Twisted times, the Schwarzian and its deformations in DSSYK

The IR dynamics of SYK is that of the Schwarzian theory, the effective theory of broken reparametrization invariance. In the double scaling limit, SYK is completely solvable by chord diagrams, whose generating functional is a bilocal Liouville theory. At low temperatures a set of modes in this description becomes soft. We interpret them as reparametrization of some twisted time coordinates, and show explicitly that they lead to the nonlinear Schwarzian theory. We further consider deformations of the theory in the double scaling limit, giving rise to diagrams with multiple species of chords, and show that the generating functional is now a Liouville theory with multiple fields. These deformations can be tracked to the IR and we discuss how they affect the Schwarzian.