Colored Group Field Theory

TL;DR

This work introduces a fermionic colored Group Field Theory with $n+1$ colors and develops a bubble-based homological framework to study the graphs directly. It proves that colored GFT graphs form abstract cellular complexes and derives minimal and maximal bubble homology groups, using spanning-tree contractions and dual arguments. It then connects Feynman amplitudes to the fundamental group of the bubble complex, showing that graphs dual to manifold space-times are constrainingly homotopically trivial and that planar 3-color bubbles are a necessary condition for triviality in 3D. Overall, the paper provides a rigorous combinatorial/topological toolkit for analyzing GFT graphs and their potential links to discretized gravity, with avenues for exploring instantons and 3D topology via bubble complexes.

Abstract

Group field theories are higher dimensional generalizations of matrix models. Their Feynman graphs are fat and in addition to vertices, edges and faces, they also contain higher dimensional cells, called bubbles. In this paper, we propose a new, fermionic Group Field Theory, posessing a color symmetry, and take the first steps in a systematic study of the topological properties of its graphs. Unlike its bosonic counterpart, the bubbles of the Feynman graphs of this theory are well defined and readily identified. We prove that this graphs are combinatorial cellular complexes. We define and study the cellular homology of this graphs. Furthermore we define a homotopy transformation appropriate to this graphs. Finally, the amplitude of the Feynman graphs is shown to be related to the fundamental group of the cellular complex.

Figures (5)

• Figure 1: nD vertex.
• Figure 2: Stranded and colored graph.
• Figure 3: A first example of 3D graph.
• Figure 4: A second example of 3D graph.
• Figure 5: A third example of a graph.

Theorems & Definitions (6)

• Definition 1
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5