Combinatorial properties of the G-degree

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 {\it G-degree} of the involved graphs, which drives the {\it $1/N$ 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 $d\ge 4$, 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.

13 pages, 2 figures. Some improvements suggested by referees, Revista Matematica Complutense, published online 28 September 2018. arXiv admin note: text overlap with arXiv:1706.07267

References in corpus (3)

• Random tensor models in the large N limit: Uncoloring the colored tensor models
• Topology in colored tensor models via crystallization theory
• G-degree for singular manifolds

Cited by in corpus (4)

• Topology in colored tensor models via crystallization theory
• Classifying compact 4-manifolds via generalized regular genus and G-degree
• Crystallizations of compact 4-manifolds minimizing combinatorially defined PL-invariants
• Compact n-manifolds via (n+1)-colored graphs: a new approach