SYK model and the Schwarzian theory

In this thesis we analized in the dail the structural properties of Sachdev-Ye-Kitaev model, in comparison to the supposed holographic partner: the nearly AdS_2 Jackiw-Teitelboim dilaton gravity black hole. The mathematical fact relating the theories is the emergence of a Schwarzian derivative-shaped action in the conformal IR limit of SYK model: the same dynamics appears holographically in the boundary of the JT gravity. The Schwarzian derivative leads us to suggest a relation between these theories and uniformization theory of Riemann surfaces.