Geometry of Chord Intertwiner, Multiple Shocks and Switchback in Double-Scaled SYK

TL;DR

This paper explores the bulk Hilbert space structure of the double-scaled SYK model, introducing intertwiners and path integral methods to analyze shockwave configurations, chaos, and the switchback effect in holographic complexity.

Contribution

It provides a systematic derivation of correlation functions with matter insertions, develops a path integral framework for multiple shockwaves, and clarifies the microscopic origin of the switchback effect in DSSYK.

Findings

Correlation functions with matter insertions are derived systematically.

Shockwave geometries in the semiclassical limit are characterized.

Sub-maximal chaos is observed at finite temperature, consistent with scramblon dynamics.

Abstract

We revisit the bulk Hilbert space interpretation of chords in the double-scaled SYK (DSSYK) model and introduce a notion of intertwiner that constructs bulk states from states with fixed boundary conditions. This leads to an isometric map that factorizes the one-particle bulk Hilbert space into a tensor product of two boundary Hilbert spaces without particle insertion. The map enables a systematic derivation of a family of correlation functions with arbitrary finite amount of matter insertions, relevant for capturing the switchback effect-a feature of holographic complexity. We further develop a path integral framework that describes multiple shockwave configurations in the semiclassical limit. For the two-body scattering processes in semi-classical regime, we show it exhibits sub-maximal chaos at finite temperature, consistent with the scramblon dynamics associated with the "fake disk" geometry. The effective "fake temperature" governing this behavior emerges from the semiclassical limit of the quantum $6 j$-symbol associated with the out-of-time-order correlator. We further analyze multi-shockwave configurations and derive precise conditions under which the switchback effect is realized, both in terms of the total chord number and the Krylov complexity of precursor operators. Our results clarify the structure of correlation functions with multiple operator insertions, their bulk interpretation in terms of shockwave geometries in the semiclassical regime, and provide a microscopic derivation of the switchback effect in the DSSYK model.