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The Loop Vertex Expansion (LVE) is a constructive technique using canonicalcombinatorial tools. It works well for quantum field theories withoutrenormalization, which is the case of the field theory studied in this paper.Tensorial Group Field Theories (TGFT) are a new class of field theoriesproposed to quantize gravity. This paper is devoted to a very simple TGFT forrank 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.

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