q-Askey Deformations of Double-Scaled SYK

TL;DR

This paper introduces q-Askey deformations of the double-scaled SYK model, exploring their bulk interpretation, spectral properties, and algebraic structures, revealing new geometric transitions and dualities.

Contribution

It constructs and analyzes a family of deformed SYK models using q-Askey polynomials, uncovering new geometric phases, algebraic classifications, and dualities with gauge theories.

Findings

Deformations encode recurrence relations of q-Askey polynomials.

Chord number relates to Einstein-Rosen bridge length at finite temperature.

Models exhibit a transition to discrete energy levels indicating a geometric change.

Abstract

We construct families of deformations of the double-scaled SYK (DSSYK) model and investigate their bulk interpretation. We introduce microscopic deformations of the SYK model which, after ensemble averaging and in the double-scaling limit, are described by a transfer matrix encoding the recurrence relations of basic orthogonal polynomials in the q-Askey scheme. For certain families of deformations in the semiclassical limit at finite temperature, the chord number (encoding Krylov complexity) corresponds to the length of an Einstein-Rosen bridge connecting an End-Of-The-World brane to an anti-de Sitter asymptotic boundary. By increasing one of the deformation parameters, the models eventually exhibit discrete energy levels, signaling a new geometric transition in sine dilaton gravity. Via the SYK-Schur duality, Krylov complexity also admits a representation-theoretic interpretation as the spread of the SU(2) spin in the index of an $\mathcal{N}=2$ SU(2) gauge theory. We study the operator algebras of the deformed theories. The algebras can be type II$_1$ or type I$_\infty$ factors, depending on the operators that are included. The entanglement entropy between the type II$_1$ algebras for a pure state manifests as an extremal surface through the Ryu-Takayanagi formula. We discuss connections between our results and the emergence of baby universes in the bulk.