The Bulk Dual of SYK: Cubic Couplings (1702.08016v2)

Abstract: The SYK model, a quantum mechanical model of $N \gg 1$ Majorana fermions $\chi_i$, with a $q$-body, random interaction, is a novel realization of holography. It is known that the AdS$_2$ dual contains a tower of massive particles, yet there is at present no proposal for the bulk theory. As SYK is solvable in the $1/N$ expansion, one can systematically derive the bulk. We initiate such a program, by analyzing the fermion two, four and six-point functions, from which we extract the tower of singlet, large $N$ dominant, operators, their dimensions, and their three-point correlation functions. These determine the masses of the bulk fields and their cubic couplings. We present these couplings, analyze their structure and discuss the simplifications that arise for large $q$.

• The paper leverages fermion correlation functions to extract the masses and cubic couplings of bulk fields.
• It employs a detailed 1/N expansion and analytic treatment of two-, four-, and six-point functions to relate operator dimensions to bulk field properties.
• The analysis, particularly in the large q limit, deepens understanding of the SYK model’s gravitational dual and its intricate interaction structure.

Overview of "The Bulk Dual of SYK: Cubic Couplings"

This paper, authored by David J. Gross and Vladimir Rosenhaus, explores the bulk dual of the Sachdev-Ye-Kitaev (SYK) model, focusing on the cubic couplings of its massive fields. The SYK model is a quantum mechanical system of $N \gg 1$ Majorana fermions with a $q$-body random interaction, exhibiting features suggestive of a gravitational dual. Although known to potentially connect with an AdS$_2$ dilaton gravity model, the precise nature of its bulk dual remains largely conjectural. The paper contributes to an ongoing program to construct this dual theory through detailed analysis of the SYK model's fermion correlation functions.

The authors leverage the solvability of the SYK model in the $1/N$ expansion to derive key bulk properties. They examine two-point, four-point, and six-point correlation functions of the fermions to extract properties of the corresponding massive, singlet scalars in the bulk. The paper primarily focuses on determining the masses and cubic couplings of these bulk fields.

Key Results

• Executable Analysis of Correlation Functions:

The authors systematically decompose and analyze the SYK fermion correlation functions. The four-point function reveals a tower of $O(N)$ singlet, bilinear operators, which are dual to massive fields in the bulk. Each operator dimension $h_n$ relates to the mass $m_n$ of the corresponding bulk field via the standard AdS/CFT dictionary, $m_n^2 = h_n(h_n-1)$.

• Evaluation of Cubic Couplings:

The paper meticulously calculates the cubic interactions among the massive fields by computing the six-point fermion correlation function, discerning contributions from "contact" and planar diagrams. Each class affects the three-point functions of the fermion bilinears distinctly, leading to exact expressions for the cubic interactions $\lambda_{n m k}$.

• Large $q$ Limit Simplification:

While the methodology applies across various parameters, significant simplification occurs for large $q$. In this regime, the fermion correlation functions and resulting bulk couplings receive a streamlined analytical treatment, offering compelling insight into the structure of the bulk dual.

Implications

This work represents a significant step in formalizing the bulk dual of the SYK model, shedding light on how concepts from holography translate into specific interactions among fields with gravitational relevance.

• Theoretical Implications:

The identification and calculation of cubic couplings enriches the understanding of the SYK model's possible gravitational dual, aligning it closely with the principles of AdS/CFT duality. The distinction between contact and planar contributions offers insight into how localized versus more distributed interactions might manifest in a gravitational context.

• Practical Implications:

Though largely theoretical, the results could inform computational approaches to similar models in condensed matter physics, where SYK-like interactions and chaos properties are useful for modeling poorly understood phenomena.

Future Directions

The paper sets the stage for further exploration into higher-order interactions and larger-point correlation functions within the SYK model, offering the tantalizing possibility of a fully realized bulk dual theory. Subsequent analysis could focus on determining quartic interactions or exploring how the effective dual bulk theory behaves when SYK parameters are varied, potentially unveiling new symmetries or simplifying structures.

Overall, Gross and Rosenhaus have provided an exhaustive, technical exploration that deepens our understanding of the bulk dual structure while advancing the framework for future investigation into this holographic model's intricacies.