A simple proof of orientability in colored group field theory

TL;DR

The paper addresses whether graphs in colored group field theory generate orientable piecewise-linear manifolds. By connecting colored GFT graphs to crystallization theory and 3-gems, it establishes an orientability criterion based on bipartiteness and shows that vacuum graphs from the colored Boulatov model yield closed orientable PL manifolds in any dimension. The main result attributes orientability to the presence of two opposite-orientation vertices and demonstrates that fusion moves preserve bipartiteness, thereby maintaining orientability. This provides a rigorous, color-driven foundation for orientable topologies in cGFT and supports the broader topological control offered by colored models.

Abstract

In this short note we use results from the theory of crystallizations to prove that color in group field theories garantees orientability of the piecewise linear pseudo-manifolds associated to each graph generated perturbatively. The origin of orientability is the presence of two interaction vertices.

Figures (3)

• Figure 1: Colored GFT red and black vertices.
• Figure 2: A gluing using a colored propagator.
• Figure 3: Fusion moves on a 4-regular 4-edge colored graph of 1-, 2- and 3- dipoles respectively.