TOPOLOGY IN COLORED TENSOR MODELS

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.