Topology in colored tensor models via crystallization theory

Abstract

The aim of this paper is twofold. On the one hand, it provides a review of the links between random tensor models, seen as quantum gravity theories, and the PL-manifolds representation by means of edge-colored graphs (crystallization theory). On the other hand, the core of the paper is to establish results about the topological and geometrical properties of the Gurau-degree (or G-degree) of the represented manifolds, in relation with the motivations coming from physics. In fact, the G-degree appears naturally in higher dimensional tensor models as the quantity driving their 1/N expansion, exactly as it happens for the genus of surfaces in the two-dimensional matrix model setting. In particular, the G-degree of PL-manifolds is proved to be finite-to-one in any dimension, while in dimension 3 and 4 a series of classification theorems are obtained for PL-manifolds represented by graphs with a fixed G-degree. All these properties have specific relevance in the tensor models framework, showing a direct fruitful interaction between tensor models and discrete geometry, via crystallization theory.

34 pages, 6 figures; improvements suggested by referee. Published online 8 January 2018 by the Journal of Geometry and Physics

References in corpus (4)

• Random tensor models in the large N limit: Uncoloring the colored tensor models
• Diagrammatics of a colored SYK model and of an SYK-like tensor model, leading and next-to-leading orders
• G-degree for singular manifolds
• Combinatorial properties of the G-degree

Cited by in corpus (18)

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• Combinatorial study of graphs arising from the Sachdev-Ye-Kitaev model
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• From Kronecker to tableau pseudo-characters in tensor models
• G-degree for singular manifolds
• Colored discrete spaces: higher dimensional combinatorial maps and quantum gravity
• Maximizing the number of edges in three-dimensional colored triangulations whose building blocks are balls
• Combinatorial properties of the G-degree
• Tensor models with generalized melonic interactions
• Classifying compact 4-manifolds via generalized regular genus and G-degree
• Crystallizations of compact 4-manifolds minimizing combinatorially defined PL-invariants
• Tensor Field Theories: Renormalization and Random Geometry
• The Non-melonic Sector of Tensor Models and Gravity
• Transition Amplitudes in 3D Quantum Gravity: Boundaries and Holography in the Coloured Boulatov Model
• Trisections of PL 4-manifolds arising from colored triangulations
• Compact n-manifolds via (n+1)-colored graphs: a new approach