Quantum groups, non-commutative AdS2, 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. AdS2 carries an action of $$ \mathfrak{sl} $$ sl (2, ℝ) ≃ $$ \mathfrak{su} $$ su (1, 1), and we argue that the symmetry of the full DS-SYK model is a certain q-deformation of the latter, namely $$ {\mathcal{U}}_{\sqrt{q}} $$ U q ($$ \mathfrak{su} $$ 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 AdS2, which has this $$ {\mathcal{U}}_{\sqrt{q}} $$ U q ($$ \mathfrak{su} $$ 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 AdS3. There are families of possibly distinct q-deformed AdS2 spaces, and we point out which are relevant for the DS-SYK model.