Quantum groups, non-commutative AdS 2, and chords in the double-scaled SYK model

Abstract

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 → 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 Poincaré disk, which underlies the single-sided Schwarzian model. AdS 2 carries an action of sl $$ \mathfrak{sl} $$ (2, ℝ) ≃ su $$ \mathfrak{su} $$ (1, 1), and we argue that the symmetry of the full DS-SYK model is a certain q-deformation of the latter, namely U q $$ {\mathcal{U}}_{\sqrt{q}} $$ ( su $$ \mathfrak{su} $$ (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 AdS 2, which has this U q $$ {\mathcal{U}}_{\sqrt{q}} $$ ( su $$ \mathfrak{su} $$ (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 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.

Keywords

• 1/N Expansion
• AdS-CFT Correspondence
• Field Theories in Lower Dimensions
• Non-Commutative Geometry