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arXiv:2610.20703

Finite-cutoff geometry from double-scaled SYK

Sergio E. Aguilar-Gutierrez

hep-thgr-qcquant-phSubmitted Oct 14, 2026• Announced Oct 16, 2026
v1 · Oct 14, 2026v2 · Oct 26, 2026
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Abstract

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 $\text{T}\overline{\text{T}}+Λ_2$ deformations, which we denote $T^2$ and $T^2+Λ_1$, 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, and interpret them 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.

Comments
58 pgs+appendices, 16 figures

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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 TT‾\text{T}\overline{\text{T}}TT and TT‾+Λ2\text{T}\overline{\text{T}}+Λ_2TT+Λ2​ deformations, which we denote T2T^2T2 and T2+Λ1T^2+Λ_1T2+Λ1​, 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, and interpret them 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.
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.
ABSTRACT 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 deformations, denoted , 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 ‐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 and 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.
We construct families of deformations of the double-scaled SYK (DSSYK) model and investigate their bulk interpretation. We introduce microscopic deformations of the SYK model which, after ensemble averaging and in the double-scaling limit, are described by a transfer matrix encoding the recurrence relations of basic orthogonal polynomials in the q-Askey scheme. For certain families of deformations in the semiclassical limit at finite temperature, the chord number (encoding Krylov complexity) corresponds to the length of an Einstein-Rosen bridge connecting an End-Of-The-World brane to an anti-de Sitter asymptotic boundary. By increasing one of the deformation parameters, the models eventually exhibit discrete energy levels, signaling a new geometric transition in sine dilaton gravity. Via the SYK-Schur duality, Krylov complexity also admits a representation-theoretic interpretation as the spread of the SU(2) spin in the index of an N=2\mathcal{N}=2N=2 SU(2) gauge theory. We study the operator algebras of the deformed theories. The algebras can be type II1_11​ or type I∞_\infty∞​ factors, depending on the operators that are included. The entanglement entropy between the type II1_11​ algebras for a pure state manifests as an extremal surface through the Ryu-Takayanagi formula. We discuss connections between our results and the emergence of baby universes in the bulk.
T2T^2T2 deformations in the double-scaled SYK model: Stretched horizon thermodynamics
We study T2^22-deformations in the DSSYK model \emph{after} performing ensemble averaging, to learn about the properties of its bulk dual geometry according to finite cutoff holography. We first derive the flow equation for generic holographic quantum mechanical systems dual to dilaton gravity theories without requiring AdS asymptotic boundaries, resulting in a new dictionary entry relating the boundary deformation parameter to the bulk radial boundary cutoff. Applying the results to the DSSYK model, we show that the T2^22 deformation can probe regions of positive and approximately constant curvature in sine dilaton gravity. The dS2_22​ stretched horizon holographic proposal by Susskind \cite{Susskind:2021esx} is recovered as the radial cutoff approaches the horizons of the bulk dual. The thermodynamic quantities of the system exhibit enhanced growth in this limit. In the deformed boundary theory, we uncover phase transitions from thermodynamically stable to unstable configurations depending on its temperature, which also scrambles information at a (hyper-)fast rate in terms of its out-of-time-ordered correlator.
A bstract In this work, we formulate the holographic dictionary for the double-scaled SYK (DSSYK) model with matter operators. Based on the two-sided Hartle-Hawking (HH) state, we derive several properties of the DSSYK model, without making assumptions about the specific dual theory, including its semiclassical thermodynamics, correlation functions, and Krylov complexity. We derive these quantities from the saddle points of the DSSYK path integral preparing the HH state. We also construct a Lanczos algorithm that simultaneously evaluates Krylov state and operator complexity in the two-sided Hamiltonian system including finite temperature effects. In the semiclassical limit, both measures are encoded in the saddle points of the path integral. They have a bulk interpretation in terms of minimal geodesic lengths in an effective AdS 2 space with matter backreaction. Different saddle points correspond to geodesic distances with different evolution, and they display different scrambling properties. We also discuss about the quantization of the bulk theory dual to the DSSYK model. At last, we formulate the double-scaled algebra [ JHEP 11 (2022) 060] in bulk terms, and the dual entry of the proper radial momentum of a bulk probe.
We investigate entanglement entropy between the pair of type II1_11​ algebras of the double-scaled SYK (DSSYK) model given a chord state, its holographic interpretation as generalized horizon entropy; particularly in the (anti-)de Sitter ((A)dS) space limits of the bulk dual; and its connection with Krylov complexity. The density matrices in this formalism are operators in the algebras, which are specified by the choice of global state; and there exists a trace to evaluate their von Neumann entropy since the algebras are commutants of each other, which leads to a notion of algebraic entanglement entropy. We match it in triple-scaling limits to an area computed through a Ryu-Takayanagi formula in (A)dS2_22​ space with entangling surfaces at the asymptotic timelike or spacelike boundaries respectively; providing a first-principles example of holographic entanglement entropy for (A)dS2_22​ space. This result reproduces the Bekenstein-Hawking and Gibbons-Hawking entropy formulas for specific entangling regions points, while it decreases for others. This construction does not display some of the puzzling features in dS holography. The entanglement entropy remains real-valued since the theory is unitary, and it depends on the Krylov spread complexity of the Hartle-Hawking state. At last, we discuss higher dimensional extensions.