A symmetry algebra in double-scaled SYK

Abstract

The double-scaled limit of the Sachdev-Ye-Kitaev (SYK) model takes the number of fermions and their interaction number to infinity in a coordinated way. In this limit, two entangled copies of the SYK model have a bulk description of sorts known as the "chord Hilbert space." We analyze a symmetry algebra acting on this Hilbert space, generated by the two Hamiltonians together with a two-sided operator known as the chord number. This algebra is a deformation of the JT gravitational algebra, and it contains a subalgebra that is a deformation of the ${\mathfrak{sl}}_2$ near-horizon symmetries. The subalgebra has finite-dimensional unitary representations corresponding to matter moving around in a discrete Einstein-Rosen bridge. In a semiclassical limit the discreteness disappears and the subalgebra simplifies to ${\mathfrak{sl}}_2$, but with a non-standard action on the boundary time coordinate. One can make the action of ${\mathfrak{sl}}_2$ algebra more standard at the cost of extending the boundary circle to include some "fake" portions. Such fake portions also accommodate certain subtle states that survive the semi-classical limit, despite oscillating on the scale of discreteness. We discuss applications of this algebra, including sub-maximal chaos, the traversable wormhole protocol, and a two-sided OPE.

70 pages

References in corpus (9)

• The Pomeron and Gauge/String Duality
• The bulk Hilbert space of double scaled SYK
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• von Neumann algebras in JT gravity
• Algebra of diffeomorphism-invariant observables in Jackiw-Teitelboim Gravity
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• Wormholes, branes and finite matrices in sine dilaton gravity
• Quantum gravity of the Heisenberg algebra
• Pole-skipping in two-dimensional de Sitter spacetime and double-scaled SYK model
• Branes in JT (super)gravity from group theory
• Operator size growth in Lindbladian SYK
• The complex Liouville string: the gravitational path integral
• Quantum Groups as Global Symmetries
• D-commuting SYK model: building quantum chaos from integrable blocks
• Ringdown in the SYK model
• Quantum group origins of edge states in double-scaled SYK
• Correlation functions for open strings and chaos
• Gravitational wavefunctions in JT supergravity
• Parisi's hypercube, Fock-space fluxes, and the microscopics of near-AdS${}_2$/near-CFT${}_1$ duality
• Scaling limits of complex Sachdev-Ye-Kitaev models and holographic geometry
• Holographic tensor network for double-scaled SYK
• Unveiling Novel Insights in Quantum Dynamics through Extended Sachdev-Ye-Kitaev Model
• Off-diagonal coupling of supersymmetric SYK model
• Schur Connections: Chord Counting, Line Operators, and Indices
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• Probing Stringy Horizons with Pole-Skipping in Non-Maximal Chaotic Systems