Quantum groups, non-commutative AdS2, and chords in the double-scaled SYK model
Micha Berkooz, Misha Isachenkov, Prithvi Narayan, Vladimir Narovlansky
Abstract
A bstract 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 $$ \mathfrak{sl} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>sl</mml:mi> </mml:math> (2 , ℝ) ≃ $$ \mathfrak{su} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>su</mml:mi> </mml:math> (1 , 1), and we argue that the symmetry of the full DS-SYK model is a certain q -deformation of the latter, namely $$ {\mathcal{U}}_{\sqrt{q}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>U</mml:mi> <mml:msqrt> <mml:mi>q</mml:mi> </mml:msqrt> </mml:msub> <mml:mspace/> </mml:math> ( $$ \mathfrak{su} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>su</mml:mi> </mml:math> (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 $$ {\mathcal{U}}_{\sqrt{q}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>U</mml:mi> <mml:msqrt> <mml:mi>q</mml:mi> </mml:msqrt> </mml:msub> <mml:mspace/> </mml:math> ( $$ \mathfrak{su} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>su</mml:mi> </mml:math> (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.