Quantum Symmetry and Geometry in Double-Scaled SYK

TL;DR

The paper identifies a quantum-group symmetry underlying the chord description of the double-scaled SYK model with matter, constructing the generators of $\mathcal{U}_q(\mathfrak{su}(1,1))$ from chord operators and proving that the one-particle sector decomposes into positive discrete-series representations. The multi-particle sector is built via the coproduct, and the quantum $R$-matrix acts as a chord-crossing operator with $q$-weighted penalties, yielding a concrete Yang–Baxter relation from quantum-group fusion. A quantum-disk realization of the representations is developed, and a novel factorization formula for the bulk gravitational wavefunction in the presence of matter is derived, together with the Schwarzian-limit analysis. The work also discusses the relation to boundary $\mathcal{U}_q(\mathfrak{sl}(2,\mathbb{R}))$ structures and outlines promising directions, including supersymmetric generalizations and links to complex Liouville string theories, for a more complete holographic picture of DSSYK.

Abstract

The emergence of the quantum $R$-matrix in the double-scaled SYK model points to an underlying quantum group structure. In this work, we identify the quantum group $\mathcal{U}_q(\mathfrak{su}(1,1))$ as a subalgebra of the chord algebra. Specifically, we construct the generators of $\mathcal{U}_q(\mathfrak{s} \mathfrak{u}(1,1))$ from combinations of operators within the chord algebra and show that the one-particle chord Hilbert space decomposes into the positive discrete series representations of $\mathcal{U}_q(\mathfrak{s} \mathfrak{u}(1,1))$. Using the coproduct structure of the quantum group, we build the multi-particle Hilbert space and establish its equivalence with previous results defined by the chord rules. In particular, we show that the quantum $R$-matrix acts as a swapping operator that reverses the ordering of open chords in each fusion channel while incorporating the corresponding $q$-weighted penalty factors. This action enables an explicit derivation of the chord Yang-Baxter relation. We further explore a realization of the quantum group generators on the quantum disk, and present a novel factorization formula for the bulk gravitational wavefunction in the presence of matter. We further discuss the relation between the $\mathcal{U}_q(\mathfrak{s} \mathfrak{u}(1,1))$ structure uncovered here and the $\mathcal{U}_q(\mathfrak{s} \mathfrak{l}(2, \mathbb{R}))$ algebra previously studied from the boundary perspective. Finally, we study the gravitational wavefunction with matter in the Schwarzian regime.