Geometrization of integrable systems with applications to holography.

Abstract

The holographic principle originates from the observation that black hole entropy is proportional to the horizon area and not, as expected, to the volume. This principle has found a concrete manifestation in the Anti-de Sitter/Conformal Field Theory (AdSD/CFTD-1) correspondence. In lower dimensions, there is also a deep link between CFT2 and the Korteweg-de Vries (KdV) hierarchy of integrable systems. Therefore, it is natural to think that there is a connection between the asymptotic structure of gravity in 3D and 2D integrable systems. Indeed, this is precisely what the “geometrization of integrable systems” performs. In a nutshell, this method consists on identifying the Lagrange multipliers of a set of boundary conditions for gravity with the polynomials that span a hierarchy of integrable systems at the boundary. In this thesis we extend the discussion to include thermal stability and phase transitions. We also construct a hierarchy of integrable systems in 2D whose Poisson structure corresponds to the Bondi-Metzner-Sachs algebra in three dimensions (BMS3). Then, we extend the geometrization of integrable systems method to describe our new hierarchy in terms of the Riemannian geometry of three dimensional locally flat spacetimes. Remarkably, we make use of this sort of flat holography to understand the entropy of the cosmological spacetime in 3D as the microscopic counting of states of a dual field theory with consistent but non-standard modular and scaling properties.