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We study the polynomial Abelian or U(1)^d Tensorial Group Field Theoriesequipped with a gauge invariance condition in any dimension d. From ouranalysis, we prove the just renormalizability at all orders of perturbation ofthe phi^4_6 and phi^6_5 random tensor models. We also deduce that the phi^4_5tensor model is super-renormalizable.

Just Renormalizable TGFT's on U(1)^d with Gauge Invariance