Emergent states and algebras from the double-scaling limit of pure states in SYK

A

bstract

Recent work has emphasized a subtlety of large- N limits in AdS/CFT: a sequence of pure states in the microscopic theory need not remain pure with respect to the emergent algebra of observables. We study this phenomenon for Kourkoulou-Maldacena (KM) states in the double-scaling limit of the SYK model, and show that their ensemble-averaged algebraic description depends crucially on which observables are retained in the limit. For fermionic operators of size p ∼ N ^1/2 , generic operators converge to the usual chord operators of double-scaled SYK. The resulting von Neumann algebra is the standard Type II ₁ factor, and the KM pure states at infinite temperature converge to the tracial state, so generic probes lose access to microscopic purity. We then identify a class of operators adapted to the KM state that also survive the double-scaling limit. Since the KM state may be viewed as a projection inside the tracial state, these operators become dressed chord creation and annihilation operators, with the dressing encoding the distance to the KM state through the bulk. Once included, the limiting algebra becomes Type I _∞ and the sequence of KM states converges to a pure state. This provides a solvable example in which adding a sufficiently state-adapted operator to the emergent algebra restores access to the purity of the underlying state.

We derive exact modified chord-diagram rules for correlators of the dressed operators, including general uncrossed 2 n -point functions and crossed four-point functions. In the semi-classical limit these correlators exhibit a novel factorization pattern, while in the Schwarzian limit they reduce to correlators of boundary primaries dressed to Schwarzian modes. We also study a deformation of the chord Hamiltonian, the emergence of bound states above a critical coupling, and its relation to JT gravity coupled to an EOW brane of general tension. We identify an emergent U (1) global symmetry in the Type I _∞ algebra as well as its violation by 1 /N corrections from double-trace wormholes. Finally, we discuss analogies with boundary algebras proposed for black hole interiors and closed universes, and suggest lessons from our construction for both.