriptions we arrived at arestrikingly different, our starting points were precisely the same, and in bothcases we restricted our attention to the same low energy limit. Therefore, weare led to believe that these might, indeed, represent two equivalent math-ematical descriptions of the same physical system. Moreover, since in bothinstances we found two decoupled sectors, one of which being free bulk su-pergravity, we are compelled to make an even stronger statement and equateN = 4 U(N) super Yang Mill in 1+3 dimensions with type IIB string theoryon AdS5 × S5.Coupling strength dichotomyIn the above discussion I showed how on one side the decoupling betweensupergravity modes and super Yang Mills gauge theory, and on the otherside the decoupling between supergravity modes and string theory modesin the near horizon limit of black p-branes was central to the AdS/CFTcorrespondence. The argument behind the decoupling involved taking thelow energy limit in both cases, which, in the black p-brane scenario, involved41.1. Holography and AdS/CFTtaking the near horizon limit and the long wavelength limit. In addition, inboth cases the supergravity approximation can only be achieved in the largeN limit of the number of branes.From the coupling parameter gYM of Yang Mills theories and the gaugegroup parameter N the ’t Hooft coupling parameter given by g2YMN can beconstructed. In order to treat Yang Mills theories with gauge group SU(N)perturbatively both the ’t Hooft coupling needs to be small, as well as thegauge field parameter N has to be large, i.e.: g2YMN  1 while N  1.Alternatively, in the context of string theory, quantum gravity effectscan only be dismissed when the spacetime curvature R is much larger thanthe string scale ls. However, from equation 1.3 and the fact that the stringcoupling gs is proportional to g2YM we conclude that working on the pertur-bative regime of SYM would require the field theory and gravity parametersto obeyg2YMN ∼ gsN ∼R4l4s 1, (1.6)where R is the radius of the AdS5 space. Conversely, ensuring that theclassical gravity description of string theory is reliable requires thatR4l4s∼ gsN ∼ g2YMN  1, (1.7)therefore, the conclusion is these two limits are incompatible. This incom-patibility is precisely what makes the AdS/CFT correspondence such a use-ful tool. It provides both a way of dealing with strongly coupled gaugetheories (albeit in the large N limit only) by exploring their classical gravityduals, or, alternatively, a way of dealing with quantum gravity by exploringits weakly coupled field theory dual.While the above mentioned limits of the duality are specially interest-ing, broadly speaking the consequences of the AdS/CFT are deep. The pre-cise equivalence between a quantum gravity theory and a lower dimensionalquantum field theory imposes strong constraints on the global and local be-haviour of these two theories. Ranging from the loss of information insidea black hole, to the phase transitions of sub-atomic matter, the knowledgethat these two, while at first sight completely different physical systems, arein fact equivalent to each other, offers a completely new approach to suchquestions and many others.51.1. Holography and AdS/CFT1.1.2 The bulk to boundary mapUnderstanding the exact connection between bulk and boundary degrees offreedom is among the most fundamental questions surrounding the AdS/CFTcorrespondence. A better understanding of such map would allow recast-ing long standing quantum gravity problems in a field theoretic language.Therefore, further investigation of the map between bulk and boundary de-grees of freedom is necessary. Optimistically, it should be possible to expressany variation of the boundary theory state in terms of a well defined per-turbation of the bulk geometry; conversely, any changes in the bulk shouldbe related to a particular deformation of the boundary state in a one to onefashion.In part I of this thesis I will improve on the current understanding ofthe relation between boundary and bulk degrees of freedom in holographicduals. By examining how information contained in a portion of the bound-ary theory is encoded in the bulk and how certain theories respond to smallperturbations, I hope to gain insight into the elusive bulk to boundary dic-tionary. To do so I will tackle this problem in a few different ways, bothfrom bulk as well as boundary perspectives.The goal is to investigate how degrees of freedom inside a subregion ofa holographic field theory correlate to those in the bulk gravity dual. Agood understanding of how to describe seemingly local degrees of freedomin gravity theories, specially in the context of the gauge / gravity duality,remains elusive. Therefore, by examining the question of what bulk regionshould contain enough information to be dual to a particular subregion ofthe boundary, I can shed some light on how degrees of freedom, both localand non local, in these dual theories relate to each other.While there are a myriad of possible ways to investigate the bulk toboundary dictionary, in this thesis I will focus on sub-regions of the fieldtheory and investigate the properties of gravity duals when only partial in-formation is available. These subregions can be constructed by consideringaccelerated bulk observers and will naturally lead to entanglement entropyand its holographic dual as I will discuss in the next section. Below I willargue that from a purely field theoretic perspective it is easy to see how en-tanglement entropy and incomplete knowledge of a theory’s state are closelyrelated. Combined with the Ryu-Takayanagi[118] proposal (also discussedbelow), I can use the holographic prescription of EE (entanglement entropy)to directly connect, and in special cases even quantify, boundary and bulkstatements regarding the amount of information contained in a given regionand how it is translated between bulk and boundary languages.61.1. Holography and AdS/CFTEntanglement entropy and partial Boundary InformationThe general picture in the AdS/CFT correspondence is that of a field theorydefined on the conformal boundary of AdS space, with each state of the fieldtheory being dual to a particular geometry of the bulk2. In its most wellunderstood form, the field theory is the large N, strongly coupled limit ofsupersymmetric U(N) Yang Mills. In this limit the gravity theory simplifiesto classical supergravity, and its dynamics is given by Einstein’s equations.Among the extensive list of progress made in the direction of better un-derstanding this duality is the proposal for a holographic dual of entangle-ment entropy. For a quantum system with Hilbert space H = HA⊗HB anddensity matrix ρ, the entanglement entropy of the subsystem A is definedas SA = −TrAρA log ρA, where ρA = TrBρ is the reduced density matrix forthe subsystem A. The holographic version of this physical quantity was pro-posed by Ryu and Takayanagi [118] and is given by the area of the minimalsurface γδA that is anchored at the boundary of A, δA, and extends into thebulk space-time3 as can be seen in figure 1.1.Since these extremal surfaces are related to entanglement entropy of theboundary subregion A, they comprise a good measure of the amount ofinformation encoded as entanglement between degrees of freedom inside Aand in its complement A¯. Naturally one should expect that these minimalareas surfaces play a role at understanding how boundary degrees of freedomare related to bulk degrees of freedom.In AdS/CFT, the expectation values of field theory operators are relatedto boundary conditions of the bulk fields, hence, given the knowledge of theexpectation values of certain operators of the field theory, one should be ableto determine the bulk field (via integration), at least very near the boundary.Obviously, if all possible expectation values are known in a Cauchy surfaceof the boundary (A∪ A¯ in the example above), full knowledge of the gravitytheory is expected, however that is not the case if only partial boundaryinformation is available.Clearly one can see that any bulk field in a region causally connectedto A¯ can not be fully determined since its boundary value problem is notwell defined. Nevertheless, while this maximum bound is clear, and while a2In general the geometrical picture will be highly quantum mechanical and a classical,well behaved metric does not usually exist.3Given the negatively curved nature of anti de-Sitter space, a minimal area surface thatis anchored at the boundary will necessarily extend into the bulk. Note, however, that thearea of such surfaces is formally infinite, nevertheless we can regularize it by imposing aIR cutoff, which, in turn, is related to the UV cutoff normally imposed on the field theoryentanglement entropy (which is also infinite).71.1. Holography and AdS/CFT-1.0-0.50.00.51.0Figure 1.1: A number of extremal surfaces anchored at the boundary of atime slice of global AdS space. The boundary region A is the hemispherebounded by the endpoints of one of the extremal surfaces γA (blue lines),and the boundary of A, δA, is the boundary circle to which the surfaceattaches to.minimum amount of bulk information is guaranteed from any set of data,the exact bulk dual to a piece of boundary data, say, the region A, is stillunknown.Accelerated bulk observersOne way of modelling an observer with limited access to boundary informa-tion, and putting the ideas outlined above to test, is to consider bulk acceler-ated observer and wedges of AdS. Similarly to Rindler wedges in Minkowskispace, Rindler wedges of AdS are the portions of the full manifold acces-sible to constantly accelerating observers. From the boundary perspective,such incomplete access to boundary data is tantamount to partially tracingover some of the field theory’s degrees of freedom, therefore Rindler AdSimmediately provides a quantitative formulation of the ideas in the previoussection.The metric describing such an accelerated observer is that of a hyper-bolic black hole in AdS. This metric has a horizon, therefore a temperatureassociated with it, and its boundary is in the same conformal class as Rindlerspace and hyperbolic spacetime. It is interesting to note that for a special81.2. Phenomenological applicationstemperature (T = 12piR), the metric of the AdS Rindler wedge is diffeomor-phic to that of pure AdS itself. That is, while from the accelerated observer’spoint of view the manifold he is embeded in has a horizon, this is nothingbut a coordinate pathology, and the analytic extension of this particularmetric simply restores AdS.This is specially relevant since the state of the total field theory is knownto be the vacuum state, hence the quantum state dual to the wedge of theaccelerated observer is the partial trace over the boundary region comple-mentary to the boundary of the Rindler AdS wedge. It can immediately beseen how entanglement entropy and its holographic formulation enter theproblem.Once the general picture between hyperbolic black holes and boundaryRindler wedges is established, the temperature of the black hole can be variedwhile the behaviour of physical observers such as the energy momentumtensor is being tracked. By doing so, considerable insight on the interplaybetween boundary entanglement and the properties of the bulk geometrycan be obtained.1.2 Phenomenological applicationsEarlier in this Introduction I discussed how the coupling of the gravity dualand its field theory counterpart are inversely related. In particular, whenthe gauge theory is strongly coupled, a regime where perturbation theorymethods are not applicable, the bulk dynamics is in its simplest form, i.e.:classical general relativity. This is arguably the single most compelling fea-ture of the AdS/CFT correspondence: not only does it provide an abstractrelation between generic field theories and gravity systems, but it also pro-vides a tool box to tackle notoriously difficult problems in the study ofstrongly coupled field theories.In addition to offering the means to study aspects of strongly coupledfield theories that are usually beyond the scope of standard quantum fieldtheory methods, the AdS/CFT correspondence enables one to go even fur-ther and easily generalize theses systems to finite temperature and finitechemical potential cases. Therefore, the range of applications of the dual-ity to phenomenological toy models is vast, including the behaviour of hightemperature quark plasmas, strongly coupled electrons, high temperaturesuperconductors, and even low energy configurations of QCD.In part II of this thesis I will address examples of some of the applicationsmentioned above, and I will show how rich even basic holographic models91.2. Phenomenological applicationscan be. A few basic concepts behind some of the systems and examplesmentioned above are introduced bellow.1.2.1 Holographic fluids and metric perturbationsStudying the behaviour of small perturbations of certain metrics in Gen-eral Relativity leads to a remarkable conformal fluid interpretation whenviewed in the context of the Gauge / Gravity duality. From the modellingof high temperature relativistic plasma [21, 123], to the emergence of Ein-stein’s Equations from boundary field theory considerations [44, 88], theapplications are plenty. The investigation of small metric perturbations isusually conducted iteratively; at each step the perturbed metric is correctedin such a way that it solves Einstein’s Equation to a desired order of ac-curacy. While certain problems allow for analytic solutions at any order, anumerical approach can come in handy.In line with the aforementioned results, small metric perturbations canbe used to investigate to what degree a slight variation of the state of theboundary theory can alter the bulk metric. In particular, certain patternsof the field theory state will inexorably lead to a pathological bulk geometry.This constrains the class of physically allowed boundary states by imposingcertain smooth geometrical conditions on the bulk theory.In this thesis I investigate what kind of information about the bulk andboundary theories can be uncovered by small metric (or, from the bound-ary perspective, state) perturbations, using both analytical and numericalmethods. In particular I focus on the relation between the boundary theoryenergy momentum tensor and the bulk metric, and how one can be used todeduce the other.1.2.2 Finite temperature in holographic field theoriesThe study of field theories at finite temperature is hard and many of themethods suitable for the zero temperature case can not be used to tacklefinite temperature questions. Luckily AdS/CFT provides a natural and rela-tively easy way of modelling a holographic field theory at finite temperature.The correspondence connects a spacetime with a black hole to a bound-ary field theory on a thermal state whose temperature is just the Hawkingtemperature of the black hole. The Hawking temperature is found from theeuclideanized bulk metricds2 = α(r)dτ2 +dr2β(r)101.2. Phenomenological applicationswith periodic τ = it coordinate and α(r+) = β(r+) = 0, by demandingregularity at the horizon, ultimately leading toT =√α′(r+)β′(r+)4pi.This strikingly simple connection between black holes and thermal statesof conformal field theories is extremely useful for building phenomenologicalmodels as well as for studying basic aspects of quantum gravity.1.2.3 Finite density in holographic field theoriesSimilar to the finite temperature case, finite chemical potential also posesa challenge to standard methods in quantum field theory. Not only is theanalytical analysis particularly hard, but even numerical approaches usinglattice techniques fail to tackle this problem.The holographic dictionary dictates that a conserved charge in the fieldtheory is dual to a massless U(1) gauge field A in the bulk. The chemicalpotential and the charge density are encoded in the asymptotic behaviourof the time-component of the gauge field asµ = At(∞), (1.8)andρ =(1d− 2)∂L∂(∂rAt)∣∣∣∣r=∞, (1.9)where r is the radial direction in the bulk, with the boundary living atr = ∞. After writing down the gravitational lagrangian, our prescriptionfor computing the charge density at a given chemical potential is to solvethe equations of motion with a fixed boundary condition for the gauge field,equation (1.8), before reading off the density using equation (1.9).1.2.4 Holographic QCDQuantum Chromodynamics is believed to display a rich phase structure atfinite temperature and chemical potential, with phase transitions associatedwith deconfinement, nuclear matter condensation, the breaking of (approx-imate) flavor symmetries, and the onset at high density of quark matterphases displaying color superconductivity. However, apart from the regimesof asymptotically large temperature or chemical potential, a direct analyticstudy of the thermodynamic properties of the theory is not possible.111.2. Phenomenological applicationsA modern route to understanding properties of strongly coupled gaugetheories, that would be otherwise inaccessible, is via the AdS/CFT corre-spondence, or gauge theory / gravity duality. A bottom-up approach canbe used to generate a holographic system (starting with a gravity action)describing a confining gauge theory that exhibit a quark-matter phase withcolour (or flavour) superconductivity at large chemical potential.12Part INature of Spacetime13Chapter 2Dual of a Density Matrix2.1 IntroductionThe AdS/CFT correspondence [1, 92] relates states of a field theory onsome fixed spacetime B to states of a quantum gravity theory for whichthe spacetime metric is asymptotically locally AdS with boundary geometryB. The field theory provides a nonperturbative description of the quantumgravity theory that is manifestly local on the boundary spacetime: for agiven spacelike slice of the boundary spacetime B, the degrees of freedom inone subset are independent from the degrees of freedom in another subset.On the gravity side, identifying independent degrees of freedom is muchmore difficult; for example, the idea of black hole complementarity [131]suggests that local excitations inside the horizon of a black hole cannot beindependent of the physics outside the horizon. It is therefore interesting toask whether we can use our knowledge of independent field theory degreesof freedom to learn anything about which degrees of freedom on the gravityside may be considered to be independent.In this chapter, we consider the following question: Given a CFT on Bin a state |Ψ〉 dual to a spacetime M with a geometrical description, andgiven a subset A of a spatial slice of B, what part of the spacetime M canbe fully reconstructed from the density matrix ρA describing the state of thesubset of the field theory degrees of freedom in A?An immediate question is why we expect there to be any region thatcan be reconstructed if we know only about the degrees of freedom on asubset of the boundary. If the map between boundary degrees of freedomand the bulk spacetime is sufficiently non-local, it could be that informationfrom every region of the boundary spacetime is needed to reconstruct anyparticular subset of M . However, there are various reasons to be more opti-mistic. It is well known that the asymptotic behavior of the fields in the bulkspacetime is given directly in terms of expectation values of local operatorsin the field theory (together with the field theory action). Equipped with142.1. Introductionthis boundary behavior of the bulk fields in some region of the boundary4and the bulk field equations, we should be able to integrate these field equa-tions to find the fields in some bulk neighborhood of this boundary region.We can also compute various other field theory quantities (e.g. correlationfunctions, Wilson loops, entanglement entropies) restricted to the region Aor its domain of dependence. According to the AdS/CFT dictionary, thesegive us direct information about nearby regions of the bulk geometry.The notion that particular density matrices can be associated with cer-tain patches of spacetime was advocated in [135, 137].5 There, it was pointedout that a given density matrix may arise from many different states of thefull system, or from a variety of different quantum systems that contain thisset of degrees of freedom as a subset. Different pure states that give rise tothe same density matrix for the subset correspond to different spacetimeswith a region in common; this common region can be considered to be thedual of the density matrix.6In the bulk of this chapter, we seek to understand in general the region ofa bulk spacetime M that can be directly associated with the density matrixdescribing a particular subset of the field theory degrees of freedom. Webegin in Section 2 by reviewing some relevant facts from field theory andarguing that the density matrix associated with a region A may be morenaturally associated with the domain of dependence DA (defined below).In Section 3, we outline in more detail the basic question considered inthe chapter. In Section 4, we propose several basic constraints on the regionR(A) dual to a density matrix ρA. In Section 5, we consider two regions thatare plausibly contained in R(A). First, we argue that z(DA), the intersectionof the causal past and causal future of DA, satisfies our constraints andshould be contained in R(A), as should its domain of dependence, zˆ(DA).7We note that in some special cases, R(A) cannot be larger than zˆ(DA).However, in generic spacetimes, we argue that entanglement observables4As we recall below, knowledge of the field theory density matrix for a spatial region Aallows us to compute any field theory quantities localized to a particular codimension-zeroregion of the boundary, the domain of dependence of A.5For an earlier discussion of mixed states in the context of AdS/CFT, see [47].6As a particular example, it was pointed out in [135, 137] that a CFT on Sd in athermal density matrix, commonly understood to be dual to an AdS/Schwarzchild blackhole, cannot possibly know whether the whole spacetime is the maximally extended blackhole; only the region outside the horizon is common to all states of larger systems forwhich the CFT on Sd forms a subset of degrees of freedom described by a thermal densitymatrix.7We denote domains of dependence in the boundary with D· (for example, DA), whiledomains of dependence in the bulk are marked with a hat .ˆ152.2. Field theory considerationsthat can be calculated from the density matrix ρA certainly allow us toprobe regions of spacetime beyond zˆ(DA).8 This motivates us to consideranother region, w(DA), defined as the union of surfaces used to calculatethese entanglement observables (defined more precisely below) according tothe holographic entanglement entropy proposal [73, 118]. We show thatw(DA) (or more precisely, its domain of dependence wˆ(DA)) also satisfiesour constraints, and that for a rather general class of spacetimes, there is asense in which R(A) cannot be larger than wˆ(DA). On the other hand, weshow that in some examples, R(A) must be larger than wˆ(DA). We concludein Section 6 with a summary and discussion.2.2 Field theory considerationsTo begin, consider a field theory on some globally hyperbolic spacetimeB, and consider a spacelike slice Σ that forms a Cauchy surface. Then,classically, the fields on this hypersurface and their derivatives with respectto some timelike future-directed unit vector orthogonal to the hypersurfacedetermine the complete future evolution of the field. Quantum mechanically,the fields on this hypersurface can be taken as the basic set of variables forquantization and conjugate momenta defined with respect to the timelikenormal vector.Now consider some region A of the hypersurface Σ. Since the field theoryis local, the degrees of freedom in A are independent from the degrees offreedom in the complement A¯ of A on Σ. Thus, the Hilbert space can bedecomposed as a tensor product H = HA ⊗ HA¯, and we can associate adensity matrix ρA = tr ((|Ψ〉〈Ψ〉))A¯ to the degrees of freedom in A. Thisdensity matrix captures all information about the state of the degrees offreedom in A and can be used to compute any observables localized to A.In fact, the density matrix ρA allows us to compute field theory observ-ables localized to a larger region DA known as the domain of dependenceof A. The domain of dependence DA is the set of points p in B for whichevery (inextensible) causal curve through p intersects A (see Figure 2.1).Classically, the region DA is the subspace of B in which the field values arecompletely determined in terms of the initial data on A. Quantum mechan-ically, any operator in DA can be expressed in terms of the fields in A aloneand therefore computed using the density matrix ρA.As can be seen from Figure 2.1, any other spacelike surface A˜ homologous8It is an open question whether these observables are enough to reconstruct the space-time beyond zˆ(DA), so we cannot say with certainty that R(A) is larger than zˆ(DA).162.2. Field theory considerationsDAΣ AA~Figure 2.1: A spacelike slice Σ of a boundary manifold B (= S1× time)with a region A and its domain of dependence DA. The same domain ofdependence arises from any spacelike boundary region A˜ homologous to Awith ∂A = ∂A˜.to A with boundary ∂A˜ = ∂A shares its domain of dependence.9 Thus, insome other quantization of the theory based on a hypersurface Σ˜ with A˜ ⊂ Σ˜,we expect that the density matrix ρA˜ contains the same information as thedensity matrix ρA. It is then perhaps more natural to associate densitymatrices directly with domain of dependence regions. This observation isimportant for our considerations below: in constructing the bulk region dualto a density matrix ρA, it is more natural to use the boundary region DA asa starting point, rather than the surface A.It is useful to note that a quantum field theory on a particular domain ofdependence can be thought of as a complete quantum system, independentof the remaining degrees of freedom of the field theory. The observables ofthis field theory are the set of all operators built from the fields on A. Thestate of the theory is specified by a density matrix ρA, which allows us tocompute any such observable. The spectrum of this density matrix, andassociated observables such as the von Neumann entropy, give additionalinformation about the system. We can interpret this in a thermodynamicway as giving information about the ensemble of pure states described bythe density matrix. Alternatively, viewing this system as a subset of a larger9To see this, we note that since A and A˜ are homologous, we can deform A into A˜ anddefine B to be the volume bound by A and A˜. Then for any point p in B, consider aninextensible causal curve through p. Such a curve must necessarily pass through A. Butit cannot pass through A twice, since A is spacelike. On the other hand, the curve mustintersect the boundary of the region B twice (on the past boundary and on the futureboundary), so it must have an intersection with A˜.172.3. The gravity dual of ρAsystem that we assume is in a pure state, we can interpret this additionalinformation as telling us about the entanglement between the degrees offreedom in our causal development region with other parts of the system.2.3 The gravity dual of ρAIn this section, we consider the question of how much information the densitymatrix ρA carries about the dual spacetime. We restrict the discussion tostates of the full system that are dual to some spacetime M with a goodclassical description. Specifically, we ask the questionQuestion: Suppose that a field theory on a spacetime B in a state |Ψ〉 hasa dual spacetime M with a good geometrical description (e.g. a solutionto some low-energy supergravity equations). How much of M can be recon-structed given only the density matrix ρA for the degrees of freedom in asubset A of some spacelike slice of the boundary?Alternatively, we can ask:Consider all states |Ψα〉 with dual spacetimes Mα that give rise to a partic-ular density matrix ρA for region A of the boundary spacetime. What is thelargest region common to all the Mαs?We recall that knowledge of the density matrix ρA allows us to calculateany field theory observable involving operators localized in the domain ofdependence DA, plus additional quantities such as the entanglement entropyassociated with the degrees of freedom on any subset of A. According tothe AdS/CFT dictionary, these observables give us a large amount of in-formation about the bulk spacetime, particularly near the boundary regionDA, so it is plausible that at least some region of the bulk spacetime can befully reconstructed from this data. We will refer to this region as R(A). Weexpect that in general the density matrix ρA carries additional informationabout some larger region G(A), but this additional information does notrepresent the complete information about G(A)−R(A).In this chapter, we do not attempt to come up with a procedure toreconstruct the region R(A); rather we will attempt to use general argumentsto constrain how large R(A) can be.182.4. Constraints on the region dual to ρA2.4 Constraints on the region dual to ρABefore considering specific proposals for R(A), it will be useful to pointout various constraints that R(A) should satisfy. First, since the densitymatrices for any two subsets A and A˜ with the same domain of dependenceD correspond to the same information in the field theory, we expect thatthe region of spacetime that can be reconstructed from ρA is the same asthe region that can be reconstructed from ρA˜. Thus we have:Constraint 1: If A and A˜ have the same domain of dependence D, thenR(A) = R(A˜).For a particular boundary field theory, the bulk spacetime will be governedby some specific low-energy field equations. We assume that we are workingwith a known example of AdS/CFT so that these equations are known. Ifwe know all the fields in some region R of the bulk spacetime M , we canuse these field equations to find the fields everywhere in the bulk domain ofdependence of R (which we denote by Rˆ). Since R(A) is defined to be thelargest region of the bulk spacetime that we can reconstruct from ρA, wemust have:Constraint 2: Rˆ(A) = R(A).Now, suppose we consider two non-intersecting regions A and B on somespacelike slice of the boundary spacetime. The degrees of freedom in A andB are completely independent, so it is possible to change the state |Ψ〉 suchthat ρB changes but ρA does not.10 Changes in ρB will generally affect theregion R(B) in the bulk spacetime, but as a consequence can also affect anyregion in the causal future J+(R(B)) or causal past J−(R(B)) of R(B). Butthese changes can have no effect on the region R(A) since this region can bereconstructed from ρA, which does not change. Thus, we have:Constraint 3: If A and B are non-intersecting regions of a spacelike sliceof the boundary spacetime, then R(A) cannot intersect J(R(B)).Here we have defined J(R) = J−(R) ∪ J+(R). Note that whatever R(B)is, it certainly includes DB so as a corollary, we can say that R(A) cannotintersect J(DB). Taking B = A¯ (i.e. as large as possible without intersectingA), we get a definite upper bound on the size of R(A): it cannot be largerthan the complement of J(DA¯).10Further, we expect that for some subset of these variations, the dual spacetime con-tinues to have a classical geometric description.192.5. Possibilities for R(A)2.5 Possibilities for R(A)Let us now consider some physically motivated possibilities for the regionR(A). An optimistic expectation is that we could reconstruct the entireregion G(A) of the bulk spacetime M used in calculating any field theoryobservable localized in DA (for example, all points touched by any geodesicwith boundary points in DA). However, this cannot be a candidate forR(A), since it is easy to find examples of non-intersecting A and B on somespacelike slice of a boundary spacetime such that geodesics with endpointsin B intersect with geodesics with endpoints in A.11 Thus, G(A)∩G(B) 6= ∅(which implies G(A) ∩ J(G(B)) 6= ∅) and so Constraint 3 is violated.A lesson here is that even if field theory observables calculated froma boundary region DA probe a certain region of the bulk, they cannotnecessarily be used to reconstruct that region. Generally, we will haveR(A) ⊂ G(A) ⊂ M , where ρA contains complete information about R(A),some information about G(A) and no information about G¯(A).2.5.1 The causal wedge z(DA)A simple region that is quite plausibly included in R(A) is the set of pointsz(DA) in the bulk that a boundary observer restricted to DA can communi-cate with (i.e. send a light signal to and receive a signal back). For example,such an observer could easily detect the presence or absence of an arbitrarilysmall mirror placed at any point in z(DA). Formally, this region in the bulkis defined as the intersection of the causal past of DA with the causal futureof DA in the bulk, z(DA) ≡ J+(DA) ∩ J−(DA), as shown in Figure 2.2.12These observations correspond to perturbing the spacetime at one point inthe asymptotic region and observing the asymptotic fields at another pointat a later time. In the field theory language, such observations correspondto calculating response functions, in which the fields are perturbed at onepoint in DA and observed at another point in DA. Such calculations canbe done using only the density matrix ρA, thus we expect that z(DA) isincluded in the region R(A).11For example, suppose we consider the vacuum state of a CFT on a cylinder and takeA and B to be the regions θ ∈ (0, pi/2) ∪ (pi, 3pi/2) and θ ∈ (pi/2, pi) ∪ (3pi/2, 2pi) on theτ = 0 slice. Then the lines of constant θ are spatial geodesics in the bulk, and the regioncovered by such geodesics anchored in A clearly intersects the region of such geodesicsanchored in B.12Recall that the causal future J+(DA) of DA in the bulk is the set of points reachableby causal curves starting in DA while the causal past J−(DA) of DA is the set of points,from which DA can be reached along a causal curve.202.5. Possibilities for R(A)Figure 2.2: Causal wedge z(DA) associated with a domain of dependenceDA.By condition 2, we can extend this expectation to the proposal thatzˆ(DA) ⊂ R(A). It is straightforward to check that zˆ(DA) also satisfiescondition 3.13 Thus, the suggestion that zˆ(DA) ⊂ R(A) is consistent withour Constraints 1, 2 and 3.The boundary of the region z(DA) in the interior of the spacetime isa horizon with respect to the boundary region DA. Thus, the statementthat we can reconstruct the region z(DA) is equivalent to saying that theinformation in DA is enough to reconstruct the spacetime outside this hori-zon. This horizon can be an event horizon for a black hole, but in generalis simply a horizon for observers restricted to the boundary region DA.In certain simple examples, it is straightforward to argue that R(A)cannot be larger than z(DA) or zˆ(DA). For example, if M is pure globalAdS spacetime and A is a hemisphere of the τ = 0 slice of the boundarycylinder, then z(DA) is the region bounded by the lightcones from the past13Suppose subsets A and B of a boundary slice do not intersect and suppose p ∈J(zˆ(DB)). Then there exists a causal curve through p that intersects zˆ(DB) and thereforeintersects some q in z(DB). If p is also in zˆ(DA), this same causal curve through p mustintersect a point r in z(DA). Thus, there is a causal curve from q in z(DB) to r in z(DA).By definition of z, we must be able to extend this curve to a causal curve connecting DAto DB . But in this situation, perturbations to the fields in DA could affect the fields inDB (or vice versa), and this would violate field theory causality.212.5. Possibilities for R(A)abcdcbadFigure 2.3: In pure global AdS, causal wedges of complementary hemispher-ical regions of the τ = 0 slice intersect along a codimension-two surface. Ingeneric asymptotically AdS spacetimes, they intersect only at the boundary.and future tips of DA and the spacetime boundary, as shown in Figure 2.3.Any point outside this region is in the causal future or causal past of theboundary region DA¯,14 so by Constraint 3 (and the consequences discussedafterwards) such points cannot be in R(A).Information beyond the causal wedge z(DA)We might be tempted to guess that R(A) = zˆ(DA) in general, but wewill now see that ρA typically contains significant information about thespacetime outside the region zˆ(DA). Consider the same example of a CFTon the cylinder with the same regions A and A¯, but now consider some otherstate for which the dual spacetime is not pure AdS. A key observation15 isthat, generically, the wedges z(DA) and z(DA¯) do not intersect, except atthe boundary of A. This follows from a result of Gao and Wald [49] that lightrays through the bulk of a generic asymptotically AdS spacetime generallytake longer to reach the antipodal point of the sphere than light rays alongthe boundary. Thus, the backward lightcone from the point a in the rightpanel of Figure 2.3 is different from the forward lightcone from point d. We14This relies on the fact that for pure global AdS, the forward lightcone from the pasttip of DA (point b in Figure 2.3) is the same as the backward lightcone from the future tipof DA¯ (point c) and the backward lightcone from the future tip of DA (point a in Figure2.3) is the same as the forward lightcone from the past tip of DA¯ (point d).15We are grateful to Veronika Hubeny and Mukund Rangamani for pointing this out.222.5. Possibilities for R(A)can still argue that R(A) cannot overlap with the region J+(DA¯)∪J−(DA¯),but the complement of this region no longer coincides with zˆ(DA). Thus, itis possible that R(A) is larger than zˆ(DA) in these general cases.16To see that the density matrix ρA typically does contain informationabout the spacetime outside the region z(DA), we can take inspiration fromHubeny [70], who argued that in many examples, the field theory observablesthat probe deepest into the bulk of the spacetime are those computed byextremal codimension-one surfaces in the bulk.According to the proposal of Ryu and Takayanagi [118] and the covariantgeneralization by Hubeny, Rangamani, and Takayanagi [73], the von Neu-mann entropy of a density matrix ρC corresponding to any spatial region Cof the boundary gives the area of a surface W (C) in the bulk defined eitheras• the extremal codimension-two surface W in the bulk whose boundaryis the boundary of C. In cases where more than one such extremalsurface exists, we take the one with minimal area, or• the surface of minimal area such that the light sheets from this surfaceintersect the boundary exactly at ∂DC .In each case, it is assumed that the surface W is homologous to C. In [73],it is argued that these two definitions are equivalent.Now, consider the surface W (A) that computes the entanglement en-tropy of the full density matrix ρA. From the second definition, it is clearthat the surface W cannot have any part in the interior of z(DA). Other-wise, the light rays from any such point would reach the boundary in theinterior of region DA, and it would not be true that the light sheet from Wintersects the boundary at ∂DA. By the same argument, the surface W (A¯)that computes the entanglement entropy of ρA¯ cannot have any part in theinterior of z(DA¯). But by the first definition, the surface W (A¯) is the sameas the surface W (A), since ∂A¯ = ∂A.17 Since z(DA) and z(DA¯) generallyhave no overlap in the bulk of the spacetime, it is now clear that the surfaceW lies outside at least one of z(DA) and z(DA¯).16As an explicit example of a spacetime with this property, we can take a static spacetimewith a spherically symmetric configuration of ordinary matter in the interior, e.g. theboson stars studied in [11].17The equivalence of these surfaces and hence their areas is consistent with the factthat for a pure state in a Hilbert space H = HA ⊗HA¯, the spectrum of eigenvalues of ρAmust equal the spectrum of eigenvalues of ρA¯. Thus, the entanglement entropies S(ρA)and S(ρA¯) must agree. We do not consider here the case where the entire theory is in amixed state.232.5. Possibilities for R(A)To summarize, the area of surface W may be computed as the von Neu-mann entropy of either the density matrix ρA or the density matrix ρA¯. Inthe generic case where z(DA) and z(DA¯) do not intersect in the bulk, thesurface W must lie outside at least one of z(DA) and z(DA¯). Thus, we cansay that either the density matrix ρA carries some information about thespacetime outside z(DA) or the density matrix ρA¯ carries information aboutthe spacetime outside z(DA¯).182.5.2 The wedge of minimal-area extremal surfaces w(DA).Based on these observations, and the observation of Hubeny that the ex-tremal surfaces probe deepest into the bulk in various examples, it is naturalto define a second candidate for the region R(A) based on extremal surfaces.The surface W (A) calculates the entanglement entropy associated withthe entire domain of dependence DA (equivalently, the largest spacelike sur-face in DA). We can also consider the entanglement entropy associated withany smaller causal development region within DA. For any such region C,there will be an associated surface W (C) (as defined above) whose area com-putes the entanglement entropy (according to the proposal). Define a bulkregion w(DA) as the set of all points contained on some minimal-area19 ex-tremal codimension-two surface whose boundary coincides with the bound-ary of a spacelike codimension-one region in DA. The area of each suchcodimension-two surface is (according to [73]) equal to the entanglemententropy of the corresponding boundary region. Thus, the region w(DA) di-rectly corresponds to the region of the bulk whose geometry is probed byentanglement observables. As we have seen, the region w(DA) generallyextends beyond the region z(DA).From the region w(DA), we can define a larger region wˆ(DA) as thedomain of dependence of the region w(DA). As discussed above, knowing thegeometry (and other fields) in w(DA) and the bulk gravitational equationsshould allow us to reconstruct the geometry in wˆ(DA).We would now like to understand whether the region wˆ(DA) obeys theconstraints outlined above. Constraints 1 and 2 are satisfied by definition.It is straightforward to show that Constraint 3 is satisfied assuming that the18Again, it is easy to check this in specific examples. For explicit examples of sphericallysymmetric static star geometries asymptotic to global AdS with A equal to a hemisphereof the τ = 0 slice, the surface W (A) lies at τ = 0 and passes through the center of thespacetime, while the regions z(DA) and z(DA¯) do not reach the center.19Here, we mean minimal area among the set of extremal surfaces with the same bound-ary.242.5. Possibilities for R(A)following conjecture holds:Conjecture C1: If DA and DB are domains of dependence for non-intersectingregions A and B of a spacelike slice of the boundary spacetime, then w(DA)and w(DB) are spacelike separated.Supposing that this holds, if p is in J(wˆ(DB)), then there exists a causalcurve through p intersecting wˆ(DB), and by definition of wˆ, this causal curvealso intersects w(DB). If p is also in wˆ(DA), then every causal curve throughp intersects w(DA). Thus, there exists a causal curve that intersects bothw(DB) and w(DA), which violates C1. We conclude that wˆ(DA) satisfiesConstraints 1, 2 and 3 assuming that Conjecture C1 holds.Aside: proving Conjecture C1While a proof (or refutation) of Conjecture C1 is left to future work, wemake a few additional comments here.For the case of static spacetimes, it is straightforward to prove a resultsimilar to C1.Let A1 and A2 be two non-intersecting regions of the t = 0 boundary slice of astatic spacetime, with B1 and B2 spacelike regions in A1 and A2, respectively.Let W (B1) and W (B2) be the minimal surfaces in the t = 0 slice of the bulkspacetime with ∂W (B1) = ∂B1 and ∂W (B2) = ∂B2. Then W (B1) andW (B2) cannot intersect.To show this, consider the part of W (B1) contained in the region of thet = 0 slice bounded by W (B2) and B2, and the part of W (B2) containedin the region of the t = 0 slice bounded by W (B1) and B1. If these twopieces have different areas, then by swapping the two pieces, either the newsurface W (B1) or the new surface W (B2) will have a smaller area thanbefore, contradicting the assumption that these were minimal-area surfaces.If the two pieces have the same area, the modified surfaces will have thesame area as before, but the new surfaces will be cuspy20, such that we candecrease the area by smoothing the cusps.In attempting a more general proof, it may be useful to note that Con-jecture C1 is equivalent to the following statement (with some mild assump-tions):20The surfaces W (B1) and W (B2) cannot be tangent at their intersection because thereshould be a unique extremal surface passing through a given point with a specified tangentplane to the surface at this point.252.5. Possibilities for R(A)Conjecture C2: For any spacelike boundary region C, the surface W (C)is spacelike separated from the rest of w(DC).To see the equivalence, assume first that C1 holds and let A = C and B = C¯.If we assume the generic case that W (C) is the same as W (C¯), then W (C) =W (B) ⊂ w(DB) must be spacelike separated from w(DA) = w(DC). Con-versely, for two disjoint regions A and B, let C be any region such thatA ⊂ C and B ⊂ C¯. By definition, we have that w(DA) ⊂ w(DC) andw(DB) ⊂ w(DC¯). Assuming again that W (C) = W (C¯), Conjecture C2 im-plies that there is a spacelike path connecting any point in w(DA) ⊂ w(DC)with any point p in W (C), and that there also exists a spacelike path con-necting any point in w(DB) ⊂ w(DC¯) with the same point p. Therefore,there is a spacelike path (through p) connecting any point in w(DA) withany point in w(DB), as required for C1.While C1 is immediately more useful, C2 might be easier to prove. Con-sider any boundary region C and any point p in w(DC). Then there exists aspacelike codimension-one region Ip in the domain of dependence DC suchthat p ∈ W (Ip). Ip can be extended to a spacelike surface AI homologouswith C, with the same boundary as C, δAI = δC . The surface which cal-culates entanglement entropy is the same for AI and C: W (AI) = W (C).Consider now a one-parameter family of surfaces S(λ), which continuouslyinterpolate between AI = S(0) and Ip = S(1), and the corresponding familyof bulk minimal surfaces W (S(λ)) interpolating between W (C) and W (Ip).It is plausible that these bulk minimal surfaces change smoothly and thattheir deformations are spacelike; following the flow, we can find a spacelikepath from p to W (C), which would complete the proof of the ConjectureC2.We leave further investigation of the general validity of C1 as a questionfor future work.2121We note here that the restriction to minimal extremal surfaces (rather than all ex-tremal surfaces) is essential for the validity of this conjecture. In static spacetimes withmetric of the form ds2 = −f(r)dt2 + dr2/g(r) + r2dΩ2 where g(0) = 1 and g(r) → r2, itis possible that extremal surfaces bounded on one hemisphere intersect extremal surfacesbounded on the other hemisphere in cases where g(r) is not monotonically increasing. Forthese examples, C1 would fail if the definition of w did not restrict to minimal surfaces.262.5. Possibilities for R(A)Possible connection between the geometry of W (A) and thespectrum of ρATo summarize the discussion so far, the region wˆ(DA) satisfies conditions 1,2 and 3 assuming that Conjecture C1 is correct. Thus, wˆ(DA) is a possiblecandidate for the region R(A). A rather nice feature of this possibility is thatwˆ(DA) intersects wˆ(DA¯) along the codimension-two surface W (A) = W (A¯)defined above. Thus, the surface W represents the information in the bulkcommon to wˆ(DA) and wˆ(DA¯). The area of this surface corresponds to thevon Neumann entropy of ρA, which is the simplest information shared byρA and ρA¯. We might then conjecture that the full spectrum of ρA (which isthe same as the spectrum of ρA¯ and represents the largest set of informationcommon to ρA and ρA¯) encodes the full geometry of the surface W (i.e. thelargest set of information common to wˆ(DA) and wˆ(DA¯)).Reconstructing bulk metrics from extremal surface areasBefore proceeding, let us ask whether it is even possible that the areas of ex-tremal surfaces with boundary in some region DA carry enough informationto reconstruct the geometry in w(DA).Consider the simple case of a 1+1 dimensional CFT on a cylinder withDA a diamond-shaped region on the boundary. Given any state for the CFT,we could in principle compute the entanglement entropy associated with anysmaller diamond-shaped region bounded by the past lightcone of some pointin DA and the forward lightcone of some other point. This would give us onefunction of four variables, since each of the two points defining the smallerdiamond-shaped region is labeled by two coordinates. Assuming the statehas a geometrical bulk dual description, the bulk geometry will be describedby a metric which consists of several functions of three variables.22 Thesefunctions allow us to determine the entanglement entropy from the geometryin the wedge w(DA) via the Takayanagi et. al. proposal, so we have a mapfrom the space of metrics to the space of entropy functions. Small changesin the geometry of the wedge w(DA) will generally affect the areas of someof the minimal surfaces, while small changes in the geometry outside thewedge will generally not affect these areas. It is at least plausible that theentanglement information could be used to fully reconstruct the geometryin the wedge in some cases, since the map from wedge geometries into theentanglement information is a map from finitely many functions of threevariables to a function of four variables, and it is possible for such a map to22We are ignoring the possible extra compact dimensions in the bulk.272.5. Possibilities for R(A)be an injection.A proven result of this form in the mathematics literature [113] is thatfor two-dimensional simple23 compact Riemannian manifolds with bound-ary, the bulk geometry is completely fixed by the distance function d(x, y)between points on the boundary (the lengths of the shortest geodesics con-necting various points). This implies that for static three-dimensional space-times, the spatial metric of the bulk constant time slices can be reconstructedin principle if the entanglement entropy is known for arbitrary subsets of theboundary. However, we are not aware of any results about the portion ofa space that can be reconstructed if the distance function is known onlyon a subset of the boundary, or of any results that apply to Lorentzianspacetimes.Cases when R(A) cannot be larger than wˆ(DA)We saw above that in special cases, z(DA) together with J(z(DA¯)) coverthe entire spacetime, so Constraint 3 is just barely satisfied for z (or zˆ). Forthese examples, if z(DA¯) is in R(A¯) then R(A) cannot possibly be largerthan z(DA). On the other hand, for generic spacetimes, we argued thatonly a portion of the spacetime is covered by z(DA) and J(z(DA¯)), leavingthe possibility that R(A) could be larger than z(DA). In these examples,extremal surfaces from A typically extend into the region not covered byz(DA) or z(DA¯) (or the causal past/future of these), and this motivated usto consider wˆ(DA) as a larger possibility for R(A).We will now see that in a much wider class of examples, wˆ(DA) togetherwith J(w(DA¯)) do cover the entire spacetime. To see this, recall that thesurfaces W (A) and W (A¯) computing the entanglement entropy of the en-tire regions A and A¯ are the same by definition, as long as A and A¯ arehomologous in the bulk.24 Now, suppose that for a one-parameter family ofboundary regions B(λ) ⊂ A interpolating between A and a point (assum-ing A is contractible), the surfaces W (B(λ)) change smoothly. Similarly,suppose that for a one-parameter family of boundary regions B′(λ) ⊂ A¯interpolating between A¯ and a point (assuming A¯ is contractible), the sur-faces W (B′(λ)) change smoothly. Then the union of all surfaces W (B(λ))and W (B′(λ)) covers an entire slice of the bulk spacetime. In this case, forany point p in the bulk spacetime, either there is a causal curve through23See [113] for the definition of a simple manifold.24The only possible exception would be the case where there are two extremal surfaceswith equal area having boundary ∂A. In this case, we might call one W (A) and the otherW (A¯).282.5. Possibilities for R(A)Figure 2.4: Different possible behaviors of extremal surfaces in sphericallysymmetric static spacetimes. Shaded region indicates w(DA) where A is theright hemisphere. The boundary of the shaded region on the interior of thespacetime is the minimal area extremal surface bounded by the equatorialSd−1.p that intersects ∪λW (B(λ)) ⊂ w(DA) or else every causal curve throughp intersects ∪λW (B′(λ)) ⊂ w(DA¯). This shows that wˆ(DA) together withJ(w(DA¯)) cover the entire spacetime.To summarize, in cases where W (B) varies smoothly with B as describedabove, we have that wˆ(DA) together with J(w(DA¯)) cover the entire space-time. Thus, by Constraint 3, with this smoothness condition, if wˆ(A¯) ⊂ R(A¯)then R(A) cannot be larger than wˆ(DA).25 While there are many examplesof spacetimes for which this smooth variation does not occur (e.g. as de-scribed in the next section), spacetimes satisfying the condition are notparticularly special.An example where R(A) is strictly larger than wˆ(DA)We have seen that wˆ(DA) is in some sense a maximally optimistic proposalfor R(A) in cases where a particular smoothness condition is satisfied orwhen w(DA) ∪ w(DA¯) includes a Cauchy surface. We will now see thatthese conditions can fail to be true in some cases, and that in these cases,R(A) must be larger than wˆ(DA) for some choice of A.Consider the simple example of static spherically symmetric spacetimeswith metric of the form ds2 = −f(r)dt2 + dr2/g(r) + r2dθ2 where g(0) = 1and g(r) → r2 for large r. For any spacetime of this form, the extremalcodimension-two surfaces bounded by spherical regions on the boundary25An alternative condition that leads to the same conclusion is that w(DA) ∪ w(DA¯)includes a Cauchy surface.292.6. Discussionwill be constant-time surfaces in the bulk that can easily be computed. Bysymmetry, there always exists an extremal surface through the center of thespacetime whose boundary is an equatorial Sd−1 of the boundary Sd. Now,moving out towards the boundary along some radial geodesic, there will bea unique extremal surface passing through each point and normal to theradial line.In some cases (e.g. pure AdS), the boundary spheres for these extremalsurfaces shrink monotonically as we approach the boundary, as shown inthe left half of Figure 2.4. However, there are other cases for which g(r) isnot monotonic where the extremal surfaces shrink in the opposite direction,then grow, then shrink again, as shown in the right half of Figure 2.4.26 Inthese cases, boundary spheres with angular radius in a neighborhood of pi/2will bound multiple extremal surfaces in the bulk. The extremal surface ofminimum area in these cases is always one that is contained within one halfof the bulk space (otherwise we could construct intersecting minimal surfacesbounding disjoint regions of the boundary). Considering only the minimalsurfaces, we find that there exists a spherical region in the middle of thespacetime penetrated by no such surface. Thus, even if we choose DA to bethe entire spacetime boundary, the region w(DA) excludes the region r < r0for some r0. In this case, we have all information about the field theory(assumed to be a pure state), so R(A) should be the entire spacetime.More generally, the region w(DA) in these cases will have a “hole” if Ais chosen to be any boundary sphere with angular radius between pi/2 andpi, as shown in Figure 2.5. Note, however, that the central region is includedin z(DA) for sufficiently large A, so z(DA) 6⊂ w(DA) in these cases.2.6 DiscussionIn this note, we have presented various consistency constraints on the regionR(A) of spacetime which can in principle be reconstructed from the densitymatrix ρA for a spatial region A of the boundary with domain of dependenceDA. We have argued that the z(DA) ≡ J+(DA) ∩ J−(DA) and its domainof dependence zˆ(DA) should be contained in R(A) and that zˆ(DA) satis-fies our consistency constraints. Since entanglement observables calculated26As an explicit example, we have considered the case of a charged massive scalar fieldcoupled to gravity, with scalar field of the form φ(r) = eiωtf(r). Spherically-symmetricconfigurations of this type with non-zero charge are known as “boson-stars” [11]. We findthat for fixed ψ(0), the metric function g(r) is monotonically increasing for sufficientlysmall values of the scalar field mass, while for sufficiently large values we can have thebehavior shown on the right in Figure 2.4.302.6. DiscussionFigure 2.5: Region w(DA) (shaded) where A is a boundary sphere of angularsize greater than pi. No minimal surface with boundary in A penetrates theunshaded middle region.from ρA correspond to extremal surfaces that typically probe a region ofspacetime beyond zˆ(DA), we have also considered the union of these sur-faces w(DA) and its domain of dependence wˆ(DA) as a possibility for R(A)that is often larger than zˆ(DA). We have seen that wˆ(DA) also satisfies ourconstraints (assuming Conjecture C1), and that if wˆ(DA) ⊂ R(A) generally,then R(A) = wˆ(DA) for a broad class of spacetimes.A false constraintThe constraints discussed in this note are essentially consistency require-ments that do not make use of details of the AdS/CFT correspondence. Itis interesting to ask whether there exist any more detailed conditions thatcould constrain the region R(A) further.It may be instructive to point out a somewhat plausible constraint thatturns out to be false. For two non-intersecting regions A and B of theboundary spacetime, it may seem that the region G(B) of the spacetimeused to construct field theory observables in B should not intersect theregion R(A) dual to the density matrix ρA. The argument might be that ifthe physics in R(A) is the bulk manifestation of information in ρA, we cannotexpect to learn anything about this region knowing only ρB. It would seemthat this would be telling us directly about ρA knowing only ρB. Perhapssurprisingly, it is easy to find an example where neither w(DA) nor z(DA)satisfies this constraint, see Figure 2.6.In the planar AdS black hole geometry, take the region A to be a ball-312.6. DiscussionFigure 2.6: Spatial t = 0 slice of w(DA) (light shaded plus dark shaded)and z(DA) (dark shaded) for a planar AdS black hole. The dashed curve isa spatial geodesic with endpoints in A¯. Knowledge of observables obtainedfrom ρA¯ alone allow us to compute the length of this geodesic.shaped region on the boundary. In this case, it is straightforward to checkthat spatial geodesics with endpoints in A¯ intersect both w(DA) and z(DA).Thus, the constraint R(A) ∩ G(A¯) = ∅ can’t be correct if z(DA) ⊂ R(A).In hindsight, it is not difficult to understand the reason. Knowledge of thedensity matrix ρA¯ allows us to reconstruct R(A¯). There could be manystates of the full theory that give rise to the same density matrix ρA¯. Forany such state with a classical gravity dual description, the dual spacetimegeometry must be such that spatial geodesics anchored in DA¯ have the samelengths as in the original spacetime we were considering. But there can bemany such spacetimes. So using the information in ρA¯, we are not learningdirectly about ρA, only about the family of density matrices ραA such thatthe pair (ρA¯, ραA) can arise from a pure state |Ψ〉 that has a geometricalgravity dual.27322.6. DiscussionFigure 2.7: The region of spacetime reconstructible from density matrices ρBand ρC (shaded, right hand side picture) is smaller than that reconstructiblefrom ρB∪C (shaded, left hand side picture). Reconstruction of R(B ∪ C)−(R(B) ∪R(C)) (interior of dotted frame outside of the two shaded regions)requires knowledge of entanglement between degrees of freedom in B andC.Figure 2.8: The region of spacetime reconstructible from density matricesρAi lies arbitrarily close to the boundary (illustrated here on a spatial slice).The ability to reconstruct the bulk geometry depends entirely on the knowl-edge of entanglement among the various boundary regions.332.6. DiscussionSpacetime emergence and entanglementThe observations in this note highlight the importance of entanglement inthe emergence of the dual spacetime. Consider a collection {Ai} of subsetson the boundary such that ∪Ai covers an entire boundary Cauchy surface.In a classical system, knowing the configuration and time derivatives of thefields in each of these regions would give us complete information about thephysical system. Quantum mechanically, however, complete informationabout the system consists of two ingredients: (i) the density matrices ρAi ,and (ii) the entanglement between the various regions.If we subdivide a set A→ {B,C} and pass from ρA → {ρB, ρC}, we loseinformation about the entanglement between B and C. In the bulk picture,the region of spacetime that we can reconstruct (for any R satisfying ourconstraints) is significantly smaller than before, as we see in Figure 2.7. Theregion of spacetime that we can no longer reconstruct corresponds to theinformation about the entanglement between the degrees of freedom in Band C that we lost when subdividing.As we divide the boundary into smaller and smaller sets Ai, we retaininformation about entanglement only at successively smaller scales, whilethe bulk space ∪R(Ai) that can be reconstructed retreats ever closer tothe boundary (Figure 2.8). Conversely, knowledge of the bulk geometryat successively greater distance from the boundary requires knowledge ofentanglement at successively longer scales.28 In the limit where Ai becomearbitrarily small, we know nothing about the bulk spacetime even if weknow the precise state for each of the individual degrees of freedom via thematrices ρAi . In this sense, the bulk spacetime is entirely encoded in theentanglement of the boundary degrees of freedom.27Don Marolf has pointed out to us that the connection between two-point functions andthe lengths of spatial geodesics has been argued to fail for spacetimes that do not satisfycertain analyticity properties [91]. It is likely that demanding such properties imposeseven stronger constraints connecting ρA and ρA¯.28A very similar picture was advocated in [132].34Chapter 3Boson Stars3.1 IntroductionIn this chapter we construct solutions of asymptotically asymptotically AdSboson stars coupled to a U(1) gauge field in 3 and 4 dimensions, computethe star’s charge and mass as functions of the scalar field central density andstudy the behaviour of extremal surfaces in these backgrounds. In particular,we determine conditions under which minimal area, codimension 2 spacelikesurfaces fail to cover the whole spacetime. To understand the motivationfor doing so, let us digress for a moment and consider some intricate openquestions regarding the holographic principle in anti de Sitter spacetimes.The holographic principle is a remarkable idea relating theories withgravity to theories without gravity [129, 133]. The best understood exam-ple of holography is the AdS/CFT correspondence which proposes a one toone correspondence between a gravitational theory on anti-de Sitter spaceand a strongly coupled, large N, conformal field theory living on the confor-mal boundary of AdS [92, 140]. This correspondence has been extensivelyused to study a variety of strongly coupled field theories and has led to theformulation of successful formalisms such as AdS/QCD and AdS/CMT 29,however, the potential use of the gauge gravity duality to tackle problemsin quantum gravity has yet to show comparable progress.The intrinsic non local aspect of the bulk theory is particularly chal-lenging when trying to understand the exact connection between bulk andboundary degrees of freedom. It stands as one of the major barriers in theway of the gauge / gravity duality fulfilling its potential of answering longstanding problems in quantum gravity by recasting them in field-theoreticlanguage. In order to improve this situation, the issue of understandingthe map between bulk and boundary degrees of freedom must be addressed.Optimistically, we should be able to express any variation of the boundarytheory state in terms of a well defined perturbation of the bulk geometry;conversely, any changes in the bulk should be related to a particular pertur-29For an extensive review, refer to [58, 101].353.1. Introductionbation of the boundary state in a one to one fashion.An important question related to establishing the dictionary betweenbulk and boundary is to understand what information about the bulk iscontained in a certain region of the boundary field theory. In other words,let a local field theory defined on the conformal boundary B of a spacetimeM be in a generic state ρ. Assume this theory has a well defined gravitydual and consider, with respect to some arbitrary time slicing, a spacelikesubregion A of the boundary. Which portion of the bulk R(A) is dual tothe reduced density matrix ρA30, is a question closely related to the issue ofcharacterizing the bulk-boundary degree of freedom map.The Ryu-Takayanagi proposal [119] suggests that the entanglement en-tropy of a sub-region A of a field theory with a well behaved, static, gravitydual, is proportional to the area of the minimal area surface anchored at theboundary δA of A. Using this proposal we can construct a family of minimalsurfaces by considering the surface dual to the entanglement entropy of ρAand all the surfaces dual to the partial traces of ρA with respect to arbitrarysubregions within A. This family of surfaces defines a region in the bulk wewill call w(A) and is a possible candidate for R(A) as proposed in [33] andfurther explored in [139]31.Despite meeting several conditions for a suitable candidate for R(A)[33, 139], when A = B, the whole boundary slice, w(B) does not coverthe entire spacetime in general. It was pointed out in [33] that there arespacetimes for which no minimal surfaces reach certain areas of the bulkthat are, nevertheless, causally connected to the boundary, leading us to theconclusion that, at least for such cases, w(A) cannot be the elusive regionR(A).An explicit example of minimal surfaces that do not cover the bulk isfound in asymptotically AdS boson star backgrounds. Boson stars are solu-tions to Einstein’s equation coupled to a complex scalar field that have at-tracted the interest of physicists for over 40 years. Beginning with the workof Kaup [83] and others [117], the fields of general relativity, cosmology, andeven particle physics have shown great interest in fully understanding thesesolitonic objects. Some of the standard reviews are [78, 89, 90, 124].In this chapter we compute the physical charge and mass of asymp-totically AdS, charged boson stars as a function of the scalar field centraldensity, investigate numerically the behaviour of extremal surfaces on these30Where ρA is the partial trace of ρ over the complement of A, A¯.31Formally we should extend the discussion to DA, the causal development of A, how-ever, for clarity and objectivity’s sake we choose to omit it in the Introduction.363.2. Equations of motionbackgrounds, and determine the conditions for which w(A) fails to fullycover the bulk, which we will call hollow phases. Furthermore we comparethe conditions for the existence of hollow phases with that of known phys-ical instabilities in four dimensional charged boson star systems, and arguethat such hollow configurations are likely unstable and therefore physicallyforbidden, thus providing further evidence that w(A) may be the correctproposal for R(A).This chapter is organized as follows: in section 2 we present the action,the metric and fields ansatz and equations of motion that follow, in additionwe discuss boundary conditions and the free parameters we are left withonce these are imposed. In section 3 we explore the relation between thestar mass, charge, the scalar field central density, and the stability of chargedboson stars. We present some numerical results, compare them to what isknown from the literature, and argue that for a certain range of parametersthe solutions we find are unstable in D ≥ 4 dimensions. In section 4 wereview and extend the above discussion as well as outline the numericalstrategy to extract information about extremal surfaces and present thereader with preliminary results. In section 5 we construct phase diagramsdisplaying the relation between the free parameters, the transition betweensolid and hollow regimes, and the transition between stable and unstableconfigurations. We conclude with a few final remarks and future directionsin section 6.3.2 Equations of motionWe start by considering the Einstein-Maxwell action with a negative cos-mological constant minimally coupled to a massive complex scalar field inD spacetime dimensions32S =18piGD∫dDx√−g[12R+(D − 2)(D − 1)2−14F 2 − |(∂i − iqAi)Ψ|2 −m2 |Ψ|2].(3.1)We want to restrict our attention to stationary, spherically symmetricconfigurations and allow for electric charges only, therefore we will let themetric be of the formds2 = −f(r)dt2 +dr2g(r)+ r2dΩD−2, (3.2)32Since the main goal of this chapter is to discuss numerical solutions of this action,the Hawking-Gibbons surface term can be neglected for it does not alter the equations ofmotion.373.2. Equations of motionand adopt the following ansatz for the scalar and gauge fieldsΨ = ψ(r)e−iωt, (3.3)A0 = A(r), (3.4)Ai 6=0 = 0. (3.5)To include the gravitational contribution from both the scalar and gaugefields we write the total energy momentum tensor asTij = TSFij + TGFij , (3.6)the sum of the scalar’s energy momentum tensorTSFij = ∇iΨ∗∇jΨ +∇iΨ∇jΨ∗ − gij(|∇Ψ|2 +m2 |Ψ|2), (3.7)where ∇i is the covariant derivative (∂i − iqAi), and the gauge field energymomentum tensorTGFij = FikFkj −14gijF2. (3.8)Additionally, note that the action 3.1 is invariant under the global U(1)rotation Ψ→ eiαΨ, implying the existence of the conserved currentJ j = igjk((∇kΨ)∗Ψ− (∇kΨ)Ψ∗), (3.9)which will act as a source for the gauge field.With the above definitions in hand we obtain four linearly independentequations of motion from the tt and rr components of Einstein’s equationRij −12gijR−(D − 2)(D − 1)2gij = Tij , (3.10)the Klein-Gordon equation∇2φ−m2φ = 0, (3.11)and Maxwell’s equations∇iFij = qJ j . (3.12)In terms of the functions defined in our ansatz, the D dimensional equa-tions of motion from the tt and rr components of Einstein’s equation are−q2r2A(r)2ψ(r)2 − 2qr2ωA(r)ψ(r)2 −12(D − 2)rD−3f(r)g′(r)−12(D − 3)(D − 2)f(r)g(r) +12(D − 2)(D − 1)r2f(r)− r2g(r)A′(r)2+12(D − 3)(D − 2)f(r)− r2f(r)g(r)ψ′(r)2 −m2r2f(r)ψ(r)2 − r2ω2ψ(r)2 = 0,(3.13)383.2. Equations of motionandq2r2A(r)2ψ(r)2 + 2qr2ωA(r)ψ(r)2 −12(D − 2)rD−3f ′(r)g(r)−12(D − 3)(D − 2)f(r)g(r) +12(D − 2)(D − 1)r2f(r)− r2g(r)A′(r)2+12(D − 3)(D − 2)f(r) + r2f(r)g(r)ψ′(r)2 −m2r2f(r)ψ(r)2 + r2ω2ψ(r)2 = 0.(3.14)While from the Klein-Gordon equation we find2q2rA(r)2ψ(r) + 2(D − 2)f(r)g(r)ψ′(r) + rg(r)f ′(r)ψ′(r)+rf(r)g′(r)ψ′(r) + 2rf(r)g(r)ψ′′(r)− 2m2rf(r)ψ(r) + 2rω2ψ(r) = 0,(3.15)and Maxwell’s equations give−2rf(r)g(r)A′′(r)− 2(D − 2)f(r)g(r)A′(r) + rg(r)A′(r)f ′(r)−rf(r)A′(r)g′(r) + 4q2rA(r)f(r)ψ(r)2 + 4qrωf(r)ψ(r)2 = 0.(3.16)These comprise a set of two first order and two second order ordinarydifferential equations and allow for a six parameter family of solutions, one ofwhich is empty AdS. However, we are not interested in any generic solutionof the above set of equations, but only in those that are asymptotically AdSand regular at r = 0.By requiring the solution to be regular at the origin, the near r = 0analysis of the equations of motion leads to the following constraintsg(0) = 1, g′(0) = 0, f(0) = f0, f′(0) = 0, (3.17)A(0) = A0, A′(0) = 0, ψ(0) = ψ0, ψ′(0) = 0, (3.18)leaving three undetermined parameters. We can use the following symmetryof the equations of motionf → γ2f, A→ γA, ω → γω, t→1γt, (3.19)to fix A(0) = 1. In addition, the value of f0 is chosen to ensure that oursolutions asymptote to global AdS, that is as r → ∞, f(r) → 1 + r2, and393.3. Mass, charge and scalar central densitysimilarly for g(r). This leaves us with one free parameter, ψ0, and the taskof studying a one parameter family of solutions of equations 6.72-6.75.However, before we move forward, we are still left with the issue ofguaranteeing that the scalar field showcases the correct asymptotic fall off.In general, the large r behaviour of ψ(r) is given byψ(r) =ψ1rλ++ψ2rλ−, (3.20)withλ± =12((D − 1)±√(D − 1)2 + 4m2), (3.21)where m2 is the scalar field mass in the lagrangian 3.1, and ψ1 and ψ2 areconstants. For m2 > 0, ψ(r) has a non normalizable term that can be set tozero by picking a solution for which ψ1 = 0, while for (D − 1)2/4 < m2 < 0both terms are normalizable and therefore allowed in principle.To pick the desired fall off of ψ(r) we will look for the lowest value ofω, the phase of Ψ(t), for which we observe ψ1 = 0. We do so by using ωas a shooting parameter and imposing the boundary condition ψ1 < 10−10.Once ω is fixed we are left with only m2 and q (the scalar charge) as freetheory parameters. Henceforth in this chapter we will numerically study theone parameter family of solutions of equations 6.72-6.75, their dependenceon the solution’s parameter ψ(0), the central density of the scalar field, aswell as on the theory’s parameters m2 and q. Later we will apply theseresults to investigate the behaviour of extremal surfaces living in charged,asymptotically AdS boson star backgrounds and how their behaviour dependon the free parameters we just discussed as well as the star’s mass, chargeand the scalar’s central density studied in detail in the next section.3.3 Mass, charge and scalar central densityBefore we start the discussion of how the solutions of equations 6.72-6.75and the families of minimal surfaces depend of the free parameters discussedabove, lets take some time to look at how the total mass and total chargeof the star are calculated and how they depend on the central density of thescalar field, ψ(0).The metric (4.29) describes a spherically symmetric body whose masscan be extracted from the asymptotic behaviour of the metric function g(r).At large r we expect the metric to approach that of a charged, massive star403.3. Mass, charge and scalar central densityin AdS with no scalar field, i.e.:g(r)r→∞−−−→ 1 + r2 −2MrD−3+Q2r2(D−2), (3.22)where M is the mass of the star and Q its charge, which is simply the totalnumber of scalar particles times the charge q, i.e.:Q = q∫dD−1xJ0√−g. (3.23)However, the existence of a scalar field allows for the possibility of scalarhair, therefore we should expect that our metric, while still approaching 3.22at large r, will receive higher order corrections. Ergo we choose to write thegeneral metric asg(r) = 1 + r2 −2M(r)rD−3+Q(r)2r2(D−2), (3.24)where the large r behaviour of the function M(r) is given byM(r) ∼M +O (1/rα) , (3.25)with α being is a positive constant, and similarly for Q(r).The behaviour of the mass and charge of a boson star as a function ofthe scalar’s central density has been extensively studied in the literatureboth for asymptotically flat and asymptotically AdS spacetimes [12, 69]. Inparticular, in AdS space, the existence of a variety of new solutions and thepresence of a zero temperature phase transition have recently been shownto exist [27, 40, 50, 69]. Furthermore the stability of boson stars has alsobeen the subject of numerous studies and found to correlate with certainaspects of the behaviour between mass, charge and central density [12, 77]as we discuss below.Since later in this chapter the question of whether the solutions we finddisplaying hollow configurations of minimal area surfaces are physically sta-ble will be particularly important, we should take a closer look at the be-haviour of the mass and charge of our solutions as a function of the centraldensity ψ(0) and compare it to what is known from the literature. Our anal-ysis here will rely solely on previously known results and educated guesses,so we will refrain from a formal study of the stability of the solutions as itlies outside the scope of this work.For D ≥ 4 dimensions it has been found that for models with no gaugefield in both zero and negative cosmological constants background, and with413.3. Mass, charge and scalar central densitya gauge field in a zero cosmological constant background, the star mass as afunction of the scalar field central density M(ψ(0)) reaches a maximum valuefor a finite ψ(0) = ψc. Even though the numerical value for the maximummass and ψc change for each case, they correspond to the threshold betweendynamically stable (ψ(0) < ψc) and unstable (ψ(0) > ψc) regimes33.The D = 3 case has attracted considerably less attention and conse-quently, to the extent of the author’s knowledge, no formal result is avail-able. Nevertheless, in the context of strongly self interacting boson stars,there is evidence for the existence of a maximum mass as well as some dis-cussion regarding the positive binding energy of these objects possibly beingan indicator of instabilities [121, 122]. Despite the existence of such partialresults we will refrain from making any statements regarding the stabilityof the three dimensional solutions.With the above discussion in mind we numerically solve equations 6.72-6.75, compute the physical mass and charge of the configurations using 3.25and 3.23 for different values of ψ(0) and compare to what is known fromthe literature. We find qualitatively similar results in D = 4 to that ofpure boson stars (without gauge field), more importantly we observe theexistence of a maximum value for M as can be seen in figure 3.1.Given the proximity of the models considered in the literature and ours,together with the qualitative agreement between results, we are led to believethat our model is also unstable past the central density threshold ψc forD ≥ 4 dimensions. We will assume this conjecture is indeed true in thereminder of this chapter and use it to understand better the conditions forthe existence of a hollow w(DA) in the upcoming sections.In D = 3 dimensions we again observe qualitatively similar results tothose found in [12] (figure 3.2). Most notably we would like to highlight theabsence of a maximum mass within the central density range investigated34as it showcases a considerably different behaviour than that expected forD ≥ 4 dimensions.33At maximum mass (ψ(0) = ψc) the pulsation equation arising from the analysis ofthe time evolution of infinitesimal radial perturbations has a zero mode indicating thatψc is a boundary between stable and unstable equilibrium configurations [77].34Our numerical solution is untrustworthy past the ψ(0) = 0.8 mark in D = 3 dimen-sions, therefore precluding us from investigating the mass versus central density relationany further.423.4. Overlapping extremal surfaces0.0 0.2 0.4 0.6 0.8 1.0 1.20.000.050.100.15Ψ0M0.000.050.100.15QFigure 3.1: Plot of the star’s mass (orange) and charge (green) versus thecentral value of the scalar field ψ0 with m2 = 0 and q = 0.2 in D = 4dimensions.3.4 Overlapping extremal surfacesOur main goal in this section is to construct extremal surfaces in the back-ground of charged, asymptotically AdS boson stars found in section 2. Be-fore we do so, it is instructive to further solidify the motivations for studyingsuch objects and how it can help shedding light on the discussion presentedin the introduction. However, to accomplish this task, we will need to em-ploy more general tools. In particular, while we started our discussion in theintroduction addressing the question of what is the bulk region R(A) dualto a given boundary region A, here we will consider instead the bulk regionR(DA) dual to the causal development of A, DA, defined as the union of allboundary points connected to A by a causal curve.A natural candidate for R(DA) would be to consider the causal wedgez(DA) = J+(DA) ∩ J−(DA)35, the region in the bulk a boundary observerrestricted to DA is causally connected to. Intuitively speaking, we expectz(DA) to be at least the minimal portion of the bulk accessible to a boundary35Where J+(DA), defined as all points accessible by causal curves arising from DA, isthe causal future of DA, while J−(DA), defined as the set of points from which DA canbe reached following a causal curve, is its causal past.433.4. Overlapping extremal surfaces0.0 0.2 0.4 0.6 0.80.00.10.20.30.40.50.6Ψ0M0.000.020.040.060.08QFigure 3.2: Plot of the star’s mass (orange) and charge (green) versus thecentral value of the scalar field ψ0 with m2 = 0 and q = 0.2 in D = 3dimensions.observer living in DA. The reason being that its complete causal connectionto DA allows for the detection of any perturbation of the bulk metric insidez(DA).While z(DA) seems to be a natural guess, and it certainly imposes alower bound on the size of R(DA), examples for which z(DA) ⊂ w(DA)while z(DA) + w(DA)36, are easy to find37 and demonstrate how the regionz(DA) alone cannot, in general, hold all the information stored in DA. Incontrast, for some cases, we can argue that all the information in DA mustlie within wˆ(DA), the domain of dependence of w(DA), for anything outsidethis bulk region will causally interact with the complement of DA, DA¯. Thiscould lead us to naively expect that wˆ(DA) imposes an upper bound onR(DA), however, one can construct explicit examples for which this mightnot be true, as we will see below.Say that we let the region DA cover the entire boundary, i.e.: DA =B, an observer within DA will have access to a full Cauchy surface and,consequently, information about the entire past and future of the boundarytheory. Since we are considering a field theory with a well defined gravity36Where w(DA) is defined in the exact same way as w(A) if we exchange A by DA.37See [33] for a thorough discussion.443.4. Overlapping extremal surfacesdual, information about the past and future of the boundary should extendto information about the past and future of the bulk theory as well. In otherwords, an observer in B that has access to all possible boundary physicalobservables should be able to fully reconstruct the dual bulk metric38.If we restrict the access of this observer to knowledge of the entanglemententropy for any arbitrary region within B, we can ask how much of the bulkhe or she can probe and, better yet, infer the geometry of [31, 33, 139]. Ifthe bulk theory has a horizon, say a spherically symmetric black hole atthe origin, it was shown in [71] that, while no extremal surface of any co-dimension (or causal lines, for that matter) can probe inside the horizon,given a fixed boundary region, co-dimension 2 surfaces probe the bulk deeperthan higher co-dimension surfaces or causal lines. Now, if instead of a blackhole at the origin we consider a boson star for example, we can ask the samequestion again: how much of the bulk spacetime can a boundary observerprobe with extremal surfaces only? Note that now the entire bulk is causallyconnected to the boundary, so we know that z(B) ⊃M, therefore, if w(DB)fails to fully cover the bulk, we will have an explicit example for which z ⊃ wwhile w + z.Given the above discussion, our goal is to search within the space ofsolutions of charged boson stars in asymptotically AdS spacetime found insection 2, for configurations for which we observe overlapping of extremalsurfaces leading to a hollow w(DA), therefore addressing the question ofwhether z(DA) ⊃ w(DA) while w(DA) + z(DA) is feasible39.Our setup is both static and asymptotically globally AdS, therefore ourboundary is a sphere (equation 4.29). We will look for surfaces that extendin all polar angles, so, to our applications, it suffice to describe them ascurves θ(r), with anchor points θ(∞) = ±θ0 corresponding to the azimuthalboundary coordinates of the start and end points (see figures 3.3, 3.4 and3.5).To determine whether the entangling surfaces reach the deepest regionsin the bulk we will analyze the behaviour of rmin(θn(r)) for multiple surfaceswith distinct boundary anchor points θn(∞) = ±θn, that is, the minimumvalue of r reached by a given entangling surface θn(r) with fixed boundary38This is demonstrably true for the case of empty AdS [56, 57, 61, 79]. Some interestingrecent discussion highlights the necessity of including non local boundary operator in theboundary observer’s toolbox [23]39Note that failure to find solutions obeying z(DA) ⊃ w(DA) while w(DA) + z(DA)does not indicate this particular phenomena is not possible, in the same way that the meremathematical existence of such configurations is not enough to undermine the candidacyof w(DA) to the position of R(DA) for these could not be physically preferred.453.4. Overlapping extremal surfacesanchor points θn.We know that ±θ(∞)rmin (the anchor points of a surface that has rmin asits deepest point) covers all values of θ as we vary rmin from zero to infinity,as a result, for every 0 < r0 <∞ there is at least one extremal surface thatobeys rmin(θ(r)) = r0 (figure 3.6).However, the situation is much different if we focus on surfaces anchoredat the same boundary points. In this case we expect that only one minimalarea surface can be anchored at each given θi (figure 3.6a). While this is truein general (with the exception of θ = pi/2), this need not be the case if weconsider extremal rather than minimal area surfaces (figure 3.6b), thereforewe should not be alarmed if for certain boundary anchor points we find thatthere are multiple distinct extremum surfaces anchored to it, in fact, we aremost interested in determining when such a phenomena happens. To do sowe will numerically investigate conditions under which there exist multipleextremal surfaces anchored at the same boundary point that do not sharethe same deepest point in the bulk.Given the above discussion it is tempting to cast the question of extract-ing bulk information from the boundary in the context of Morse theory.As discussed above, the existence of degenerate extremal surfaces led toan inconsistency regarding the amount of bulk information accessible to aboundary observer, in particular we argued for the possibility of regionsin the bulk inaccessible to extremal surfaces. Similarly, the existence ofconjugate points can be shown (for some specific riemannian manifolds) toobstruct the construction of a Morse function. Therefore, by connecting thetwo, we are led to believe that the existence of degenerate extremal surfacesmay preclude us from inferring the geometry (or probing the topology) atleast in some bulk region near such surfaces.Numerical treatment of extremal surfacesIn order to apply the numerical results we found in sections 2 and 3 weneed an equation describing extremal surfaces anchored to spheres on theboundary of the spacetime with metric 4.29. Such equation can be found byminimizing the area of these surfaces given by the integral of the inducedmetric on the surface in question. In the case of a surface anchored to asphere with the background metric given by equation 4.29, the integral over463.4. Overlapping extremal surfaces-1.0-0.50.00.51.0Figure 3.3: A plot of the Penrose diagram of a time slice of multiple extremalsurfaces on a four dimensional, asymptotically AdS, charged boson bosonstar background in global coordinates for m2 = 0, q = 0.1, and ψ(0) = 0.2.From top to bottom we have θ = ±0.251pi,±0.355pi,±0.446pi,±0.5pi. In thisparticular case the central density of the scalar field is below the thresholdψh, therefore there are no degenerate extremal surfaces (see figure 3.6), inother words, we observe a solid w(DA).the induced metric is simplyA = Vol(SD−3)∫dr (sin θ(r))D−3 rD−3√1g(r)+ r2(dθdr)2, 40 (3.26)40The area obtained from equation 3.26 is infinite, however it can be regularized by tak-ing the difference A−A0, where A0 is just equation 3.26 calculated on a fixed backgroundmetric g0 with the appropriate function θg0(r) and the same boundary conditions, i.e.:same boundary anchor points. In this chapter, while computing numerical values for thearea of extremal surfaces was not our goal, as discussed below we did do it as a consistencycheck of some of the statements made in [33]. When doing so we set the reference metricto be that of pure AdS in global coordinates, that is g0(r) = 1 + r2, found θg0 by solving473.4. Overlapping extremal surfaces-1.0-0.50.00.51.0Figure 3.4: Again, a plot of the Penrose diagram of a time slice of multipleextremal surfaces on a four dimensional, asymptotically AdS, charged bosonboson star background in global coordinates for m2 = 0, q = 0.1, andψ(0) = 1.2. However, in this example, the scalar central density is above thethreshold ψh and we observe the existence of degenerate extremal surfacesfor a range of boundary anchor points θ (see figure 3.6). In particular, foranchor points θ = ±pi/2 there are three solutions two of which (blue line)lie on top of each other, have minimal area and do not penetrate the dashedsmall circle, while the third (red line) corresponds to a non minimal areaextremal surface. As discussed in this chapter, no minimal area surfacepenetrates the deepest bulk points within the small dashed circle.from which follows, by an extremization procedure, that a minimal areasurface described by the curve θ(r) must obey the following second orderdifferential equationequation 3.27 (which greatly simplifies in this case), and computed the area difference.483.4. Overlapping extremal surfaces-1.0-0.50.00.51.0Figure 3.5: A similar plot as figures 3.3 and 3.4 highlighting the behaviourof all three extremal surfaces anchored at the same boundary points θ =±0.49pi. Although two of the solutions (red lines) penetrate the dashedcircle, these are not the minimal area and therefore are not part of thew(DA) set.−rD−1θ′(r) sinD−3(θ(r))(− g′(r)g(r)2 + 2r2θ′(r)θ′′(r) + 2rθ′(r)2)2(1g(r) + r2θ′(r)2)3/2+(D − 1)rD−2θ′(r) sinD−3(θ(r))√1g(r) + r2θ′(r)2+(D − 3)rD−1θ′(r)2 cos(θ(r)) sinD−4(θ(r))√1g(r) + r2θ′(r)2−(D−3)rD−3 cos(θ(r)) sinD−4(θ(r))√1g(r)+ r2θ′(r)2+rD−1θ′′(r) sinD−3(θ(r))√1g(r) + r2θ′(r)2= 0.(3.27)493.4. Overlapping extremal surfacesUsing the numerical solutions we found in section 2, in particular thefunction g(r) in equation 4.29, we are able to solve equation 3.27 numericallyand fully determine the function θ(r) in any spacetime dimension D. Ourgoal is to find solutions that asymptote to our desired boundary anchorpoints, ie.: θ(∞) = ±θ0. To do so we solved equation 3.27 subjected to theboundary conditionsθ(r0) = 0, and θ′(r0) =∞, 41 (3.28)and used r0 as a shooting parameter to meet the condition θ(∞) = θ0. Bysetting θ(r0) = 10−6 and θ′(r0) = 103, we increased r0 starting from 10−10and were able to tune θ(∞) = θ0 to an accuracy of 10−5. Once a solutionwas found we kept increasing r0 until another solution was found or ourcode could no longer solve the equations. We later mirrored the solutions togenerate the negative θ side of the curve, rotated it, and changed coordinatesto Penrose coordinates in order to generate figures 3.3, 3.4 and 3.5.The above described method allow us to acquire quantitative informationregarding the behaviour of extremal surfaces in charged, asymptotically AdSboson star backgrounds. However, as we should expect, the solutions wefind, even for fixed boundary conditions, are not unique, in fact, our goal is toinvestigate the conditions for the existence of degenerate solutions (multiplesolutions for fixed boundary points).From now on we shall refer to a particular solution of equation 3.27,θ(r), as an extremal surface, moreover, since we are only considering surfacesanchored at spheres, ±θ(∞) is enough to characterize the boundary anchorpoints of the extremal surface determined by θ(r).To investigate the possibility of the existence of degenerate extremalsurfaces, we use our numerical method to search for different solutions θi(r)for which θi(∞) = θ0, ∀i, while rmin(θi(r)) 6= rmin(θi′(r)) for at least onei′. In other words, we look for extremal surfaces anchored at the sameboundary points that do not share a common deepest bulk point rmin(θi(r))(figures 3.4 and 3.5).The existence of such multiplicity of extremal surfaces with commonboundary anchor points will in general preclude most of them from havingminimal area and, as argued in [33], can be used to show that no familyof minimal area surfaces can cover the bulk in its entirety. Nevertheless,as a consistency check, we computed numerically the regularized area forsurfaces anchored at various boundary points and confirmed that, indeed,41Note how this condition follows from the spherical symmetry of the space-time inconsideration.503.5. Phase diagramswhen more than one solution exists (with the same θ(∞)), the one with thebigger rmin has the smallest area.Therefore, our strategy to determine whether our charged boson starsolutions display such hollow phases is to look for an extremal surface θh(r)that obeys θh(∞) = pi/2 while having rmin(θh) 6= 0 (figure 3.4) since, fromthe spacetime symmetry, we know that there exist one extremal surface forwhich θ(∞) = pi/2 with rmin(θ) = 0.As a warming up exercise we fix the value of the parameters m2 and qand, as we vary the central density of the scalar field ψ0, we observe thesystem to transition between solid and hollow phases as seen in figure 3.6.Note how on figure 3.6a there is only one solution extending into the bulkand reaching a specific rmin for each θ, whereas on figure 3.6b there is morethan one value of rmin for a given boundary anchor point (a fixed θ0) nearthe r = 0 region.0.0 0.2 0.4 0.6 0.8 1.0 rmin0.00.10.20.30.40.50.6Q HΠ radL(a) ψ(0) = 0.20.0 0.2 0.4 0.6 0.8 1.0 rmin0.00.10.20.30.40.50.6Q HΠ radL(b) ψ(0) = 1.2Figure 3.6: A comparison of two different central densities for the plot ofrmin(θ(r)) in four dimensions, both cases with m2 = 0 and q = 0.1. It is clearhow for anchor points roughly between 0.17pi < θ < pi/2 there exist threedistinct extremal surfaces with different values of rmin, however, only one ofthem has minimal area. See figures 3.3, 3.4 and 3.5 for specific examples.Our goal for the next section is to determine the precise value ψh of ψ(0)for which this transition occur as a function of the parameters m2 and q and,in the four dimensional case, compare it to ψc, the central density thresholdbetween stable and unstable configurations.3.5 Phase diagramsIn this section continue to apply the numerical solutions and quantities foundin sections 2 and 3 to explore in further detail the conditions for which weobserve solid and hollow phases. Since we are dealing with a one parameter513.5. Phase diagramsfamily of solutions and have two free theory parameters, we should be ableto construct a three dimensional phase diagram and find a two dimensionalsurface separating solid and hollow phases. To numerically accomplish thistask we start by fixing q while varying m2. For each value of m2 and q welook for the lowest value of ψ(0) for which we observe multiple values ofrmin(θ0) for the same, fixed, θ0 and find a line separating the two regions,we then repeat this process for multiple different values of q.As seen in figures 3.7 and 3.8, we observe a similar behaviour for boththree and four dimensional cases. Is is clear that the threshold value ofψ(0) decreases as we increase either m2 or q, and a maximum, finite value isattained as the scalar mass approaches the BF bound and the charge goesto zero.-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.20.00.20.40.60.81.0m2Ψ 0Figure 3.7: Critical scalar field central density separating solid and hollowconfigurations for D = 3 dimensions with q = 0.1 in blue, q = 0.2 in lightblue (dashed) and q = 0.3 in green (dotted). In all cases a central densityvalue below the line correspond to solid solutions, while above it lies thehollow regime.As discussed earlier in this chapter, four dimensional boson stars areknown to be unstable when the central scalar density rises above a criticalvalue ψc in numerous different setups. Since we just established the existenceof ψh, the threshold between solid and hollow configurations, we want to523.5. Phase diagrams-2.0 -1.5 -1.0 -0.5 0.00.00.20.40.60.81.01.21.4m2Ψ 0Figure 3.8: Critical scalar field central density separating solid and hollowconfigurations for D = 4 dimensions with q = 0.1 in blue, q = 0.2 in lightblue (dashed) and q = 0.3 in green (dotted). In all cases a central densityvalue below the line correspond to solid solutions, while above it lies thehollow regime.compare it to ψc so we can determine whether the hollow solutions we findare in fact physically permitted.Stability of AdS charged boson stars and hollowed phases.The stability of boson stars has been a subject of intensive study in the pastdecades, while focus has been given to boson stars in flat spaces, similarresults exist in AdS space. Despite subtle changes between the flat, curved,self interacting or charged cases, it is well known that four dimensional bosonstars reach a maximum mass value for a finite central density ψc and areunstable past this point. The nature of the instability and how it dependson the various variations of the model, while important on their own, arenot within the scope of this study, for us it suffices to know that the valueof ψ(0) = ψc for which the stars mass as a function of scalar central density,M(ψ(0)), is maximum represents the threshold between stable and unstableregimes and that this seems to be universal across different types of boson533.5. Phase diagramsstars42.In order to numerically determine ψc we fix q and m2 and search for thehighest value of the star mass M as a function of ψ(0). Similarly to thephase diagrams above (figures 3.7 and 3.8), we find, for each value of q, aline dividing stable and unstable regimes in the m2 vs. ψ0 plane. We observethat, within the range of parameters we studied, we always have ψc < ψh (asseen in figures 3.9 and 3.10), indicating that the hollow solutions we found,while being perfectly fine in a mathematical sense, do not correspond to aphysically preferred phase and suffer from dynamical instabilities.-2.0 -1.5 -1.0 -0.5 0.00.00.20.40.60.81.01.21.4m2Ψ 0Figure 3.9: A phase diagram displaying both transitions found in D = 4dimensions (stable → unstable and solid → hollow) with q = 0.1. Thered line (below) is the stability threshold, a value of ψ0 above it (the lightblue region) renders a dynamically unstable configuration. The blue line(above), once again, represents the transition between solid and hollow con-figurations. It is clear from this figure how, in the range studied, only solidconfigurations are physically allowed.42For the interested reader we direct you the reviews [124] and [90] and the work [12]for a lengthy discussion on boson star instabilities and more.543.6. Final comments-2.0 -1.5 -1.0 -0.5 0.00.00.20.40.60.81.01.21.4m2Ψ 0Figure 3.10: A phase diagram comparing the central densities ψc and ψh asa function of m2 for different values of q. The dotted lines are for q = 0.3,the dashed lines for q = 0.2, while the solid lines for q = 0.1. The warmcoloured lines (below) correspond to transition between stable and unstableconfigurations, while the cold coloured lines (above) transition between solidand hollow phases. As we lower the charge q both ψc and ψh increase,however their difference remains roughly unchanged, highlighting how hollowsolutions are unstable for all range of parameters investigated.3.6 Final commentsIn this chapter we numerically investigated the behaviour of extremal, codi-mension 2, spacelike surfaces in charged, asymptotically AdS, boson starbackgrounds. Our main goal was to establish the conditions for which fami-lies of minimal area spacelike surfaces anchored on the boundary fail to fullycover the bulk of the spacetime. As discussed in the Introduction, this studywas motivated by recent ideas regarding a possible connection between theholographic description of entaglement entropy and the gravity dual of areduced density matrix as discussed in [33].We observed that the relation between the star’s mass as well the thestar’s charge and the central density of the scalar field in four dimensionsbehave much like what is known for both neutral boson stars in AdS and553.6. Final commentscharged boson stars in flat space. Notably, the existence of a maximummass for a finite ψ(0) = ψc strongly hints towards the presence of a stabilitythreshold and can be used to infer the physical feasibility of the hollowsolutions we were so interested in. In three dimensions we found resultsakin to what is known in the literature for other types of boson stars, inparticular, we observed a behaviour similar as the one found in [12] for 2+1dimensional neutral boson stars in asymptotically AdS spacetime.Our analysis of the behaviour of extremal surfaces with fixed boundarypoints led us to the conclusion that, both in three and four dimensions,for fixed m2 and q there is a maximum value for the central density ofthe scalar field ψh for which the minimal area surfaces reach every pointin the bulk space (figures 3.7 and 3.8). Therefore one should expect thatcharged boson stars with a high enough ψ(0) could provide a clear obstaclein the way of w(DA) being a universal candidate for R(DA). However wesaw that, at least in four dimensions, there is good evidence indicating thatsolutions with ψ(0) ≥ ψh are unstable (figure 3.9). Extrapolating well knownresults in the literature for both boson stars with and without gauge fieldsin flat space, and boson stars without gauge field in AdS space, we findthat for given m2 and q there is a threshold central density value ψc forwhich the solutions cease to be stable if ψ(0) > ψc. Remarkably, in thefour dimension case in question, we found that for every pair of m2 and q,ψc < ψh, i.e.: solutions for which w(DB) fail to cover the entire bulk and, inparticular, z(DA) ⊃ w(DA) while w(DA) + z(DA), are physically unstable.Unfortunately, to the extent of this author’s knowledge, much less in knownabout the stability of three dimensional boson stars, therefore precluding usfrom saying anything about the stability of both regimes we found.We believe the results found in this work support some of the ideas dis-cussed in [33] and further explored in [23, 31, 139]. The unstable characterof hollowed solutions strengthens the proposal of w(DA) as a good candi-date for R(DA) and complements other recent works on the subject. Wealso believe that a deeper understanding of extremal surfaces on chargedboson stars backgrounds can serve as a fruitful test ground for numerousholographic ideas including, but not restricted to, the holographic entangle-ment entropy, the holographic dual of a density matrix, zero temperaturequantum phase transitions [69], etc.56Chapter 4Rindler Quantum Gravity4.1 Introduction and summaryAccording to the AdS/CFT correspondence [1, 92], asymptotically globallyAdS spacetimes in certain quantum theories of gravity have an exact de-scription as states of a conformal field theory on Sd. In this chapter, weshow (see Section 4.2) that the same asymptotically AdS spacetimes maybe described alternatively as entangled states of a pair of CFTs on hyper-bolic space. This description in terms of hyperbolic space CFTs is preciselyanalogous to the description of Minkowski space field theory states in termsof entangled states of the field theory on two complementary Rindler wedges.In particular, if we focus on one of the Hd CFTs, the degrees of freedom livein a density matrix, and this density matrix describes physics in a wedgeof the dual spacetime accessible to an accelerated observer, as shown inFigure 4.1.The description of pure AdS in terms of the hyperbolic space theories isthe specific entangled state43|0global〉 =1Z∑ie−piRHEi |ELi 〉 ⊗ |ERi 〉 , (4.1)where RH is the curvature length scale of the hyperbolic space and |Ei〉are energy eigenstates of the hyperbolic space CFTs. For this state, eachhyperbolic space CFT is described by a thermal density matrix with temper-ature (2piRH)−1, similar to the Rindler description of the Minkowski spacevacuum.44 State (4.1) has precisely the same form as the state of a pairof CFTs on Sd that corresponds to the maximally extended eternal AdS-Schwarzschild black hole [75, 93]. This is no coincidence: thermal states ofthe Hd CFT correspond to asymptotically AdS black holes with boundary43Here, and throughout this chapter, we use∑to denote both discrete and continuoussums over states.44The fact that the reduced density matrix associated with the boundary of a singlewedge of pure AdS maps to a thermal density matrix for the CFT in hyperbolic space wasdemonstrated recently in [29], which formed part of the inspiration for this work.574.1. Introduction and summaryFigure 4.1: A pair of accelerating observers in pure global AdS. The space-time region accessible to each is a wedge whose boundary geometry can bechosen as Hd ×R. Each wedge has a dual description as a thermal state ofa CFT on this Hd ×R boundary geometry. The full spacetime is describedby an entangled state of the two Hd CFTs.geometry Hd ×R [41] and the choice of temperature T = (2piRH)−1 is spe-cial in that it corresponds to a “topological” black hole that is locally pureAdS.If a Rindler wedge of pure AdS is described by a thermal density ma-trix, it is interesting to ask about the spacetime interpretation of the “mi-crostates” contributing to this ensemble, i.e. the microstates of the topolog-ical black hole. We argue (see Sections 4.3 and 4.4) that typical pure statesof the hyperbolic space CFT are dual to spacetimes that are almost indis-tinguishable from a Rindler wedge of pure AdS away from the horizon, buthave the horizon replaced by some type of singularity where a geometricaldescription of the spacetime ceases to exist.45 The description in equa-tion (4.1) of pure AdS may then be given a spacetime interpretation as inFigure 4.2: a quantum superposition of disconnected singular wedges yieldsthe connected global AdS spacetime.46 This description suggests strongly45This similar to the “fuzzball” proposal of Mathur for black hole states; see [96] for areview.46This provides another explicit example of the idea [135–137] that connected space-times emerge by entangling degrees of freedom in the non-perturbative description. Basedon these observations, Mathur has argued [97, 99] that asymptotically flat spacetime couldbe understood as a quantum superposition of fuzzball geometries associated with Rindler584.1. Introduction and summaryΣ pi =i iFigure 4.2: Quantum superposition of microstate geometries yielding pureAdS spacetime. Each choice of complementary Rindler wedges leads to adifferent decomposition of AdS into a superposition of disconnected space-times.that the physics of AdS space outside the wedges (lightly shaded region inFigure 4.2) is encoded in the information about how the two Hd CFTs areentangled with each other.The role of the microstate geometries is rather different for pure AdS ascompared with an ordinary black hole. For black holes formed from collapse,the physical state is a pure state, one of the microstates of the black hole,and the black hole geometry may be understood as giving a coarse-graineddescription of the physics. For pure AdS, the microstates have little to dowith the physical spacetime. For these microstates, spacetime ends wherethe Rindler horizon would have been, while in the physical spacetime, theRindler wedge is smoothly connected to a larger spacetime. The latterproperty is linked to the fact that the hyperbolic space CFT degrees offreedom are highly entangled with some other degrees of freedom. Thus,in describing pure AdS, it is crucial that the degrees of freedom of thehyperbolic space CFT are entangled with the other degrees of freedom, i.e.that they are genuinely described by a density matrix.To highlight the importance of this entanglement, we consider in Sec-tion 4.5 a concrete realization of the “disentangling experiment” discussedin [135–137]. Varying the temperature parameter in the state (4.1) awayfrom β = 2piRH changes the degree of entanglement between the two hyper-bolic space CFTs (or the two halves of the sphere in the original picture)in a particular way. In this case, we can describe exactly what happens tothe geometry: for any temperature T , the corresponding global geometry isthe maximally extended hyperbolic space black hole with that temperature.From these explicit geometries, we can look specifically at what happensmicrostates. Our AdS discussion here provides a concrete realization of Mathur’s sugges-tions.594.1. Introduction and summaryFigure 4.3: Static observer in de Sitter space (left) and accelerated observerin AdS. Both have access to only a portion of the full spacetime, bounded(on one side in the AdS case) by a horizon with a thermal character.to a spatial slice of the spacetime as we vary the temperature. As the en-tanglement decreases, we find that the asymptotic regions correspondingto the two halves of the sphere become further apart and that the area ofsurfaces separating the two sides decreases, as argued on general grounds in[135–137].Lessons for cosmological spacetimesThe physics of accelerated observers in AdS spacetimes shares many quali-tative features with the physics of observers in cosmological spacetimes withaccelerated expansion. In Figure 4.3 (right), we have depicted an AdS ob-server with constant acceleration. The worldline for this observer startsand ends on the AdS boundary. This observer can communicate with (sendlight signals to and receive light signals back from) only a portion of thefull global AdS spacetime shown by the shaded region in the figure. Wesee that this shaded region has a very similar character to the static patchaccessible to a geodesic observer in de Sitter spacetime, shown on the left inFigure 4.3. Both regions are bounded in the bulk spacetime by a horizon.Both observers see geodesic objects accelerating away from them towardsthe horizon. Finally, both horizons have a thermal character, emitting Un-ruh/Hawking radiation characteristic of some particular temperature.These similarities give hope that some of the observations in this chaptermay be helpful in understanding how to generalize AdS/CFT to provide anon-perturbative description of quantum gravity in cosmological settings.In this context, it is interesting that we have given a precise description (viaa density matrix for a subset of degrees of freedom) of patches accessibleto particular observers in a complete model of quantum gravity. Like staticpatches in de Sitter space, these patches are bounded by observer-dependent604.2. A Rindler description of asymptotically global AdS spacetimeshorizons with an associated observer-dependent horizon area. In the de Sit-ter case, this observer-dependence obfuscates the proper interpretation ofthe entropy associated with this horizon area. In our present example, theinterpretation of the observer-dependent entropy is clear: a single spacetimecan be represented in many different ways as an entangled state of two sub-sets of degrees of freedom. Different choices of the subsets correspond todifferent patches (or different observers), and the the entropy associated tothe horizon area in a patch measures the entanglement between the sub-sets. Alternatively, the entropy can be viewed as a count of microstates:the density matrix describing the subset of degrees of freedom associatedwith a single patch can be viewed as an ensemble of pure states, and eachof these has a dual interpretation as a microstate geometry that is similarto the patch away from the horizon. It seems possible that all of thesecomments might apply equally well to de Sitter space or other cosmologi-cal spacetimes.47 Some additional discussion on generalizing AdS/CFT tocosmological spacetimes is found in Section 4.6.4.2 A Rindler description of asymptoticallyglobal AdS spacetimesIn the study of quantum fields on curved spacetime (or “semiclassical” quan-tum gravity), much of the physics of field theory on black hole backgroundsor on spacetimes with cosmological horizons can be understood by consid-ering field theory on Rindler space, related to the physics experienced byaccelerating observers in Minkowski space. It is interesting then to askwhether it is possible to describe precisely the physics accessible to an ac-celerated observer in a fully quantum mechanical description of gravity. Inthis section, we shall see that for asymptotically globally AdS spacetimes de-scribed by states of a CFT on Sd, there is an alternate dual description thatis precisely analogous to the Rindler description of field theory on Minkowskispace.4.2.1 Asymptotically AdS spacetimes as entangled states oftwo hyperbolic space CFTsConsider an asymptotically globally AdS spacetime dual to a pure state|ΨSd〉 of some CFT on Sd ×Rt. For any point P on the boundary cylinder,we can consider the region DP consisting of all points on the boundary which47Our discussion here is similar to recent comments of Mathur in [98].614.2. A Rindler description of asymptotically global AdS spacetimesP DRDLDRDLFigure 4.4: Conformal map from the boundary of a Poincare patch toMinkowski space. Region DL (solid) maps to one Rindler wedge ofMinkowski space, while the dotted region, DR, maps to the other wedge.The Poincare patch boundary DP is the region bounded by dashed lines.are not timelike separated from P ; for pure AdS, this forms the boundary of aPoincare patch. By a conformal transformation (reviewed in the appendix),the region DP can be mapped to Minkowski space; associated to this, wehave a map from states of the Sd CFT to states of the CFT on Minkowskispace.48Now, consider two complementary Rindler wedges of Minkowski space,regions R = {x1 ≥ 0, |t| < x1} and L = {x1 ≤ 0, |t| < |x1|} for some choiceof coordinates. Under the map from DP , these regions are the images of twocomplementary “diamond-shaped” regions,49 as shown in Figure 4.4. Anystate of the CFT on Minkowski space can be represented as an entangledstate of the quantum field theories on the separate Rindler wedges R andL. For example, the Minkowski space vacuum is described in terms of field48Since the region DP includes a complete spatial slice of the boundary cylinder (aboundary Cauchy surface) knowing the state of the fields on DP is the same informationas knowing the fields on the entire boundary cylinder; thus, the map |ΨSd〉 → |ΨRd〉 isan isomorphism. Care must be taken in choosing the appropriate boundary conditions forthe fields on Minkowski space.49Each of these regions is the intersection of the interior of the future light cone ofsome point pi with the interior of the past light-cone of a point pf in the future of pi.Alternatively, the regions are domain of dependence of a ball-shaped subset of some spatialslice of the boundary cylinder.624.2. A Rindler description of asymptotically global AdS spacetimestheories on the complementary Rindler wedges by the entangled state|0M 〉 =1Z∑ie−βEi2 |ELi 〉 ⊗ |ERi 〉 . (4.2)where |ELi 〉 and |ERi 〉 are corresponding eigenstates of the Rindler Hamilto-nians on the two wedges (boost generators in the full Minkowski space).By another conformal transformation (reviewed in the appendix) theRindler wedges R and L can each be mapped to Hd× time, where Hd is thed-dimensional hyperbolic space with metricds2 = du2 +R2H sinh2 uRHdΩ2d−1 . (4.3)Thus, the entangled state of the field theory on two Rindler wedges mapsto an entangled state of the pair of CFTs on hyperbolic space. Under theconformal transformations to Hd × Rt, the Rindler Hamiltonian in eachwedge maps to the Hamiltonian generating time translations in Hd × Rt.Thus, the state (4.2) describing pure global AdS spacetime maps to the state|0〉Sd =1Z∑ie−piRHEi |ELi 〉Hd ⊗ |ERi 〉Hd . (4.4)of the pair of CFTs on Hd × R. Here, one finds that the temperatureparameter β takes on the specific value 2piRH .In the CFT on Sd, the state corresponding to pure global AdS spacetimeis clearly quite special: it is the energy eigenstate of the CFT Hamiltonianwith the lowest possible energy. In the alternate description, the state (4.4)is not in any sense a minimum energy state for the Hamiltonian of eitherHd CFT. However, states of the form (4.4) have the maximum amount ofentanglement entropy for a given energy expectation value.50We will see below that states of the Hd×Hd CFT without entanglementcorrespond to states of the Sd CFT with a singular stress-energy at thelightlike boundaries of the two regions that map to the two hyperbolic spaces.By the AdS/CFT dictionary, singularities in the stress-energy tensor can beassociated with deformations of the metric which violate the asymptoticallyAdS boundary conditions. Thus, while we can associate a state of the Hd×Hd CFT to any asymptotically globally AdS spacetime (for a theory dualto a CFT on Sd), general states of the Hd ×Hd CFT correspond to a moregeneral class of spacetimes.50This is true for the state (4.4) at any temperature, but within this set of states, theone with T = (2piRH)−1 is the only one with asymptotically global AdS asymptotics, aswe will see explicitly in Section 4.4.634.2. A Rindler description of asymptotically global AdS spacetimes4.2.2 The description of a single Rindler wedgeSo far, we have shown that any state of a CFT on Sd × Rt can be repre-sented as an entangled state of a pair of hyperbolic space CFTs. This givesan alternate description of asymptotically global AdS spacetimes. We willnow see that the information contained in the individual hyperbolic spaceCFTs corresponds (roughly) to the information accessible to a pair of com-plementary accelerating observers in the bulk. Thus, we can think of thehyperbolic space picture as giving a Rindler description of asymptoticallyglobal AdS spacetimes.Consider first the case of pure global AdS, described in the hyperbolicspace CFT picture as the entangled state (4.4). In this state, the reduceddensity matrix for each individual CFT is the thermal density matrix corre-sponding to temperature 1/(2piRH).51 Generally, thermal states of a CFTon hyperbolic space are dual to asymptotically locally AdS black hole solu-tions with boundary geometry Hd, discussed in detail in [41] and reviewedin Section 4.5 below.52 However, for the particular temperature 1/(2piRH),such a black hole is locally pure AdS. For such a black hole, the region out-side the horizon corresponds exactly with a “Rindler wedge” of pure AdS,the region accessible to an accelerating observer whose worldline starts andends at the past and future tips of the diamond-shaped region that maps tohyperbolic space, as shown in Figure 6.15.53We will now argue that the density matrix for the single Hd CFT gen-erally does not carry any information about the region beyond this wedge.We recall that the two copies of Hd ×Rt are related by conformal transfor-mations to two complementary diamonds (which we refer to as DR and DL)on the cylindrical boundary of global AdS. For the full CFT on Sd, thereare many pure states that give rise to precisely the same density matrix forthe region DR. For these states, the fields in the complementary region DLgenerally differ, and such differences can affect any point in the bulk in thecausal past or causal future of DL, as we see in Figure 6.15. Thus, there aremany states of the full field theory for which the density matrix for DR isthe same but the physics in the region J+(DL) ∪ J−(DL) differs. We con-51This has been derived directly in [29].52There is no analog of the Hawking-Page transition here, though we have a qualitativechange in the causal structure of the maximally extended solutions at T = 1/(2piRH).For temperatures below (2piRH)−1 the black holes have a causal structure similar toReissner-Nordstrom AdS black holes while for temperatures higher than (2piRH)−1 thecausal structure is similar to Schwarzschild AdS black holes.53Here, an “accessible” point is one for which the observer can send a light signal toand receive a light signal back from that point.644.2. A Rindler description of asymptotically global AdS spacetimesL DRDFigure 4.5: Wedges of pure AdS spacetime. Field theory observables in DR(the shaded part of the boundary cylinder) probe the bulk region J+(DR)∩J−(DR) (the shaded region of the bulk). Any point in this region can receivea light signal (blue line) from and send a light signal to DR. Physics outsidethis region can be altered by changes on the boundary that do not affect thestate of the fields in DR. One trajectory along which such changes propagateis shown in red.clude that the density matrix associated with the region DR (equivalently,the density matrix for the corresponding Hd CFT) knows only about thecomplement of J+(DL) ∪ J−(DL). But for pure AdS, this is exactly theregion J+(DR) ∩ J−(DR) outside the horizon of the hyperbolic black holewith temperature 1/(2piRH).To summarize, the density matrices for the two hyperbolic space CFTsdescribe the physics in two complementary Rindler wedges of pure AdS.The individual density matrices generally do not have information aboutthe spacetime regions beyond the respective Rindler horizons. The addi-tional information that comes from knowing the full state as compared withknowing the two density matrices is the information about how the degreesof freedom in the two CFTs are entangled with each other. Thus, we can saythat the physics in the region outside the two Rindler wedges is describedby the entanglement between the two hyperbolic space CFTs.54In this section, we have focused on the case of pure global AdS spacetime.More general asymptotically AdS spacetimes correspond to other entangled54The arguments in this section apply equally well to Schwarzschild-AdS spacetimesdual to an entangled state of two Sd CFTs. They suggest that a single Sd CFT in athermal density matrix knows only about the region outside the horizon of the black hole.Knowing anything about physics behind the horizon requires knowledge of both CFTs.654.3. The microstates of a Rindler wedge of AdSstates of the two hyperbolic space CFTs. The question of what region ofthese spacetimes is associated with the density matrix for a single hyperbolicspace CFT (and more generally, what region of the spacetime dual to a state|Ψ〉 of a CFT onM can be reconstructed from the density matrix for any spa-tial subset of degrees of freedom of the CFT) was considered by the presentauthors recently in [32] (and by others in [24, 72]). There, we argued thatthe identification of the wedges J+(DR)∩J−(DR) and J+(DL)∩J−(DL) asthe duals of the density matrices associated with DR and DL is somewhatspecific to pure AdS. For more general spacetimes (with matter), the causalwedges J+(DR) ∩ J−(DR) and J+(DL) ∩ J−(DL) do not intersect in thebulk, but the density matrices associated with DR and DL carry informationabout larger wedges w(DR) and w(DL) that generally do intersect.4.3 The microstates of a Rindler wedge of AdSThe state (4.4) that describes pure AdS in the Hd×Hd picture has exactlythe same form as the states of a CFT on Sd × Sd that describe maximallyextended Schwarzschild-AdS black hole spacetimes [75, 93]. Specifically,in both cases the degrees of freedom of the two CFTs are entangled suchthat each “factor” theory is in a thermal state. The basic reason for thissimilarity is that (as we have seen) pure global AdS itself can be understoodas a particular type of black hole.We have argued that the thermal density matrix for a single CFT is dualto the region outside the horizon of the black hole. As usual, the (suitablyregularized) area of the black hole horizon can be identified with the (regu-larized) entropy of the density matrix. For the full theory with a second CFTon Hd, this entropy would naturally be viewed as an entanglement entropy,measuring the extent to which the degrees of freedom are entangled witheach other. But in discussions of black hole physics, such an entropy is morecommonly viewed as a thermodynamic entropy counting microstates con-tributing to the ensemble described by the density matrix. Since the “blackhole” in our case is a patch of pure AdS spacetime, it may seem odd to talkabout its microstates. However, the thermal density matrix for the Hd CFTcan certainly be viewed as an ensemble of pure states and AdS/CFT sug-gests that these pure states should have some dual gravity interpretation.The goal of this section is to understand better these microstate geometries.664.3. The microstates of a Rindler wedge of AdSPDRL DL DLFigure 4.6: Subregion D (shaded) of boundary D of a Rindler wedge ofAdS. The complement of D in D is shown dotted. (a) Regions D and Don the boundary of AdS. The corresponding bulk regions are also shown.W is a surface in the bulk whose area computes the entanglement entropyof the fields in D. (b) D is mapped to portion of a Rindler wedge. (c) Dis mapped to a finite portion of the infinite hyperbolic plane.Interpreting the hyperbolic black hole microstatesRecall that the Hd CFT can be viewed as the theory on a Rindler wedgeof a Minkowski space forming the boundary of a Poincare patch. Pure AdScorresponds to the Minkowski space vacuum and, as usual, the descriptionof the fields on one Rindler wedge is via a thermal state. In this picture,the microstates are pure energy eigenstates of the Rindler space field theory,most of which are typical states in the ensemble described by the densitymatrix. For such typical states, we expect that almost any macroscopicobservable will be nearly identical to the corresponding observable in thethermal state. Field theory observables (e.g. correlation functions, Wilsonloops, entanglement entropies) tell us about the geometry of the dual space-time, which suggests that the gravity dual of one of these typical microstatesshould be almost identical to the gravity dual of the thermal state itself. Letus try to understand this in more detail.674.3. The microstates of a Rindler wedge of AdSMicrostate geometries look like the Rindler wedge of pure AdSaway from the horizonConsider a domain of dependence region D in the boundary spacetime,55which is slightly smaller than the boundary D of our Rindler wedge (seeFigure 4.6). In the map from D to hyperbolic space times time, the regionD maps to a finite region of Hd × R. Thus, we can think of the degreesof freedom in D as forming a small subset of the full set of degrees offreedom in the hyperbolic space field theory. For a typical microstate in somethermodynamic ensemble, the reduced density matrix for a small subsetof degrees of freedom should be nearly the same as the reduced densitymatrix that arises from the thermal state itself [52, 114]. In fact, given theexact reduced density matrix ρD(T ) arising from the thermal state of thehyperbolic space CFT, there should be many pure states of the full theoryfor which the reduced density matrix on D is exactly ρD(T ). The reasonis that for any density matrix ρ =∑pi|Ai〉〈Ai| we can always choose a purestate |Ψ〉 =∑pi|Ai〉 ⊗ |Bi〉 in a theory with a sufficiently large numberof added degrees of freedom such that the reduced density matrix for thesmaller system is exactly ρ. Here, we certainly have enough degrees offreedom, since there is an infinite volume of hyperbolic space outside thefinite region that is the image of D under the conformal transformationfrom D to Hd × R.56 Thus, restricting to any subregion D of D, there isno way in general that we can distinguish a pure microstate on D from themixed state on D dual to pure AdS. According to [24, 32, 72], this meansthat the bulk region associated with the boundary D will be the same asfor pure AdS.5755The “domain of dependence” of a spatial region A is the set of all points p such thatevery inextensible causal curve through p passes through A. The boundary D of a Rindlerwedge of pure AdS is the domain of dependence of a ball-shaped subset of a sphericalconstant-time slice of the boundary cylinder (e.g. a hemisphere of the t = 0 sphere). Theregion D can be taken as the domain of dependence of a slightly smaller ball.56Note, however, that if a UV cutoff is imposed on the original field theory on Sd, therewill be a limit to how large the region D can be such that we can still choose a purestate on D to exactly reproduce the density matrix ρD arising from the vacuum of theSd CFT.57For a general microstate, the density matrix ρD will not necessarily be exactly thesame as the one arising from the thermal state, but typically it will be almost identical.The corresponding bulk region should then be almost indistinguishable from pure AdS.684.3. The microstates of a Rindler wedge of AdSRindler horizon replaced by something singular in microstategeometriesLet us now consider what happens at the boundary of the region D. Aftera conformal transformation that takes the region D to a Rindler wedge ofMinkowski space, this boundary becomes the Rindler horizon. For any purestate of quantum field theory on Rindler space (considered together with apure state of the quantum field theory on the complementary Rindler wedgesuch that we have some state of the full Minkowski space field theory), weexpect that the stress-energy tensor is singular on the Rindler horizon. Thiswas shown for the state |0L〉 ⊗ |0R〉 of a free scalar field theory in [112]and we demonstrate it more generally for any product state |ΨL〉 ⊗ |ΨR〉 inSection 4.4. We also show in Section 4.5.3 that the state |0L〉 ⊗ |0R〉 gives asingular stress-energy tensor on the Rindler horizon for any conformal fieldtheory. We expect this conclusion to extend to any product state.58 Thus,while the field theory observables for a pure state on D can be arbitrarilyclose to the vacuum observables away from the Rindler horizon, the behaviorat this horizon (i.e. the boundary of D) is drastically different.What is the bulk interpretation of this? We have seen above that the bulkregion of the microstate spacetime associated with any smaller region D willbe almost indistinguishable from a wedge of pure AdS. On the other hand,as D grows to become the full region D, drastic differences appear, withvarious observables diverging as we hit the boundary of D. This suggeststhat the bulk spacetime dual to a typical microstate of the Hd CFT inthe T = (2piRH)−1 thermal ensemble differs significantly from the Rindlerwedge of pure AdS at the horizon. These differences can only occur at thehorizon of the Rindler wedge region, since we have seen that any smallerwedge should be almost indistinguishable from AdS. This suggests that thedual of a microstate of the T = (2piRH)−1 hyperbolic black hole should looklike a Rindler wedge of AdS, but with the bulk horizon replaced by sometype of singularity, probably of a non-geometric character.More evidence for such singular behavior comes from considering thebehavior of entanglement observables in the field theory and their proposedgravity dual description. We recall that according to the proposal of Ryu andTakayanagi [118] (and the covariant generalization [73]), the von Neumannentropy of the density matrix associated with some spatial region A of a58This is consistent with a result from algebraic quantum field theory that finite sub-regions of a quantum field theory do not admit pure states [28, 46]. This implies thatstarting from a product state in e.g. a lattice regularized theory, and taking a continuumlimit must lead to some singular behavior at the interface between the regions.694.3. The microstates of a Rindler wedge of AdSquantum field theory with weakly curved holographic dual is equal to thearea of an extremal surface W in the dual spacetime such that the boundaryof W is the same as the boundary of A (as in figure 4.6). For a microstateof the Hd CFT, we expect that the von Neumann entropy associated with asmaller region D should be very similar to that for the same region in thethermal state. On the other hand, for the entire region D, the von Neumannentropy for the thermal state dual to a wedge of pure AdS is equal to the areaof the Rindler horizon, while the von Neumann entropy of a pure microstateis zero. For the microstate, this implies either that the surface W has zeroarea, or that the Ryu-Takayanagi formula no longer applies (e.g. becausethe relevant region of spacetime no longer has a weakly curved geometricaldescription). In either case, it appears that the metric on the boundaryof the Rindler wedge is replaced by something singular (or non-geometric)when we pass from the black hole geometry (i.e. the wedge of pure AdS) tothe microstate geometry.Connections to previous workOur observations here illustrate and elaborate on an observation in [135–137]about the emergence of spacetime in AdS/CFT. There, it was pointed out,based on the example of the eternal AdS black hole, that a classically con-nected spacetime can arise from quantum superpositions of spacetimes withtwo disconnected components. This phenomenon is apparent in the presentsetup: we have argued that typical microstates in the Hd CFT thermal en-semble correspond to spacetimes similar to a Rindler wedge of AdS, but withthe horizon replaced by something singular or non-geometrical. In the state(4.4) describing pure AdS, we have a superposition of states |ELi 〉 ⊗ |ERi 〉,each of which can be interpreted as a disconnected pair of these microstatespacetimes. The quantum superposition (4.4) represents pure global AdSspacetime, giving rise to the picture in Figure 4.2. Compared to the earlierobservations [135–137], a new feature is that (for this particular case) themicrostate geometries contributing to the superposition are almost identicalto pure AdS in their interior, but end rather abruptly at the place wherethe horizon would be in the connected version of the spacetime. Also, sincethere are many possible choices for complementary Rindler wedges in AdS,this example highlights the fact that there are many ways to decompose agiven spacetime into a superposition of disconnected spacetimes.Our conclusions about the geometry of Rindler microstates are simi-lar to the “fuzzball” proposal of Mathur in that black hole microstates aregeometries for which the horizon of the black hole has been replaced by704.4. Rindler space resultssomething else. In our case, the “something else” may simply be some kindof lightlike singularity at which spacetime ends.59 Mathur has specificallyproposed [97, 99] (based on a flat space limit of the observations in [136]about eternal AdS black holes) that a Rindler wedge of asymptotically flatspace should have fuzzball microstates, and that empty space can be rep-resented as a quantum superposition of disconnected geometries consistingof a pair of these Rindler fuzzballs. Our results confirm Mathur’s sugges-tions in detail for the closely related case of empty AdS space. In this case,we have been able to give an explicit description of the Rindler-space the-ories (the hyperbolic space CFTs) and a (somewhat indirect) descriptionof the fuzzball-geometries (the gravity duals of specific microstates of thesehyperbolic CFTs).4.4 Rindler space resultsIn this section, we consider a free massless scalar field theory on 1+1-dimensional Minkowski space and its alternative description based on thefields in a pair of complementary Rindler wedges. We prove that if a statefactorizes into left and right components, i.e. if it is not entangled, itsenergy-momentum diverges on the Rindler horizon. This result suggeststhat “AdS microstates” discussed in the previous section are singular on thehorizon of the AdS Rindler wedge. Interestingly, a divergent stress-energyon the boundary violates the AdS asymptotics, so the microstates cannoteven be said to be asymptotically AdS! In the next section we complementthis calculation with evidence that a state without entanglement representsa spacetime whose two parts have pinched off and disconnected from oneanother.Consider a scalar field φ in two-dimensional Minkowski spacetime. Wewould like to divide this spacetime into a left and a right Rindler wedge.Defining U = t − z and V = t + z, the right Rindler wedge is given byU < 0 < V . Because the dynamics of the left- and right-moving modes isindependent and identical, we focus below on the right-moving sector, whosedynamics is independent of V .59Note however, that some of the arguments we have used here are specific to thehyperbolic space CFT. In a similar discussion with Hd replaced by Sd, we could argue ina similar way that the eternal black hole geometry behind the horizon has no relevancefor black hole microstate spacetimes, but not that the microstates spacetimes are exactlythe same as the black hole but with an abrupt end where the horizon would be.714.4. Rindler space resultsA complete set of right-moving Rindler modes isφλ,R(U) = Θ(−U)1√4piλ(−aU)iλ/a , (4.5)φλ,L(U) = Θ(U)1√4piλ(aU)−iλ/a , (4.6)in terms of which, the field φ has an expansion:φ(U) =∫ ∞0dλ(bλ,Rφλ,R + b†λ,Rφ∗λ,R)+∫ ∞0dλ(bλ,Lφλ,L + b†λ,Lφ∗λ,L).(4.7)The relationship between the Minkowski vacuum and the Rindler vacuumcan be written as|0,Mink〉 = U|0, L〉|0, R〉 , (4.8)where U =∏λ>0exp(− arctan(e−piλ/a)(b†λ,Rb†λ,L − bλ,Rbλ,L)).For definiteness, we focus attention on the UU -component of the stress-energy tensor. Taking the expectation value of the stress-energy tensor inthe Minkowski vacuum as our reference point, we find that the stress-energyin the Rindler vacuum isTRLUU (U) = 〈0, L|〈0, R|(∂Uφ(U))2|0, L〉|0, R〉 − 〈0,Mink|(∂Uφ(U))2|0,Mink〉 (4.9)= −2∫ ∞0dλ[β2λ(|∂Uφλ,R|2 + |∂Uφλ,L|2) + 2αλβλRe (∂Uφλ,L∂Uφλ,R)],where αλ = epiλ/aβλ = (1− e−2piλ/2)−1/2 are the Rindler coordinates Bogoli-ubov coefficients.At nonzero U , only the first term in equation (4.9) is non-zero. Usingequations (4.5,4.6), we get that TRLUU (U 6= 0) = −1/(48piU2). To study thestress energy tensor at U = 0, we follow the approach in [112], and regularizethe operator by replacing the modes φλ,R/L in equation (4.9) with φλ,R/Lgiven byφλ,R(U) =1√4piλ(a(U − i))iλ/a − (a(U + i))iλ/aepiλ/a − e−piλ/a, (4.10)φλ,L(U) =1√4piλepiλ/2 (a(U − i))−iλ/a − e−piλ/2 (a(U + i))−iλ/aepiλ/a − e−piλ/a.(4 11)724.4. Rindler space resultsPDRPDLFigure 4.7: Position of cuts in the U-plane for formulas (4.10) and (4.11).These expressions are valid and finite on the entire real line when we placethe cuts as shown Figure 4.7. Such placement of cuts implies, for example,that for U < 0,(a(U− i))iλ/a = ((e−ipi)(−a(U− i)))iλ/a = epiλ/a (−a(U− i))iλ/a . (4.12)This ensures that φλ,R/L approach φλ,R/L for → 0. We can now show that∫∞−∞ dU TRLUU is positive and diverges for small  like −1. Thus, the Rindlervacuum has a divergent stress-energy tensor on the boundary at U = 0.We now demonstrate that it is not possible to cancel this singularity inthe stress-energy tensor at the horizon in a general separable state:60|Ψ〉 =∞∑k=1∞∑ni=1∫ ( k∏i=1dλi)f(n1,...,nk)(λ1, . . . , λk)(k∏i=1(b†λi,L)ni)|0〉L(4.14)⊗∞∑k=1∞∑ni=1∫ ( k∏i=1dλi)g(n1,...,nk)(λ1, . . . , λk)(k∏i=1(b†λi,R)ni)|0〉R .The expectation value of the stress-energy tensor in this state is equal tothat of the Rindler vacuum plus the additional contributionTΨUU = 〈Ψ|(∂Uφ(U))2|Ψ〉 − 〈0, L|〈0, R|(∂Uφ(U))2|0, L〉|0, R〉= 〈Ψ| : (∂Uφ(U))2 : |Ψ〉 , (4.15)60This state is more general than a tensor product of Fock space states of the left/rightRindler wedges, which takes the form∞∑n=1∫ ( n∏i=1dλi)fn(λ1, . . . , λn)(n∏i=1b†λi,L/R)|0〉L/R . (4.13)However, the typical states contributing to a finite temperature ensemble at infinite volumeare of this more general form.734.4. Rindler space resultswhere :: indicates normal ordering of Rindler raising and lowering operators.ThenTΨUU =∫ ∞0dσ1∫ ∞0dσ2[〈Ψ|bσ1,Lbσ2,L|Ψ〉 ∂Uφσ1,L∂Uφσ2,L + c.c. (4.16)+ 2 〈Ψ|b†σ1,Lbσ2,L|Ψ〉 ∂Uφ∗σ1,L∂Uφσ2,L (4.17)+ 〈Ψ|bσ1,Rbσ2,R|Ψ〉 ∂Uφσ1,R∂Uφσ2,R + c.c.(4.18)+ 2 〈Ψ|b†σ1,Rbσ2,R|Ψ〉 ∂Uφ∗σ1,R∂Uφσ2,R (4.19)+ 2 〈Ψ|bσ1,Lbσ2,R|Ψ〉 ∂Uφσ1,L∂Uφσ2,R + c.c.(4.20)+ 2 〈Ψ|b†σ1,Lbσ2,R|Ψ〉 ∂Uφ∗σ1,L∂Uφσ2,R + c.c.].(4.21)Terms (4.17) and (4.19) are non-negative everywhere, for example∫ ∞0dσ1∫ ∞0dσ2〈Ψ|b†σ1,Lbσ2,L|Ψ〉∂Uφ∗σ1,L∂Uφσ2,L =∣∣∣∣∫ ∞0dσ ∂Uφσ,L bσ,L|Ψ〉∣∣∣∣2(4.22)and therefore their contribution to the stress-energy tensor cannot cancelthe Rindler vacuum singularity at U = 0, which is also positive.For terms (4.16), (4.18) and (4.20), consider the behavior of the regular-ized stress-energy tensor near U = 0,∫ δ−δ dUTΨUU for small δ. For concrete-ness, we focus on the term (4.16), the argument for the other terms beingsimilar. Under the substitution U = x, we see that as → 0∫ δ−δdU ∂Uφσ1,L∂Uφσ2,L → −1−iσ1−iσ2 × (smooth function of σ1 and σ2) .(4.23)Due to the rapidly oscillating factor −iσ1−iσ2 , integrating over σ1 and σ2in equation (4.16) will give lim→0 ∫ δ−δ dUTΨUU = 0. The locus where theoscillations cancel, σ1 + σ2 = 0, lies outside the region of integration σ1 >0, σ2 > 0, so even if 〈Ψ|bσ1,Lbσ2,L|Ψ〉 were to contribute a delta functionδ(σ1 − σ2) to the integrand, the integral would remain zero.The remaining term, (4.21), could give a non-zero contribution, if 〈Ψ|b†σ1,Rbσ2,L|Ψ〉contributed δ(σ1−σ2), as the rapidly oscillating term takes the form ±i(σ1−σ2).However, the separable form of our state |Ψ〉 does not allow for such a delta-function term in 〈Ψ|b†σ1,Rbσ2,L|Ψ〉.Thus any separable state has a divergent stress-energy tensor on theboundary at U = 0. To cancel the singularity in the stress-energy tensoron the boundary, we need an entangled state. As an example, consider the744.5. Effects of disentangling on geometryMinkowski vacuum written in the following suggestive and convenient form|Mink〉 =∏λ>0[1√Zλ∞∑n=0e−pinλ/a(b†λ,R)n√n!(b†λ,L)n√n!]|0, R〉|0, L〉 , (4.24)such that each wedge is in a thermal density matrix with the Rindler temper-ature T = a/2pi. In this state, 〈Mink|bσ1,Rbσ2,L|Mink〉 = δ(σ1−σ2)e−piσ1/a/(1−e−2piσ1/a) = δ(σ1 − σ2) βσ1ασ1 . Since T|Mink〉UU = −TRLUU , this delta-functioncontribution must cancel the divergence at U = 0 precisely.We have demonstrated than the singularity in the stress-energy tensorat U = 0 can only be cancelled in a state with entanglement between theright and the left Rindler wedge. Our discussion in Section 4.3 indicatesthat it should be possible to cancel the Rindler vacuum stress-energy tensorin the interior of a Rindler wedge by adding Rindler quanta to the Rindlervacuum. To complete our discussion, we will now show that we can achievethis to any desired accuracy with only a single quantum. Consider:|Ψ1〉 =∫dλf(λ)b†λ,L|0〉L|0〉R , (4.25)wheref(λ) =e−(λ−λ0)2/(2∆2)√2pi∆. (4.26)At nonzero U we get (approximately, with ∆ small enough)TΨ1UU = 2∣∣∣∣∫dλf(λ)∂UφL,λ∣∣∣∣2=λ02pia2U2e−∆2(ln(aU))2/a2 . (4.27)For (aU) in the interval [e−a/∆, ea/∆], TΨ1UU is approximatelyλ02pia2U2 . Byadjusting ∆ and λ appropriately, we can therefore construct a state witharbitrarily small total stress-energy tensor TΨ1UU + TRLUU inside the shadedregion in Figure 4.6(b).4.5 Effects of disentangling on geometryWe have seen that entanglement is crucial for the description of pure AdSspace in terms of a pair of hyperbolic space CFTs. To further highlight this,we consider in this section the effects on the bulk geometry of changing theamount of entanglement between the degrees of freedom in the two theories,which correspond to the two halves of the sphere in the original picture. This754.5. Effects of disentangling on geometryprovides an explicit example of the “disentangling experiment” proposed in[136].As we have seen, the “Rindler” description of pure global AdS space isgiven by the state|0M 〉 =1Z∑ie−βEi2 |ELi 〉 ⊗ |ERi 〉 (4.28)with temperature chosen as β = 2piRH . In this state, the degrees of free-dom in the two hyperbolic space CFTs are entangled with each other. Theclaim in [136] was that if we change the state so that this entanglement isdecreased, the dual spacetime should pinch off in the sense that the area ofthe bulk minimal surface separating the two halves should decrease and thedistance between points in the two asymptotic regions should increase.In the present context, we can decrease (or increase) the entanglementbetween the two sides by lowering (or raising) the temperature in the state(4.28).61 In this case, each separate CFT on Hd will be in a thermal state,corresponding in the bulk to an asymptotically AdS black hole (brane) withboundary geometry Hd. These black holes were described and interpreted inthe AdS/CFT context in [41]. The full state (4.28) corresponds to the max-imally extended version of these black holes with two asymptotic regions.By studying the geometry of a spatial slice of these black hole spacetimesas a function of β, we will see that the qualitative expectations in [136] areprecisely realized in this explicit example.4.5.1 Review of the hyperbolic black holesIn d+ 2 spacetime dimensions, the hyperbolic black hole geometry for tem-perature T is described by the metricds2 = −f(r)dt2 +dr2f(r)+r2l2(dHd)2 (4.29)withf(r) =r2l2−µrd−1− 1 . (4.30)61The entanglement can be quantified by the entanglement entropy S = −tr ( ) ρR ln ρR.This entropy is divergent, but can be regulated by introducing an ultraviolet cutoff in thetheory. In this case, the difference between the entanglement entropy for two differentstates of the theory (considering the same pair of complementary spacetime regions) shouldbe finite and regulator independent as the cutoff is removed.764.5. Effects of disentangling on geometryThis has temperatureβ =4pil2r+dr2+ − l2(d− 1), (4.31)where r+ is the horizon radius defined by f(r+) = 0. The case µ = 0corresponds to the “topological black hole” that is a patch of pure AdS space.Both positive and negative values of µ are allowed, with the constraint thatµ > µext = −2d− 1(d− 1d+ 1) d+12ld−1 . (4.32)These coordinates cover the region exterior to the horizon, but the spacetimecan be extended in the usual way to include a second asymptotic region (ormore in the case µ < 0). The causal structure is similar to the Schwarzschild-AdS black hole for µ > 0 and to the Reissner-Nordstrom AdS black hole forµ < 0.4.5.2 Geometrical effects of changing the temperature /entanglementWe would now like to compare the geometries for different values of µ (whichcontrols the temperature/entanglement). Note that the boundary geometryis fixed; for all values of µ the metric takes the asymptotic form:ds2 = −r2l2dt2 +l2r2dr2 +r2l2(dHd)2 (4.33)Thus, we can match the various spacetimes asymptotically by identifyingpoints with the same t, r, and Hd coordinates in the region of large r.Distance across the spacetimeFirst, we ask how the distance across the spacetime from one asymptoticregion to the opposite one depends on µ. Of course, the distance is infinite,but its deviation from the µ = 0 case of pure AdS is finite and well defined.To compute this, we can choose some cutoff distance R. Then the distanceacross the spacetime on the t = 0 slice (corresponding to the τ = 0 slice inglobal coordinates) at the origin of the hyperbolic space is2∫ Rr+dr√f(r). (4.34)774.5. Effects of disentangling on geometry-1 1 2 3 4 5ΜΜext-2-11234Regularized DistanceFigure 4.8: Numerical integration of regularized distance for d=1, 2, 3, 4(orange, blue, red, green, respectively).Subtracting off the result for µ = 0 (with the same cutoff R) and taking thelimit as R→∞ givesL(µ)−L(0) = 2∫ ∞r+(µ)dr1√r2l2 −µrd−1 − 1−1√r2l2 − 1−2∫ r+(µ)ldr√r2l2 − 1(4.35)This is finite, since the integrand in the first integral behaves as 1/rd+1 forlarge r. For d = 1, we have explicitly that∆L = −l ln(1 + µ). (4.36)Thus, the two sides of the spacetime get further apart as the entanglementbetween the corresponding degrees of freedom decreases. The same con-clusion holds for other values of d as indicated by a numerical evaluationof equation (4.35) (see Figure 4.8). These results are consistent with thegeneral expectations in [136].Areas of minimal surfacesWe can similarly look at the areas (i.e. d-dimensional volumes) of minimalsurfaces in the spacetime. We first consider the surface r = r+ that dividesthe spacetime in half and forms the horizon of the hyperbolic black hole.The area of this is infinite, but we can look at the area per unit field theoryvolume as a function of µ. This is proportional to rd+, where r+ is relatedto µ by µ = rd−1+ (r2/l2− 1) (monotonic for µ > µext). Thus, the area of thesurface separating the two halves of the space increases monotonically as weincrease the entanglement (e.g. for d = 1, we get Area ∝√µ+ 1).784.5. Effects of disentangling on geometryWe can also look at the areas of other t = 0 spacelike surfaces thatapproach smaller spherical regions of the boundary. These extremize theactionA = Vol(Sd−1)∫dr rd−1 sinhd−1u(r)√√√√ 1r2l2 −µrd−1 − 1+ r2(dudr)2.(4.37)For d = 1, the action simplifies toL =∫dr√l2r2 − l2(1 + µ)+ r2(dudr)2. (4.38)In this case, the path u(r) must satisfyddrr2 dudr√l2r2−l2(1+µ) + r2(dudr)2= 0 (4.39)Assuming dr/du = 0 at some r = rmin, we havedudr=1rl√r2 − l2(1 + µ)rmin√r2 − r2min(4.40)Setting the origin of hyperbolic space u = 0 at this rmin, we find that theasymptotic value of u (which we call u0) as r →∞ is:u0 =ln rmin+l√1+µrmin−l√1+µ2√1 + µ(4.41)Inverting the relationship, we obtain:rmin = l√1 + µ coth (u0√1 + µ) (4.42)From expression (4.38), the length of such a curve in the region r < R ofspacetime is:l(µ,R) = 2l∫ Rrmindr√r2 − l2(1 + µ)r√r2 − r2min(4.43)= 2l ln√R2 − r2min +√R2 − l2(1 + µ)√r2min − l2(1 + µ)794.5. Effects of disentangling on geometry-1 1 2 3 4 5ΜΜext-0.50.51.01.5Regularized AreaFigure 4.9: Numerical integration of regularized distance with u0 = 1 ford=1, 2, 3, 4 (orange, blue, red, green, respectively).We can again subtract off the value for µ = 0 and take the limit as R→∞to obtain the finite result:l(µ)− l(0) = 2l lnsinh(u0√1 + µ)sinh(u0)√1 + µ. (4.44)We see that the area separating the two regions decreases as we lowerthe temperature (hence decreasing the entanglement entropy). The higher-dimensional versions can be tackled numerically, and we see that essentiallythe same pattern is repeated for all cases (Figure 4.9). These results, to-gether with the distance across spacetime, provides a realization of the ideasin [136], that is, as entanglement entropy decreases, the two wedges of space-time pinch off from each other.4.5.3 CFT on Sd interpretation of the Hd states at differenttemperaturesAt temperature T = (2piRH)−1, the state (4.28) maps back to the vacuumstate of the Sd CFT, so the energy density is constant on the sphere (equal tothe Casimir energy density). For other temperatures, the energy density isspatially constant and time-independent in the hyperbolic space picture, butnot in the Sd description. In this section, we determine explicitly the stress-energy tensor on Sd for the state corresponding to (4.28) at an arbitrarytemperature.For the states corresponding to hyperbolic black holes at various temper-atures, the stress-energy tensor in the dual CFT on hyperbolic space withmetricds2 = −dT 2 +R2(du2 + sinh2u dΩ2d−1) (4.45)804.5. Effects of disentangling on geometryis given by [41]〈Tµν〉 =116piGl(d +µld−1)diag(−d, 1, . . . , 1), (4.46)whered =2(d!!)2(d+ 1)!d(4.47)for odd d and zero for even d. For these states, we can map back to statesof the CFT on Sd. In this case, we have the metric on the region D isconformally related to the hyperbolic space metric:gsphereµν = e2φghypµν (4.48)Hence, we conclude that [22](〈Tsphereαβ〉µ − 〈Tsphereαβ〉µ=0) = e−(d+1)φ(〈Thypαβ〉µ − 〈Thypαβ〉µ=0) ,(4.49)where the µ = 0 state corresponds to the vacuum of the field theory on thesphere.Starting from the metric (4.45) for hyperbolic space times time, thechange of coordinatestan(τ/R) =sinh(T/R)coshu(4.50)tan θ =sinhucosh(T/R)(4.51)givesds2 = e2φ(− dτ2 +R2(dθ2 + sin2 θ dΩ2d−1))(4.52)withe2φ =cos2(τ/R) + sin2 θcos2(τ/R)− sin2 θ. (4.53)In these coordinates, the stress tensor is〈Tµν〉 =116piGl(d +µld−1) (diag(1, 1, . . . , 1)− (d+ 1) M), (4.54)whereM =11−tan2θ tan2(τ/R)sinθ sin(τ/R) cosθ cos(τ/R)1−sin2θ−sin2(τ/R)− sinθ sin(τ/R) cosθ cos(τ/R)1−sin2θ−sin2(τ/R)11−cot2θ cot2(τ/R)0 (4.55)814.5. Effects of disentangling on geometryUsing (4.49), we find that the stress tensor for the corresponding state ofthe CFT on the domain of dependence of the half-sphere (i.e. the region D)with metricds2 = −dτ2 +R2(dθ2 + sin2 θ dΩ2d−1)(4.56)is〈Tsphereαβ(µ)〉 − 〈Tsphereαβ〉vac =µ16piGld(cos2(τ/R) + sin2 θcos2(τ/R)− sin2 θ)(d+1)/2 (1− (d+ 1)M). (4.57)As an example, the energy density (minus the Casimir energy) is given byT 00 − T 00vac =µ16piGld(cos2(τ/R) + sin2 θcos2(τ/R)− sin2 θ)(d+1)/2(d+ 11− tan2θ tan2(τ/R)− 1)(4.58)This diverges on the lightlike boundary of the causal development region ofthe half-sphere, τ/R = ±(pi/2− θ). At τ = 0 the energy densityT 00 − T 00vac =dµ16piGld(1 + sin2 θ1− sin2 θ)(d+1)/2(4.59)diverges at the equator θ = pi/2 and this singularity propagates forward andbackward in time along the light sheets. We note also that for µ < 0 theτ = 0 energy density is negative away from the equator.62 However, sincethe total energy on the sphere must be higher than for the vacuum stateof the CFT, there must be a singular positive contribution to the energydensity at the equator such that the total energy on the sphere (relative tothe vacuum energy) is positive.While we have focused in this section on some CFT with a gravity dual,the stress-energy tensor for an arbitrary conformal field theory at finite tem-perature on Hd×R is determined by homogeneity and conformal invariance(tracelessness) to be〈Tµν〉 = f(2piRHT ) diag(−d, 1, . . . , 1), (4.60)62The negative energy density here is a well-known possibility, which illustrates howthe weak and null energy conditions may be violated. The negative energy should bethought of as a Casimir-type vacuum energy. Certain inequalities restrict the extent ofsuch negative energy densities. They can be used to prove averaged versions of the energyconditions in certain situations (see e.g. [45]).824.6. Comments on generalization to cosmological spacetimesproportional to (4.46) that was our starting point. Thus, the result (4.57)holds in general, with the replacementµ16piGld→ f(2piRHT )− f(1) . (4.61)In particular, except for the special temperature T = (2piRH)−1 that cor-responds to the vacuum state on Sd, the stress energy tensor is singular atthe boundary of the domain of dependence of the half-sphere.4.6 Comments on generalization to cosmologicalspacetimesIn the introduction, we recalled various qualitative similarities between Rindlerpatches of AdS and patches accessible to observers in cosmological space-times. Based on these similarities, it seems plausible that the description ofphysics inside a cosmological horizon should be in terms of a density matrixfor some degrees of freedom. However, both the details of the patch geome-try and the local spacetime dynamics are different in cosmological examples.In this section, we offer a few comments on how the holographic descriptionmight be modified in going from the case of accelerated observers in AdS tothe case of observers in cosmological spacetimes63.A characteristic feature of asymptotically AdS spacetimes not presentin the cosmological examples is the AdS boundary. The patches accessibleto an observer in de Sitter space or other homogeneous spacetimes withaccelerated expansion are bounded by the cosmological horizon and havefinite spatial volume. In the AdS case, all the patches we have described haveinfinite spatial volume since they include a boundary region. We know thatthe boundary region is tied to the UV degrees of freedom in the field theory.Thus, we might guess that patches of AdS without the boundary region aredescribed by a reduced density matrix for a subset of field theory degreesof freedom that excludes the UV degrees of freedom; such density matriceshave been considered recently in [15]. For a CFT on Sd, excluding the UVdegrees of freedom (e.g. spherical harmonic modes of the fields with angularmomenta above a certain cutoff) leaves us with a finite number of degrees offreedom, those of a large N matrix model with a finite number of matrices.Thus, a description for finite volume patches might be via mixed statesfor a large N matrix model, where these matrix model degrees of freedomare entangled with (and perhaps interacting with) other degrees of freedom63For some other approaches to this question, see [6, 16, 48, 51, 128].834.6. Comments on generalization to cosmological spacetimesassociated with the rest of the spacetime.64 A very similar conclusion wasreached by Susskind in [130] for independent reasons.At a more detailed level, in order to describe local bulk physics char-acteristic of a spacetime with positive, rather than negative cosmologicalconstant, we should expect that the Hamiltonian associated with time evo-lution in some patch should be different from one describing patches of AdS.Short of providing a specific suggestion here, we only observe that for a par-ticular geodesic trajectory in AdS, flat, and de Sitter space, other geodesictrajectories respectively accelerate towards, move away at constant velocity,or accelerate away from this trajectory. In the context of matrix models,these behaviors can be put in “by hand” at the classical level by choosingpositive, zero, or negative mass-squared terms for bosonic degrees of free-dom. Thus, a completely speculative suggestion would be that the type ofmatrix model whose mixed states would describe physics in a patch of aspacetime with accelerated expansion might involve negative mass squaredterms for the bosonic matrices.65 We caution, however, that quantum ef-fects typically dominate the effective potential in a matrix model; only forvery special theories, typically with significant cancellations in the effectivepotential due to supersymmetry, do we expect any kind of dual spacetimepicture to emerge. For an alternate (and more in-depth) discussion on howto modify CFT physics in order to describe de Sitter or FRW (rather thanAdS) dynamics, see [38, 39].64We are not suggesting that arbitrarily small or localized patches of spacetime canbe associated with some particular degrees of freedom, only that certain patches may beassociated with certain mixed states of a model with a finite number of degrees of freedom.65A slightly more concrete motivation of this suggestion is as follows. Starting fromthe N = 4 SYM theory on S3, a particular way to truncate to the IR degrees of freedomis to keep only the lowest spherical harmonic modes. This can be done in a way thatpreserves maximal supersymmetry, and the result is the Plane Wave Matrix Model, whichhas positive mass for all bosonic degrees of freedom. The density matrix for this model thatarises starting from the vacuum of N = 4 SYM and tracing out the rest of the degreesof freedom should describe a patch of pure AdS. For flat spacetime, the most concreteproposals for dual descriptions involve limits of models for which the bosonic potentialhas many flat directions (e.g. the BFSS matrix model). It is from these flat directions(preserved at the quantum level) that the asymptotic flatness of the dual spacetime issupposed to emerge. Thus, our naive suggestion is realized in specific models for the AdSand flat cases.84Part IIApplications of AdS/CFT85Chapter 5Holographic Fluids andMetric Perturbations5.1 IntroductionIn the AdS / CFT correspondence the energy momentum tensor of theboundary field theory corresponds to the metric tensor of the bulk theory.In practise, this means that the expectation value of the field theory energymomentum tensor is given by the fall off of the bulk metric field; equivalently,〈Tµν〉, together with the boundary metric, comprise a full set of boundaryconditions for the bulk metric field.Since the metric field is closely connected to the boundary stress ten-sor, we immediately wonder about what sorts of metrics are dual to certaingeneric types of stress tensors. Furthermore, we should expect that by con-straining the types of field theories we want to consider e.g.:, by restrictingto those that are solutions of some arbitrary set of equations, we would beleft with a much smaller set of bulk duals. In particular, we can consider aminimalistic approach and impose only the conservation equations Tµµ = 0and ∂µTµν = 0 on the boundary theory and investigate what sorts of con-straints will arise on the bulk geometry. This has been a topic of intensestudy in the past few years and many results exist. Notably, it is knownthat the gravity dual of a field theory with an energy momentum tensor thatobeys the hydrodynamics equations and has well defined temperature andvelocity fields must be non pathological: however, if any of these conditionsis relaxed, it is not known what should be expected.In this chapter we will use the gauge / gravity duality and fluid / gravitycorrespondence to look in detail at how small perturbations of the boundaryfield theory stress tensor are reflected in the bulk metric. To accomplish thiswe will add small perturbations to a well known black brane solution dualto a relativistic fluid at finite temperature. These metrics perturbations willnaturally generate corrections to the boundary energy momentum tensorwhich, in turn, will be constrained by the conservation equations. Therefore,865.2. Fluid / gravity correspondencethe perturbations considered will be constrained to those that render theboundary stress tensor traceless and divergenceless to first order. Once themost general perturbation of this kind is written we will see how to useEinstein’s Equations to first order in the perturbation parameter to findboth analytical and numerical metric corrections.5.2 Fluid / gravity correspondenceThe above mentioned perturbative approach, when applied to perturbationsin the long wavelength limit, leads to the well known fluid / gravity corre-spondence. Certain gravity solutions known as black branes can be general-ized to describe the behaviour of plasmas for which both temperature andvelocity fields are a function of space-time coordinates. These generalizedblack branes are not solutions of Einstein’s equations, but can be correctedin a perturbative fashion, order by order, such that the corrected metricdoes, indeed solve Eintein’s equations up to the desired order [21, 134].What distinguishes these fluid / gravity correspondence models from acompletely generic perturbation is the long wavelength approximation, insuch limit the conservation equations lead directly to the well known rel-ativistic hydrodynamic equations, including higher derivatives corrections.Equivalently, the energy momentum tensor of the dual field theory [14] is pre-cisely that of a relativistic, conformal fluid with temperature T and propervelocity uµ, with the higher derivative corrections introducing less than per-fect characteristic such as viscosity, compressibility and so on [126]. It is,therefore, qualitatively written asTµν ∝ T d (ηµν + (d− 1)uµuν) +O (∂u) .Before we can move to a broader discussion, in the remainder of thissection we will see the fluid / correspondence in more detail.5.2.1 Long wavelength limitOften, under the right conditions, the macroscopic behaviour of a many par-ticle system is remarkably different than its microscopic behaviour. Whilethe true number of degrees of freedom thermodynamical systems have isusually of order 1023 or larger, the macroscopic system may be described bya handful of physical quantities. Although we do not fully understand thetransition between the IR and UV behaviour, we know certain conditions875.2. Fluid / gravity correspondencethe systems must obey in order for such a macroscopic description to beavailable.Clearly, if a system of many particles changes too fast ( say, the averagekinetic energy of particles within a certain finite region ) it may be hard,or even impossible to assign a meaningful macroscopic physical quantity(in the current example, temperature) that characterizes the whole systemor (macroscopic) parts of it. Therefore, it is natural to expect that, whilevariations of the tentative macroscopic quantities are allowed, these shouldbe characterized by a length scale large enough so that the quantity canstill be well defined, in other words, they should vary slowly. The precisemeaning of slowly varying is dependent on the system in question, below wediscuss an appropriate definition for our current problem.In the case of fluids, our intuition tell us that not only temperature, butalso the fluid’s velocity, are the basic quantities necessary to macroscopicallydescribe its behaviour. Again, both temperature and velocities are allowedto change, in other words, we expect them to be functions of both positionand time, nevertheless, in order to properly define these two quantities wewill require that they are slowly varying. Formally speaking, we can onlyemploy the fluid approximation for a system in which the scale at whichthe temperature varies is small compared to the temperature it self, andsimilarly for the velocity field, i.e.:∂TT 1, and∂uu 1, (5.1)equivalently, if L is the typical length at which the temperature is varying,than we must impose LT  1.For any many-particle system for which temperature and velocity fieldscan be defined (obeying the above conditions), we can make use of thefluid approximation to describe it. This is not only true for relativistic fieldtheories, but, as we will see, can also be used to study holographic fieldtheories and boosted black branes [21, 134].5.2.2 The Fefferman-Graham expansionFrom the AdS/CFT dictionary we know that the bulk metric field is con-nected to the energy momentum tensor of the boundary theory. More pre-cisely, the expectation value of the boundary field theory energy momentumtensor is given by a particular term of the fall off of the bulk metric. At thispoint we can wonder about whether this is finite or physically meaningful atall, since the energy momentum tensor is a notoriously problematic quantity885.2. Fluid / gravity correspondencein conventional field theory. In the context of AdS/CFT, the procedure ofcalculating it [14] includes a normalization step66 usually taking the emptyAdS space as the zero value for the field theory stress tensor and computingthe difference.Quantitatively, the relation between bulk metric and boundary energymomentum tensor is made evident when the bulk metric is written in theFefferman-Graham gauge,ds2 =l2z2(dz2 + gµν(x, z)dxµdxν), (5.2)where l is the AdS radius and the AdS boundary is at z = 0. When writtenin this form, we can expand the metric gµν in series around z = 0,g(x, z) = g(0)(x)+z2g(2)(x)+ · · ·+zdg(d)(x)+h(d)(x)zd log z2 +O(zd + 1),(5.3)and immediately read off the boundary field theory’s energy momentumtensor from the zd coefficient [34]〈Tµν〉 =d16piGNg(d)µν +Xµν [gn], (5.4)where Xµν [gn] is related to the conformal anomalies of the boundary CFT.In what follows we will restrict our discussions to a four dimensional bulkgravity theory (1 + 2d boundary CFT), for which the term Xµν [gn] is zero.5.2.3 Fluid / gravity correspondenceSince its first appearance [21] the fluid / gravity correspondence has come along way, and not only our understanding of it has increased dramatically,but also the number of examples and models multiplied considerably. Inits original form, this correspondence was a parallel between generalizedboosted black branes subjected to small corrections and field theories withfluid-like energy momentum tensors with higher derivative corrections.From the AdS/CFT dictionary we know that black branes (black holeswith a flat horizon) in four dimensional anti de Sitter space with metricds2 = 2dvdr − r2f(br)dv2 + r2d~x2, (5.5)wheref(r) = 1−1r3, b =34piT,66Equivalently, the addition of a counter term to the gravity action.895.2. Fluid / gravity correspondencewith T the temperature of the black brane and v = t+∫drr2f(r) a ingoing nullcoordinate, are dual to a strongly coupled plasma at finite temperature T .This metric can be generalized by a Lorentz transformation and still remaina solution of Einstein’s equations. Written again in Eddington-Finkelsteincoordinates, the boosted black brane metric with proper velocityu0 =1√1− ~β2, ui =βi√1− ~β2(5.6)isds2 = −2uµdxµdr − r2f (br)uµuνdxµdxν + r2Pµνdxµdxν , (5.7)where Pµν = ηµν + uµuν is the projection operator to directions orthogonalto uµ.The above metric can be further generalized to a position dependentversion (uµ → uµ(x), and T → T (x) in equation 5.7) that, while not anexact solution of Einstein’s equations, can be corrected in such a way tobecome a solution up to a certain perturbation other (again, in the longwave length limit). Schematically we haveg(0)(x)→ g(0)(x) + g(1)(x) + 2g(2)(x) + · · · ,such thatEE(g(0)) = 0 +O(), (5.8)EE(g(0) + g(1)) = 0 +O(2), (5.9)· · ·where g(0)(x) is the position dependent version of 5.7, g(i) is the most generalcovariant correction that can be added to the ansatz 5.7 that obeys thesymmetries of the problem and EE stands for Einstein’s equations appliedto the metric in question.When imposed on these generalized black brane solutions, the conserva-tion equations lead precisely to the hydrodynamic equations for the bound-ary theory together with higher derivative corrections. With AdS/CFTtechniques [14] the energy momentum tensor for the boundary theory canbe found to beTµν =12(4piT3)3(ηµν + 3uµuν) +O (∂u) , (5.10)which is precisely what we would expect for a relativistic perfect fluid withhigher derivatives corrections.905.3. Conservation equations and the stress energy tensor5.3 Conservation equations and the stress energytensorWe can try to generalize the discussion above and go beyond the long wave-length limit. We saw how under the right assumptions the conservationequations led directly to the relativistic hydrodynamics equations. Never-theless, we expect any physical system to obey such general physical laws,so it is interesting to ask what sort of general behaviours about a systemcan we infer by imposing the conservation equations only.The above question is even more interesting in the context of holographicfield theories since it immediately extends to a gravitational systems as well.Therefore, we should expect that by investigating this question we can alsogain knowledge of a gravitational system reacts to small perturbations andhow the metric fall off must behave if this system is to obey the conservationequations.5.3.1 A simple exampleWe wish to study the metric dual to the most general boundary stress tensorand, in particular, what conditions guarantee its regularity and what kindof information from each side can we extract if we know either the energymomentum tensor of the boundary dual theory, or the bulk metric.Since the boundary stress tensor is determined by the fall off of the bulkmetric, we should be able to infer information about the bulk metric fromthe kind of boundary energy momentum tensor it induces; conversely, weshould also be able to predict certain characteristics of the dual field theoryenergy momentum tensor by looking at the bulk metric.To illustrate this discussion let us consider a simple example of a fieldtheory in (1 + 2) dimensions with a constant and diagonal — regularized —energy momentum tensorTµν =a+ b 0 00 a 00 0 b , (5.11)with a and b constants. When b = a this is simply a stationary relativis-tic fluid at finite temperature dual to a black brane with metric given byequation 5.5 and temperatureT =34pi(2a)13 .915.3. Conservation equations and the stress energy tensorThe conservation equations imposed on this boundary stress tensor leadtrivially toTµµ = 0, and ∇µTµν = 0. (5.12)Therefore, as far as the conservation equations go, the theory in question isperfectly acceptable, however, its holographic dual says otherwise. In orderto see this we must first find the appropriate dual metric that has the correctasymptotic fall off. Given the simple nature of this particular ansatz we areable to find a fully non linear solution which not only tell us the whole storybut also servers as a bench mark for perturbations around similar metrics.By writing down a generic metric in the GF form and using the sym-metries of the problem to simplify the ansatz, we are able to find that themetric dual to a CFT with equation (5.11) as stress tensor is given byds2 =1z2(dz2−A(z)B(z)−a−bdt2 +A(z)B(z)adx2 +A(z)B(z)bdy2), (5.13)withA(z) =(1−a2 + ab+ b248z6) 23, (5.14)andB(z) =1 +√a2+ab+b248 z31−√a2+ab+b248 z323√3a2+ab+b2. (5.15)Notice that this metric has a horizon at z =(48a2+ab+b2) 16.When a = b this horizon is regular and the metric is smooth across it, inparticular, the square of the Riemann tensor, Rabcd, is finite at the horizon.However, if a 6= b this is no longer the case, the horizon is now singular,not only the metric components become pathological, but also the squareof the Riemman tensor diverges at the former horizon surface67; in otherwords, the bulk metric has a naked singularity at z =(48a2+ab+b2) 16for aconstant, diagonal and anisotropic Tµν , despite it being a solution of thehydrodynamics equations.This result indicates that the conservation equations alone can not be thewhole story when searching for CFT states with well behaved gravity duals67The full expression for the square of the Riemann tensor, while perfectly calculable, isfar too long and adds little to the present discussion. Below we will return to this problemin a perturbative fashion which is enough to illustrate the divergence of (Rabcd)2.925.3. Conservation equations and the stress energy tensor(even in the case of energy momentum tensors as simple as the one above);it is clear from this example that they fail to exclude the existence of nakedsingularities. In addition we also see that highly non trivial informationabout the geometry is encoded in the boundary CFT state. It is known thatrelativistic fluids with a definite four velocity and temperature are dual tonon pathological space times — black branes — however we were able toshow how these conditions can not be relaxed too indiscriminately.Since the ultimate goal should be to analyze more generic types of bound-ary stress tensors, which can only be done perturbativelly, it is interestingto study the above result in the perturbative regime as well, in this way weare able to identify what sorts of phenomena are available at first, second,or higher orders in the perturbation parameter. This is easily done just bysetting b = a+ , where  is a small parameter.By expanding the metric components is powers of  we findA(z)B(z)a = A(z)B(z)a|=0 + Fx(z) +O(2), (5.16)whereFx(z) =(az3 − 4) (44−az3 −12)2/3 ((az3 + 4)log(84−az3 − 1)− 2az3)12a 3√16− a2z6,andA(z)B(z)b = A(z)B(z)a|=0 − Fy(z) +O(2), (5.17)whereFy(z) =(az3 − 4) (44−az3 −12)2/3 ((az3 + 4)log(84−az3 − 1)+ 2az3)12a 3√16− a2z6,which clearly showcases hints of pathological behaviour already at first order.The expansion of the square of Riemann tensor, however, givesRabcdRbcda =24(a4z12 + 16a3z9 + 224a2z6 + 256az3 + 256)(az3 + 4)4−3072(az6(az3 − 4))(az3 + 4)5+2048z6(az3 − 4)4 (az3 + 4)6FR(z)2+O(3), (5.18)whereFR(z) = a6z18− 16a5z15 + 320a4z12− 960a3z9 + 4480a2z6− 2048(az3 − 1),935.4. Discussionfrom which we can conclude that the curvature divergence at the horizoncan only be seen at second or higher orders in the perturbation expansion,which indicates that while a second order analysis is necessary to bettercharacterize the geometry of space time, many of its aspects can be inferredfrom first order only.5.3.2 GeneralizationNow that we understand better the simplest case of a energy momentumtensor that solves hydrodynamics equations and has a non well behaveddual, we could in principle try to generalize it. Ideally we would like tostudy a completely generic perturbation around the black brane geometrygµν(z, x) = gBBµν (z, x) + hµν(z, x). (5.19)Such perturbation is naturally connected to a deformation of the boundarystress tensorTµν = TBBµν + δTµν . (5.20)To quantitatively address the above equations we can solve Einstein’sEquation perturbatively using (5.19) as our ansatz while imposing (5.20)(together with the conservation equations) as a boundary condition on thesolutions.With a solution in hand we can look at the behaviour of the metric func-tion hµν(z, x) near the horizon and compare it to the exact case discussedabove, this would allow us to determine whether a particular boundary stresstensor leads to a bulk naked singularity, the blow up or vanishing of metriccomponents, and etc.In practice, the procedure outlined above cannot be applied to a com-pletely generic situation. What we could do, however, is focus on certainsimpler ansatz, such as eikxhµν(z) and try to determine conditions for wellbehaved bulk geometries dual to these configurations. Many different ap-proaches could be used to explore this question further, and a thoroughanalysis of this ansatz will be left for future work.5.4 DiscussionIn this chapter we discussed how to use the gauge / gravity duality and fluid/ gravity correspondence to look in detail at how small perturbations of theboundary field theory stress energy tensor are reflected on the bulk metric.945.4. DiscussionWhen the perturbations are in the long wavelength limit we know fromthe literature that the conservation equations lead to the well known rela-tivistic hydrodynamics equations, and the energy momentum tensor of thefield theory is that of a relativistic conformal fluid with higher order correc-tions.However, we were interested in knowing how far we could go if the longwavelength limit was given up. In other words, what sort of information andconstraints on both the field theory as well as the gravity bulk arise fromimposing the conservation equations on a generic space-time or field theorywith a given generic energy momentum tensor.What we saw was that, even for a very simple example, some interest-ing, non trivial, behaviours could arise. In particular, we looked in detailat a field theory with a constant and traceless energy momentum tensor(equation (5.11)). For this system we were able to find an exact solutionfor the bulk metric (equation (5.13)) that asymptotes to the desired stressenergy tensor, this allowed us to explore this example closely and investigatehow the relation between the constants a and b was reflected on the bulkgeometry.What we saw was that when a 6= b, even perturbatively, the bulk ge-ometry was plagued with a naked singularity. When considering a = b+ ,with   1 we saw that the naked singularity only made itself evident atsecond order in  (when the Riemann tensor divergence appeared), despitethe metric tensor itself showcasing signs of pathological behaviour alreadyat first order.We finished by glancing over how the simple case considered could begeneralized. Unfortunately fully generalizing this approach is not viable,however some special cases may be within reach of numerical methods. Fromour discussion above we believe that in the case of a perturbative analysis,looking at the metric components at first order may provide strong evidencefor the existence or not, of space-time pathologies.95Chapter 6Density versus ChemicalPotential in HolographicField Theories6.1 IntroductionThe AdS/CFT correspondence [53, 92, 140], which conjectures the equiv-alence of a gravity theory in d + 1 dimensions and a gauge theory in ddimensions, has become a valuable tool for the study of strongly coupledfield theories. Using the correspondence, many questions about quantumfield theories may be phrased in the context of a gravity theory; in the limitof strong coupling, certain previously intractable field theory calculationsare mapped to relatively simple classical gravity computations.Holography and finite densityOne difficult regime of strongly coupled field theory that gauge / gravityduality is particularly suited to study is that of finite charge density. Here,lattice techniques fail due to the ‘sign problem’: at finite chemical potential,the Euclidean action becomes complex which results in a highly oscillatorypath integral. We can avoid this difficulty by mapping the problem to agravity dual using the AdS/CFT dictionary. According to the dictionary,in order to have a global U(1) symmetry in the field theory, one needs toinclude a U(1) gauge field in the gravity bulk. The charge density andchemical potential are encoded in the asymptotic behaviour of the gaugefield. At strong coupling in the field theory, the bulk theory is well describedby classical gravity, and one may solve the classical equations of motion onthe gravity side to study the field theory at finite density.Given this relatively simple access to finite density configurations, wemight hope that some physically realistic strongly interacting systems maybe approximately described by a holographic dual. In this case, qualitativefeatures of the holographic theory would carry over to the exact theory. It966.1. Introductionwould be useful to characterize the types of finite density field theories thathave a dual formulation and admit this type of study.In this chapter, we seek to answer this question from the perspective ofthe holographic theory. Specializing to holographic probes, in which fieldsare considered as small fluctuations on fixed gravitational backgrounds, westudy systems with the minimal structure of a conserved charge and findthe ρ − µ relations that are possible in the field theory duals. We attackthis problem by first deriving constraints on the relationship based on gen-eral grounds before studying several specific examples of holographic fieldtheories.Summary of resultsIn our study, we observe that, at large densities, the field theory dual to asubstantial class of gravity models can be described by a power law relationof the form68ρ = cµα. (6.1)Firstly, we look to understand the constraints on the the ρ − µ rela-tionship from the point of view of the field theory, using local stabilityand causality. Usually, results here depend on the particular form of thefree energy. In all cases with ρ − µ behaviour (6.1), local thermodynamicstability places the condition α > 0 on the exponent. In general, for a the-ory at low temperature, we may write the particular free energy expansionf ∝ −µα+1 − aµβT γ , with γ > 0 and a > 0, with corresponding chargedensity ρ ∝ (α+ 1)µα + aβµβ−1T γ . Combined, local stability and causalitydemand that α ≥ 1 and γ > 1.Next, we consider Born-Infeld and Maxwell actions for the gauge fieldin a generic background. Under mild assumptions, in both cases, the powerα is constrained. For the Born-Infeld action, the conditionα > 1 (Born-Infeld action) (6.2)arises,69 while, for the Maxwell action, the power law coefficient is fixed toα = 1. (Maxwell action) (6.3)68Here and throughout, α refers to the power in this form of ρ− µ relationship.69Naively, we could construct systems for which α ≤ 1, however, in these situations, thecontribution of the constant charge density to the total energy diverges, consequently wecan not say that there is a power law relation. This divergence signals a breakdown of theprobe approximation rendering these systems outside the scope of this chapter. Noticethat α > 1 is consistent with the bound derived from stability and causality.976.1. IntroductionInterestingly, these conditions are in agreement with those derived fromfield theory considerations, giving rise to the same range of possible valuesof α. In summary, all power law relationships consistent with stability andcausality can be realized in simple probe gauge field setups by varying thebackground metric.To see which values of α arise for backgrounds corresponding to specificmodels, we explore a variety of 3 + 1 Poincare´-invariant holographic fieldtheories dual to Dp-Dq brane systems and ‘bottom-up’ models with gaugeand scalar fields. The former have been used, for example, in studies of holo-graphic systems with fundamental matter [86, 95, 105, 120, 138], producingmany features of QCD, including confinement,70 chiral symmetry breaking,and thermal phase transitions [2, 13, 20, 87]. Bottom-up, phenomenologicalmodels have been studied in various model-building applications includingsuperconductors71 [58–60, 64, 66, 67] and superfluids [10, 18, 62].In the Dp-Dq systems, table 6.1, a variety of powers α in the range1 < α ≤ 3 are realized, respecting the α > 1 constraint. Note that theseresults only involve the Born-Infeld action and neglect couplings of the braneto other background spacetime fields.Probe braned = 4 d = 5Background branes D9 D8 D7 D6 D5 D4 D8 D7 D6D3 3 3 3D4 5/2 2 3/2 3 5/2D5 2 2D6 3/2Table 6.1: The power α in the relationship ρ ∝ µα at large ρ for 3 +1 dimensional field theories dual to the given brane background with theindicated probe brane, with d− 1 shared spacelike directions. For d = 5 thetheory is considered to have a small periodic spacelike direction while forbackground Dp branes with p > 3, the background is compactified to 3 + 1dimensions.In the phenomenological probe models, table 6.2, in all cases except one(the probe gauge field in the black hole background), the dominant power70It was recently pointed out that the usual identification of the black D4 brane as thestrong coupling continuation of the deconfined phase in the field theory is not valid [94].71A top-down realization of a gauge / gravity superconductor has been found in [7].986.1. Introductionα is determined by conformal invariance, since we consider asymptoticallyAdS backgrounds.72 Since µ and T are the only dimensionful parameters,the density must take the form ρ = µd−1h(T/µ), where the underlyingspace has d spacetime dimensions. At large µ and fixed T , we can expandh to see that µd−1 dominates the ρ − µ relationship. In systems with onesmall periodic spacelike direction, the dominant power α is larger than thecorresponding theory without a periodic direction since, at large densities,on the scale of the distance between charges, the theory is effectively higherdimensional.73 Our study of bottom-up models also includes an analysisof the gravity models in the full backreacted regime. As seen in table 6.2,the power law α in these cases is also determined by the same conformalinvariance argument.In these bottom-up models we are more interested in the detailed be-haviour at intermediate values of µ. It is found that, in general, whenthe scalar field condenses in the bulk, the corresponding field theory is in adenser state than that without the scalar field. As well, the field theory dualto the gauge field and scalar field in the soliton background is in a denserstate than that dual to the same fields in the black hole background. In thesystems with a scalar field, at large µ, the ρ−µ relationship is well fit by theform ρ = c(q,m2)µα,74 where q and m2 are the charge and mass-squaredof the scalar field. While the power α is fixed by the conformal invariance,we find that the scaling coefficient c(q,m2) increases with increasing q ordecreasing m2.72Different power laws can arise for holographic theories on different backgrounds, suchas Lifshitz spacetimes. However, these will not be considered here.73The phase transition that holographic theories with a periodic direction undergo asthe density increases was studied in [81].74In the probe cases we can scale q to 1, leaving c = c(m2).996.1. IntroductionRegime Background Fields d = 4 d = 5probeblack holeφ 1 1φ, ψ 3 4soliton φ, ψ 4backreactedblack holeφ 3 4φ, ψ 3 4soliton φ, ψ 4Table 6.2: The power α in the relationship ρ ∝ µα at large ρ for 3 + 1dimensional field theories dual to the given gravitational background withthe stated fields considered in either the probe or backreacted limits. φ isthe time component of the gauge field, ψ is a charged scalar field, and d isthe number of spacetime dimensions. For d = 5 the theory is considered tohave a small periodic spacelike direction.OrganizationIn section 6.2, we discuss some possible general examples of finite densityfield theories and attempt to establish bounds on the ρ − µ relationshipby imposing thermodynamical constraints on these systems. In section 6.3we briefly introduce holographic chemical potential and find, for Maxwelland Born-Infeld types of action, under mild assumptions, to what extentthey reproduce the relationship found in 6.2. In section 6.4 we investigatethe probe limit of both top-down and bottom-up theories; first we studyDp-Dq systems, then we move to gauge and scalar fields in both black holeand soliton (with one extra periodic dimension) backgrounds. Section 6.5extends the analysis of the bottom-up models to include the backreaction ofthe fields on the metric.Relation to previous workSome of the results presented in this chapter have appeared previously inthe literature. Finite density studies for probe brane systems have appearedfor the Sakai-Sugimoto model [20, 63, 84, 116], the D3-D7 system [9, 42, 43,86, 105], and the D4-D6 system [100]. The bottom-up models we considerare naturally studied at finite chemical potential (see, for example, [58] forthe black hole case and [68] for the soliton dual to a 2 + 1 dimensional fieldtheory) due to the presence of the gauge field.Our work focusses on the ρ− µ relation at large chemical potential over1006.2. CFT thermodynamicsa broad class of theories that are dual to 3+1 dimensional field theories. Wefind, on very general grounds, constraints on the ρ−µ relation in holographicmodels constructed from Maxwell and Born-Infeld actions. Additionally, weuse thermodynamical considerations to constrain the ρ−µ relation from thefield theory point of view and find that these constraints are in agreementwith those derived holographically. Further, we extend the analysis in theabove references to the large density regime and include additional examples,collecting the results of a large range of models.6.2 CFT thermodynamicsIn this section, by appealing to local thermodynamic stability and causalityin the field theory, we attempt to establish generic constraints satisfied bythe coefficient α from a purely field theory stand point. The results foundhere will lay ground for our intuition when approaching this problem fromthe holographic side.Generic system at large chemical potentialIn order to study the density and chemical potential from the field theoryperspective, we begin with a general ansatz for the free energy of a hypothet-ical system. In the large density limit, we expect that the chemical potentialwill dominate the expression, so we may write75f ∝ −µα+1 − aµβT γ + . . . , (6.4)where the dots denote corrections higher order in T/µ. For a positive,imposing a positive entropy density s = −(∂f/∂T )|µ > 0 implies γ > 0,consistent with the second term being subleading in the low temperatureexpansion.Considering the field theory as a thermodynamical system and imposinglocal stability demands that [36]76χ =(∂ρ∂µ)T> 0, (6.5)andCρ = T(∂s∂T)ρ= −T∂2f∂T 2−(∂2f∂T∂µ)2 1∂2f∂µ2 > 0. (6.6)75Recall ρ = −(∂f/∂µ)T so that, again, ρ ∝ µα.76χ is the charge susceptibility and Cρ is the specific heat at constant volume.1016.2. CFT thermodynamicsApplying these to (6.4) in the T/µ → 0 limit gives the constraints α > 0and γ > 1.Examining the speed of sound vs of our system also allows us to establisha constraint. To ensure causality, we impose0 ≤ vs ≤ 1, (6.7)with the speed of sound given by [62]v2s = −[(∂2f∂T 2)ρ2 +(∂2f∂µ2)s2 − 2(∂2f∂T∂µ)ρs](sT + ρµ)[(∂2f∂T 2)(∂2f∂µ2)−(∂2f∂T∂µ)2] , (6.8)where ρ and s are the charge and entropy densities. For γ > 1, this impliesthe stronger bound of α ≥ 1. This is the same bound as derived in section6.3 from consideration of the bulk dual of field theories. It is interestingthat it arises from very general circumstances in both cases.Zero temperatureIn the zero temperature limit of ansatz (6.4) only the first term survives, sothat f ∝ −µα+1. In this case, the only condition for local stability is givenby equation (6.5), which trivially leads to ρ ∝ µα with α > 0. Computingthe speed of sound and enforcing causality leads again to α ≥ 1.General conformal theoryFor a conformal field theory in d spacetime dimensions, the most generalfree energy density isf = −µdg(Tµ), (6.9)where g(x) is an arbitrary dimensionless function. Local stability dependson the details of the function g, and a general statement is not possible atthis point. To ensure causality, we compute equation (6.8), finding the speedof propagation to bev2s =1d− 1, (6.10)from which it follows directly that a conformal theory obeys requirement(6.7) only in dimension d ≥ 2. This result is trivial, as sound waves are notpossible if there are no spacelike dimensions to propagate in.1026.3. General holographic field theories at finite densityFree fermionsAs an example, we will compute the ρ− µ relationship for a system of freefermions. In the grand canonical ensemble, the partition function for spin1/2 particles of charge q in a 3 dimensional box and subjected to a largechemical potential isZ(µ, T ) =∏~n(1 + e−β(E~n−µq)), (6.11)where the product is over available momentum levels. The partition functionfor antiparticles follows with the replacement q → −q so we include antipar-ticles by considering the total partition function Z˜(µ, T ) = Z(µ, T )Z(−µ, T ).Passing to the continuum limit, approximating the fermions as massless, andsetting q = 1, the resultant charge density isρ =µ33pi2+µT 23. (6.12)The dominant power in this case is the same as is expected in a genericconformal field theory.6.3 General holographic field theories at finitedensityIt was shown in the previous section how local stability and causality leadto α ≥ 1. In this section, under mild assumptions, we investigate the Born-Infeld and Maxwell actions in the large µ regime and observe to what extentthey fall under the general results from section 6.2.6.3.1 Finite densityTo find constraints on the ρ − µ relation in holographic field theories, webegin by studying very general systems with the minimal structure of aconserved charge. The holographic dictionary gives that a conserved chargein the field theory is dual to a massless U(1) gauge field A in the bulk [103].If the gauge field is a function only of the radial coordinate r, the chemicalpotential and the charge density are encoded in the behaviour of A asµ = At(∞) (6.13)andρ = −∂SE∂At(∞), (6.14)1036.3. General holographic field theories at finite densitywhere SE is the Euclidean action evaluated on the saddle-point and thederivative is taken holding other sources fixed. As discussed in [86], anequivalent expression for the charge density is77ρ =(1d− 2)∂L∂(∂rAt), (6.15)where the normalization of ρ has been chosen for later convenience. Afterwriting down the gravitational lagrangian, our prescription for computingthe charge density at a given chemical potential is to solve the equations ofmotion with a fixed boundary condition for the gauge field, equation (6.13),before reading off the density using equation (6.15).6.3.2 Gauge field actionsTo include a gauge field in our AdS/CFT construction, we simply includeit in the bulk action. Two gauge field lagrangians that have appeared inholographic studies are the Maxwell and the Born-Infeld lagrangians. Typi-cally, the Maxwell action is used in bottom-up holographic models while theBorn-Infeld action appears in the study of brane dynamics. Below, in sec-tion 6.4 we will consider holographic models using both types of lagrangians.However, much insight can be gained by investigating these actions undergeneric conditions. Therefore, in this section, we study general versions ofthese two lagrangians, at fixed temperature and large chemical potential, inthe probe approximation.78 Interpreting our results using (6.13) and (6.15),we will develop some constraints for the ρ − µ relationship in holographictheories described by these actions.The Maxwell actionConsider a gauge field described by the Maxwell action∫ √−gF 2 in a generalbackground of the formds2 = gFTµν (r)dxµdxν + grr(r)dr2. (6.16)If we assume homogeneity in the field theory directions and consider a purelyelectrical gauge field (keeping only its time-component), the lagrangian is77Generically, At is a cyclic variable, so that the conjugate momentum is conserved, andwe may evaluate this expression at any r.78In the probe approximation, we assume there is no backreaction on the gravity metric.This is enforced in this case by studying the gauge field lagrangian on a fixed backgroundgeometry.1046.3. General holographic field theories at finite densitysimplyL = g(r) (∂rAt)2 , (6.17)for some function g(r). From this we findρ =(2d− 2)g(r)∂rAt. (6.18)In the systems considered below, the spacetime either has a horizon orsmoothly cuts off at some radius rmin. The value of the gauge field at thispoint is a boundary condition for the problem. Below, At(rmin) is eitherzero or a constant, neither of which affect the ρ − µ behaviour; we takeAt(rmin) = 0 here. Integrating (6.18), we findµ = ρ(d− 22)∫ ∞rmindrg(r). (6.19)Provided the integral is finite, we haveρ ∝ µ. (6.20)Thus, for any holographic field theory with the gauge field described onlyby the Maxwell lagrangian in a fixed metric we have α = 1.The Born-Infeld actionThe Born-Infeld action is the non-linear generalization of Maxwell electro-dynamics and is the appropriate language in which to describe the dynamicsof gauge fields living on branes. Assuming homogeneity in the field theorydirections, so that the gauge potential varies only with the radial direction,these systems are governed by an action of the generic form79L =√g(r)− h(r)(∂rAt)2, (6.21)where again, we take At to be the only non-zero part of the gauge field. Thecharge density is given by the constant of motionρ =(1d− 2)h(r)∂rAt(r)√g(r)− h(r)(∂rAt)2. (6.22)79g(r) and h(r) are arbitrary functions; g(r) is not related to the previous discussion.1056.4. Holographic probesHere, we assume that the gauge field is sourced by a charged black holehorizon at r+.80 Euclidean regularity of the potential At fixes its value atthe horizon as At(r+) = 0 [86]. Then, we can integrate to findµ =∫ ∞r+dr√g(r)h(r)(d− 2)ρ√h(r) + (d− 2)2ρ2. (6.23)To extract the large ρ behaviour, we split the integral at Λ  1. Forρ Λ, the integral from r+ to Λ approaches a constant, while the functionsin the integral from Λ to ∞ can be approximated by their large r forms,which will be denoted with a ∞ subscript. The expression for the chemicalpotential now becomesµ ≈∫ Λr+dr√g(r)h(r)+∫ ∞Λdr√g∞(r)h∞(r)(d− 2)ρ√h∞(r) + (d− 2)2ρ2. (6.24)The ρ dependence of µ comes from the second term. If g∞(r)/h∞(r) ≈ r2mand h∞(r) ≈ rn, by putting x = r/ρ2/n we find thatµ ∼ ρ(2+2m)/n∫ ∞r+ρ2/ndxxm√xn + 1. (6.25)The convergence of the integral here requires that n/(2 + 2m) > 1, resultingin the relationshipρ ∝ µα with α > 1, (6.26)where the power α depends on the specific bulk geometry.6.4 Holographic probesWith the general constraints of the previous sections in hand, we move onto study particular holographic field theories in the probe approximation,to see which specific values of α are realized. Here, we study two commonprobe configurations that have arisen in previous holographic studies. Theseare extensions of the actions considered in section 6.3. First, we examineprobe branes in the black brane background using the Born-Infeld action.80To have a non-trivial field configuration, a source for the gauge field in the bulk isrequired. In the low temperature, horizon-free versions of these models, this source isgiven by lower dimensional branes wrapped in directions transverse to the probe branes[142].1066.4. Holographic probesThen, we move on to the phenomenological perspective, in which we writedown an effective gravity action without appealing to the higher dimensionalstring theory. In this approximation, using the Maxwell action, we lookat the gauge field in both the planar Schwarzschild black hole and solitonbackgrounds, with and without a coupling to a scalar field.In both cases, in the systems we consider, the only sources in the fieldtheory are the temperature T and chemical potential µ. Below, we fix Tand work at large µ (such that µ/T  1). In this regime, we look for arelationship ρ ∝ µα + . . . , where the dots denote terms higher order in T/µ.6.4.1 Probe branes and the Born-Infeld actionIn the systems we will consider here, the background consists of Nc D-branes;in the largeNc limit, these branes are replaced with a classical gravity metric.In this regime, fundamental matter is added by placing Nf probe branes inthe geometry [82].The brane actionAssuming that the background spacetime metric Gµν is given, the actiongoverning the dynamics of a single Dq probe brane is the Born-Infeld actionS ∝∫dq+1σe−φ√−det(gab + 2piα′Fab). (6.27)Here, latin indices refer to brane coordinates and greek indices denote space-time coordinates, while Xµ(σa) describes the brane embedding. gab is theinduced metric on the probe brane given by gab = ∂aXµ∂bXνGµν , Fab isthe field strength for the U(1) gauge field on the brane, and φ is the dilatonfield. Following the previous discussion, the only component of the gaugefield we choose to turn on is At, additionally, we assume it depends onlyon the radial coordinate r, At = At(r). Considering that the probe braneis extended in the r direction and the spacetime metric is diagonal, thelagrangian simplifies toL ∝ e−φ√−det(gab)(1 +(∂rAt)2gttgrr), (6.28)1076.4. Holographic probeswhere we rescaled At to absorb the 2piα′ term. In the notation of equation(6.21), we can writeg(r) = −det(gab)e−2φ, (6.29)h(r) =det(gab)e−2φgttgrr. (6.30)The backgroundFor Nc Dp branes, at large Nc, the high temperature background is the blackDp brane metric, given by81ds2 = H−1/2(−fdt2 + d~x2p) +H1/2(dr2f+ r2dΩ28−p), (6.31)withH(r) =(Lr)7−p, f(r) = 1−(r+r)7−p, eφ = gsH(3−p)/4. (6.32)L is the characteristic length of the space, while gs is the string coupling.This metric has a horizon at r = r+.Our probe Dq brane is fixed to share d − 1 spacelike directions withthe Dp branes. If p > d − 1, the fundamental matter propagates on a ddimensional defect and we may consider the extra p−(d−1) directions alongthe background brane to be compactified, giving an effective d dimensionalgauge theory at low energies. Alternatively, we can build a d−1 dimensionalgauge theory by compactifying one or more of the directions shared by theprobe and background branes. Below, we will study field theories that areeffectively 3 + 1 dimensional using both methods.We stipulate that the Dq probe brane wraps an Sn inside the S8−pand extends along the radial direction r. These quantities are related byq = d+ n. The induced metric on the Dq brane isds2 = H−1/2(−fdt2 + d~x2d−1) +(η(r) +H1/2f)dr2 +H1/2r2dΩ2n, (6.33)whereη(r) = ∂rXµ∂rXνGµν −Grr. (6.34)81More details on this solution can be found in [95].1086.4. Holographic probesCalculating equations (6.29) and (6.30) gives82g(r) = r2nfH12 (p+n−d−3)(η(r) +H1/2f), (6.35)h(r) = r2nH12 (p+n−d−2), (6.36)from which (6.23) gives the chemical potentialµ =∫ ∞r+dr(d− 2)ρ√r2n(Lr)( 7−p2 )(p+n−d−2) + (d− 2)2ρ2√fη(r)H1/2+ 1. (6.37)Now, η(r) will be some combination of (∂rχi)2, where the χi denote thedirections of transverse brane fluctuations. By writing down the equationsof motion we can observe that ∂rχi = 0 is a solution, in which case theprobe brane goes straight into the black hole along the radial direction r.This describes the high temperature, deconfined regime; we set η(r) = 0 inthe following.For large ρ we findρ ∝ µ14 [(p−7)(p−d−2)+(p−3)(q−d)], (6.38)so that for the probe brane systems,α =14[(p− 7)(p− d− 2) + (p− 3)(q − d)]. (6.39)As above, α is constrained as α > 1 for convergence of the integral. If α ≤ 1,the contribution of the constant charge density to the total energy diverges,signalling a breakdown of the probe approximation. At this point, we canuse equations (6.38) and (6.39) to investigate what type of ρ−µ behaviourscan arise from Dp-Dq brane constructions.Example: the Sakai-Sugimoto modelThe well-known Sakai-Sugimoto model [120] consists of Nf probe D8-D8branes in a background of Nc D4 branes compactified on a circle. We havep = 4, q = 8, and d = 4. Putting these numbers into (6.38) yieldsρ ∝ µ5/2, (6.40)consistent with previous results [20, 116].82We leave the constant factors of gs from eφ out of the lagrangian, as our goal here isjust the power law dependence.1096.4. Holographic probesρ− µ in 3 + 1 dimensional probe brane theoriesEquation (6.39) determines the dominant power law behaviour in all Dp-Dq configurations relevant to 3 + 1 dimensional field theory. As discussedabove, we can set the number of shared probe and background directions tobe d − 1 = 3 or put d − 1 = 4 and demand one of the the spacelike shareddirections to be periodic; see table 6.1 for the results. The power α = 3 isan upper bound for the 3 + 1 dimensional probe brane gauge theories wehave considered.Our calculation above involves only the Born-Infeld action for the probebrane and in particular neglects any possible Chern-Simons terms that ap-pear due to the coupling between the brane and a spacetime tensor field.The Chern-Simons term is important in the D4-D4 system, for example[138].6.4.2 Bottom-up models and the Einstein-Maxwell actionWe now turn our attention to bottom-up AdS/CFT models in the proberegime. To construct a phenomenological gauge / gravity model, we beginwith a theory of gravity with a cosmological constant, such that the geometryis asymptotically AdS. To study the relationship between charge density andchemical potential in the dual field theory, we demand that there must bea gauge field in the bulk. At this point, our model has the ingredients forus to compute our desired result. But, one may ask what type of extensionsare possible. Motivated by superconductivity and superfluidity studies, wewill consider also a charged scalar field in our gravity theory. Adding ascalar field alters the dynamics of the system, notably resulting in differentphases [54, 85]. When the scalar field takes on a non-zero expectation value,this breaks the U(1) gauge symmetry in the bulk and corresponds to thepresence of a U(1) condensate in the boundary theory.The particular model we study is the Einstein-Maxwell system with acharged scalar field:S =∫dd+1x√−g{R+d(d− 1)L2−14FµνFµν − |∂µψ − iqAµψ|2 − V (|ψ|)}.(6.41)Different dual field theories may be obtained by considering this action indifferent regimes and with different parameters. Below, we make the follow-ing ansatz for the gauge and scalar fields:A = φ(r)dt, ψ = ψ(r). (6.42)1106.4. Holographic probesThe r component of Maxwell’s equations will give that the phase of thecomplex field ψ is constant, so without loss of generality we take ψ real. Forthe remainder of the study, we choose units such that L = 1 and considerthe potential V (ψ) = m2ψ2.The probe limitTo get the probe approximation for the system described by (6.41), werescale ψ → ψ/q and A → A/q before taking q → ∞ while keeping theproduct qµ fixed (to maintain the same A − ψ coupling). The gauge andscalar fields decouple from the Einstein equations and we study the fields ina fixed gravitational background.The background is governed by the actionS =∫dd+1x√−g {R+ d(d− 1)} . (6.43)One solution here is the planar Schwarzschild-AdS black hole, given byds2bh = (−fbh(r)dt2 + r2dxidxi) +dr2fbh(r), (6.44)withfbh(r) = r2(1−rd+rd), (6.45)where r+ is the black hole horizon. Below, we consider two systems in theSchwarzschild-AdS background: the probe gauge field, and the probe gaugeand scalar fields.Computing µ and ρIf the kinetic term for the gauge theory on the gravity side is the Maxwelllagrangian,L =14√−gFµνFµν , (6.46)then for an asymptotically AdS space the field equation for the time com-ponent of the gauge field isφ′′ +d− 1rφ′ + · · · = 0, (6.47)1116.4. Holographic probeswhere ′ denotes an r derivative and . . . denotes terms that have higherpowers of 1/r. The solution isφ(r) = φ1 +φ2rd−2+ . . . . (6.48)Recalling that φ(∞) = µ determines that φ1 = µ, while we can plug (6.48)into (6.46) and compute, using (6.15), that φ2 = ρ. We have thatφ(r) = µ−ρrd−2+ . . . , (6.49)so that in practice, below, we just have to read off the coefficients of theleading and next to leading power of 1/r to find the chemical potential andthe charge density.The scalar fieldSolving the scalar field equation at large r in an asymptotically AdS spaceresults in the behaviourψ =ψ1rλ−+ψ2rλ++ . . . , (6.50)whereλ± =12{d±√d2 + 4m2}. (6.51)For m2 near the Breitenlohner-Freedman (BF) bound [25, 26], in the range−(d−1)2/4 ≥ m2 ≥ −d2/4, the choice of either ψ1 = 0 or ψ2 = 0 results in anormalizable solution [85]. For m2 > −(d− 1)2/4, ψ1 is a non-normalizablemode and ψ2 is a normalizable mode. For the cases with the scalar field,we define our field theory by taking ψ1 = 0, so that we never introduce asource for the operator dual to the scalar field.The probe gauge fieldHere, we study the probe gauge field, without the scalar field, in the Schwarzschild-AdS background (6.44). The equation of motion for φ isφ′′ +d− 1rφ′ = 0. (6.52)Regularity at the horizon demands that φ(r+) = 0 and the AdS/CFTdictionary gives φ(∞) = µ, leading toφ(r) = µ(1−rd−2+rd−2). (6.53)1126.4. Holographic probesThen, applying (6.49), we haveρ = µrd−2+ . (6.54)The horizon r+ depends only on the temperature, T = r+d/4pi,83 so this isa linear relationship between ρ and µ, in accordance with (6.20).Adding a scalar fieldWe now turn on the scalar field in (6.41), and consider the dynamics in theSchwarzschild-AdS background (6.44).The field equations becomeψ′′ +(f ′bhfbh+d− 1r)ψ′ +(q2φ2f2bh−m2fbh)ψ = 0, (6.55)φ′′ +d− 1rφ′ −2q2ψ2fbhφ = 0. (6.56)At this point, we can scale q to 1 by scaling φ and ψ, and so m is the onlyparameter here.The coupling allows the gauge field to act as a negative mass for thescalar field. At small chemical potentials, ψ = 0 is the solution. As weincrease µ, the effect of the gauge field on the scalar field becomes largeenough such that the effective mass of the scalar field drops below the BFbound of the near horizon limit of the geometry, so that a non-zero profilefor ψ is possible, and we have a phase transition to the field theory state withbroken U(1) symmetry. A smaller (more negative) squared mass results ina smaller critical chemical potential, at which the scalar field turns on.Using a simple shooting method, for d = 4 we numerically solve equations(6.55, 6.56) and arrive at the relationshipρ = cpbh(m2)µ3, (6.57)where cpbh(m2) is a scaling constant that depends on the mass of the scalarfield. The coupling to the scalar field has resulted in the larger power (α = 3)in the scaling of ρ. A smaller squared mass corresponds to a larger value ofcpbh and, for a given chemical potential, is dual to field theory with a highercharge density. In figure 6.1, we can see the existence of a denser state whenthe scalar field turns on as well as the relative relation between the mass ofthe scalar field and the charge density in the field theory.83For a Euclidean metric ds2 = α(r)dτ2 + dr2β(r) with periodic τ = it coordinate andα(r+) = β(r+) = 0, regularity at the horizon demands that the temperature (the inverseperiod of τ) be given by T =√α′(r+)β′(r+)/4pi.1136.4. Holographic probes10 20 301510020050010002000Μ  TѐT3Figure 6.1: Charge density versus chemical potential for the probe gaugeand scalar fields, section 6.4.2, on a log-log scale. The thick dashed line isfor the system with no scalar field for which, analytically, ρ ∝ µ. At a criticalchemical potential, depending on the mass of the scalar field, configurationswith non-zero scalar field become available. The thin dotted line is a modelpower law ρ ∝ µ3, as described in equation (6.57). From left to right, thethick solid lines are for scalar field masses m2 = −15/4, −14/4, −13/4, and−3. A more negative scalar field mass results in a denser field theory stateat a given chemical potential.The soliton probeMotivated by recent work [19, 68, 107], we now add more structure to thebulk theory in the form of an extra periodic dimension. To model a 3 + 1dimensional field theory, we set d = 5 and stipulate that this includes oneperiodic spacelike coordinate w of length 2piR. At energies much less thanthe scale set by this length, E  1/R, the dual field theory will be effectively3+1 dimensional. The extra dimension sets another scale for the field theory1146.4. Holographic probesand enables a richer phase structure in the system.84With the extra periodic direction, there is another solution to the back-ground described by (6.43). This is the AdS-soliton, given as the double-analytic continuation of the Schwarzschild-AdS solution (6.44):ds2sol = (r2dxµdxµ + fsol(r)dw2) +dr2fsol(r), (6.58)withfsol = r2(1−r50r5). (6.59)Here, r0 is the location of the tip of the soliton. For regularity, it is fixed bythe length of the w dimension asr0 =25R. (6.60)By computing the free energy of the systems, it can be shown that the soli-ton background dominates over the black hole background for small enoughtemperatures and chemical potentials. As the temperature or chemical po-tential is increased, there is a first order phase transition to the black hole,which is the holographic version of a confinement / deconfinement transition.For zero scalar field, the soliton can be considered at any temperatureand chemical potential; the period of the Euclidean time direction defines thetemperature while φ = µ = constant is a solution to the field equations. Inthis case, ρ = 0 and we do not have an interesting ρ−µ relation. Consideringa non-zero scalar field provides a source for the gauge field and allows non-trivial configurations.In the soliton background (6.58), the equations of motion areψ′′ +(f ′solfsol+4r)ψ′ +(q2φ2r2fsol−m2fsol)ψ = 0, (6.61)φ′′ +(f ′solfsol+2r)φ′ −2q2ψ2fsolφ = 0. (6.62)As in the black hole case, at this point we can set q = 1 by scaling the fields.84The phase diagram including both black hole and soliton solutions, was studied in[68] for a 2+1 dimensional field theory in the context of holographic superconductors andin [19] for a 3 + 1 dimensional field theory in the context of holographic QCD and coloursuperconductivity.1156.5. ρ− µ in backreacted systemsAfter numerically integrating, we haveρ = cpsol(m2)µ4. (6.63)Compared to the black hole case, above, we find a larger power of µ. At largedensities, the average distance between charges becomes small compared tothe size R of the periodic direction. In this limit, the system becomeseffectively higher dimensional and so we would expect a larger power α inthe ρ− µ relationship. The numerics were consistent with this result.As can be seen in figure 6.2, a more negative mass squared results ina smaller critical chemical potential and a denser field theory state at agiven chemical potential. This is as expected by comparing the structure ofthe equations to those in the black hole case. Further, at a given chemicalpotential, the soliton solution corresponds to a denser field theory state thanthe black hole solution with the same scalar field mass.6.5 ρ− µ in backreacted systemsDespite our analysis in section 2 relying on the probe approximation, it isinteresting to ask how much of a difference allowing for backreaction on thebottom-up models could make to the ρ − µ relation and the bounds foundpreviously. Henceforth we generalize the bottom-up model introduced insection 6.4.2 and allow for the backreaction of the gauge and scalar field onthe metric. Recall that the action isS =∫dd+1x√−g{R+ d(d− 1)−14FµνFµν − |∂µψ − iqAµψ|2 −m2ψ2}.(6.64)We start by studying the well-known Reissner-Nordstrom-AdS (RN-AdS)solution to the Einstein equation, in which ψ = 0. Later, we allow thescalar field to acquire a non-zero profile and investigate its consequences onthe ρ−µ profile. We finish with the investigation of the backreacted versionof the solitonic solution.6.5.1 Charged black holesThe backreacted solution with no scalar field is the planar RN-AdS blackhole, given byds2 = (−fRN(r)dt2 + r2dxidxi) +dr2fRN(r), (6.65)1166.5. ρ− µ in backreacted systems6 8 10 12 14 160100200300400500Μ  TѐT4Figure 6.2: Charge density versus chemical potential for the probe gaugeand scalar fields in the soliton background, section 6.4.2, and the d = 5black hole background, section 6.4.2. The thin dashed line is the probegauge field in the black hole background for which, analytically, ρ ∝ µ. Thethick solid lines are the soliton results (from left to right, the squared massof the scalar field is −22/4, −5, −18/4, and −4) while the thick dashed linesare the black hole results (again, from left to right, m2 = −22/4, −5, −18/4,and −4). Each of the thick lines approaches the power law ρ ∝ µ4, equation(6.63). At a given chemical potential, the soliton background gives a fieldtheory in a denser state.with85fRN(r) = r2(1−(1 +(d− 2)µ22(d− 1)r2+)rd+rd+(d− 2)µ22(d− 1)r2(d−2)+r2(d−1)). (6.66)The gauge potential isφ(r) = µ(1−rd−2+rd−2), (6.67)85We parametrize this solution in terms of the location of the horizon r+ and theasymptotic value of the gauge field (the chemical potential µ) instead of the usual choicesof the charge and mass of the black hole.1176.5. ρ− µ in backreacted systemsso that, using (6.49), we have ρ = µrd−2+ . Here, the horizon r+ can beexpressed as a function of the temperature and chemical potential throughthe Hawking temperatureT =14pi(dr+ −(d− 2)2µ22(d− 1)r+). (6.68)Eliminating r+ in favour of ρ and µ in (6.68), we may solve for ρ to findρ =((d− 2)22d(d− 1)) d−22µd−1[(2(d− 1)d) 12 2piT(d− 2)µ+√1 +8pi2(d− 1)T 2d(d− 2)2µ2]d−2.(6.69)Notice that the dominant power in the ρ−µ relationship is µd−1, as expectedin a d dimensional conformal field theory. For d = 4, the particular large µexpansion isρ =16µ3 +pi√6µ2T +12pi2µT 2 +14√32pi3T 3 + . . . . (6.70)6.5.2 Hairy black holesIf we turn on the scalar field, an analytic solution to the equations of motionis no longer possible and we turn to numerical calculation. We take as ourmetric ansatzds2 = −g(r)e−χ(r)dt2 +dr2g(r)+ r2(dxidxi), (6.71)where g(r) will be fixed to have a zero at r+, giving a horizon. We arrive atthe following equations of motion:ψ′′ +(g′g−χ′2+d− 1r)ψ′ +1g(q2φ2eχg−m2)ψ = 0, (6.72)φ′′ +(χ′2+d− 1r)φ′ −2q2ψ2gφ = 0, (6.73)χ′ +2rψ′2d− 1+2rq2φ2ψ2eχ(d− 1)g2= 0, (6.74)g′ +(d− 2r−χ′2)g +reχφ′22(d− 1)+rm2ψ2d− 1− dr = 0. (6.75)1186.5. ρ− µ in backreacted systemsThe first two equations can be derived via the Euler-Lagrange equationsfor φ and ψ, while the final two equations are the tt and rr components ofEinstein’s equation.In this system, as in the probe case, section 6.4.2, at small chemicalpotentials the scalar field is identically zero. As we increase the chemi-cal potential above a critical value, the system undergoes a second orderphase transition to a state with non-zero scalar field. When the scalar fieldcondenses, the corresponding field theory is in a denser state at the samechemical potential than for the system without scalar field.We solve the equations numerically for d = 4, to yield the result, in thephase with the scalar field,ρ = cbh(q,m2)µ3. (6.76)As we increase the charge or decrease the mass squared of the scalar field, thecritical chemical potential, at which the scalar condenses, decreases, whilethe scaling coefficient cbh increases. The scaling coefficient cbh(q,m2) is, inall cases we checked, larger than the coefficient of the µ3 term in the AdS-Reissner-Nordstrom black hole, equation (6.70), indicating that the densityscales faster with the chemical potential when the scalar field is present.When we include metric backreaction for the black hole, the dominantpower in the ρ − µ relationship is greater than the probe case when thereis no scalar field and is the same as the probe case when there is a scalarfield, indicating that, at least for the systems considered, the bounds foundfor the ρ− µ behaviour apply to the backreacted cases as well.6.5.3 Backreacted solitonMotivated by the form of the soliton background (6.58) we choose the metricansatzds2 =dr2r2B(r)+ r2(eA(r)B(r)dw2 − eC(r)dt2 + dxidxi), (6.77)where we constrain B(r0) = 0 so that the tip of the soliton is at r0. Thefield and Einstein equations giveψ′′ +(6r+A′2+B′B+C ′2)ψ′ +1r2B(e−C(qφ)2r2−m2)ψ = 0, (6.78)φ′′ +(4r+A′2+B′B−C ′2)φ′ −2ψ2q2φr2B= 0, (6.79)1196.6. DiscussionB′(4r−C ′2)+B(ψ′2 −12A′C ′ +e−Cφ′22r2+20r2)++1r2(e−C(qφ)2ψ2r2+m2ψ2 − 20)= 0, (6.80)C ′′ +12C ′2 +(6r+A′2+B′B)C ′ −(φ′2 +2(qφ)2ψ2r2B)e−Cr2= 0, (6.81)A′ =2r2C ′′ + r2C ′2 + 4rC ′ + 4r2ψ′2 − 2e−Cφ′2r(8 + rC ′). (6.82)We solve equations (6.78-6.81) numerically with asymptotically AdSboundary conditions before integrating (6.82) to find A.86 The results areconsistent with a ρ− µ relationship of the formρ = csol(q,m2)µ4. (6.83)As in the probe case, the effective higher dimension of the space dictatesthe power in the relationship. The dependence of csol(q,m2) on q and m2is as in the backreacted black hole case, section 6.5.2. Like the black holewith scalar field, the backreacted soliton with scalar field gives the samedominant power α as the corresponding probe case.6.6 DiscussionIn this chapter we studied the ρ−µ relation for a variety of holographic fieldtheories, and set conditions for physically consistent relationships based onlocal stability and causality. We observed that all of the examples consideredare well modelled by a power law ρ = cµα in the large µ regime and that noneof them fail to satisfy any of the general constraints stablished in sections 6.3and 6.2. Except for the case of a probe gauge field in the Schwarzschild-AdSblack hole background, the power α in all the bottom-up models obeyed thegeneric dimensional argument discussed in the introduction, as can be seenin table 6.2. This resulted in a larger power for the models with an extraperiodic dimension. The brane constructions, table 6.1, displayed a largervariety of power laws, with the range 1 < α ≤ 3, where α depended on theparticular dimensions of the probe and background branes.86More details on the numerical process can be found in [19].1206.6. DiscussionThe study of bottom-up models led to the conclusion that, in general,the presence of a non-zero profile for the scalar field in the bulk inducesa larger charge density on the boundary. In most cases, this change wasrealized as an increase of the scaling coefficient c while the power law waskept unaltered. The only exception was the probe Einstein-Maxwell case,section 6.4.2. Here, in the absence of a scalar field, the probe Maxwellfield enjoys its standard linear equations of motion, and naturally we find alinear ρ−µ relationship. With a non-zero scalar field, the power law becomesρ ∝ µd−1, as expected for the underlying CFT. In systems with an extraperiodic direction, the numerical results displayed in figure 6.2 support theconclusion that at a given (large enough) chemical potential, the solitonicphase is denser than the corresponding black hole phase.Despite our attempt to survey a large variety of holographic models, wedo not claim to have presented a complete report and we do not discard thepossibility of finding different ρ − µ relations in other types of bottom-upand top-down models. For example, one generalization would be to includeNf > 1 flavour branes in the Dp-Dq systems; this has been shown to changethe power α in the relation [116]. It would be interesting to extend thisstudy to other classes of systems and to see how the results compare tothose given here.121Chapter 7Color Super Conductivity7.1 IntroductionBackgroundQuantum Chromodynamics is believed to display a rich phase structureat finite temperature and chemical potential, with phase transitions asso-ciated with deconfinement, nuclear matter condensation, the breaking of(approximate) flavor symmetries (which are exact in generalizations withequal quark masses and/or massless quarks), and the onset at high densityof quark matter phases displaying color superconductivity [3, 4] (for reviewssee for example [5, 102, 106, 115, 127]). However, apart from the regimesof asymptotically large temperature or chemical potential, a direct analyticstudy of the thermodynamic properties of the theory is not possible.Even using numerical simulations, only the physics at zero chemical po-tential is currently accessible, since at finite µ the Euclidean action becomescomplex, and the resulting oscillatory path integral cannot reliably be sim-ulated using standard Monte-Carlo techniques.87 Current proposals for thephase diagram of QCD and related theories are largely based on qualitativearguments and phenomenological models. While these provide a plausiblepicture, it is possible that they miss important features of the physics. Itwould certainly be satisfying to have examples of theories similar to QCDin which the full phase diagram could be explored directly.The holographic approachA modern route to understanding properties of strongly coupled gaugetheories, that would be otherwise inaccessible, is via the AdS/CFT cor-respondence, or gauge theory / gravity duality. This suggests that cer-tain quantum field theories (usually called “holographic theories”), generallywith large-rank gauge groups, are equivalent to gravitational systems. By87Note however that the regime of small µ/T may be accessible numerically via a per-turbative expansion.1227.1. Introductionthis correspondence, calculations of physical observables in the field theoryare mapped to gravitational calculations; in many cases difficult strongly-coupled quantum mechanical calculations in the field theory (such as thoserequired to understand the thermodynamic properties of QCD) are mappedto relatively simple classical gravity calculations. Optimistically, it maythen be possible to find a theory qualitatively similar to QCD for which thephysics at arbitrary temperature and chemical potential can be understoodexactly via simple calculations in a dual gravitational system.By now, there are well-known examples in gauge-theory / gravity dualityfor which the field theory shares many of the qualitative features of QCD(see, for example [120]). Further, many of these theories have been studiedat finite temperature and chemical potential, revealing phase transitionsassociated with deconfinement, chiral symmetry breaking, meson melting,and the condensation of nuclear matter. However, to date, most of thetheories that can be studied reliably using dual gravity calculations havethe restriction that the number of flavors is kept fixed in the large Nc limit.In such theories, the physics at large chemical potentials is known to bequalitatively different than in real QCD. For example, at asymptoticallylarge chemical potential, theories with large Nc and fixed Nf are believed toexhibit an inhomogeneous “chiral density wave” behavior [35, 125], ratherthan the homogenous quark matter phases predicted for finite Nc and Nf . Inorder to find examples of holographic theories which most closely resemblereal QCD at finite chemical potential, one should therefore attempt to findexamples of calculable gravitational systems corresponding to theories withfinite Nf/Nc. This situation presents some technical challenges, as we nowreview.In the well-known examples of holographic gauge theories, the addition offlavor fields in the field theory corresponds to adding D-branes on the gravityside [82]. Quarks correspond to strings which have one endpoint on theseD-branes, while mesons correspond to the quantized modes of open stringswhich begin and end on the branes. The configurations of these D-branesin theories with finite Nf and large Nc are determined by finding action-minimizing configurations of the branes on a fixed background geometry.On the other hand, in order to have Nf of order Nc in a large Nc theory,we need a large number of these flavor branes, and these will back-reacton the spacetime geometry itself. For Nf ∼ Nc, there are as many degreesof freedom in the flavor fields as there are in the color fields (gauge fieldsand adjoints), so it is natural to expect that the back-reaction will be sosignificant that in the final description the flavor branes themselves will becompletely replaced by a modified geometry with fluxes (in the same way1237.1. Introductionthat the branes whose low-energy excitations give rise to the adjoint degreesof freedom do not appear explicitly in the gravity dual description of thefield theory).There has been significant progress in understanding the back-reactionof flavor branes, with some fully-back reacted analytic solutions available(for a review see [110]), but so far, there has not been enough progress tofully explore the phase structure of a QCD-like theory with finite Nf/Nc.In particular, as far as we are aware, color superconductivity phases havenot been identified previously in holographic field theories.88Quark matter from the bottom upIn this chapter, we aim to come up with a holographic system describing aconfining gauge theory that does exhibit a quark-matter phase with colorsuperconductivity at large chemical potential. However, motivated by recentcondensed matter applications of gauge/gravity duality (see, for example[54]), we will avoid many of the technical challenges described above bytaking what is known as a “bottom up” approach. Rather than working ina specific string theoretical model which takes into account the back-reactionof flavor branes, we will make an ansatz for the ingredients necessary for sucha model to describe the relevant physics. We study the simplest possiblegravitational theory with this minimal set of features, with the hope that itcaptures the qualitative physics of interest. We will indeed find that eventhis simple theory exhibits many of the expected features.IngredientsWe wish to construct a gravitational theory to provide a holographic de-scription of a four-dimensional confining gauge theory on Minkowski spacewith Nf ∼ Nc flavors. On the gravity side, the Minkowski space will ap-pear as the fixed boundary geometry of our spacetime, but we must haveat least one extra dimension corresponding to the energy scale in the fieldtheory. Since the field theory has a scale (the QCD or confinement scale),the asymptotic behavior of the solution must exhibit an additional scalerelative to the asymptotically AdS geometries that appear in gravity dualsof conformal field theories. In the simplest examples of gravity duals forconfining gauge theories, this scale is provided by the size of an additional88However, see [30] for a possible manifestation of the related color-flavor locking phasein a holographic system.1247.1. Introductioncircular direction in the geometry.89 Thus, we will work with a gravitationalsystem in six dimensions whose boundary geometry is R3,1 × S1. We willassume that the asymptotic geometry is locally Anti-de-Sitter space, so theconfining gauge theory we consider arises from a five-dimensional conformalfield theory compactified on a circle. When we study the theory at finitetemperature, there will be an additional circle in the asymptotic (Euclidean)geometry, the Euclidean time direction whose period is 1/T .The gauge theories we are interested in have at least one other con-served current, corresponding to baryon (or quark) number. By the usualAdS/CFT dictionary, this operator corresponds on the gravity side to a U(1)gauge field in the bulk. The asymptotic value of the time component forthis gauge field corresponds to the chemical potential in our theory, whilethe asymptotic value of the radial electric flux corresponds to the baryoncharge density in the field theory. For a given chemical potential, the min-imum action solution will have some specific value for the flux, allowing usto relate density and chemical potential.The color superconductivity phases believed to exist at large density inQCD and related theories are usually characterized by condensates of theform 〈ψψ〉, bilinear in the quark fields ψ, which spontaneously break theU(N) gauge symmetry, and the U(1)B global symmetry. Naively, we wouldwant to model such operators by a bulk charged scalar field corresponding tothe condensate. However, bulk fields always correspond to gauge-invariantoperators, while by definition the ψψ bilinears which break the gauge sym-metry are not gauge-invariant (in fact, there is no way to make a singletfrom two fundamental fields, except in the case of SU(2)). Additionally, thesimplest gauge-invariant operators charged under U(1)B involve N ψ fieldsand have dimension of order N , thus our holographic dual theory shouldhave no light scalar fields charged under the U(1)B gauge field.The correct way to understand the condensation of the ψψ bilinears isas an example of spontaneously broken gauge symmetry (as in the Higgsmechanism), rather than as a phase transition characterized by some gauge-invariant order parameter. Nevertheless, the transition to color supercon-ductivity can be characterized by the discontinuous behavior of gauge-invariantoperators, which are of the form ψψ(ψψ)†. Such operators are gauge invari-ant and neutral under the U(1)B, and therefore should correspond to anuncharged scalar field in the bulk with dimension of order 1.9089There are other possibilities here, as we mention briefly in the discussion section.90As emphasized to us by Andreas Karch, a gauge invariant operator of the formO4 = ψψ(ψψ)† can be written as a sum of terms OαOα where each Oα ∼ (ψ†ψ)α isgauge invariant (and α represents flavor/Lorentz indices). Thus, O4 is something like a1257.1. IntroductionCombining everything so far, we want to study gravity in six dimensionswith negative cosmological constant and boundary geometry R3,1×S1 witha U(1) gauge field and a neutral scalar field. The simplest action for thissystem is91∫d6x√−g{R+20L2−14F 2 − |∂µψ|2 −m2|ψ|2}, (7.1)where we include one tunable parameter, the mass m of the scalar field,which determines the dimension of the corresponding operator in the dualfield theory. More generally, we could consider other potentials for the scalarfield, or a more complicated action (e.g. with a Chern-Simons term or ofBorn-Infeld type) for the gauge field, but we restrict here to this simplestpossible model.92ResultsStarting with the model (7.1), we have explored the phase structure by min-imizing the gravitational action for specific values of temperature (corre-sponding to the asymptotic size of the Euclidean S1 direction) and chemicalpotential (corresponding to the asymptotic value of A0). Our results for thephase diagrams are shown in figures 7.1,7.2,7.3. For small µ, we find a con-fined phase at low-temperature and a deconfined phase at high temperature,with the scalar field uncondensed in each case. However, increasing µ at zerotemperature, we find (setting LAdS = 1) for −254 ≤ m2 ≤ −5 a transition toa phase with nonzero scalar condensate (on a geometry with horizon) andfinite homogeneous quark density, as expected for a color superconductiv-ity phase. Increasing the temperature from zero, we find a transition backto the deconfined phase at a remarkably low temperature; for example, atm2 = −6, the critical temperature at which superconductivity disappears isT/µ ∼ .00006333 .double-trace operator. In a large N theory, factorization of correlators implies that theexpectation value of O4 can be calculated classically from the Oα expectation values (upto 1/N corrections). Thus, discontinuous behavior of O4 should be directly related todiscontinuous behavior in the simpler gauge-invariant operators Oα (which also have nobaryon charge), so it may be more appropriate to think of the scalar field in our model asbeing dual to one of these simpler operators.91Since we will also consider the case of a charged scalar field, we have written the actionusing standard normalizations for a complex scalar, but we will take the scalar to be realin the uncharged case.92For another approach to modeling the QCD phase diagram by an effective holographicapproach, see for example [36, 55].1267.1. Introduction0.20.30.40.5 Tdeconfined00.10.2 00.511.522.5μconfinedsuperconductingFigure 7.1: Phase diagram of our model gauge theory with m2 = −6,R = 2/5. Region in dashed box is expanded in next figure.The tendency for the scalar field to condense at low temperatures for therange of masses above can be understood in a simple way, as explained forexample in [60, 74]. In d + 1-dimensional anti-de Sitter space with anti-deSitter radius L, the minimum mass for a scalar field to avoid instability ism2BF = −d2/(4L2). The minimum action solution for large chemical poten-tial in the absence of any scalar field is a planar Reissner-Nordstrom blackhole solution with one of the isometry directions periodically identified. Inthe limit of zero temperature, the near horizon region of this black hole hasgeometry AdS2 × R4, with the radius of the AdS2 equal to L2 = L/√20.Thus, in the near-horizon region, there will be an instability toward conden-sation of the scalar field if m2 < −1/(4L22) = −5/L2. We thus have a range(setting L = 1) of −25/4 ≤ m2 ≤ −5 for which the scalar field tends tocondense in the near-horizon region but is stable in the asymptotic region.Numerical simulations verify that we indeed have scalar field condensationfor precisely this range of masses.While there is no guarantee that the gravitational system we study has alegitimate field theory dual, “top-down” gravitational systems correspondingto fully consistent field theories must have the same basic elements (usuallywith additional fields and a more complicated Lagrangian). The fact thatthe expected physics emerges even in our stripped-down version suggests1277.1. Introduction0.00010.0002Tdeconfinedconfined00.0001 012µsuperconductingFigure 7.2: Phase diagram of our model gauge theory with m2 = −6,R = 2/5. Region in dashed box is expanded in next figure.0.00010.0002Tdeconfined00.0001 1.73231.73251.7327µconfinedsuperconductingFigure 7.3: Phase diagram of our model gauge theory with m2 = −6,R = 2/5. The dashed curve represents the phase boundary in theory withouta scalar field.1287.2. Basic setupthat quark-matter phases will be found also in the complete models, onceback-reaction effects are under control. Optimistically, qualitative featuresthat we find in the bottom-up model (such as the extremely low transi-tion temperature between superconducting and deconfined phases) may bepresent also in more complete holographic theories. In this case, our simplemodel may provide novel qualitative insights into fully consistent QCD-liketheories.Charged scalarWhile less relevant to color superconductivity, it is also interesting to explorethe physics of our model when we make the scalar field charged under thegauge field. In this case, the scalar field corresponds to a gauge-invariantoperator in the field theory that is charged under the U(1) associated withA, and the kinetic term for the scalar field is modified in the usual way as∂µψ → ∂µψ − iqAµψ. As we have argued above, this symmetry cannot beU(1)B, but could be another flavor symmetry, such as isospin in a model withtwo or more flavors. The flavor superconductivity associated with mesoncondensation was studied previously in the holographic context (with finiteNf ), for example in [7, 8, 17]. Our results are qualitatively similar to theones obtained in those studies, and we leave more detailed comparison forfuture work.In section 5 below, we determine the phase diagram for various valuesof q and m. The same system was studied for the 2+1 dimensional casein [68] and originally in [107] for the case of large q. The application therewas to holographic insulator/superconductor systems, but the intriguingresemblance of the phase diagrams in those papers to QCD phase diagramspartially motivated the present study.7.2 Basic setupIn this chapter, we consider holographic field theories with a conserved cur-rent Jµ, assumed to be a baryon current (or isospin current when we considercharged scalar fields) and some gauge-invariant operator O whose conden-sation indicates the onset of (color or flavor) superconductivity. We wouldlike to explore the phase structure of the theory for finite temperature T andchemical potential µ; that is, we would like to find the phase that minimizesthe Gibbs free energy density g = e − Ts − µρ, where e, s, and ρ are theenergy density, entropy density, and charge density in the field theory. We1297.2. Basic setupcan also ask about the values of e, s, ρ, and 〈O〉 as a function of temperatureand chemical potential.As discussed in the introduction, our holographic theories are defined bya dual gravitational background which involves a metric, U(1) gauge field,and scalar field, with a simple action∫d6x√−g{R+20L2−14F 2 − |∂µψ|2 −m2|ψ|2}. (7.2)We choose coordinates (t, x, y, z) for the non-compact field theory directions,w for the compact field theory direction, and r for the radial direction. Wetake boundary conditions for which the asymptotic (large r) behavior of themetric isds2 →( rL)2 (−dt2 + dx2 + dy2 + dz2 + dw2)+(Lr)2dr2 ,where w is taken to be periodic with period R. To study the theory at finitetemperature, we take the period of τ = it in the Euclidean solution to be1/T .The equations of motion constrain the gauge field to behave asymptoti-cally asAν = aν −jν3r3+ . . . .Since Aν is assumed to be the field corresponding to the conserved baryoncurrent operator Jν , in the field theory, the usual AdS/CFT dictionarytells us that aν is interpreted as the coefficient of the Jν in the Lagrangian(i.e. an external source for the baryon current) while jµ is interpreted asthe expectation value of baryon current for the state corresponding to theparticular solution we are looking at. To study the theory at finite chemicalpotential µ without any external source for the spatial components of thebaryon current, we want to takeaν = (µ, 0, 0, 0) .The scalar field equations of motion imply that asymptoticallyψ =ψ1rλ−+ψ2rλ++ · · · , (7.3)whereλ∓ =12(d∓√d2 + 4m2) .1307.2. Basic setupThe holographic field theories we consider are defined by assuming ψ1 = 0.In this case, λ+ gives the dimension of the operator dual to ψ.93 In this case,ψ2 (which will be different for solutions corresponding to different states ofthe field theory) gives us the expectation value of the operator O in the fieldtheory.By the AdS/CFT correspondence, the field theory free energy corre-sponds to the Euclidean action of the solution. Thus, to investigate thefield theory state which minimizes free-energy for given T and µ, we need tofind the gravitational solution with boundary conditions given above whichminimizes the Euclidean action. Note that we only consider solutions withtranslation invariance in t, x, y, z, and w. It would be interesting to investi-gate the possibility of inhomogeneous phases (or at least the stability of oursolutions to inhomogeneous perturbations) but we leave this as a questionfor future work.Calculating the actionIn order to obtain finite results when calculating the gravitational action fora solution, it is important to include boundary contributions to the action.In terms of the Lorentzian metric, gauge field and scalar, the fully regulatedexpression that we require is [54]S = limrM→∞[−∫r<rMdd+1x√−g{R+d(d− 1)L2−14F 2 − |Dµψ|2 −m2|ψ|2}+∫r=rMddx√−γ{−2K +2(d− 1)L−1Lλ−|ψ|2}],whereλ− =d2−12√d2 + 4m2 .Here, γ is the metric induced on the boundary surface r = rM , and K isdefined asK = γµν∇µnν ,where nµ is the outward unit normal vector at r = rM . The scalar countert-erm here is the appropriate one assuming that our boundary condition is tofix the coefficient of the leading term in the large r expansion of ψ. Sincewe are setting this term to zero, it turns out that the counterterm vanishesin the rM →∞ limit.93For a certain range scalar field masses in the range −d2/4 ≤ m2 ≤ −d2/4 + 1, it isalso consistent to define a theory by fixing ψ2 = 0. In this case, the dimension of the dualoperator is λ−. We consider this case briefly in section 4.2.1317.2. Basic setupFor all cases we consider, the metric takes the formds2 =r2L2dx2i + g00(r)dt2 + grr(r)dr2 + gww(r)dw2 . (7.4)Assuming the Einstein equations are satisfied, we can show (by subtractinga term proportional to the xx component of the equation of motion) thatthe integrand in the first term may be written as a total derivative withrespect to r−√−g{R+d(d− 1)L2−14F 2 − |Dµψ|2 −m2|ψ|2}= ∂r(2rgrr√−g).Usingnµ = (0, . . . , 0,√grr) ,we haveK = γµν∇µnν= γµν{−Γrµνnr}= γµν{12grr∂gµν∂r√grr}=12√grrγµν∂γµν∂r=1√grr∂ ln(√−γ)∂rso that√−γ(−2K) = −2√grr∂√−γ∂r.Our final expression for the action density isS/Vd =2rgrr√−g∣∣∣∣rMr0+{−2√grr∂√−γ∂r+2(d− 1)L√−γ}r=rM. (7.5)Action in terms of asymptotic fieldsIt is convenient to rewrite the expression (7.5), in terms of the asymptoticexpansion of the fields. For the ansatz (7.4), and the boundary conditionsappropriate to our case, we findgtt = −r2 +g(3)ttr3+ . . . ,1327.3. Review: ψ = 0 solutionsgrr =1r2+g(7)rrr7+ . . . ,gww = r2 +g(3)wwr3+ . . . ,ψ =ψ(3)r3+ . . . ,φ = µ−ρ3r3+ . . . .Inserting these expansions into our expression above for the action we findthat (assuming the term at r = r0 vanishes)S = 5g(3)ww + 4g(7)rr − 5g(3)tt .However, using the equations of motion, we find that g(3)ww + g(7)rr − g(3)tt = 0,so we can simplify to:S = −g(7)rr . (7.6)Numerically, it can be a bit tricky to read off g(7)rr because there is also a1/r8 term in the expansion of grr. But using the equations of motion, wecan findg(8)rr =34(7 +m2)(ψ(3))2 .From this, it follows that the combination−r7grr(r) + r5 −34(7 +m2)r5ψ2(r)behaves like−g(7)rr +O(1/r3) .So, we can numerically evaluate the action by takingS ≈ −r7∗grr(r∗) + r5∗ −34(7 +m2)r5∗ψ2(r∗) ,where r∗ is taken to be large but not too close to the cutoff value.7.3 Review: ψ = 0 solutionsWe begin by considering the solutions for which the scalar field is set tozero.1337.3. Review: ψ = 0 solutions7.3.1 AdS Soliton solutionAt zero temperature and chemical potential, the simplest solution with ourboundary conditions is pure AdS with periodically identified w. However,assuming antiperiodic boundary conditions for any fermions around the wcircle, there is another solution with lower action. This is the AdS soliton[65], described by the metric (setting L = 1)ds2 = r2(−dt2 + dx2 + dy2 + dz2 + f(r) dw2)+dr2r2f(r), (7.7)wheref(r) = 1−r50r5. (7.8)As long as we choose the period 2piR for w such thatr0 =25R(7.9)the solution smoothly caps off at r = r0. This IR end of the spacetimecorresponds in the field theory to the fact that we have a confined phasewith a mass gap. The fluctuation spectrum about this solution correspondsto a discrete spectrum of glueball states in the field theory.Starting from this solution, we can obtain a solution valid for any tem-perature and chemical potential, by periodically identifying the Euclideantime direction and setting A0 = µ everywhere. Using (7.6) we find that theaction for this solution isSsol = −r50 = −(25R)5.The negative value indicates that this solution is preferred over the pureAdS solution with action zero.7.3.2 Reissner-Nordstrom black hole solutionFor sufficiently large temperature and/or chemical potential, the AdS solitonis no longer the ψ = 0 solution with minimum action. The preferred solutionis the planar Reissner-Nordstrom black hole, with metricds2 = r2(−dt2f(r) + dx2 + dy2 + dz2 + dw2)+dr2r2f(r), (7.10)1347.4. Neutral scalar field: color superconductivitywheref(r) = 1−(1 +3µ28r2+)r5+r5+3µ2r6+8r8, (7.11)the scalar potential isφ(r) = µ(1−r3+r3),and w is periodically identified as before.This solution has a horizon at r = r+. The temperature of the solu-tion (determined as the inverse period of the Euclidean time for which theEuclidean solution is smooth) is given in terms of r+ byT =14pi(5r+ −9µ28r+). (7.12)From (7.6), we find that the action for this solution isSRN = −r5+(1 +38µ2r2+).Thus, we find that the black hole solution has lower action than the solitonforr+(1 +38µ2r2+) 15>25R,where r+ is determined in terms of T and µ by (7.12). This defines a curvein the T −µ plane that begins on the µ = 0 axis at T = 1/(2piR) and curvesdown to the T = 0 axis at µ = 219/10/(51/234/5R) ≈ 4.3547/(2piR), as shownin figure 7.4.As usual, the existence of a horizon in this solution indicates that thecorresponding field theory state is in a deconfined phase [141].In the next sections, we consider solutions with nonzero scalar field.We will find that for large µ there exist solutions with nonzero scalar fieldthat have lower action than the solutions we have considered, so the phasediagram of figure 7.4 will be modified.7.4 Neutral scalar field: color superconductivityIn the case of a neutral scalar field, our simple model has no explicit sourcefor the gauge field in the bulk, so homogeneous solutions with a non-trivial1357.4. Neutral scalar field: color superconductivity0.20.30.40.5 TReissner-Nordstrom(deconfined)00.10.2 00.511.522.5μAdS soliton(confined)Figure 7.4: Phase diagram without scalar field, in units where R = 2/5.static electric field (corresponding to a non-zero baryon number density inthe field theory) necessarily have a horizon from which the flux can emerge94.To look for solutions of this form, we consider the ansatz95ds2 = −g(r)e−χ(r)dt2 +dr2g(r)+ r2(dw2 + dx2 + dy2 + dz2) ,At = φ(r) ,ψ = ψ(r) .The scalar and Maxwell’s equations that follow from the action (7.2) areψ′′ +(4r−χ′2+g′g)ψ′ −m2gψ = 0 , (7.13)φ′′ +(4r+χ′2)φ′ = 0 , (7.14)94In a more complete model, the source might be provided by some non-perturbativedegrees of freedom in the theory, such as the wrapped D-branes that give rise to baryonsin the Sakai-Sugimoto model.95We could have considered a more complicated ansatz, with an extra undeterminedfunction in front of dw2. However, it is plausible that as for the ψ = 0 solution, theminimum action solution for the case where the w circle does not contract in the bulk isa periodic identification of the solution with non compact w and rotational invariance inthe x, y, z, w directions.1367.4. Neutral scalar field: color superconductivitywhile the Einstein equations are satisfied ifχ′ +rψ′22= 0 , (7.15)g′ +(3r−χ′2)g +reχφ′28+m2rψ24− 5r = 0 . (7.16)These have two symmetries:ψ˜(r) = ψ(ar) , φ˜(r) =1aφ(ar) , χ˜(r) = χ(ar) , g˜(r) =1a2g(ar) ,(7.17)arising from the underlying conformal invariance, andχ˜ = χ+ ∆ , φ˜ = e−∆2 φ . (7.18)We would like to find solutions with a horizon at some r = r+. Theelectric potential must also vanish at the horizon, and we are looking forsolutions for which the leading falloff ψ1 in (7.3) vanishes for the scalar.Also, multiplying the first equation (7.13) by g and evaluating at r = r+ fixesψ′(r+) in terms of ψ(r+) and g′(r+). Altogether, our boundary conditionsareg(r+) = 0 , φ(r+) = 0 , χ(∞) = 0 , ψ1 = 0 ,andψ′(r+) =8m2ψ(r+)40r+ − 2m2r2+ψ2(r+)− r+eχ(r+)(φ′(r+))2.The remaining freedom to choose r+ and φ′(r+) leads to a family of solutionswith different T and µ. Explicitly, we haveµ = φ(∞) , T =14pig′(r+)e−χ(r+)/2 .Note that solutions with the same T/µ are simply related by the scalingsymmetry (7.17).7.4.1 Numerical evaluation of solutionsTo find solutions in practice, we can make use of the symmetries (7.17) toinitially set r+ = 1 and χ(0) = 0 and solve the equations with boundaryconditionsg(1) = 0 , χ(0) = 0 , φ(1) = 0 , φ′(1) = E0 , ψ(1) = ψ0 ,1377.4. Neutral scalar field: color superconductivityandψ′(1) =8m2ψ040− 2m2ψ20 − E20.We can integrate the φ and χ equations explicitly to obtainχ(r) = −∫ r0dr˜12r˜(∂ψ∂r)2,φ(r) = E0∫ r1dr˜r˜4e−12χ(r˜) ,leaving the remaining equationsψ′′ + (4r+rψ′24+g′g)ψ′ −m2gψ = 0 ,g′ +3gr+gr4ψ′2 +E208r7+m2rψ24− 5r = 0 .We use E0 as a shooting parameter to enforce ψ1 = 0, and find onesolution for each ψ0. From these solutions, we apply the symmetry (7.18)with ∆ = −χ(∞) to restore χ(∞) = 0 and finally use the symmetry (7.17)to scale to the desired temperature or chemical potential.Using this method, we find that solutions exist for scalar mass in therange −25/4 ≤ m2 ≤ 5, which is exactly the range of masses for which thescalar is stable in the asymptotic region but unstable in the near-horizonregion.96 For a given m2 in this range, solutions exist in the region T/µ <γ(m2), where γ(m2) is a dimensionless number depending on m2 (which weevaluate in the next section). The value of γ(m2) is remarkably small forall m2 in the allowed range. For example, with m2 = −6 (not particularlyclose to the limiting value m2 = −5), we have γ ≈ .00006333. It wouldbe interesting to understand better how this small dimensionless numberemerges since the setup has no small parameters. From the bulk point ofview it is presumably related to the warping between IR and UV regions ofthe geometry.97 From the boundary viewpoint, the low critical temperaturemay be explained by the BKL scaling [74, 76, 80] near a quantum criticalpoint98.96Solutions of this form were first found in lower dimensions in [60]. The zero-temperature limit of such solutions were considered in [67].97By considering the alternate quantization mentioned in section 2 and fine-tuning themass so that the dual operator has the smallest possible dimension consistent with uni-tarity in the dual field theory, we can obtain γ as large as 0.0151, so even under the mostfavorable circumstances, the critical T/µ is quite small.98We thank D.T. Son for pointing this out to us.1387.4. Neutral scalar field: color superconductivityFor a given T and µ, we can use (7.6) to evaluate the action for thesolution and compare this with the action for the soliton and/or Reissner-Nordstrom solution with the same T and µ. We find that the action forthe new solutions is always less than the action for the Reissner-Nordstromsolutions, and is also less than the action for the soliton solutions for chem-ical potential in a region µ > µc(T ). Thus, the solutions with scalar fieldrepresent the equilibrium phase in the region T/µ < γ, µ > µc(T ), as shownin figures 7.1- 7.3 above.The transition between the deconfined and superconducting phases issecond order, while the transition between confined and superconductingphases is first order. The place where these phase boundaries meet repre-sents a triple point for the phase diagram where the three phases (confined,deconfined, superconducting) can coexist.7.4.2 Critical temperatureFor fixed m2, the value of ψ(0) in the solutions increases from zero at T/µ =γ, diverging as T/µ → 0. Since ψ is small everywhere near T/µ = γ, thecritical value of T/µ will be the value where the ψ equation, linearizedaround the Reissner-Nordstrom background, has a solution with the correctboundary conditions. Thus, we consider the equationψ′′ + (4r+g′g)ψ′ −m2gψ = 0 , (7.19)where (setting r+ = 1)g(r) = r2 −(1 +3µ28)1r3+3µ28r6,and find the value µ = µc for which the equation admits a solution withboundary conditions ψ(1) = 1 (we are free to choose this), ψ′(1) = m2/g′(1)and the right falloff (ψ1 = 0) at infinity.99The choice r+ = 1 implies that T = (5 − 9µ2/8)/(4pi), so we haveγ = (5 − 9µ2c/8)/(4piµc). The results for γ(m2) are plotted in figure 7.5.For comparison, we also considered the theory defined with the alternate99To obtain a very accurate result, we first find a series solution ψlow near r = 1 withψ(1) = 1 (we are free to choose this) and ψ′(1) = m2/g′(1) and find a series solution ψhighfor large r with the correct fall-off (ψ1 = 0) at infinity. Starting with ψlow and ψ′low atsome r = r1 where the low r series solution is still very accurate, we then numericallyintegrate up to r = r2 where the large r series is very accurate and then find µ for whichψ′num(r2)/ψnum(r2) = ψ′high/ψhigh.1397.4. Neutral scalar field: color superconductivity0.00010.011-6.5-6-5.5-5m21E-121E-101E-080.0000010.0001(T/μ) CFigure 7.5: Critical T/µ vs m2 of neutral scalar (filled circles). Massis above BF bound asymptotically but below BF bound in near-horizonregion of zero-temperature background solution in the range −6.25 ≤ m2 <−5. Unfilled circles represent critical values in the theory with alternatequantization of the scalar field, possible in the range −6.25 ≤ m2 < −5.25.quantization (ψ∞2 = 0) of the bulk scalar field (mentioned in section 2). Aswe see in figure 7.5, the critical temperatures are somewhat larger in thiscase, but still much smaller than 1 relative to µ.7.4.3 Properties of the superconducting phaseIn the superconducting phase, it is interesting to ask how the charge den-sity and free energy behave as a function of chemical potential. Since thesolutions (as for the planar RN-black hole solutions) are trivially related tosolutions where the w direction is non-compact, and since the underlyingtheory has a conformal symmetry, physical quantities in this phase (or inthe RN phase) behave as µnF (T/µ) for some non-trivial function F and apower n.100 At the critical value of T/µ, we have a second order transi-tion from the RN phase to the phase with scalar, so the free energy and itsderivatives, and other physical quantities such as the density, are continuous100If the solutions instead depended on the circle direction in a non-trivial way, we mighthave a general function of RT and Rµ.1407.5. Charged scalar field: flavor superconductivityacross the transition. Thus, the relevant function F in these cases will bethe same for the two phases across the transition. We find that the functionF for either the charge density or the free energy changes very little betweenthe very small value of T/µ where the the transition occurs and the T → 0limit. Thus, to a good approximation, we find that the density and freeenergy behave in the superconducting phase in the same way as for the zerotemperature limit of the RN phase. For R=2/5, we haveρ ≈ 0.320µ4 ,whileG ≈ −.064µ5 .In both cases, the behavior at large µ is governed by the underlying 4+1dimensional conformal field theory.7.5 Charged scalar field: flavor superconductivityIn this section, we generalize our holographic model to the case where thescalar field is charged under the gauge field in the bulk. As we discussed inthe introduction, this implies that the dual field theory includes some low-dimension gauge-invariant operator with charge, so the charge in this caseis more naturally thought of as some isospin-type charge (since the smallestgauge-invariant operators carrying baryon charge have dimensions of orderN).A significant qualitative difference in this case is that a scalar field con-densate acts as a source for the electric field in the bulk, so it is possible tohave solutions with no horizon carrying a finite charge density in the fieldtheory. This gives the possibility of a fourth phase in which the scalar fieldcondenses in the soliton background.To obtain the action for the charged scalar case, we begin with the action(7.2) and make the replacement ∂µψ → ∂µψ − iqAµψ. The results of theprevious section correspond to q = 0.7.5.1 Low-temperature horizon free solutions with scalarAbove some critical value of µ, there exist horizon-free geometries with ascalar field condensate. The solutions may be parameterized by the magni-tude of the scalar at the IR tip of the geometry, and we will find a singlesolution for each such value. To determine these geometries, we need to take1417.5. Charged scalar field: flavor superconductivityinto account back-reaction on the metric. The most general solution withthe desired properties can be described by the ansatzds2 = r2(eA(r)B(r)dw2 + dx2 + dy2 + dz2 − eC(r)dt2) +dr2r2B(r),At = φ(r) ,ψ = ψ(r) , (7.20)where we demand A(∞) = C(∞) = 0 and B(∞) = 1. As for the solitongeometry, we expect that the w circle is contractible in the bulk so thatB(r0) = 0 for some r0. For the geometry to be smooth at this point, theperiodicity of the w direction must be chosen so that2piR =4pie−A(r0)/2r20B′(r0). (7.21)Starting from the action (7.2) with scalar derivatives replaced by covari-ant derivatives, the scalar and Maxwell equations are:ψ′′ +(6r+A′2+B′B+C ′2)ψ′ +1r2B(e−C(qφ)2r2−m2)ψ = 0 , (7.22)φ′′ +(4r+A′2+B′B−C ′2)φ′ −2ψ2q2φr2B= 0 . (7.23)Following [68], we find that the Einstein equations give:A′ =2r2C ′′ + r2C ′2 + 4rC ′ + 4r2ψ′2 − 2e−Cφ′2r(8 + rC ′), (7.24)C ′′ +12C ′2 +(6r+A′2+B′B)C ′ −(φ′2 +2(qφ)2ψ2r2B)e−Cr2= 0 , (7.25)B′(4r−C ′2)+B(ψ′2 −12A′C ′ +e−Cφ′22r2+20r2)+1r2(e−C(qφ)2ψ2r2+m2ψ2 − 20)= 0 . (7.26)These equations have two scaling symmetries,ψ˜(r) = ψ(ar) , φ˜(r) =1aφ(ar) , A˜(r) = A(ar) ,B˜(r) = B(ar) , C˜(r) = C(ar) , (7.27)andC˜ = C + ∆ , φ˜ = e∆2 φ . (7.28)1427.5. Charged scalar field: flavor superconductivity-10.911.11.21.31.4Sμ-2Sq=2Figure 7.6: Action vs chemical potential for soliton with scalar solutions,taking m2 = −6 and q = 2.Numerical evaluation of solutionsTo find solutions, we first use the scaling symmetries to fix r0 = 1 andC(r0) = 0. For each value of ψ(1), we use φ(1) as a shooting parameter,choosing the value so that ψ has the desired behavior for large r. From thesolution obtained in this way, we can use (7.28) with ∆ = −C(∞) to obtainthe desired boundary condition C(∞) = 0 in the rescaled solution. From(7.21), we see that the choice r0 = 1 corresponds to a periodicity for the wdirection equal to2piR =4pie−A(1)/2B′(1). (7.29)which will generally be different for solutions corresponding to different val-ues of ψ(1). In order to obtain solutions corresponding to our chosen valueR = 2/5 (such that the action for the soliton solution is -1) we use the scal-ing (7.27), taking a = B′(1)/5e−A(∞)/2. After all the scalings, we calculatethe chemical potential and action (making use of (7.6)) asµ = φ(∞) , S = [B] 1r5.The action is plotted against chemical potential for various values of qin figures 7.6, 7.7, and 7.8 taking the example of a mass just above the BFbound, m2 = −6.1437.5. Charged scalar field: flavor superconductivity011.11.21.31.41.51.61.71.81.92Sμ-2-1Sq = 1.3Figure 7.7: Action vs chemical potential for soliton with scalar solutions,taking m2 = −6 and q = 1.3.00.50.70.91.11.31.51.71.92.1μ-2-1 Sq=1.2Figure 7.8: Action vs chemical potential for soliton with scalar solutions,taking m2 = −6 and q = 1.2.1447.5. Charged scalar field: flavor superconductivityWe find that for large enough values of q, the chemical potential increasesmonotonically and the action decreases monotonically as we increase ψ(r0).This implies that we have a second order transition to the superconductingphase at a critical value, which can be determined by a linearized analysis(see appendix A) to be µ ≈ 1.0125/q.Below q ≈ 1.35, the chemical potential is no longer monotonic in ψ(r0).We see that for q = 1.3, this results in a second order phase transition atµ ≈ 1.558, followed by a first order phase transition at µ ≈ 1.616 (takingR = 2/5). For smaller q (e.g. q = 1.2 in figure 7.8), we simply have afirst order transition to the superconducting phase at a value of chemicalpotential that is less than the value for the solution with infinitesimal scalarfield. All of these results are completely analogous to the lower-dimensionalresults of [68].7.5.2 Hairy black hole solutionsAt high temperatures, the w circle is no longer contractible, and we assumethat (as for the solutions without scalar field) the solution can be obtained byperiodic identification of a solution with boundary R4,1 instead of R3,1×S1.Thus, we take the ansatzds2 = −g(r)e−χ(r)dt2 +dr2g(r)+ r2(dw2 + dx2 + dy2 + dz2) ,At = φ(r) ,ψ = ψ(r) .The scalar and Maxwell’s equations areψ′′ +(4r−χ′2+g′g)ψ′ +1g(eχq2φ2g−m2)ψ = 0 , (7.30)φ′′ +(4r+χ′2)φ′ −2q2ψ2gφ = 0 , (7.31)while the Einstein equations are satisfied ifχ′ +rψ′22+reχq2φ2ψ22g2= 0 , (7.32)g′ +(3r−χ′2)g +reχφ′28+m2rψ24− 5r = 0 . (7.33)1457.5. Charged scalar field: flavor superconductivityThese have two symmetries:ψ˜(r) = ψ(ar) , φ˜(r) =1aφ(ar) , χ˜(r) = χ(ar) , g˜(r) =1a2g(ar) ,(7.34)andχ˜ = χ+ ∆ , φ˜ = e−∆2 φ . (7.35)As we did for q = 0, we would like to find solutions with a horizon at somer = r+. The electric potential must also vanish at the horizon, and we arelooking for solutions for which the leading falloff ψ1 in (7.3) vanishes for thescalar. Also, multiplying the first equation (7.30) by g and evaluating atr = r+ fixes ψ′(r+) in terms of ψ(r+) and g′(r+). Altogether, our boundaryconditions areg(r+) = 0 , φ(r+) = 0 , χ(∞) = 0 , ψ1 = 0 ,andψ′(r+) =8m2ψ(r+)40r+ − 2m2r2+ψ2(r+)− r+eχ(r+)(φ′(r+))2.The remaining freedom to choose r+ and φ′(r+) leads to a family of solutionswith different T and µ. Explicitly, we haveµ = φ(∞) , T =14pig′(r+)e−χ(r+)/2 .Solutions with the same T/µ are simply related by the scaling symmetry(7.34).Numerical evaluation of solutionsTo find solutions in practice, we can make use of the symmetries (7.34, 7.35)to initially set r+ = 1 and χ(0) = 0 and solve the equations with boundaryconditionsg(1) = 0 , χ(0) = 0 , φ(1) = 0 , φ′(1) = E0 , ψ(1) = ψ0 ,andψ′(1) =8m2ψ040− 2m2ψ20 − E20.We use E0 as a shooting parameter to enforce ψ1 = 0, and find onesolution for each ψ0. From these solutions, we apply the symmetry (7.35)with ∆ = −χ(∞) to restore χ(∞) = 0 and finally use the symmetry (7.34)to scale to the desired temperature or chemical potential.1467.5. Charged scalar field: flavor superconductivity0.0 0.5 1.0 1.5 2.0Μ0.00.10.20.30.40.5TFigure 7.9: Phase diagram for m2 = −6 and q = 2. Clockwise from theorigin, the phases correspond to the AdS soliton (confined), RN black hole,black hole with scalar, and soliton with scalar.7.5.3 Phase diagramsAt a generic point in the phase diagram, we can have up to four solutions(AdS soliton, planar RN black hole, soliton with scalar, black hole withscalar), or more in cases where there is more than one solution of a giventype.To map out the phase diagram, we evaluate the action for the varioussolutions using the methods of section 2. The equilibrium phase correspondsto the solution with lowest action. The phase diagrams for q = 1.3 and q = 2(in the case m2 = −6) are shown in figures 7.9 and 7.10/7.11).For large q, the condensation of the scalar field occurs in a region of thephase diagram where the back-reaction is negligible, so the phase diagrammay be understood here (for µ ∼ 1/q) by treating the gauge field and scalaron a fixed background (the Schwarzschild black hole). The resulting phasediagram is shown in figure (7.12).1477.5. Charged scalar field: flavor superconductivity0.0 0.5 1.0 1.5 2.0Μ0.00.10.20.30.40.5TFigure 7.10: Phase diagram for m2 = −6 and q = 1.3. Clockwise from theorigin, the phases correspond to the AdS soliton (confined), RN black hole,black hole with scalar, and soliton with scalar.1.5 1.6 1.7 1.8 1.9 2.0Μ0.000000.000050.000100.000150.00020TFigure 7.11: Small temperature region of phase diagram for m2 = −6 andq = 1.3. Dashed line represents a first order transition within the solitonwith scalar phase.1487.6. Discussion1 2 3 4Μq0.20.40.60.8TFigure 7.12: Phase diagram for large q, m2 = −6.7.6 DiscussionIn this note, we have investigated the phase structure for a simple class ofholographic systems which we have argued have the minimal set of ingredi-ents to holographically describe the phenomenon of color superconductivity.Even in these simple models, we find a rich phase structure with featuressimilar to the conjectured behavior of QCD at finite temperature and baryonchemical potential. It would be useful to verify the thermodynamic stability(and also the stability towards gravitational perturbations) of the phasesthat we have identified. This could indicate regions of the phase diagramwhere we have not yet identified the true equilibrium phase for the model,for example since our ansatz might be too symmetric.We have calculated some of the basic thermodynamic observables, butit would be interesting to investigate more fully the physical properties ofthe various phases and establish more definitively a connection between thephase we find at large µ and small temperature and the physics of colorsuperconductivity.Apart from the ψψψ†ψ† condensate that we can see directly using theingredients of our model, there are various other features that character-ize a color superconductivity phase [5]. Typically, the breaking of gaugesymmetry is accompanied by some breaking of exact or approximate flavorsymmetries. Thus, the superconducting phase has a low-energy spectrumcharacterized by Goldstone bosons or pseudo-Goldstone bosons associatedwith the broken flavor symmetries, together with massive vector bosons as-1497.6. Discussionsociated with the spontaneously broken gauge symmetry. It would thereforebe interesting to analyze the spectrum of fluctuations in our model to com-pare with these expectations.A caveat related to looking for features associated with the global flavorsymmetries (and their breaking) in our model is that we may not haveincluded enough ingredients in our bottom-up approach for all these featuresto be present. In simple models where the flavor degrees of freedom areassociated with probe branes, there are explicit gauge fields in the bulk dualto the global symmetry current operators. However, in fully back-reactedsolutions (appropriate for studying Nf ∼ Nc), these branes are replacedby a modified geometry with additional fluxes (for an explicit example ofsuch solutions, see [37]). In these solutions (which we are trying to modelin our approach), it is less clear how to identify the global symmetry groupfrom the gravity solution, but presumably it has to do with some detailedproperties of the geometry. Thus, it is possible that the Goldstone modesassociated with broken flavor symmetries correspond to fluctuations in somefields (e.g. form-fields) that we have not included.The color superconducting condensate also breaks the global baryonnumber symmetry, so there should be an associated Goldstone boson re-lated to the phase of the condensate, and associated superfluidity phenom-ena. In other holographic models with superfluidity, the condensate is dualto a charged scalar field in the bulk and the Goldstone mode is related tofluctuations in the phase of this field. However, as we mentioned in theintroduction, the baryon operator has dimension of order N , so we do notexpect a light charged scalar field in the bulk. In a more complete top-downmodel, the baryon operator may be related to some non-perturbative de-grees of freedom (such as D-branes) in the bulk, and it may be necessary tohave a model with these degrees of freedom included in order to directly seethe Goldstone mode from the bulk physics. Related to these observations,it may be interesting to probe our model with D-branes (put in by hand),in order to make the relation to microscopic physics more manifest, and tohelp gain a better understanding of the phenomenological parameters of ourmodel.There are a number of variants on the model that would be interestingto study. First, the breaking of scale-invariance, implemented in our modelby the varying circle direction in the bulk, could be achieved in other ways,replacing gww with a more general scalar field, as in the model of [36]. Inthe setup of that paper, the transition between confined and deconfinedphases was found to exhibit crossover behavior at small chemical potential,a feature expected in the real QCD phase diagram and expected generally1507.6. Discussionfor massive quarks with sufficiently large Nf/Nc. It would be interesting tolook for an even more realistic holographic model by incorporating featuresof the model we have studied here and the model of [36].It would also be interesting to look at the effects of a Chern-Simons termfor the bulk gauge field. In [104] and [111], it was shown that such a term(with sufficiently large coefficient) gives rise to an instability toward inho-mogeneous phases, perhaps associated with the chiral density wave phasebelieved to exist at large density in QCD with Nc  Nf [35, 125]. It isinteresting to investigate the interplay between these inhomogeneous insta-bilities and the superconducting instabilities discussed in the present chap-ter. It would also be interesting to consider more general actions (such asBorn-Infeld) for the gauge field, interaction terms for the scalar field in thebulk, or other couplings between the scalar field and gauge field.Finally, once the technical challenges of writing down fully back-reactedsolutions for top-down models of holographic QCD with Nf Nc have beenovercome, it will be interesting to see whether the basic features we find hereare manifested in the more complete string-theoretic models. If certain fea-tures are found to be universal, these might taken as qualitative predictionsfor the QCD phase diagram, or at least motivate an effort to understandwhether these features are also present in the phase diagram of real-worldQCD.151Chapter 8ConclusionIn this thesis I showed how the AdS/CFT correspondence can not only beused to address deep questions in quantum gravity and the nature of space-time, but also as a powerful tool to construct strongly coupled field theorymodels ranging from relativistic fluids to QCD-like theories.Part I of this thesis was focused on a fundamental question regardingthe holographic principle, i.e.: characterizing the precise map between bulkand boundary degrees of freedom. At first glance, simply by realizing howfundamentally different the physics of these two sides seem, I argued thatsuch a map is highly non trivial, most likely non local and even possibly notwell behaved. However, as discussed in chapters 2, 3, and 4, there are goodreasons to believe that, while still non trivial, such map does follow certaingeneral rules and it is not arbitrarily non local.By direct use of entanglement entropy on the field theory side, and itsholographic realization on the gravity side, I was able to set an upper limiton the size of the bulk region dual to a portion of the boundary. Thisbulk region, I believe, should contain enough information to allow for thereconstruction of every physical observer within the causal development ofthe region of interest on the boundary, as discussed in chapter 2. It isinteresting to point out that, as indicated by the scaling of entropy in space-times with gravity, quantum gravity seems to be highly redundant in itsinformation storing, a fact that is not only supported by the holographicprinciple, but directly present in my discussion on the bulk to boundarymap and partial information retrieval.Moreover, I addressed how a potential counter argument mentioned inchapter 2, arising from high central densities boson stars, was avoided bynoticing that these objects are believed to be unstable and, therefore, phys-ically forbidden — as argued in chapter 3.In addition, in chapter 4 I provided a detailed discussion of Rindlerwedges of AdS space, their mathematical formulation as hyperbolic blackholes, and how entanglement is a fundamental piece for the existence of aholographic space-time. In this chapter it was also observed how entangle-ment between boundary degrees of freedom had a dramatic influence on the152Chapter 8. Conclusionshape and even topology of the gravity dual. Moreover, I argued how themicro states of certain black holes can be understood as isolated patchesof space-time with pathological horizons — as indicated by their divergentenergy momentum tensors — that, nonetheless, when ensembled togethercan lead to a smooth, larger, space-time.In part II of this thesis I shifted focus to some practical uses of theAdS/CFT correspondence. I started by looking at how small perturbationsof the metric field of certain gravity solutions can have a direct interpretationas a relativistic fluid on the field theory side. This result is known as thefluid / gravity correspondence and can be used in a perturbative manner tocompute higher order corrections to the relativistic Navier-Stokes equation,as well as to directly study the dynamics of such metric perturbations solelyfrom the gravity theory point of view.In chapter 5, with the hope of capturing a more general set of solutions,subject only to the most basic physical constraints, I went further and re-laxed the conditions imposed on the perturbations usually considered in thefluid/ gravity correspondence literature. What I observed was how certainfeatures of the bulk metric had a direct and dramatic consequence on thetypes of field theory solutions we obtained on the boundary. Conversely,I showed how demanding that the boundary theory only obeyed the basicconservations laws was not enough to ensure a well behaved bulk dual; these,in turn, are most likely related to out of equilibrium, or unstable boundarysolutions that are out of the scope of the techniques employed.Next, in chapter 6, I discussed how generic holographic field theories canbe constructed from a relatively simple gravity model in a bottom up aswell as a top down approach. Not only I was able to define such theories,but the extension to finite temperature and finite chemical potential —both notoriously hard in conventional field theory — was both natural and(relatively) easy to be dealt with.In this chapter I discussed both numerical and analytical analysis of a va-riety of holographic models and was able to identify their major differencesand similarities. For models based on Dp-Dq brane systems I computedthe exact relation between charge and chemical potential in the field the-ory and found that it reproduced some well known results in the literature.Additionally, I generalized the results to an arbitrary Dp-Dq brane ansatz.For the bottom up approach I opted for a numerical analysis of the den-sity versus chemical potential relation in different dimensions, with differentfield contents and with or without back reaction, the results of which weresummarized in table 6.2.Finally, as an extension of the bottom up approach discussed in chapter153Chapter 8. Conclusion6, in chapter 7 I analyzed a holographic model that showcases a much richerphase diagram, describing, at least qualitatively, the phenomenon of coloursuper conductivity predicted by QCD.To accomplish this I built a bottom up holographic model that includeda neutral scalar field coupled to gravity. This model not only has the wellknown Hawking-Page transition in the bulk — from an AdS soliton to aReissner-Nordstrom black hole — which in the context of holographic fieldtheory is interpreted as a confinement - deconfinement transition, but I alsoobserved a second phase transition, since now not only temperature, butalso chemical potential is a free solution parameter.Moreover, the discussion was extended to charged scalar fields coupled togravity — which I argued could potentially model the phenomena of flavoursuperconductivity — and uncovered an even richer phase diagram. Thisphase new diagram was shown to be highly dependent on the parameters ofthe theory, and changed dramatically when the mass or charge of the scalarfield were changed.In this thesis I showed how the AdS/CFT correspondence can addressdeep ingrained issues of quantum gravity, as well as be used to build realisticmodels of strongly coupled field theories. 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JHEP,07:006, 1998.otsep1000165Appendix AAppendix to Chapter 4A.1 Coordinate transformationsIn this appendix, we show that a conformal transformation between theboundary of some Poincare patch and Minkowski space maps diamond-shaped regions as in Figure 4.4 to complementary Rindler wedges of Minkowskispace. We also argue that there is another conformal transformation thatmaps one of these diamond-shaped regions to hyperbolic space times time.Starting with the cylinder Sd ×R in coordinatesds2 = −dT 2 + dR2 + sin2RdΩ2d−1, (A.1)the change of coordinatestanT ±R2= t± r (A.2)followed by the conformal transformationds2 →14((r − t)2 + 1) ((r + t)2 + 1)ds2 (A.3)takes the region DP = {−pi < T < pi,R < pi−|T |} to Minkowski space withmetricds2 = −dt2 + dr2 + r2dΩ2d−1 . (A.4)The region DP forms the boundary of a Poincare patch in AdS. If pi andpf are any points on the past and future boundaries of DP (the past andfuture tips of a diamond-shaped region as in Figure 4.4), then the forwardand backward lightcones from pi and pf divide DP into four regions, asin Figure 4.4. After the transformation to Minkowski space, the space isstill divided into four regions by lightcones, but since pi and pf map tothe infinite past and infinite future, these lightcones become intersectinglightlike planar hypersurfaces. After a Poincare transformation, these canbe mapped to the surfaces x = t and x = −t that bound Rindler wedgesof Minkowski space, with the region D(pi, pf ) (the region bounded by the166A.1. Coordinate transformationsforward lightcone from pi and the backward lightcone from pf ) mapping toone of the wedges. As an example, the transformation above, without anyadditional Poincare transformation, maps the domain of dependence of theθ < pi/2 hemisphere of the T = 0 sphere to the Rindler wedge x > 0, |t| < x.To map D(pi, pf ) to Hd × R using a conformal transformation, we cancombine a map D(pi, pf ) to a Rindler wedge of Minkowski space as above,with a map back to a region |T | < T0, R < T0−|T | (the causal developmentof a ball in the T = 0 sphere), with a third conformal transformation (givenexplicitly in [29]) to Hd × R. We note in particular [29] that in the mapfrom the Rindler wedge to Hd × R, the Rindler Hamiltonian maps to thegenerator of time translations.167Appendix BAppendix to Chapter 7B.1 Large charge limitIn this appendix, we analyze the case of large q. This is particularly simple,since in this limit, the back-reaction of the scalar and the gauge field on themetric go to zero in the region of the phase diagram where transitions tothe superconducting phases occur. Explicitly, we can show that in the limitq → ∞ with qµ fixed, the gauge field and scalar field decouple from theequations for the metric, but still give rise to a nontrivial phase structure.To investigate this, we need only consider the scalar field and gauge fieldequations on the fixed background spacetimes corresponding to low tem-peratures (the soliton geometry) and high temperatures (the Schwarzschildblack hole).Low TemperatureStarting from the action (7.2) for the scalar field and gauge field on thesoliton background (7.7), we find that the equations of motion are (settingL = 1)φ′′ +(f ′f+4r)φ′ −2q2r2fψ2φ = 0 ,ψ′′ +(f ′f+6r)ψ′ +q2r4fφ2ψ −m2r2fψ = 0 ,where f is defined in (7.8).These equations have two scaling symmetries related to the conformalsymmetry of the boundary field theory and to the absence of back-reactionin our large charge limit. Given a solution (φ(r), ψ(r), r0, q,m), we can checkthat the scaling(φ(r), ψ(r), r0, q,m)→ (βφ(αr), βαψ(αr),r0α,qαβ,m)168B.1. Large charge limitsends solutions to solutions. For our calculations, we will use this to setr0 = q = 1.Multiplying these equations by f and taking the limit r → r0 = 1, wefind that regular solutions must obeyφ′(1) =2ψ2(1)φ(1)5,ψ′(1) =ψ(1)5(m2 − φ2(1)).We have two remaining parameters, ψ(0) and φ(0). One of these can be fixedby demanding that the “non-normalizible” mode of ψ vanishes at infinity,while different values of the remaining parameter correspond to differentvalues of µ.Employing numerics, we find that for a fixed value of m2, there is somecritical value of µ above which solutions with a condensed scalar field exist.In order to determine the critical value µc(m2), we use the fact that thefield values go to zero as we approach the critical µ from above. Thus, atthe critical µ, the equations above linearized around the background solutionφ = µ should admit a solution with the correct boundary conditions. Thelinearized equations decouple from each other, so we need only study the ψequation. This becomesψ′′ +(6r5 − 1r(r5 − 1))ψ′ +r(µ2 −m2r2)r5 − 1ψ = 0 .We can take ψ(1) = 1 without loss of generality, so the boundary conditionfor ψ′ becomesψ′(1) =15(m2 − µ2) .Given m2, we now find µ2 by demanding that the leading asymptotic mode(ψ1) of ψ vanishes. Our results for the critical µ as a function of m2 areshown in figure B.1.High temperatureThe high temperature geometry relevant to the limit of large q with µq fixedis the µ → 0 limit of the Reissner-Nordstrom geometry (7.10), which givesthe planar AdS-Schwarzschild black hole (with one of the spatial directionscompactified). This is the relevant background for T > 1/(2piR).Explicitly, we haveds2 = r2(−dt2f(r) + dx2 + dy2 + dz2 + dw2)+dr2r2f(r),169B.1. Large charge limitwheref(r) = 1−r5+r5.Here, r+ is related to the temperature byr+ =4piT5.The equations of motion in this background areψ′′ +(f ′f+6r)ψ′ +q2r4f2φ2ψ −m2r2fψ = 0 .The equations have the same scaling symmetry as before, so we can set r+ =q = 1 for numerics. Here, the choice r+ = 1 corresponds to T = 1/(2piR),where R is the radius chosen in the previous section by setting r0 = 1. Inthis case, the boundary conditions areφ(1) = 0 , ψ′(1) =m2L2ψ(1)5.To determine the physics at other temperatures, we can fix q and R and usethe scaling to adjust the temperature.For any values of parameters, we have a solutionψ = 0 , φ(r) = µ(1−1r3) .corresponding to the pure Reissner-Nordstrom background in the probelimit.As in the low temperature phase, we find a critical value µc = F (m2)(or, restoring temperature dependence, µc = TTcF (m2)) for each choice ofm2, above which there is another solution with nonzero ψ. This critical µmay again be determined by a linearized analysis, from which we obtain theequationψ′′ +(6r5 − 1r(r5 − 1))ψ′ +(µ2(r3 − 1)2r4(r5 − 1)2−m2r3r5 − 1)ψ = 0 .We can set ψ(1) = 1 without loss of generality, and this requiresψ′(1) =m25.These can be solved numerically to find F (m2), and our results (with thelow temperature results) are plotted in figure B.1.A sample phase diagram, for the case m2 = −6 is shown in figure 7.12.170B.1. Large charge limitFigure B.1: Critical values of µq vs m2 for scalar condensation in largeq limit. The top curve is the critical value for µ in black hole phase (justabove the transition temperature), while the bottom curve is the critical µin low temperature phase.B.1.1 Order of phase transitions in the probe limitTo complete this section, we verify analytically that the action for solutionswith scalar field in the probe limit is always less than the correspondingunperturbed solution. In this limit we neglect the gravity back reaction ofthe gauge fields and scalar. The on-shell action in this approximation isgiven byST d=∫dd+1x√−ggttgrrA′2t2. (B.1)We have used the fact that the scalar action is quadratic and vanishes on-shell once the boundary value of scalar is kept to zero [10]. Writing the actionin this simple form gives us information about the relative free energy of thedifferent phases.The solution for At in the superconducting phase may be written asASt = A0t + δAt , (B.2)where δAt → 0 in the IR region of the bulk and near the boundary. A0t isthe value of At in the normal phase. Then, from eq. (B.1) we getSnewT dV=SoldT dV+ 2∫dr√−ggrrgtt∂rA0t∂r(δAt) +∫√−ggrrgtt(δAt)′22dr .(B.3)171B.2. Critical µ for solutions with infinitesimal charged scalarThe cross term between A0t and δAt vanishes after integrating by parts andthen using the eom of A0t . HenceδS = Snew − Sold = (TdV )∫√−ggrrgtt(δAt)′22dr < 0asgtt < 0. (B.4)Therefore if a phase with non-trivial scalar condensate exists it will alwayshave a lower free energy than the normal phase and the associated transitionwill be of second order.The introduction of gravity may give rise to a positive term in the on-shell action and the nature of phase transition may change.B.2 Critical µ for solutions with infinitesimalcharged scalarTo find the critical µ at which solutions with infinitesimal scalar field exist,we find the value of µ for which the linearized scalar equation about the ap-propriate background admits a solution with the right boundary conditionsat infinity.At low temperatures, this gives (setting r0 = 1)ψ′′ +(g′g +4r)ψ′ + 1g(q2φ2r2 −m2)ψ ,g(r) = r2 − 1r3 , φ = µ ,while for the RN black hole background (setting r+ = 1) we haveψ′′ +(g′g +4r)ψ′ + 1g(q2φ2g −m2)ψ ,g(r) = r2 −(1 + 3µ28)1r3 +3µ28r6 ,φ = µ(1− 1r3).More general values of r0 or r+ can be restored by the scaling symmetry.For m2 = −6, we find a critical value of µ in the low-temperature casegiven by µlowq = 5.089/(2piR). At high temperatures, the critical solutionsexist T/µ when has a critical value as plotted in figure B.2.172B.2. Critical µ for solutions with infinitesimal charged scalar0.010.1100.20.40.60.811.21.41.61.82(T/μ)c0.000010.00010.001(T/μ)cqFigure B.2: Critical T/µ vs charge q for condensation of m2 = −6 scalarfield in Reissner-Nordstrom background.173

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