Applications and Fundamentals of Holography

In this thesis we explore the fascinating topic of Holography for various systems. The intuitive idea of Holography is that one has at hand two dual descriptions of the physical reality at different number of dimensions much like in the usual holograms, where one projects a three dimensional object onto a two dimensional screen. In the Prologue we provide a self-contained introduction to the basic physical concepts and techniques behind the holographic duality: Large N, Gauge theories and String theory.The first part of the thesis comprises of chapters 2 and 3, where we describe in detail how holography can be applied to realistic condensed matter systems and subsequently analyze in detail the electromagnetic properties of strongly interacting Weyl semimetals via the use of a dual Holographic gravitational model. In particular we study the effect of interactions in the electrical conductivity and we discover some novel effects such as an anomalous magnetic moment stemming from these strong interactions. In the second part of this thesis, chapters 4 and 5, we study the best understood example of Holography - that between Matrix Models and non-critical String Theory - where analytical calculations are possible on both sides of the duality. Our physical motivation is to study Cosmology in two dimensions, through the use of an S1/Z2 orbifold of Euclidean time with a subsequent analytic continuation. In the Matrix Quantum mechanics model the extra spacelike dimension is provided through the matrix eigenvalues and we find that it is possible to describe the initial and final wavefunctions of this toy-Cosmology that capture the presence of a closed string condensate at the beginning and end of time (Big-Bang – Big Crunch). In the third part of the thesis, we revisit Gerard 't Hooft's approach of describing a unitary S-matrix for a black hole that takes into account the backreaction of ingoing particles to the outgoing ones, in the light of 't Hooft's partial wave expansion in spherical harmonics. We use a model of waves scattering in an inverted oscillator potential– that exactly reproduces the unitary black hole S-matrix for all spherical harmonics; each partial wave having a 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 bedistributed among the many partial waves of the four dimensional black hole. This then opens the possibility of a microscopic understanding of the horizon dynamics. In the Epilogue, we conclude with some possible future avenues related to the construction of such a microscopic model