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arXiv:2602.06113

Deforming the Double-Scaled SYK & Reaching the Stretched Horizon From Finite Cutoff Holography

Sergio E. Aguilar-Gutierrez

hep-thgr-qcquant-phSubmitted Feb 5, 2026• Announced Feb 9, 2026• 0 citations
v1 · Feb 5, 2026v2 · Feb 26, 2026v3 · May 9, 2026v4 · Aug 6, 2026v5 · Aug 20, 2026
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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 $\text{T}\bar{\text{T}}(+Λ_2)$ deformations, denoted $T^2(+Λ_1)$, as special cases. In general, the deformation mixes the chord basis of the Hilbert space in the seed theory, which we order through a modification of the Lanczos algorithm. The resulting chord number in the ordered basis represents a wormhole length at a finite cutoff in the bulk. We study the thermodynamic properties of the deformed theory; the evolution of $n$-point correlation functions with matter chords; the growth of complexity of the Hartle-Hawking state; and the entanglement entropy between the double-scaled algebras 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 $T^2$ and $T^2+Λ_1$ 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.

Comments
61+40 pgs, 16 figs, 1 tab

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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 a modification of the Lanczos algorithm. The resulting chord number in the ordered basis represents a wormhole length at a finite cutoff in the bulk. We study the thermodynamic properties of the deformed theory; the evolution of nnn-point correlation functions with matter chords; the growth of complexity of the Hartle-Hawking state; and the entanglement entropy between the double-scaled algebras 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 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, which we interpret 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.
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.
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.
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.
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.