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Quantum groups, non-commutative $AdS_2$, and chords in the double-scaled SYK model

Micha Berkooz, Mikhail Isachenkov, Prithvi Narayan, Vladimir Narovlansky

TL;DR

The paper develops a quantum-group–driven framework for the double-scaled SYK model, showing that its transfer-matrix and chord dynamics encode a q-deformed AdS2 symmetry, specifically u_{su(1,1)}(su(1,1)) at finite q. By reducing AdS3 to AdS2 and constructing lattice/noncommutative AdS2 spaces, the authors realize the DS-SYK Hamiltonian as a Casimir of a quantum group, and connect the short-time (chord) dynamics to Liouville/Schwarzian physics in the q→1 limit. They introduce several q-deformed AdS2 constructions (and q-Lobachevsky spaces), analyze eigenfunctions and q-Fourier transforms, and argue for a no-bound-state truncation to recover the physically meaningful transfer-matrix. The work links DS-SYK to non-commutative geometry and quantum groups, offering a concrete 2D geometric lift of the DS-SYK dynamics with potential implications for holography and quantum gravity in deformed AdS backgrounds.

Abstract

We study the double-scaling limit of SYK (DS-SYK) model and elucidate the underlying quantum group symmetry. The DS-SYK model is characterized by a parameter $q$, and in the $q\rightarrow 1$ and low-energy limit it goes over to the familiar Schwarzian theory. We relate the chord and transfer-matrix picture to the motion of a ``boundary particle" on the Euclidean Poincar{é} disk, which underlies the single-sided Schwarzian model. $AdS_2$ carries an action of $\mathfrak{s}\mathfrak{l}(2,{\mathbb R}) \simeq \mathfrak{s}\mathfrak{u}(1,1)$, and we argue that the symmetry of the full DS-SYK model is a certain $q$-deformation of the latter, namely $\mathcal{U}_{\sqrt q}(\mathfrak{s}\mathfrak{u}(1,1))$. We do this by obtaining the effective Hamiltonian of the DS-SYK as a (reduction of) particle moving on a lattice deformation of $AdS_2$, which has this $\mathcal{U}_{\sqrt q}(\mathfrak{s}\mathfrak{u}(1,1))$ algebra as its symmetry. We also exhibit the connection to non-commutative geometry of $q$-homogeneous spaces, by obtaining the effective Hamiltonian of the DS-SYK as a (reduction of) particle moving on a non-commutative deformation of $AdS_3$. There are families of possibly distinct $q$-deformed $AdS_2$ spaces, and we point out which are relevant for the DS-SYK model.

Quantum groups, non-commutative $AdS_2$, and chords in the double-scaled SYK model

TL;DR

The paper develops a quantum-group–driven framework for the double-scaled SYK model, showing that its transfer-matrix and chord dynamics encode a q-deformed AdS2 symmetry, specifically u_{su(1,1)}(su(1,1)) at finite q. By reducing AdS3 to AdS2 and constructing lattice/noncommutative AdS2 spaces, the authors realize the DS-SYK Hamiltonian as a Casimir of a quantum group, and connect the short-time (chord) dynamics to Liouville/Schwarzian physics in the q→1 limit. They introduce several q-deformed AdS2 constructions (and q-Lobachevsky spaces), analyze eigenfunctions and q-Fourier transforms, and argue for a no-bound-state truncation to recover the physically meaningful transfer-matrix. The work links DS-SYK to non-commutative geometry and quantum groups, offering a concrete 2D geometric lift of the DS-SYK dynamics with potential implications for holography and quantum gravity in deformed AdS backgrounds.

Abstract

We study the double-scaling limit of SYK (DS-SYK) model and elucidate the underlying quantum group symmetry. The DS-SYK model is characterized by a parameter , and in the and low-energy limit it goes over to the familiar Schwarzian theory. We relate the chord and transfer-matrix picture to the motion of a ``boundary particle" on the Euclidean Poincar{é} disk, which underlies the single-sided Schwarzian model. carries an action of , and we argue that the symmetry of the full DS-SYK model is a certain -deformation of the latter, namely . We do this by obtaining the effective Hamiltonian of the DS-SYK as a (reduction of) particle moving on a lattice deformation of , which has this algebra as its symmetry. We also exhibit the connection to non-commutative geometry of -homogeneous spaces, by obtaining the effective Hamiltonian of the DS-SYK as a (reduction of) particle moving on a non-commutative deformation of . There are families of possibly distinct -deformed spaces, and we point out which are relevant for the DS-SYK model.
Paper Structure (39 sections, 200 equations, 6 figures)

This paper contains 39 sections, 200 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The transfer-matrix method for double-scaled SYK
  3. A crash review of the derivation
  4. Evaluating the moments: the transfer-matrix and the chord Hilbert space
  5. The transfer-matrix
  6. The Hilbert space
  7. The $q\rightarrow 1$ limit
  8. The transfer-matrix and $AdS_2$ spacetime
  9. The Liouville form of the Schwarzian
  10. The Liouville action from $AdS_3$ and motion on $AdS_2$
  11. The discrete trajectories of the boundary particle in the transfer-matrix approach
  12. The boundary particle trajectory in chord number
  13. The initial and final state, and the carving out of $AdS_2$
  14. A hands-on approach to non-commutative $AdS_2$
  15. The $\mathcal{U}_{q^{1/2}}(\mathfrak{s}\mathfrak{u}(1,1))$ algebra
  16. ...and 24 more sections

Figures (6)

  • Figure 1: An example of a chord diagram, and the emergence of a bulk at large chord number.
  • Figure 2: An example of a chord diagram contributing to $m_k$ with $k=6$.
  • Figure 3: The possibilites at each node.
  • Figure 4: A contribution to the overlap of $|C_1C_0C_2C_0\rangle$ and $|C_0C_0C_1C_2\rangle$.
  • Figure 5: Equal Poincaré time slicing of the disk.
  • ...and 1 more figures