Dynamical actions and q-representation theory for double-scaled SYK

TL;DR

This paper demonstrates that double-scaled SYK amplitudes can be derived from a quantum particle on the quantum group SU_q(1,1), establishing a connection to q-representation theory and boundary path integrals.

Contribution

It constructs the representation theory and path integral formulation for a particle on SU_q(1,1), linking DSSYK amplitudes to quantum group structures and boundary gravitational theories.

Findings

Representation matrices reproduce DSSYK wavefunctions and correlators.

Constructed a q-analog of Schwarzian and Liouville boundary path integrals.

Evidence suggests the theory is a sine dilaton gravity capable of describing AdS and dS quantum gravity.

Abstract

We show that DSSYK amplitudes are reproduced by considering the quantum mechanics of a constrained particle on the quantum group SU$_q(1,1)$. We construct its left-and right-regular representations, and show that the representation matrices reproduce two-sided wavefunctions and correlation functions of DSSYK. We then construct a dynamical action and path integral for a particle on SU$_q(1,1)$, whose quantization reproduces the aforementioned representation theory. By imposing boundary conditions or constraining the system we find the $q$-analog of the Schwarzian and Liouville boundary path integral descriptions. This lays the technical groundwork for identifying the gravitational bulk description of DSSYK. We find evidence the theory in question is a sine dilaton gravity, which interestingly is capable of describing both AdS and dS quantum gravity.