A Cordial Introduction to Double Scaled SYK

TL;DR

This paper reviews the double scaled SYK model, highlighting its exact solvable properties, algebraic structures, and connections to quantum gravity, offering insights into non-commutative geometries and universality classes.

Contribution

It provides a comprehensive overview of the double scaled SYK model, including exact solutions, algebraic frameworks, and links to quantum gravity and non-commutative spaces.

Findings

Exact spectrum and correlation functions derived

Connection to Schwarzian quantum mechanics established

Quantum group structure suggests non-commutative $AdS_2$ geometry

Abstract

We review recent progress regarding the double scaled Sachdev-Ye-Kitaev model and other $p$-local quantum mechanical random Hamiltonians. These models exhibit an expansion using chord diagrams, which can be solved by combinatorial methods. We describe exact results in these models, including their spectrum, correlation functions, and Lyapunov exponent. In a certain limit, these techniques manifest the relation to the Schwarzian quantum mechanics, a theory of quantum gravity in $AdS_2$. More generally, the theory is controlled by a rigid algebraic structure of a quantum group, suggesting a theory of quantum gravity on non-commutative $q$-deformed $AdS_2$. We conclude with discussion of related universality classes, and survey some of the current research directions.