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Bi-Local Holography in the SYK Model

Antal Jevicki, Kenta Suzuki, Junggi Yoon

TL;DR

This work develops a bi-local, replica-based collective field formulation of the SYK model to establish explicit $1/N$ Feynman rules in terms of bi-local propagators and vertices. It demonstrates how these rules reproduce large-$N$ Schwinger-Dyson structure and can be interpreted as Witten-type diagrams for an AdS$_2$ bulk, with a dynamical time variable implemented to fix reparametrization zero modes via a Schwarzian action. The analysis provides a concrete mapping from bi-local fluctuations to AdS$_2$ bulk modes, including a detailed treatment of the propagator and cubic interactions, and discusses how higher-point interactions fit into the bulk picture. The framework lays groundwork for further explorations of AdS locality, finite-temperature extensions, and a more complete holographic dictionary in the SYK context.

Abstract

We discuss large $N$ rules of the Sachdev-Ye-Kitaev model and the bi-local representation of holography of this theory. This is done by establishing $1/N$ Feynman rules in terms of bi-local propagators and vertices, which can be evaluated following the recent procedure of Polchinski and Rosenhaus. These rules can be interpreted as Witten type diagrams of the dual AdS theory, which we are able to define at IR fixed point and off.

Bi-Local Holography in the SYK Model

TL;DR

This work develops a bi-local, replica-based collective field formulation of the SYK model to establish explicit Feynman rules in terms of bi-local propagators and vertices. It demonstrates how these rules reproduce large- Schwinger-Dyson structure and can be interpreted as Witten-type diagrams for an AdS bulk, with a dynamical time variable implemented to fix reparametrization zero modes via a Schwarzian action. The analysis provides a concrete mapping from bi-local fluctuations to AdS bulk modes, including a detailed treatment of the propagator and cubic interactions, and discusses how higher-point interactions fit into the bulk picture. The framework lays groundwork for further explorations of AdS locality, finite-temperature extensions, and a more complete holographic dictionary in the SYK context.

Abstract

We discuss large rules of the Sachdev-Ye-Kitaev model and the bi-local representation of holography of this theory. This is done by establishing Feynman rules in terms of bi-local propagators and vertices, which can be evaluated following the recent procedure of Polchinski and Rosenhaus. These rules can be interpreted as Witten type diagrams of the dual AdS theory, which we are able to define at IR fixed point and off.

Paper Structure

This paper contains 12 sections, 113 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Replica Collective Field Theory
  3. 1/N Diagrams
  4. Two-point Function
  5. AdS Bulk
  6. Time as a Dynamical Variable
  7. The Propagator
  8. Conclusion
  9. Higher Loop Diagrams
  10. Two- and Three-point Function
  11. Evaluation of the Time Coordinate Action
  12. Identity

Figures (6)

  • Figure 1: One-loop vacuum diagrams and their $n$-dependence.
  • Figure 2: Two-point function of (diagonal) bi-local fluctuation. The double line represents the full propagator of SYK model.
  • Figure 3: Three-point function of (diagonal) bi-local fluctuation. Different colors (red, blue, black) of double lines are used for non-planar diagrams. They intersect only at the four-point vertices.
  • Figure 4: First kind of two-loop vacuum diagrams and their $n$-dependence.
  • Figure 5: Second kind of two-loop vacuum diagrams and their $n$-dependence.
  • ...and 1 more figures