From chords and complexity to dynamical wormholes with matter: Towards a bulk double-scaled (SYK) algebra

Abstract: We formulate a bulk description of the double-scaled algebra of the DSSYK model \cite{Lin:2022rbf}. Based on the Hartle-Hawking (HH) state with matter operators, we derive several properties of the DSSYK model, without making assumptions about the specific dual theory, including its semiclassical thermodynamics, correlation functions, and Krylov complexity. These quantities are found from the saddle points of the DSSYK path integral preparing the HH state. We also construct a Lanczos algorithm that simultaneously evaluates Krylov state and operator complexity in the two-sided Hamiltonian system including finite temperature effects. In the semiclassical limit, both measures are encoded in the saddle points of the path integral. They have a bulk interpretation in terms of minimal geodesic lengths in an effective AdS$_2$ space with matter backreaction. Different saddle points correspond to geodesic distances with different evolution, and they display different scrambling properties. This allows us to quantize the bulk theory dual to the DSSYK model. At last, we deduce the boundary/bulk dictionary of the double-scaled algebra, and the dual entry of the proper radial momentum of a bulk probe.