Resumen:
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.