Quartic Tensor Models

TL;DR

This paper studies quartic tensor models, focusing on their mathematical properties, map expansions, and introducing enhanced models with novel perturbative behaviors, advancing understanding of tensor models in quantum gravity.

Contribution

It provides rigorous results on analyticity and bounds for quartic tensor models and introduces enhanced models with distinct perturbative characteristics.

Findings

Proved analyticity and bounds for tensor model cumulants

Rewritten quartic models as multi-matrix models via intermediate field representation

Introduced and analyzed enhanced models with new perturbative behavior

Abstract

Tensor models are measures for random tensors. They generalise matrix models and were developed to study random geometry in arbitrary dimension. Moreover, they are strongly connected to quantum gravity theories as additionally to the standard bare-bones models, they encompass the field theoretical approach to loop quantum gravity known as group field theory. In the present thesis, we focus on the restricted case of quartic tensor models, for which a far greater number of rigorous mathematical results have been proven. Quartic models can be re-written as multi-matrix models using the intermediate field representation, and their perturbative expansions can be written as series expansions over combinatorial maps. Using a variety of map expansions, we prove analyticity results and useful bounds for the cumulants of various tensor models : the most general standard quartic model at any rank and the simplest renormalisable tensor field theory at rank 3. Then, we introduce a new class of models, the enhanced models, which perturbative expansions display new behaviour, different to the so called melonic behaviour that characterise most known tensor models so far.