Quantum groups, non-commutative $Ad{S}_2$, and chords in the double-scaled SYK model

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 $q\to 1$ 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. $Ad{S}_2$ carries an action of $\mathfrak{sl}(2,\mathbb{R})\simeq \mathfrak{su}(1,1)$, and we argue that the symmetry of the full DS-SYK model is a certain $q$-deformation of the latter, namely ${\mathcal{U}}_{\sqrt{q}}(\mathfrak{su}(1,1))$. We do this by obtaining the effective Hamiltonian of the DS-SYK as a (reduction of) particle moving on a lattice deformation of $Ad{S}_2$, which has this ${\mathcal{U}}_{\sqrt{q}}(\mathfrak{su}(1,1))$ 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 $Ad{S}_3$. There are families of possibly distinct $q$-deformed $Ad{S}_2$ spaces, and we point out which are relevant for the DS-SYK model.

70 pages, 6 figures

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