In one word the core subject of this thesis is holography. What we mean by holography broadly is the mapping of a gravitational theory in D dimensions to a quantum mechanics system or quantum field theory in one less dimension

In chapter 1, we give a basic and self-contained introduction of the various physical notions that one needs to be familiar with in order to read this thesis.

In chapter 2,we revisit the old black hole S-matrix construction and its new partial wave expansion of ’t Hooft. Inspired by old ideas from non-critical string theory and the c = 1 Matrix Quantum Mechanics, we formulate the scattering in terms of a quantum mechanical model –of waves scattering off- inverted harmonic oscillator potentials– that exactly reproduces the unitary black hole S-matrix for all spherical harmonics; each partial wave corresponds to an inverted harmonic oscillator with ground state energy that is shifted relative to the s-wave oscillator. Identifying a connection to two dimensional string theory allows us to show that there is an exponential degeneracy in how a given total initial energy may be distributed among many partial waves of the four dimensional black hole.

In chapter 3, we study Matrix Quantum Mechanics (MQM) on the Euclidean time orbifold S^{1}/Z_{2}. Upon Wick rotation to Lorentzian time and taking the double-scaling limit this theory provides a toy model for a big-bang/big-crunch universe in two dimensional non-critical string theory where the orbifold fixed points become cosmological singularities. We derive the partition function both in the canonical and grand canonical ensemble in two different formulations and demonstrate a matching between them. We pinpoint the contribution of twisted states in both of these formulations either in terms of bi-local operators acting at the end-points of time or of branch-cuts on the complex plane. Finally we discuss some interesting features of the partition function and the possibility of realising it as a *tau*- function of an integrable hierarchy.

In chapter 4, we investigate quantum critical points in a 2+1- dimensional gauge theory at finite chemical potential and magnetic field. The gravity dual is based on four dimensional N = 2 Fayet-Iliopoulos gauged supergravity and the solutions we consider –that are constructed analytically– are extremal, dyonic, asymptoticallyAdS4 black branes with a nontrivial radial profile for the scalar field and extremal, magnetically charged, asymptotically AdS4 acceptable singular solutions with a nontrivial radial profile for the scalar field. We discover a line of second order fixed points at a critical value of the magnetic field between the dyonic black brane and an extremal “thermal gas” solution with a singularity of good-type, according to the acceptability criteria of Gubser. The dual field theory is a strongly coupled nonconformal field theory at finite charge and magnetic field, related to the ABJM theory deformed by a triple trace operator. We find similarities between the behaviour of the VeV of the scalar operator under B and that of the quark condensate in 2 + 1- dimensional Nambu-Jona-Lasinio models.