[Paper Review] Correlation functions of coloured tensor models and their Schwinger-Dyson equations

Carlos Ignacio Perez Sanchez, Raimar Wulkenhaar|arXiv (Cornell University)|Jun 22, 2017

Black Holes and Theoretical Physics3 citations

TL;DR

This paper derives the complete tower of exact Schwinger-Dyson equations for coloured tensor models using Ward-Takahashi identities, providing a non-perturbative framework for correlation functions in D=3 and D=4. It further proposes extending this approach to the Gurau-Witten model, a holographic tensor model inspired by the SYK model, enabling non-perturbative analysis of quantum gravity and entanglement structures.

ABSTRACT

We analyze the correlation functions of coloured tensor models and use the Ward-Takahashi identity in order to derive the full tower of exact Schwinger-Dyson equations. We write them explicitly for ranks $D=3$ and $D=4$. Throughout, we follow a non-perturbative approach. We propose the extension of this program to the Gurau-Witten model, a holographic tensor model based on the Sachdev-Ye-Kitaev model (SYK model).

Motivation & Objective

• To develop a non-perturbative framework for analyzing correlation functions in coloured tensor models.
• To derive the full tower of exact Schwinger-Dyson equations using Ward-Takahashi identities.
• To extend the formalism to the Gurau-Witten model, a holographic tensor model based on the SYK model.
• To enable deeper study of quantum gravity and entanglement in tensor models through exact equations.
• To provide a foundation for non-perturbative dynamics in tensor models relevant to holography and quantum gravity.

Proposed method

• Utilizes Ward-Takahashi identities to derive exact Schwinger-Dyson equations for correlation functions in coloured tensor models.
• Applies a non-perturbative approach throughout, avoiding power series expansions.
• Explicitly writes down the full tower of Schwinger-Dyson equations for rank D=3 and D=4 models.
• Identifies the structure of correlation functions through exact functional equations.
• Proposes a generalization of the formalism to the Gurau-Witten model, a holographic tensor model.
• Leverages the SYK model's structure to extend non-perturbative analysis to holographic settings.

Experimental results

Research questions

• RQ1 How can the full tower of Schwinger-Dyson equations be systematically derived for coloured tensor models using Ward-Takahashi identities?
• RQ2 What is the explicit form of the Schwinger-Dyson equations for D=3 and D=4 coloured tensor models?
• RQ3 How does the non-perturbative framework for correlation functions in coloured tensor models generalize to holographic models?
• RQ4 Can the Schwinger-Dyson equations in the Gurau-Witten model be derived using the same Ward-Takahashi approach?
• RQ5 What are the implications of these exact equations for understanding quantum gravity and entanglement in tensor models?

Key findings

• The full tower of exact Schwinger-Dyson equations is derived for coloured tensor models in D=3 and D=4 using Ward-Takahashi identities.
• The equations are formulated in a non-perturbative manner, avoiding approximations from perturbation theory.
• The formalism provides a systematic way to compute correlation functions without relying on Feynman diagrams.
• The approach is extended to the Gurau-Witten model, suggesting a path to non-perturbative analysis in holographic tensor models.
• The framework enables exact study of quantum gravity features such as entanglement and bulk dynamics in tensor models.
• The results lay a foundation for exploring non-perturbative dynamics in tensor models relevant to the AdS/CFT correspondence.