Bounding bubbles: the vertex representation of 3d Group Field Theory and the suppression of pseudo-manifolds

TL;DR

This paper introduces a vertex-based representation of 3D Group Field Theory that simplifies the analysis of Feynman graphs, leading to new bounds on amplitudes and suppression of pseudo-manifold configurations.

Contribution

It develops a novel vertex-dependent GFT representation that clarifies the structure of bubbles and provides optimal bounds on amplitudes, reducing pseudo-manifold contributions.

Findings

Derived new scaling bounds for regularized amplitudes.

Showed suppression of pseudo-manifold configurations.

Provided optimal bounds organized by bubble genus.

Abstract

Based on recent work on simplicial diffeomorphisms in colored group field theories, we develop a representation of the colored Boulatov model, in which the GFT fields depend on variables associated to vertices of the associated simplicial complex, as opposed to edges. On top of simplifying the action of diffeomorphisms, the main advantage of this representation is that the GFT Feynman graphs have a different stranded structure, which allows a direct identification of subgraphs associated to bubbles, and their evaluation is simplified drastically. As a first important application of this formulation, we derive new scaling bounds for the regularized amplitudes, organized in terms of the genera of the bubbles, and show how the pseudo-manifolds configurations appearing in the perturbative expansion are suppressed as compared to manifolds. Moreover, these bounds are proved to be optimal.