Quantum groups, non-commutative AdS2, and chords in the double-scaled SYK model

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 Poincare disk, which underlies the single-sided Schwarzian model. AdS(2) carries an action of sl(2, R) similar or equal to, su(1, 1), and we argue that the symmetry of the full DSSYK model is a certain q-deformation of the latter, namely U-vq(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-vq(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.