Mathematics of the Holographic Principle

The holographic principle, which states that quantum gravity in a given spacetime region admits an equivalent description in terms of a quantum system without gravity on its boundary, is a very promising candidate to lay the foundations of our understanding of quantum gravity. However, a precise general formulation of this principle, as well as its domain of applicability, are yet to be understood. This thesis explores the foundations of holography from a mathematical point of view. In particular, the theory of von Neumann algebras is exploited to understand features of the emergence of spacetime in holography, leveraging tools from quantum error correction in infinite dimensions as well as results on harmonic analysis. Some connections between hyperbolic geometry and recent developments in holography are also elucidated, an approach to the factorization of holographic theories based on Hopf algebras is developed, and a puzzle regarding the description of a closed universe within the context of the AdS/CFT correspondence is put forward.