Quantum symmetry and geometry in double-scaled SYK
摘要
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 𝒰q(𝔰𝔲(1, 1)) as a subalgebra of the chord algebra. Specifically, we construct the generators of 𝒰q(𝔰𝔲(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 𝒰q(𝔰𝔲(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 also discuss the relation between the 𝒰q(𝔰𝔲(1, 1)) structure uncovered here and the 𝒰q(𝔰𝔩(2, ℝ)) algebra previously studied from the boundary perspective. Finally, we study the gravitational wavefunction with matter in the Schwarzian regime.