Colored Tensor Models - a Review (1109.4812v3)

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.

• The paper presents colored tensor models as an extension of matrix models to higher dimensions, detailing the 1/N expansion and topological classification.
• It meticulously surveys methodological frameworks, including Gaussian measures, Feynman graph expansions, and Schwinger-Dyson equations, to compute quantum observables.
• The review discusses implications for quantum gravity and phase transitions, highlighting melonic graphs and continuum limits as key insights.

Colored Tensor Models: A Review

This paper offers an extensive review of colored tensor models, depicting their relevance as a potent computational tool in the exploration of random geometry in dimensions beyond two. Authored by Razvan Gurau and James P. Ryan, the article encapsulates the breadth of mathematical and theoretical constructs integral to colored tensor models.

Colored tensor models inherit their foundational principles from matrix models, which describe fluctuating two-dimensional surfaces. A key element in both frameworks is the encoding of theories into combinatorial entities such as Feynman graphs that represent topological spaces. These models encompass features like a $1/N$ expansion — a vital tool establishing how graph amplitudes scale — Schwinger–Dyson equations, and classical solutions. However, colored tensor models extend these concepts to higher dimensions, primarily utilizing tensors instead of matrices.

Concepts and Tools

The narrative unfolds with an introduction to quantum field theories characterized as probability measures for random functions, detailing their correlation functions and partition functions, pivotal in computation and understanding of physical observables. The paper explores Gaussian measures — the cornerstone of quantum field theories — employed to calculate these observables through perturbative expansions leading to Feynman graphs.

Subsequent sections introduce the specifics of colored tensor models, each defined by a collection of random fields indexed by colors. The rigor of these models is solidified as they illustrate a strong parallel to crank up the ancestry of matrix models, enriched now by the introduction of higher-dimensional cellular complexes through the bubbling mechanism.

Topological and Combinatorial Structure

The review uniquely dissects the topology of colored graphs, underscoring their nested structure composed of bubbles, from $0$-cells (vertices) up to $D$-cells. These models encapsulate a $D$-dimensional cellular complex structure, dual to finitely abstract simplicial complexes — pseudo-manifolds which incorporate a broader scope beyond matrix models.

The combinatorics embedded within these models are further exemplified through structures such as dipole moves — pivotal in manipulating graphs — and jackets, Riemann surfaces that assert topological equivalence and aid the definition of graph degree. Notably, the degree helps classify graphs and establishes the $1/N$ expansion pivotal in elucidating the statistical behavior of random spaces.

Tensor Models

Engaging with tensor models, the paper transitions from abstract formulations to specific examples — notably the independent identically distributed (i.i.d.) model — revealing its amplitude dependence solely on graph degree. The $1/N$ expansion unfolds into combinatorial and topological variants, providing a structured methodology to comprehend the unique scaling behavior of graphs.

Critical Behavior

The critical behavior section is a profound exploration into the continuum limit, where melonic graphs reveal an intriguing correspondence with colored rooted $(D+1)$-ary trees. Such mappings enable the resummation of the leading order series and extrapolate a phase transition to a continuum theory of large spaces. Despite their spherical topology, the precise geometry of these spaces remains an open field of inquiry.

Schwinger-Dyson Equations and Classical Solutions

The paper progresses into the space of Schwinger-Dyson equations, offering a purely algebraic vision — transforming traditional quantum equations of motion into life-like framework representations. The pursuit of classical solutions marks a return to foundational aspects, where the discussion centers around splitting degrees of freedom and analyzing quantum fluctuations around explicit classical solutions.

Conclusion and Future Directions

With a forward-looking intent, the authors extrapolate colored tensor models beyond their initial formulations — threading applications through quantum gravity, matrix models, and statistical physics. While melonic graphs maintain dominance in amplitude scaling, broader theoretical constructs promise a multiplicity of critical behaviors and universal principles, paving the pathway for potential future endeavors in both theoretical refinement and applicability.

The paper clearly demarcates the terrain of colored tensor models as vast, balancing between meticulous mathematical constructs and speculative theoretical applications, ensuring a comprehensive resource for researchers immersed in advancing quantum field theory and random geometry.