Quantum Symmetry and Geometry in Double-Scaled SYK

TL;DR

This paper uncovers a quantum group structure within the double-scaled SYK model, linking the quantum R-matrix to the quantum group f(f(1,1)) and exploring its implications for the model's algebraic and gravitational properties.

Contribution

It identifies the quantum group f(f(1,1)) as a subalgebra of the chord algebra and constructs its generators, revealing new algebraic insights into the SYK model and its gravitational wavefunctions.

Findings

Quantum group f(f(1,1)) is embedded in the chord algebra.

The quantum R-matrix acts as a swapping operator with q-weighted factors.

Derived a factorization formula for the gravitational wavefunction with matter.

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.