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

TL;DR

This work reframes the amplitudes of double-scaled SYK in terms of the representation theory of the quantum group SU_q(1,1), constructing explicit left- and right-regular representations and imposing gravitational-like boundary constraints to reproduce DSSYK data. It then builds a continuum action for a particle on SU_q(1,1) whose quantization yields the constrained representations, leading to q-Liouville and q-Schwarzian boundary theories that play the role of UV-complete boundary descriptions and suggest a first-order bulk dual via a Poisson-Sigma/dilaton gravity with a sine potential. The results provide a concrete, group-theoretic path to connect DSSYK with a 2d gravitational bulk and offer a framework to explore both AdS2 and dS2 regimes, as well as potential supersymmetric generalizations. The formalism clarifies how discrete sampling in the quantum group setting naturally encodes chord-like bulk discretization and yields amplitudes matching known DSSYK correlators, while highlighting avenues for factorization, Liouville comparison, and future UV-complete bulk constructions.

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.