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Peer-reviewedCC BYJournal of High Energy Physics

Quantum symmetry and geometry in double-scaled SYK

  • Jeremy van der Heijden1
  • Erik Verlinde2
  • Jiuci Xu3
  1. 1University of British Columbia
  2. 2University of Amsterdam
  3. 3University of California
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Abstract

A bstract 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 𝒰 q (𝔰𝔲(1, 1)) as a subalgebra of the chord algebra. Specifically, we construct the generators of 𝒰 q (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 𝒰 q (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 𝒰 q (𝔰𝔲(1, 1)) structure uncovered here and the 𝒰 q (𝔰𝔩(2 , ℝ)) algebra previously studied from the boundary perspective. Finally, we study the gravitational wavefunction with matter in the Schwarzian regime.

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Cite

BibTeXRIS
BibTeX
@article{heijden2026quantum,
  title = {{Quantum symmetry and geometry in double-scaled SYK}},
  author = {Jeremy van der Heijden and Erik Verlinde and Jiuci Xu},
  journal = {Journal of High Energy Physics},
  year = {2026},
  month = may,
  doi = {10.1007/jhep05(2026)148},
  url = {https://doi.org/10.1007/jhep05(2026)148},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Quantum symmetry and geometry in double-scaled SYK
AU  - Jeremy van der Heijden
AU  - Erik Verlinde
AU  - Jiuci Xu
PY  - 2026
DA  - 2026/05/13
JO  - Journal of High Energy Physics
DO  - 10.1007/jhep05(2026)148
UR  - https://doi.org/10.1007/jhep05(2026)148
AB  - A bstract 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 𝒰 q (𝔰𝔲(1, 1)) as a subalgebra of the chord algebra. Specifically, we construct the generators of 𝒰 q (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 𝒰 q (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 𝒰 q (𝔰𝔲(1, 1)) structure uncovered here and the 𝒰 q (𝔰𝔩(2 , ℝ)) algebra previously studied from the boundary perspective. Finally, we study the gravitational wavefunction with matter in the Schwarzian regime.
SN  - 1029-8479
SN  - 1126-6708
SN  - 1127-2236
ER  -