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 theory on 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.

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 theory on 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.

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 theory on 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. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

Holographic dualities are now an established tool in the study of universal properties of strongly coupled field theories. Yet, theories without translational symmetry are still poorly understood in this context. In this dissertation, we investigate three new approaches to this challenging problem. The first part of the dissertation concerns a class of phenomenological holographic models in which momentum relaxation can be achieved without breaking translational symmetry in the dual geometry. In particular, we focus on an example in which the dual geometry is similar to anti-de Sitter (AdS) Brans-Dicke theory. We study the thermodynamic and transport properties of the model and show that for strong momentum relaxation and low temperatures the model has insulator-like behaviour. In the second part, we go beyond the effective description and consider holographic theories which explicitly break translational symmetry. From the perspective of gravity, these theories translate to geometries that vary explicitly in the boundary space-like coordinates. We refer to these geometries as 'inhomogeneous' and investigate two approaches to study them. The first is motivated by the question: "what happens to a homogeneous geometry when coupled with a field varying randomly in space?". Starting from an AdS geometry at zero or finite temperature, we show that a spatially varying random Maxwell potential drives the dual field theory to a non-trivial infra-red fixed point characterised by an emerging scale invariance. Thermodynamic and transport properties of this disordered ground state are also discussed. The second is motivated by the complementary question: "how does a random geometry affect a probe field?". In the weak disorder limit, we show that disorder induces an additional power-law decay in the dual correlation functions. For certain choices of geometry profile, this contribution becomes dominant in the infra-red, indicating the breaking of perturbation theory and the possible existence of a phase transition induced by disorder. The third and last part of this dissertation switches from the gravity to the field theoretical side of the duality. We discuss the Sachdev-Ye-Kitaev (SYK) model, a disordered many-body model with distinctive black hole-like properties. We provide analytical and numerical evidence that these holographic properties are robust against a natural one-body deformation for a finite range of parameters. Outside this interval, this system undergoes a chaotic-integrable transition.

In the first part of this thesis, we study the duality of Wilson loops and M2-branes in AdS₄/CFT₃. We focus on supersymmetric M-theory solutions on AdS₄xY₇ that have a superconformal dual description on S³ = ?AdS⁴. We will find that the Hamiltonian function h_M for the M-theory circle plays an important role in the duality. We show that an M2-brane wrapping the M-theory circle is supersymmetric precisely at the critical points of h_M, and moreover the value of this function at those points determines the M2-brane actions. Such a configuration determines the holographic dual of a Wilson loop for a Hopf circle in S³. We find agreement in large classes of examples between the Wilson loop and its dual M2-brane and also that the image h_M(Y₇) determines the range of support of the eigenvalues in the dual large N matrix model, with the critical points of h_M mapping to points where the derivative of the eigenvalue density is discontinuous. We will then move away from the three-sphere and construct gravity duals to a broad class of N=2 supersymmetric gauge theories defined on a general class of three-manifold geometries. The gravity backgrounds are based on Euclidean self-dual solutions to four-dimensional gauged supergravity. As well as constructing new examples, we prove in general that for solutions defined on the four-ball the gravitational free energy depends only on the supersymmetric Killing vector. Our result agrees with the large N limit of the free energy of the dual gauge theory, computed using localisation. This constitutes an exact check of the gauge/gravity correspondence for a very broad class of gauge theories defined on a general class of background three-manifold geometries. To further verify that our gravitational backgrounds are indeed dual to field theories on their boundaries, we compute Wilson loops and their M2-brane duals in this general setting. We find that the Wilson loop is given by a simple closed formula which depends on the background geometry only through the supersymmetric Killing vector field. The supergravity dual M2-brane precisely reproduces this large N field theory result. This constitutes a further check of AdS₄/CFT₃ for a very broad class of examples.

Motivated by the AdS/CFT correspondence, new supersymmetric solutions to Type IIB and Type IIA supergravity are presented. These solutions contain $AdS_5$ or $AdS_4$ factors and are generated using T-duality symmetries of supergravity. The technique used to generate these solutions consists of performing a series of non-Abelian and Abelian T-dualities, sometimes with coordinate shifts in-between, to Freund-Rubin type seed backgrounds. An added bonus of the gauge fixing procedure inherent in non-Abelian T-Duality is the freedom to generate backgrounds with extra free parameters, some examples of which are presented. Aspects of the dual field theories of these new solutions are analyzed using holography techniques. The supersymmetry of these new backgrounds is also discussed. In addition to supergravity backgrounds with AdS, the study of generalized Calabi-Yau manifolds in the context of flux compactifications is briefly reviewed. The particular case of the resolved cone over $Y^{p,q}$ and its admission of generalized SU(3) structure solutions is examined. Contrary to geometries with $AdS$ factors, whose field theory duals are conformal field theories, these types of geometries can be phenomenologically interesting to study, as their gauge theory duals are minimally supersymmetric and confining, thus they could someday help aid our understanding of strongly-coupled QCD (Quantum Chromodynamics).

The holographic principle has become an extraordinary tool in theoretical physics, most notably in the form of the Anti-deSitter Conformal Field Theory (AdS/CFT) correspondence, in which classical gravitational degrees of freedom in N-dimensions are related quantum field theory degrees of freedom in N − 1 dimensions in the limit of a large number of fields. Here we present an account of the AdS/CFT correspondence, also known as the gauge/gravity duality, from its origins in the large N 'tHooft expansion, up to Maldacena's proposal that type IIB string theory in the presences of D-branes at low energy is dual to an N = 4, d = 4, U(N) super Yang-Mills on AdS5 × S5 . We begin with an extensive review of (super)string theory including D-branes. We then present the general formulation of the AdS/CFT in the supergravity background of AdS5 × S5 , along with several examples of how it is used in terms of the identification of bulk fields with operators on the bound- ary of a CFT. We move on to discuss two applications of the gauge/gravity duality. The first is the application of the holographic gauge/gravity correspondence to the QCDk-string. The second applies the AdS/CFT formalism to a Kerr black hole solution embedded in 10-dimensional heterotic sting theory. These two applications of the holographic gauge/gravity duality comprise the original work presented here. We follow with summaries and discussions of the background material, the original work, and future investigations.

The AdS/CFT correspondence and its extensions to more general gauge/gravity dualities have provided a powerful framework for the study of strongly coupled gauge theories. This thesis explores properties of a large class of thermal strongly coupled gauge theories using the gravity dual. In order to bring the holographic framework closer to Quantum Chromodynamics (QCD), we study theories with matter in the fundamental representation. In particular, we focus on the holographic dual of SU(Nc) supersymmetric Yang-Mills theory coupled to Nf<<Nc flavours of fundamental matter at finite temperature, which is realised as Nf Dq-brane probes in the near horizon (black hole) geometry of Nc black Dp-branes. We explore many aspects of these Dp/Dq brane systems, often focussing on the D3/D7 brane system which is dual to a four dimensional gauge theory. We study the thermodynamics of the Dq-brane probes in the black hole geometry. At low temperature, the branes sit outside the black hole and the meson spectrum is discrete and possesses a mass gap. As the temperature increases, the branes approach a critical solution. Eventually, they fall into the horizon and a phase transition occurs. At large Nc and large 't Hooft coupling, we show that this phase transition is always first order. We calculate the free energy, entropy and energy densities, as well as the speed of sound in these systems. We compute the meson spectrum for brane embeddings outside the horizon and find that tachyonic modes appear where this phase is expected to be unstable from thermodynamic considerations. We study the system at non-zero baryon density nb and find that there is a line of phase transitions for small nb, terminating at a critical point with finite nb. We demonstrate that, to leading order in Nf/Nc, the viscosity to entropy density ratio in these theories saturates the conjectured universal bound. Finally, we compute spectral functions and diffusion constants for fundamental matter in the high temperature phase of the D3/D7 theory.

The AdS/CFT correspondence and its extensions to more general gauge/gravity dualities have provided a powerful framework for the study of strongly coupled gauge theories. This thesis explores properties of a large class of thermal strongly coupled gauge theories using the gravity dual. In order to bring the holographic framework closer to Quantum Chromodynamics (QCD), we study theories with matter in the fundamental representation. In particular, we focus on the holographic dual of SU(Nc) supersymmetric Yang-Mills theory coupled to Nf<<Nc flavours of fundamental matter at finite temperature, which is realised as Nf Dq-brane probes in the near horizon (black hole) geometry of Nc black Dp-branes. We explore many aspects of these Dp/Dq brane systems, often focussing on the D3/D7 brane system which is dual to a four dimensional gauge theory. We study the thermodynamics of the Dq-brane probes in the black hole geometry. At low temperature, the branes sit outside the black hole and the meson spectrum is discrete and possesses a mass gap. As the temperature increases, the branes approach a critical solution. Eventually, they fall into the horizon and a phase transition occurs. At large Nc and large 't Hooft coupling, we show that this phase transition is always first order. We calculate the free energy, entropy and energy densities, as well as the speed of sound in these systems. We compute the meson spectrum for brane embeddings outside the horizon and find that tachyonic modes appear where this phase is expected to be unstable from thermodynamic considerations. We study the system at non-zero baryon density nb and find that there is a line of phase transitions for small nb, terminating at a critical point with finite nb. We demonstrate that, to leading order in Nf/Nc, the viscosity to entropy density ratio in these theories saturates the conjectured universal bound. Finally, we compute spectral functions and diffusion constants for fundamental matter in the high temperature phase of the D3/D7 theory.

This dissertation covers two di erent but related topics: the construction of new black hole solutions and the study of the microscopic origin of black hole entropy. In the solution part, two di erent sets of new solutions are found. The rst concerns a Plebanski-Demianski type solution in the ve-dimensional pure Einstein gravity, and the second concerns a three-charge (two of which equal) two-rotation solution to the ve-dimensional maximal supergravity. Obtaining new and interesting black hole solutions is an important and challenging task in studying general relativity and its extensions. During the past decade, the solutions become even more important because they might nd applications in the study of the gauge/gravity duality, which is currently in the central stage of the quantum gravity research. The Kerr/CFT correspondence is a recently propose example of the gauge/gravity duality. In the entropy part, we explicitly show that the Kerr/CFT correspondence can be applied to all known extremal stationary and axisymmetric black holes. We improve over previous works in showing that this can be done in a general fashion, rather than testing di erent solutions case by case. This e ort makes it obvious that the common structure of the near horizon metric for all known extremal stationary and axisymmetric black holes is playing a key role in the success of the Kerr/CFT correspondence. The discussion is made possible by the identi cation of two general ans atze that cover all such known solutions.