A strong interaction is known to exist between edge-colored graphs (which encode PL pseudo-manifolds of arbitrary dimension) and random tensor models (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 tensor models context. In the present paper - by making use of combinatorial properties concerning Hamiltonian decompositions of the complete graph - we prove that, in any even dimension d greater or equal to 4, the G-degree of all bipartite graphs, as well as of all (bipartite or non-bipartite) graphs representing singular manifolds, is an integer multiple of (d-1)!. As a consequence, in even dimension, the terms of the 1/N expansion corresponding to odd powers of 1/N are null in the complex context, and do not involve colored graphs representing singular manifolds in the real context. In particular, in the 4-dimensional case, where the G-degree is shown to depend only on the regular genera with respect to an arbitrary pair of "associated" cyclic permutations, several results are obtained, relating the G-degree or the regular genus of 5-colored graphs and the Euler characteristic of the associated PL 4-manifolds.

Combinatorial properties of the G-degree / Casali, M. R.; Grasselli, L.. - In: REVISTA MATEMATICA COMPLUTENSE. - ISSN 1139-1138. - 32:(2019), pp. 239-254. [10.1007/s13163-018-0279-0]

Combinatorial properties of the G-degree

M. R. Casali
;
L. Grasselli
2019

Abstract

A strong interaction is known to exist between edge-colored graphs (which encode PL pseudo-manifolds of arbitrary dimension) and random tensor models (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 tensor models context. In the present paper - by making use of combinatorial properties concerning Hamiltonian decompositions of the complete graph - we prove that, in any even dimension d greater or equal to 4, the G-degree of all bipartite graphs, as well as of all (bipartite or non-bipartite) graphs representing singular manifolds, is an integer multiple of (d-1)!. As a consequence, in even dimension, the terms of the 1/N expansion corresponding to odd powers of 1/N are null in the complex context, and do not involve colored graphs representing singular manifolds in the real context. In particular, in the 4-dimensional case, where the G-degree is shown to depend only on the regular genera with respect to an arbitrary pair of "associated" cyclic permutations, several results are obtained, relating the G-degree or the regular genus of 5-colored graphs and the Euler characteristic of the associated PL 4-manifolds.
28-set-2018
32
239
254
Combinatorial properties of the G-degree / Casali, M. R.; Grasselli, L.. - In: REVISTA MATEMATICA COMPLUTENSE. - ISSN 1139-1138. - 32:(2019), pp. 239-254. [10.1007/s13163-018-0279-0]
Casali, M. R.; Grasselli, L.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11380/1166815
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