Colored Tensor Models - a Review

TL;DR

Colored tensor models are a promising extension of matrix models for studying random geometry in higher dimensions, sharing many properties and offering new insights into topological and geometric structures.

Contribution

This review provides a comprehensive introduction to colored tensor models, highlighting their properties, recent developments, and future research directions in higher-dimensional quantum geometry.

Findings

Share properties with matrix models, including Feynman graphs encoding topological spaces

Possess a 1/N expansion and continuum large volume limit

Exhibit Schwinger-Dyson equations forming a Lie algebra

Abstract

Colored tensor models have recently burst onto the scene as a promising conceptual and computational tool in the investigation of problems of random geometry in dimension three and higher. We present a snapshot of the cutting edge in this rapidly expanding research field. Colored tensor models have been shown to share many of the properties of their direct ancestor, matrix models, which encode a theory of fluctuating two-dimensional surfaces. These features include the possession of Feynman graphs encoding topological spaces, a 1/N expansion of graph amplitudes, embedded matrix models inside the tensor structure, a resumable leading order with critical behavior and a continuum large volume limit, Schwinger-Dyson equations satisfying a Lie algebra (akin to the Virasoro algebra in two dimensions), non-trivial classical solutions and so on. In this review, we give a detailed introduction of colored tensor models and pointers to current and future research directions.