Schwarzian theories and the Sachdev-Ye-Kitaev model

In this thesis we analyze an interesting recurrent pattern of symmetry-induced emergence of the Schwarzian derivative in modern Theoretical Physics. In particular we study the IR modes of the SYK model, analyzing them in a functional integral approach recently proposed by Belokurov and Schavgulidze, in which the Schwarzian derivative plays a central role. Then we move to a novel formulation of the Quantum Hamilton-Jacobi equation, proposed by by M. Matone and A.E. Faraggi. Starting from geometrical assumptions it retrieves important features of Quantum Mechanics, such as energy quantization. The picture we give unveils an intertwining between functional integration, quantum mechanics and geometry, based upon the Schwarzian derivative.