A quantum observable which received renewed attention recently is entanglement entropy. It's application ranges over several fields in physics, from condensed matter physics to general relativity. In this dissertation we study entanglement entropy for quantum field theories in the presence of defects and singularities.

We study entanglement entropy using the framework of AdS/CFT correspondence. We focus on entangling surfaces across ball-shaped regions for systems outside their ground state. Quantum field theories in the presence of defects are considered first. These are the six-dimensional $(2,0)$ theory in the presence of Wilson surfaces and the four-dimensional $\cN = 4$ super-Yang-Mills theory in the presence of surface defects of the disordered type. Their holographic entanglement entropy is calculated applying the Ryu-Takayanagi prescripstion on their holographic duals, which are eleven-dimensional supergravity (M-theory) solutions for the former and ten-dimensional type IIB supergravity solutions for the latter. Other holographic observables are computed as well: the holographic stress tensor and the expectation value of the defect (operator). For the disordered defects, an alternative expression for the additional entanglement entropy due to the defect (in terms of expectation values) is derived, adapting the method of Lewkowycz and Maldacena for Wilson loops. The two entanglement entropies agree up to an additional term, the origin of which may be attributed to the conformal anomaly of even dimensional defects as we discuss.

The holographic entanglement and free energy is computed for five-dimensional super conformal field theories, starting from their holographic supergravity duals. Although the supergravity solutions possess singularities, these do not obstruct our calculations. The expected relation between the two observables is verified. This supports the supergravity solutions as holographic duals and gives the first quantitative results for five-dimensional superconformal field theories.

We study three aspects of gauge-gravity duality. First, we explore holographic models of

conformal field theories with boundary by way of holographic renormalization group flows.

Second, we propose an extension and application of the covariant holographic entangelement

entropy proposal to warped anti-de-Sitter spacetimes. Third, we exhibit the existence of

higher-spin black holes with Lifshitz asymptotics in the Chern-Simons formulation of higher

spin gravity.

In this dissertation, we discuss half-BPS solutions of gauged supergravity that are holographic realizations of conformal line defects and interfaces. These solutions are constructed by taking a suitable ansatz for the geometry, consisting of a warped product of an AdS spacetime and a sphere, if necessary, over a line, and solving the supersymmetry variations. Quantities such as one-point functions in the presence of the defect and the entanglement entropy are calculated holographically.

In chapter 1, we review the AdS/CFT correspondence and holography. In chapter 2, we relate two different formulations of AdS6 solutions in type IIB supergravity. In chapter 3, we construct solutions in six-dimensional F(4) gauged supergravity that are dual to line defects. In chapter 4, we construct solutions in four-dimensional N = 2 gauged supergravity that are dual to line defects, obtained by a double-analytic continuation of BPS black hole solutions. In chapter 5, we construct Janus solutions in three-dimensional N = 8 gauged supergravity that are dual to interface CFTs.

This dissertation summarizes my research in the Lifshitz higher spin Chern-Simons theory and its relation to the integrable system KdV hierarchy as a Ph.D. candidate at UCLA. In Chapter 1, I briefly review the higher spin gravity theory and introduce the Chern-Simons theory as a realization of the Vasiliev theory in three dimensional spacetime. In Chapter 2, I review the KdV hierarchies. In Chapter 3, I discuss how to construct a solution to the Chern-Simons theory which yields a spacetime that exhibits Lifshitz scaling, I also calculate the boundary charge algebra and show the asymptotic Lifshitz symmetry is realized in terms of it. In Chapter 4, I reveal the relation between the Lifshitz Chern-Simons theory and the KdV hierarchies (in the non-supersymmetric case), a proof of the general correspondence is also given using the Drinfeld-Sokolov formalism. In Chapter 5, I work out the supersymmetric extension of this correspondence in a particular case, with the boundary charge algebra of the supersymmetric Chern-Simons theory and the second Hamiltonian structure of the super KdV identified. In Chapter 6, I discuss on the results of my study and possible directions of future research.

In this dissertation, we will discuss two different holographic constructions. First, we construct a Penrose limit for the type IIB AdS6 � S2 � Σ solutions of [1, 2, 3]. These solutions are dual to 5d SCFT which can often be described in terms of long quiver gauge theories. The resulting plane wave spacetime takes a universal form and the worldsheet action of the Green-Schwarz string is quadratic in the light cone gauge allowing us to obtain the spectrum of string excitation.

Next, we present a solution of d = 7 N = 4 gauged supergravity which is dual to co- dimension 2 defect living in a six-dimensional SCFT. Regularity conditions are satisfied by a one parameter family of solutions. We then uplift these to eleven dimensions where they can be understood as Lin-Lunin-Maldacena solutions. Using their electrostatic formulation, we find an infinite family of regular solutions describing holographic defects. We compute holographic observables in both the seven-dimensional and eleven-dimensional theories.

In this dissertation, we explore 1/2-BPS conformal defects that are holographically realized as a warped product of anti-de Sitter spacetime and a circle in gauged supergravity. These solutions can be obtained as the double analytic continuation of BPS black holes with hyperbolic horizons. Observables including the expectation value of the defect and one-point functions of fields in the presence of the defect are calculated.

In Chapter 1, we present a brief review of the AdS/CFT correspondence and its supergravity approximation together with an introduction to conformal defects. In Chapter 2, we construct a singular spacetime in pure D = 4, N = 2 gauged supergravity dual to a 1/2-BPS conformal line defect. In Chapter 3, we show that the coupling of vector multiplets to the previous solution is capable of removing the singularity and present several examples. In Chapter 4, we construct solutions in D = 5, N = 4 gauged supergravity dual to 1/2-BPS surface operators in N = 2 superconformal field theories.