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Space-Time in the SYK Model

Sumit R. Das, Animik Ghosh, Antal Jevicki, Kenta Suzuki

TL;DR

The paper tackles the problem of identifying the bulk spacetime dual to the Euclidean SYK model. It demonstrates that a non-local Radon-type transform on bi-local $n$-point Green's functions maps to Euclidean AdS$_2$ with leg factors, signaling coupling to additional bulk states akin to discrete 2D string states. By formulating these Leg transformations as a canonical map in phase space, the work clarifies how Euclidean bulk geometry can be reconciled with the SYK bi-local data and provides a framework for incorporating extra bulk degrees of freedom. The results point toward a richer, nonlocal bulk reconstruction, potentially extending to a three-dimensional perspective and connections to $c=1$ string theory.

Abstract

We consider the question of identifying the bulk space-time of the SYK model. Focusing on the signature of emergent space-time of the (Euclidean) model, we explain the need for non-local (Radon-type) transformations on external legs of $n$-point Green's functions. This results in a dual theory with Euclidean AdS signature with additional leg-factors. We speculate that these factors incorporate the coupling of additional bulk states similar to the discrete states of 2d string theory.

Space-Time in the SYK Model

TL;DR

The paper tackles the problem of identifying the bulk spacetime dual to the Euclidean SYK model. It demonstrates that a non-local Radon-type transform on bi-local -point Green's functions maps to Euclidean AdS with leg factors, signaling coupling to additional bulk states akin to discrete 2D string states. By formulating these Leg transformations as a canonical map in phase space, the work clarifies how Euclidean bulk geometry can be reconciled with the SYK bi-local data and provides a framework for incorporating extra bulk degrees of freedom. The results point toward a richer, nonlocal bulk reconstruction, potentially extending to a three-dimensional perspective and connections to string theory.

Abstract

We consider the question of identifying the bulk space-time of the SYK model. Focusing on the signature of emergent space-time of the (Euclidean) model, we explain the need for non-local (Radon-type) transformations on external legs of -point Green's functions. This results in a dual theory with Euclidean AdS signature with additional leg-factors. We speculate that these factors incorporate the coupling of additional bulk states similar to the discrete states of 2d string theory.

Paper Structure

This paper contains 6 sections, 80 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Question of Dual Spacetime
  3. Transformations and Leg Factors
  4. Green's Functions and Leg Factors
  5. Conclusion
  6. Unit Normalized EAdS/dS Wave Functions

Figures (1)

  • Figure 1: The de Sitter potential $V_{{\rm dS}_2}$ has bound states and scattering states. On the other hand, the Euclidean AdS potential $V_{{\rm AdS}_2}$ has only scattering modes.