SYK collective field theory as complex Liouville gravity

Abstract

We establish a precise relationship between the $GΣ$ collective field theory of the double scaled SYK model and the worldsheet theory of the complex Liouville string a.k.a. sine dilaton gravity. The relationship is similar to the lightcone gauge in critical string theory, and to what transpires when we gravitationally dress to an observer in gravity: one of the Liouville fields plays the role of a dynamical clock with respect to which the second Liouville field evolves. This other Liouville field is identified with the collective field of SYK, which thus acquires a direct gravity interpretation. The relevant 2D worldsheet geometry is that of a disk with specific crosscap and FZZT boundary conditions, as deduced from the $GΣ$ formulation. We compute the CLS amplitude on this geometry and find that this coincides with the DSSYK partition function. We indicate how our results can be lifted to 3D gravity, previewing upcoming work. An outflow of our results is that physical operators of DSSYK are mapped to holonomy operators (Verlinde lines) of complex Liouville theory on the crosscap geometry, which in turn have a 3D representation in terms of line operators in 3D de Sitter gravity. We show that the partition function of SYK can be represented as the expectation value of a circular gravitational Wilson line on $\mathbb{RP}^3$ (a.k.a. elliptic 3D de Sitter space).