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

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