From a “geometric topology” point of view, the theory of manifold representation by means of edge-colored graphs has been deeply studied since 1975 and many results have been achieved: its great advantage is the possibility of encoding, in any dimension, every PL d-manifold by means of a totally combinatorial tool. Edge-colored graphs also play an important rôle within colored tensor models theory, considered as a possible approach to the study of Quantum Gravity: the key tool is the G-degree of the involved graphs, which drives the 1/N expansion in the higher dimensional tensor models context, exactly as it happens for the genus of surfaces in the two-dimensional matrix model setting. Therefore, topological and geometrical properties of the represented PL manifolds, with respect to the G-degree, have specific relevance in the tensor models framework, show-ing a direct fruitful interaction between tensor models and discrete geometry, via edge-colored graphs. In colored tensor models, manifolds and pseudomanifolds are (almost) on the same footing, since they constitute the class of polyhedra represented by edge-colored Feynman graphs arising in this context; thus, a promising research trend is to look for classification results concerning all pseudomanifolds represented by graphs of a given G-degree. In dimension 4, the goal has already been achieved - via singular 4-manifolds - for all compact PL 4-manifolds with connected boundary up to G-degree 24. In the same dimension, the existence of colored graphs encoding different PL mani-folds with the same underlying TOP manifold, suggests also to investigate the ability of tensor models to accurately reflect geometric degrees of freedom of Quantum Gravity.
TOPOLOGY IN COLORED TENSOR MODELS / Casali, Maria Rita; Cristofori, Paola; Grasselli, Luigi. - (2023). (Intervento presentato al convegno Convegno internazionale "Geometric Topology, Art, and Science" tenutosi a Modena, 8 giugno 2023 - Reggio Emilia, 9-10 giugno 2023 nel 8-10 giugno 2023).
TOPOLOGY IN COLORED TENSOR MODELS
Maria Rita Casali;Paola Cristofori;Luigi Grasselli
2023
Abstract
From a “geometric topology” point of view, the theory of manifold representation by means of edge-colored graphs has been deeply studied since 1975 and many results have been achieved: its great advantage is the possibility of encoding, in any dimension, every PL d-manifold by means of a totally combinatorial tool. Edge-colored graphs also play an important rôle within colored tensor models theory, considered as a possible approach to the study of Quantum Gravity: the key tool is the G-degree of the involved graphs, which drives the 1/N expansion in the higher dimensional tensor models context, exactly as it happens for the genus of surfaces in the two-dimensional matrix model setting. Therefore, topological and geometrical properties of the represented PL manifolds, with respect to the G-degree, have specific relevance in the tensor models framework, show-ing a direct fruitful interaction between tensor models and discrete geometry, via edge-colored graphs. In colored tensor models, manifolds and pseudomanifolds are (almost) on the same footing, since they constitute the class of polyhedra represented by edge-colored Feynman graphs arising in this context; thus, a promising research trend is to look for classification results concerning all pseudomanifolds represented by graphs of a given G-degree. In dimension 4, the goal has already been achieved - via singular 4-manifolds - for all compact PL 4-manifolds with connected boundary up to G-degree 24. In the same dimension, the existence of colored graphs encoding different PL mani-folds with the same underlying TOP manifold, suggests also to investigate the ability of tensor models to accurately reflect geometric degrees of freedom of Quantum Gravity.File | Dimensione | Formato | |
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