Constructive Tensorial Group Field Theory I: The $U(1)-T^4_3$ Model

TL;DR

This paper applies the Loop Vertex Expansion to a simple tensorial group field theory with U(1) symmetry, demonstrating its Borel summability and providing a constructive approach to quantum gravity models.

Contribution

It introduces a constructive analysis of a rank three tensorial group field theory using LVE, showing Borel summability without renormalization.

Findings

The $U(1)$-$T^4_3$ model has no ultraviolet divergence.

The model is proven to be Borel summable in its coupling constant.

LVE effectively constructs the theory without renormalization.

Abstract

The Loop Vertex Expansion (LVE) is a constructive technique using canonical combinatorial tools. It works well for quantum field theories without renormalization, which is the case of the field theory studied in this paper. Tensorial Group Field Theories (TGFT) are a new class of field theories proposed to quantize gravity. This paper is devoted to a very simple TGFT for rank three tensors with U(1)group and quartic interactions, hence nicknamed $U(1)$-$T^4_3$. It has no ultraviolet divergence, and we show, with the LVE, that it is Borel summable in its coupling constant.