Dynamical actions and q-representation theory for double-scaled SYK

Abstract

We show that DSSYK amplitudes are reproduced by considering the quantum mechanics of a constrained particle on the quantum group SU${}_q(1,1)$. We construct its left-and right-regular representations, and show that the representation matrices reproduce two-sided wavefunctions and correlation functions of DSSYK. We then construct a dynamical action and path integral for a particle on SU${}_q(1,1)$, whose quantization reproduces the aforementioned representation theory. By imposing boundary conditions or constraining the system we find the $q$-analog of the Schwarzian and Liouville boundary path integral descriptions. This lays the technical groundwork for identifying the gravitational bulk description of DSSYK. We find evidence the theory in question is a sine dilaton gravity, which interestingly is capable of describing both AdS and dS quantum gravity.

51 pages, v2: matches published version

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• Krylov spread complexity as holographic complexity beyond JT gravity
• The dilaton gravity hologram of double-scaled SYK
• Tridiagonal Hamiltonians modeling the density of states of the Double-Scaled SYK model
• Wormholes, branes and finite matrices in sine dilaton gravity
• Quantum gravity of the Heisenberg algebra
• Complexity = Anything Can Grow Forever in de Sitter
• Branes in JT (super)gravity from group theory
• The complex Liouville string: the gravitational path integral
• Quantum group origins of edge states in double-scaled SYK
• Horizon causality from holographic scattering in asymptotically dS${}_3$
• Gravitational wavefunctions in JT supergravity
• Holographic central charge for sine dilaton gravity
• Holographic tensor network for double-scaled SYK
• Probing sine dilaton gravity with flow central charge
• Unveiling Novel Insights in Quantum Dynamics through Extended Sachdev-Ye-Kitaev Model
• Minimal Factorization of Chern-Simons Theory -- Gravitational Anyonic Edge Modes
• Fiducial observers and the thermal atmosphere in the black hole quantum throat
• Asymptotics of Weil-Petersson volumes and two-dimensional quantum gravities