Investigation on the holographic principle

The holographic principle asserts that any given codimension two spacelike surface limits the information content of adjacent regions. We first review various entropy bounds which lead to the formulation of this conjecture, putting great emphasis on the UV-IR connection. We propose to use noncommutative field theory as a toy model to study the holographic mapping mechanism. In particular, we investigate how the fundamental dipole structure emerges in noncommutative gauge theories by using matrix formulation. The momentum dependent growing behavior of the dipoles can provide a simple way to map the bulk degrees of freedom onto the boundary. In the context of the AdS/CFT correspondence, which is the best known example of a holographic theory, we study the thermodynamics of N = 4 supersymmetric Yang-Mills theory at two-loop level and compare the result to the supergravity calculation. This provides an excellent example to illustrate the idea of strong/weak duality. Questions about a possible large N phase transition still remain unsolved.