2017/04/30 by María Rita Casali, Maria Rita Casali, Paola Cristofori +3 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Combinatorics #Degree (music) #Dimension (graph theory) #Geometry #Mathematics #Noncommutative and Quantum Gravity Theories #Physics #Pure mathematics #Representation (politics) #Surface (topology) #Tensor (intrinsic definition) #Topological and Geometric Data Analysis #Topology (electrical circuits) #hep-th #math-ph #math.CO #math.GT #math.MP #msc:57M15 #msc:57N10 #msc:57N13 #msc:57Q15 #msc:57Q25 #msc:83E99
paper · pdf · doi:10.1016/j.geomphys.2018.01.001
34 pages, 6 figures; improvements suggested by referee. Published online 8 January 2018 by the Journal of Geometry and Physics
openalex publication_date 2018/01/08 · arxiv created 2018/03/07 · arxiv updated 2018/03/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.