Colored Group Field Theory

TL;DR

This paper introduces a new fermionic colored group field theory, analyzing its graph topologies, proving their cellular complex structure, and relating graph amplitudes to fundamental groups, advancing topological understanding in quantum gravity models.

Contribution

It proposes a fermionic colored group field theory and systematically studies its graph topologies, establishing their cellular complex structure and topological properties.

Findings

Graphs are combinatorial cellular complexes.

Cellular homology and homotopy are defined for these graphs.

Graph amplitudes relate to the fundamental group of the 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.