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.
Highlights
The problem of gravity quantization is a well-known and deeply investigated issue in the community of theoretical and mathematical physicists
In this paper we show that the G-degree of a closed 3-manifold is nothing but its gem-complexity
It is to be noted that the Gurau degree shares with the gem-complexity the finiteness property stated in the above theorem, while for the regular genus the same property does not hold for d = 3 and it is unknown in higher dimension
Summary
The problem of gravity quantization is a well-known and deeply investigated issue in the community of theoretical and mathematical physicists. Non-orientable) d-pseudomanifold: the vertices of the graph represent the d-simplices of P and the colored edges of the graph describe the pairwise gluing in P of the (d−1)-faces of its maximal simplices (the graph becomes the dual 1-skeleton of P ) In this framework, many results have been achieved during the last 40 years; noteworthy are the classification results obtained in dimensions 3 and 4 with respect to the PL-manifold invariants regular genus and gem-complexity, introduced and investigated in GEM theory with geometric topology aims (see for example [10] for the 3-dimensional case, [12] and [14] for the 4-dimensional one).
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