Relativistic and non-relativistic holography

Holography is one of the most interesting developments in String theory in the past few years. In this thesis, we discuss various aspects of the correspondence at the phenomenological and conceptual levels. At the phenomenological level, holographic duals of strong coupling QCD-like boundary theories is constructed using intersecting D-branes configurations Dq-Dp-Dp, in which the Dp???s are probe branes under Dq???s background geometry. Confinement, chiral symmetry phase transitions and their relation are carefully studied for various values of q, p and other parameters of the system. In terms of discrete symmetry, introducing torsion by adding the Nieh-Yan term (with space-time dependent coefficient) into 3+1 Einstein-Hilbert gravity is postulated to correspond to parity symmetry breaking in the boundary. A solitonic solution, which we call the torsion vortex as it resembles in many ways the Abrikosov vortex, is found in the bulk and believed to be dual to the parity breaking vacuum of the boundary theory. At the conceptual level, we deliberately tackle the boundary condition, variational principle and renormalization problems in non-relativistic holography. The systematic boundary condition formalism of Skenderis and Van Rees is extended to scalar field in Schr??dinger space-time. In the same background, variational principle and renormalization procedures are constructed for fermions and vector fields, paying careful attention to the representation theory of the Schr??dinger group. In any cases, 2-point functions for the dual operators are computed for various values of the parameter spaces