Weighting bubbles in group field theory

TL;DR

This paper introduces a modified group field theory model that incorporates extra indices to track bubbles in Feynman diagrams, revealing new symmetries and topological invariants related to algebraic structures.

Contribution

It proposes a novel GFT model with additional indices encoding bubble structures, leading to new symmetries and topological invariants in three dimensions.

Findings

New symmetries interpreted as vertex translations

Extra indices encode semi-simple algebra structures

Associative algebra choice yields bubble-based topological invariants

Abstract

Group field theories (GFT) are higher dimensional generalizations of matrix models whose Feynman diagrams are dual to triangulations. Here we propose a modification of GFT models that includes extra field indices keeping track of the bubbles of the graphs in the Feynman evaluations. In dimension three, our model exhibits new symmetries, interpreted as the action of the vertex translations of the triangulation. The extra field indices have an elegant algebraic interpretation: they encode the structure of a semi-simple algebra. Remarkably, when the algebra is chosen to be associative, the new structure contributes a topological invariant from each bubble of the graph to the Feynman amplitudes.