Finite-cutoff geometry in double-scaled SYK: SYK deformations, branes and CMC-Krylov complexity

We develop a finite-cutoff holographic description of the double-scaled SYK (DSSYK) model. We construct microscopic deformations of Kourkoulou-Maldacena and thermofield-double SYK states whose double-scaling limits reproduce a lower-dimensional analogue of \(\text{T}\overline{\text{T}}\) and \(\text{T}\overline{\text{T}}+\Lambda_2\) deformations, which we denote $T^2$ and $T^2+\Lambda_1$, respectively. The ensemble-averaged theory describes wormhole states in finite-cutoff sine dilaton gravity, and finite-cutoff anti-de Sitter and de Sitter geometries in the low- and high-energy regimes, respectively. The analysis includes both perturbative and non-perturbative solutions to the energy spectrum flow equation, and the regions behind and beyond the respective horizons. Only the perturbative branch admits physical quantum states. We study how to incorporate end-of-the-world (ETW) branes in the finite cutoff holographic dictionary. The ETW brane leads to an additional boundary theory deformation, which leads to black hole or eternal traversable wormhole geometries depending on the existence of normalizable bound states. We extend the finite cutoff length evaluations for constant mean curvature (CMC) slices, which we interpret as complexity in the boundary theory. We discuss possible extensions with matter. Our results provide a microscopic framework for studying the finite-cutoff boundaries, CMC slices and branes in the holographic dictionary of DSSYK.