Towards a full solution of the large N double-scaled SYK model

Abstract

We compute the exact, all energy scale, 4-point function of the large N doublescaled SYK model, by using only combinatorial tools and relating the correlation functions to sums over chord diagrams. We apply the result to obtain corrections to the maximal Lyapunov exponent at low temperatures. We present the rules for the non-perturbative diagrammatic description of correlation functions of the entire model. The latter indicate that the model can be solved by a reduction of a quantum deformation of SL(2), that generalizes the Schwarzian to the complete range of energies.

JT gravity, KdV equations and macroscopic loop operators

“…It would be very significant if we can find the exact eigenvalue density ρ(E). It is argued in [46,47] that the eigenvalue density of the SYK model is closely related to the q-Hermite polynomials. It would be interesting to see if the double scaling limit of the q-Hermite polynomials has some connection to the exact eigenvalue density ρ(E) of the JT gravity case.…”

An exact quantization of Jackiw-Teitelboim gravity

“…Another interesting direction is to study the role of q-deformations for the 2d gauge theory associated to a non-compact group, which have played an important role in the case of compact groups [72]. Such a deformation is also relevant from the boundary perspective, where [73] have shown that correlation functions in the large-N double-scaled limit of the SYK model can be described in terms of representations of q-deformed SU (1, 1).…”

Defects in Jackiw-Teitelboim quantum gravity

“…There has been a recent explosion of interest in the so-called Sachdev-Ye-Kitaev models, see e.g. [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. These 0+1d quantum models of all-to-all interacting fermions contain 1+1d gravitational features such as black holes and chaotic behavior.…”