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arXiv:2312.00880 · Journal of High Energy Physics

Matter correlators through a wormhole in double-scaled SYK

Kazumi Okuyama

Published Feb 21, 2024• 11 citations
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Abstract

A bstract We compute the two-point function of matter operators in the double-scaled SYK (DSSYK) model, where the two matter operators are inserted at each end of the cylindrical wormhole. We find that the wormhole amplitude in DSSYK is written as a trace over the chord Hilbert space. We also show that the length of the wormhole is stabilized in the semi-classical limit, by the same mechanism worked for the JT gravity case.

DOI
10.1007/jhep02(2024)147

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A bstract We compute the two-point function of matter operators in the double-scaled SYK (DSSYK) model, where the two matter operators are inserted at each end of the cylindrical wormhole. We find that the wormhole amplitude in DSSYK is written as a trace over the chord Hilbert space. We also show that the length of the wormhole is stabilized in the semi-classical limit, by the same mechanism worked for the JT gravity case.
A bstract We consider matter correlators in the double-scaled SYK (DSSYK) model. It turns out that matter correlators have a simple expression in terms of the doubled Hilbert space H⊗H\mathcal{H}\otimes \mathcal{H}H⊗H , where H\mathcal{H}H is the Fock space of q -deformed oscillator (also known as the chord Hilbert space). In this formalism, we find that the operator which counts the intersection of chords should be conjugated by certain “entangler” and “disentangler”. We explicitly demonstrate this structure for the two- and four-point functions of matter operators in DSSYK.
We derive a large-N wormhole saddle in the Sachdev-Ye-Kitaev (SYK) model for the variance of the thermal one-point function of a heavy operator across realizations of the microscopic couplings. We propose that this saddle describes a double trumpet geometry. In the double-scaling limit, we express the wormhole as a transition amplitude in an emergent bulk Hilbert space of chords connecting the two boundaries. In the Schwarzian limit, the chords proliferate and the wormhole defines a quantum transition amplitude in the open-universe Hilbert space of Jackiw-Teitelboim (JT) gravity. Comparing the SYK wormhole amplitude with its counterpart in JT gravity coupled to matter provides a direct test of the heavy-operator randomness hypothesis in SYK. The microscopic SYK couplings specify the α-parameters of the wormhole. For a fixed realization of the couplings, we compute the observable in the corresponding α-state. We provide the bulk wavefunction of the associated half-wormhole, which exhibits erratic correlations in the length basis that do not self-average. We extend the bulk description to the Hilbert space of a closed universe. We point out that the correlation chords from which the wormhole geometry emerges structurally resemble local observer worldlines.
A bstract In holography, gauging symmetries of the boundary theory leads to important modifications in the bulk. In this work, we study constraints to gauge symmetry sectors in the chord Hilbert space of the double-scaled SYK (DSSYK) with matter, and we connect them to different proposals of its bulk dual. These sectors include chord parity symmetry, corresponding to End-Of-The-World (ETW) branes and Euclidean wormholes in sine dilaton gravity; and relative time-translations in a doubled DSSYK model (resulting from a single DSSYK with an infinitely heavy matter chord) used in de Sitter holography. We define and evaluate partition functions and thermal correlation functions of the ETW brane and Euclidean wormhole systems in the boundary theory. We deduce the holographic dictionary by matching geodesic lengths in the bulk with the spread complexity of the parity-gauged DSSYK. The Euclidean wormholes of fixed size are perturbatively stable, and their baby universe Hilbert space is non-trivial only when matter is added. We conclude studying the constraints in the path integral of the doubled DSSYK. We derive the gauge invariant operator algebra of one of the DSSYKs dressed to the other one and discuss its holographic interpretation.
A bstract Sine-dilaton gravity has been proposed as the holographic dual of the double scaled SYK (DSSYK) model. In this work, we examine this duality by deriving general matter correlation functions directly from the bulk perspective. A novel technique we develop is to treat matter lines in the bulk as end-of-the-world (EOW) branes, allowing us to implement a splitting and gluing procedure on the bulk spacetime. This geometric procedure gives rise to the structure of two-sided multi-particle wormhole Hilbert space. In contrast to previous Hilbert space constructions in DSSYK, the length basis in our case factorizes across the subregions, whereas the energy basis acquires a non-local, state-dependent structure determined by the EOW brane quantization in each subregion. Different choices of splitting correspond to distinct representations of the same Hilbert space, all of which are equivalent for physical observables. Based on this framework, we compute general correlation functions, including the OTOC, and show that they exactly reproduce the DSSYK results obtained from chord diagrams. By calculating the OTOC in different representations, we uncover a new identity for the 6j-symbol of the quantum group Uq(su(1,1)) {\mathcal{U}}_q\left( su\left(1,1\right)\right) Uq​(su(1,1)) U q su 1 1 . Finally, the resulting wormhole Hilbert space enables us to go beyond the disk to compute matter correlation functions on the double trumpet and to include bulk matter loops.
We study the properties of the double-scaled SYK (DSSYK) model under chord Hamiltonian deformations based on finite cutoff holography for general dilaton gravity theories with Dirichlet boundaries. The formalism immediately incorporates a lower-dimensional analog of TTˉ(+Λ2)\text{T}\bar{\text{T}}(+Λ_2)TTˉ(+Λ2​) deformations, denoted T2(+Λ1)T^2(+Λ_1)T2(+Λ1​), as special cases. In general, the deformation mixes the chord basis of the Hilbert space in the seed theory, which we order through the Lanczos algorithm. The resulting Krylov complexity for the Hartle-Hawking state represents a wormhole length at a finite cutoff in the bulk. We study the thermodynamic properties of the deformed theory; the growth of Krylov complexity; the evolution of nnn-point correlation functions with matter chords; and the entanglement entropy between the double-scaled algebras of the DSSYK model for a given chord state. The latter, in the triple-scaling limit, manifests as the minimal codimension-two area in the bulk following the Ryu-Takayanagi formula. By performing a sequence of T2T^2T2 and T2+Λ1T^2+Λ_1T2+Λ1​ deformations in the upper tail of the energy spectrum in the deformed DSSYK, we concretely realize the cosmological stretched horizon proposal in de Sitter holography by Susskind. We discuss other extensions with sine dilaton gravity, end-of-the-world branes, and the Almheiri-Goel-Hu model.
In holography, gauging symmetries of the boundary theory leads to important modifications in the bulk. In this work, we study constraints to gauge symmetry sectors in the chord Hilbert space of the double-scaled SYK (DSSYK) with matter, and we connect them to different proposals of its bulk dual. These sectors include chord parity symmetry, corresponding to End-Of-The-World (ETW) branes and Euclidean wormholes in sine dilaton gravity; and relative time-translations in a doubled DSSYK model (resulting from a single DSSYK with an infinitely heavy matter chord) used in de Sitter holography. We define and evaluate partition functions and thermal correlation functions of the ETW brane and Euclidean wormhole systems in the boundary theory. We deduce the holographic dictionary by matching geodesic lengths in the bulk with the spread complexity of the parity-gauged DSSYK. The Euclidean wormholes of fixed size are perturbatively stable, and their baby universe Hilbert space is non-trivial only when matter is added. We conclude studying the constraints in the path integral of the doubled DSSYK. We derive the gauge invariant operator algebra of one of the DSSYKs dressed to the other one and discuss its holographic interpretation.