Building the Holographic Dictionary of the DSSYK from Chords, Complexity & Wormholes with Matter

TL;DR

This paper develops a holographic dictionary for the DSSYK model with matter, deriving thermodynamics, correlation functions, and complexity measures from saddle points, and connecting them to bulk geodesics and algebraic structures.

Contribution

It introduces a holographic dictionary for DSSYK with matter, linking saddle point analysis to bulk geodesics, complexity, and algebraic structures without relying on a specific dual theory.

Findings

Derived thermodynamics and correlation functions from saddle points.

Constructed a Lanczos algorithm for Krylov complexity at finite temperature.

Connected complexity measures to bulk geodesic lengths in AdS$_2$ with matter.

Abstract

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 in bulk terms, and the dual entry of the proper radial momentum of a bulk probe.