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We study the functional renormalization group of a three-dimensionaltensorial Group Field Theory (GFT) with gauge group SU(2). This model generates(generalized) lattice gauge theory amplitudes, and is known to beperturbatively renormalizable up to order 6 melonic interactions. We consider aseries of truncations of the exact Wetterich--Morris equation, which retainincreasingly many perturbatively irrelevant melonic interactions. Thistensorial analogue of the ordinary local potential approximation allows toinvestigate the existence of non-perturbative fixed points of therenormalization group flow. Our main finding is a candidate ultraviolet fixedpoint, whose qualitative features are reproduced in all the truncations we havechecked (with up to order 12 interactions). This may be taken as evidence foran ultraviolet completion of this GFT in the sense of asymptotic safety.Moreover, this fixed point has a single relevant direction, which suggests thepresence of two distinct infrared phases. Our results generally support theexistence of GFT phases of the condensate type, which have recently beenconjectured and applied to quantum cosmology and black holes.

Asymptotic safety in three-dimensional SU(2) Group Field Theory: evidence in the local potential approximation