Symmetry Sectors in Chord Space and Relational Holography in the DSSYK

TL;DR

The paper develops a boundary-centric framework to gauge symmetries in the chord Hilbert space of the DSSYK model with matter, producing distinct symmetry sectors that correspond to ETW branes, Euclidean wormholes, and relative time translations in a doubled DSSYK setup. By imposing Dirac-type constraints, it derives ASC-type Hamiltonians, constructs Hartle-Hawking states, analyzes path-integral saddle points, and computes disk and cylinder correlators, establishing a holographic dictionary that equates wormhole geodesics with boundary spread complexity $\lambda\mathcal{C}(t)$. It further shows Euclidean wormholes are perturbatively stable with a one-dimensional baby-universe Hilbert space (absent correlated matter) and derives multi-boundary partition functions, linking to alpha-states in the baby-universe framework. The doubled DSSYK construction is connected to dS$_3$ holography via a relational, clock-based algebra of observables, offering a boundary realization that intersects with, and clarifies, proposals based on sine dilaton gravity and NV-type dS$_3$ models. Overall, the work provides a solvable setting in which gauged boundary symmetries yield a coherent boundary-to-bulk map, relational holography, and a concrete operational path to exploring higher-genus wormholes and closed-universe holography.

Abstract

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.

Figures (8)

• Figure 2: Bulk dual of the DSSYK model with constraints: (a) a one-sided ETW brane and (b) two-sided symmetric ETW brane (glued together through the orange arrows). An AdS$_2$ black hole background has an ETW brane (orange) with tension $m_{\rm ETW}$ (\ref{['eq:dictionary 3']}), dual to a heavy operator $\hat{\mathcal{O}}_\Delta$. The red line denotes a minimal length geodesic curve (which defines the distance $L_{\rm ETW}=\lambda~\mathcal{C}(t)$ (\ref{['eq:dictionary 1']})) starting the asymptotic boundary at time $u$ in the bulk coordinates (\ref{['eq:effective geometry']}) (with $\tau=\mathrm i u+\beta/2$) and reaching the ETW brane.
• Figure 3: Comparison plot between the spread complexity of the (a) one-sided brane system (\ref{['eq:length ETW brane']}), and (b) a two-sided symmetric brane system (\ref{['eq:length Z2 wormhole length']}). We are taking $\lambda=0.001$, $\lambda\Delta=1$, $J=1$, and $\theta$ decreases from $\theta=\pi/2$ (orange) in multiples of $0.3$ from top to bottom at late times. In both cases, there is a early time parabolic growth of Krylov complexity, and late time growth. Note the different rates of growth depending on the energy $E(\theta)$ of the solution.
• Figure 4: (a) Half wormhole wavefunction (\ref{['eq:half wormhole ETW']}), and (b) double trumpet partition function (\ref{['eq:double trumpet']}). $b$ is the circumference of the curve in the boundary, representing the origin of the constrained Hamiltonian (\ref{['eq:ASC Hamiltonian']}). The boundary parameters $X$ and $Y$ are shown in (\ref{['eq:XY 1s']}) and (\ref{['eq:Z2 sym condition']}) for the one and two-sided brane models respectively.
• Figure 5: Cylinder two-point correlation function (\ref{['eq:cylinder amplitude']}) of the gauged DSSYK Hamiltonian (\ref{['eq:ASC Hamiltonian']}). $\hat{\mathcal{O}}_{\Delta}$ (blue) represent the matter chord insertions.
• Figure 6: Effective action in the sum (\ref{['eq:final cylinder amplitude']}) at fixed length $\ell=\lambda n$ for the one-sided (1s) (\ref{['eq:XY 1s']}), and the two-sided (2s) symmetric (\ref{['eq:Z2 sym condition']}) ETW brane models. $\ell$ takes values (a) $\sim\mathcal{O}(1)$, and (b) $\sim\mathcal{O}(10^2)$. The chosen parameters are $J\beta=-1$, $\Delta_w=1$, $\lambda=10^{-4}$, and the different values of $\Delta$ shown in the figure. Very similar results are obtained for other parameters. The asymptotic evolution shows that (\ref{['eq:final cylinder amplitude']}) always has a global minimum.