Mark Van Raamsdonk

The AdS/CFT correspondence is a far-reaching equivalence between theories of quantum gravity in spacetimes with negative cosmological constant, such as anti-de Sitter (AdS) space, and lower-dimensional, non-gravitational quantum systems, such as conformal field theories (CFTs). In this thesis, we will use a version of AdS/CFT applicable to boundary conformal field theories (BCFTs) to investigate the physics of supersymmetric gauge theories, and to develop holographic models for cosmology and black hole physics. We make frequent use of an ansatz for holographic BCFT wherein AdS spacetime ends on a surface called an end-of-the-world (ETW) brane, and of the Ryu-Takayanagi (RT) formula for holographic entanglement entropy. We first study the N = 4 supersymmetric Yang-Mills (SYM) theory on a half-space, with boundary conditions preserving scale invariance and half of the original supersymmetry. We calculate a conjectured renormalization group (RG) monotone called boundary F for the most general such boundary conditions using the RT formula. In some cases, we perform an exact calculation using supersymmetric localization, and find exact agreement for the leading large N term as a function of the 't Hooft coupling. Next, we introduce a toy model for cosmological physics in the framework of AdS/CFT, wherein a 4D cosmology resides on an ETW brane propagating behind the horizon in a black hole microstate. We study the time-dependent physics of the behind-the-horizon region in such microstates, finding that it can often be probed by the time-dependence of entanglement entropy for sufficiently large CFT subsystems. We investigate the plausibility of obtaining localized 4D gravity on the ETW brane in both effective and microscopic versions of this model. Last, we consider a doubly-holographic model of a radiating black hole, and apply the RT formula to analyze the time-dependence of the fine-grained entropy of its radiation. We obtain an analogue of the Page curve consistent with unitarity due to a phase transition between RT surfaces, after which the radiation system encodes part of the black hole interior.

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The AdS/CFT dictionary connects bulk gravitational physics in an asymptotically anti-de Sitter (AdS) background to the quantum dynamics of a conformally invariant field theory (CFT), defined on the asymptotic boundary. In this thesis, we explore, extend and apply this dictionary for a boundary CFT (BCFT), with a particular focus on the physics of black holes. In a BCFT, a physical boundary is added to the CFT. The simplest dual geometry is an asymptotically AdS spacetime cut off by a physical surface called an end-of-the-world (ETW) brane, homologous to the BCFT boundary. For certain highly symmetric configurations of the BCFT called boundary states, symmetry constrains these ETW branes to a one-parameter family labelled by extrinsic curvature. By placing the CFT boundary in Euclidean time, we construct a one-parameter family of black hole microstates in arbitrary dimension. In two dimensions, we analytically calculate minimal surfaces in the bulk geometry and show that, for some parameter regimes, they pierce the black hole horizon and become disconnected. According to the Hubeny-Rangamani-Ryu-Takayanagi (HRT) formula, these surfaces compute entanglement entropy in the field theory. We find a set of conditions which ensure that the microscopic entanglement entropy, arising from a correlator of twist operators, agrees with the bulk result. In particular, the BCFT reproduces the phase transition between connected and disconnected minimal surfaces. This twist correlator can be immediately evaluated on any conformally related background, giving access to entanglement entropy in a number of other contexts of physical interest. By analytically continuing the thermofield state of a half-line, we arrive at a simple toy model of an evaporating black hole, where the phase transition in minimal surface corresponds to information escaping from the interior. For the BCFT on an interval, this transition can be recast in terms of the performability of a certain set of quantum tasks. We use related techniques to constrain the operator content of BCFTs dual to spacetimes with a sharp brane in arbitrary dimension. We discover they are finely tuned, with a fragile causal structure which becomes “fuzzy” under small changes to the spectrum.

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The AdS/CFT correspondence relates quantum gravity in asymptotically AdS spacetimes to conformal field theories living on the boundary of that spacetime. Here, we initiate a new perspective on the AdS/CFT correspondence. In particular we study relativistic quantum tasks, which are quantum computations with inputs and outputs occurring at specified spacetime locations. We note that the AdS/CFT correspondence implies the same tasks are possible in quantum gravity in AdS spacetimes as are possible in the dual CFT. Using this, we find a relationship between the existence of overlaps in certain light cones in the bulk spacetime and entanglement in the boundary CFT. This complements the usual perspective on geometry and entanglement in AdS/CFT, which relates minimal surfaces in AdS to entanglement in the CFT. Further, we point out various instances where this bulk/boundary tasks relationship implies novel statements about tasks.

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In this thesis we discuss applications of quantum information theoretic concepts toquantum gravity and the low-energy regime of quantum field theories.The first part of this thesis is concerned with how quantum information spreadsin four-dimensional scattering experiments for theories coupled to quantum electrodynamicsor perturbative quantum gravity. In these cases, every scattering processis accompanied by the emission of an infinite number of soft photons or gravitons,which cause infrared divergences in the calculation of scattering probabilities.There are two methods to deal with IR divergences: the inclusive and dressedformalisms. We demonstrate that in the late-time limit, independent of the method,the hard outgoing particles are entangled with soft particles in such a way that thereduced density matrix of the hard particles is essentially completely decohered.Furthermore, we show that the inclusive formalism is ill-suited to describe scatteringof wavepackets, requiring the use of the dressed formalism. We construct theHilbert space for QED in the dressed formalism as a representation of the canonicalcommutation relations of the photon creation/annihilation algebra, and argue that itsplits into superselection sectors which correspond to eigenspaces of the generatorsof large gauge transformations.In the second part of this thesis, we turn to applications of quantum informationtheoretic concepts in the AdS/CFT correspondence. In pure AdS, we find anexplicit formula for the Ryu-Takayanagi (RT) surface for special subregions in thedual conformal field theory, whose entangling surface lie on a light cone. Theexplicit form of the RT surface is used to give a holographic proof of Markovicityof the CFT vacuum on a light cone. Relative entropy of a state on such specialsubregions is dual to a novel measure of energy associated with a timelike vector flow between the causal and entanglement wedge. Positivity and monotonicity ofrelative entropy imply positivity and monotonicity of this energy, which yields aconsistency conditions for solutions to quantum gravity.

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The gauge / gravity duality, or holographic correspondence, is a theoretical tool that allows the description of strongly coupled field theory through a dual classical gravity theory. In this thesis, we advance the use of numerical methods in applications of the holographic correspondence to the study of strongly coupled field theories in three situations.Firstly, we study the relationship between chemical potential and charge density across myriad examples of Lorentz invariant 3+1 dimensional holographic field theories with the minimal structure of a conserved charge. Solving for the classical gravitational configurations dual to the field theories and extracting the charge density and chemical potential, we enumerate the relationships that can exist in a wide range of holographic theories.Secondly, we study the spontaneous formation of inhomogeneous (striped) order, a phenomenon that has been observed in the cuprates, in a 2+1 dimensional strongly coupled field theory. By numerically solving the equations of motion using finite difference techniques, we construct the full nonlinear striped black brane solutions that provide the gravity dual to this field theory. We evaluate the thermodynamics and show that the system undergoes a second order phase transition to the striped phase as the temperature is lowered.Finally, we apply the holographic correspondence to study particular aspects of quantum chromodynamics (QCD). First, we develop a phenomenological holographic model to describe the colour superconductivity phase of QCD, which is believed to exist at large quark density. We construct the phase diagram for our model, which includes confined, deconfined, and superconducting phases. In a separate project, we revisit the construction of the baryon in the Sakai-Sugimoto model of holographic QCD. In this model, gauge field configurations on the probe D8 flavour branes with non-trivial topological charge (instantons) correspond to baryons in the dual field theory. In order to extend previous studies, we relax an assumption of spherical symmetry and, utilizing pseudospectral methods, numerically construct the deformed instanton in the bulk. Compared to previous studies, we find significantly more realistic values for the mass and size of the baryon.

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In the first part of this thesis we explore the entanglement structure of relativistic field theories in momentum space. We discuss a Wilsonian path integral formulation and a perturbative approach. Using perturbation theory we obtain results for specific quantum field theories. These are understood through scaling and decoupling properties of field theories. Convergence of the perturbation theory taking loop diagrams into account is also discussed. We then discuss the entanglement structure in systems where Lorentz invariance is broken by a Fermi surface. The Fermi surface helps the convergence of perturbation theory and entanglement of modes near the Fermi surface is shown to be amplified, even in the presence of a large momentum cutoff. In the second part of this thesis we explore the connection between entanglement and gravity in the context of the AdS/CFT correspondence. We show that there are certain thermodynamic-like relations common to all conformal field theories, which when mapped via the AdS/CFT correspondence to the bulk are tantamount to Einstein's equations, to lowest order in the metric.

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The holographic principle connects theories with gravity to lower dimensional theories without gravity. Notably, the AdS/CFT correspondence — the first concrete realization of the holographic principle — provides a one to one map between string theory in Anti de Sitter space, and a strongly coupled, large N, SU(N) super Yang-Mills gauge theory in one less dimension.In this thesis, within the context of holographic field theories, I improve on the current understanding of the map between gravity (bulk) and gauge theory (boundary) degrees of freedom. Furthermore, I explore some of the applications of the AdS/CFT correspondence to the study of strongly coupled field theories.I study the map between bulk and boundary degrees of freedom mainly by trying to determine what is the gravity dual of a subset of the boundary field theory. In the process of doing so I show how extremal surfaces, entanglement entropy, hyperbolic black holes, and boson stars are fundamental tools in this quest.Next, I explore a few examples of direct applications of the correspondence as a model building device. I discuss how AdS/CFT can be used to construct quasi realistic strongly coupled physical systems ranging from relativistic fluids to plasmas and high temperature superconductors. Finally, I compare some of the results obtained in this thesis with known standard field theory results.

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While varied applications of gauge/gravity duality have arisen in literature from studies of condensed matter systems including superconductivity to studies of quenched Quantum Chromodynamics (QCD), this thesis focuses on applications of the duality to holographic QCD-like field theories and to inflationary model that uses a QCD-like field theory.In particular the first half of the thesis examines a holographic QCD-like field theory with scalar quarks closely related to the Sakai-Sugimoto model of holographic QCD. The behaviour of baryons and mesons in the model is examined to find a continuous mass spectrum for the mesons, and a baryon that can identified with a topological charge. It then slightly modifies the theory to further study the behaviour of holographic field theories.The second half of the thesis presents a novel model for early Universe inflation, using an SU(N) gauge field theory as the inflaton. The inflation model is studied at both weak coupling and strong coupling using the gauge/gravity duality. The robustness of model’s predictions to exciting multiple inflationary fields beyond the single field of its original proposal, and its possible role in breaking the supersymmetry of the Minimal Supersymmetric Standard Model is also explored.

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In this thesis we summarize three of our work using the gauge/gravity dualities to build effective theories for strongly coupled phenomena. First, we study the thermodynamics of large N pure 2+1 dimensional Yang-Mills theory on a small spatial S². By studying the effective action for the Polyakov loop order parameter, we show analytically that the theory has a second order deconfinement transition. Our results together with extrapolation of lattice QCD results imply a critical radius in the phase diagram where the deconfinement transition switches from second order to first order. We show that the point at the critical radius and temperature can be either a tricritical point with universal behavior or a triple point separating three distinct phases.Second, we study a model of holographic QCD at zero temperature and finite chemical potential. We find that as the baryon chemical potential is increased, the system transitions to a nuclear matter phase characterized by a condensate of instantons on the probe D-branes in the string theory dual. The electrostatic interactions between the instantons cause the condensate to expand towards the UV with increasing chemical potential, giving a holographic version of the expansion of the Fermi surface. We also find possible explanation of the ``chiral density wave'' instability in large N QCD. We argue that the model can be used to make semi-quantitative predictions of the binding energy per nucleon for nuclear matter in QCD.Third, we consider an Abelian Higgs model placed in an AdS black hole background. Such model has been shown to exhibit superconductor like transitions. In the superconducting phase the system shows infinite DC conductivity. This suggests the possibility of turning on a time independent supercurrent. In this paper we study such supercurrent solutions and the associated phase diagram. We find a critical point in the phase diagram where the second order superconducting transition becomes first order. Supercurrent solutions are well studied in condensed matter systems. We find some qualitative agreement with known results.

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Gauge/gravity duality is an important tool for learning about strongly coupled gauge theories. This thesis explores a set of examples of this duality in which the field theories have SU(2|4) supersymmetry and discrete sets of vacuum solutions. Specifically, we use the duality to propose Lagrangian definitions of type IIA Little String Theory on S⁵ as double-scaling limits of the Plane-Wave Matrix Model, maximally supersymmetric Yang-Mills theoryon R x S² and N=4 supersymmetric Yang-Mills theory on R×S³/Zk. We find the supergravity solutions dual to generic vacua of the Plane-Wave Matrix Model and maximally supersymmetric Yang-Mills theory on R×S².We use the supergravity duals to calculate new instanton amplitudes for the Plane-Wave Matrix Model at strong coupling. Finally, we study a natural coarse-graining of the vacua, and find that the associated geometries are singular. We define an entropy functional that vanishes for regular geometries, is non-zero for singular geometries, and is maximized by the thermal state.

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