Functional renormalisation group approach for tensorial group field theory: a rank-3 model

Abstract

Abstract:We set up the Functional Renormalisation Group formalism for Tensorial Group Field Theory in full generality. We then apply it to a rank-3 model over U(1) 3 , endowed with a kinetic term linear in the momenta and with nonlocal interactions. The system of FRG equations turns out to be non-autonomous in the RG flow parameter. This feature is explained by the existence of a hidden scale, the radius of the group manifold. We investigate in detail the opposite regimes of large cut-off (UV) and small cut-off (IR) of the FRG equations, where the system becomes autonomous, and we find, in both case, Gaussian and non-Gaussian fixed points. We derive and interpret the critical exponents and flow diagrams associated with these fixed points, and discuss how the UV and IR regimes are matched. Finally, we discuss the evidence for a phase transition from a symmetric phase to a broken or condensed phase, from an RG perspective, finding that this seems to exist only in the approximate regime of very large radius of the group manifold, as to be expected for systems on compact manifolds.

Group field theory in dimension4−ϵ

“…However the reader should be reminded that the equations we used to determine this fixed point are valid only in the deep ultraviolet sector. Hence one should expect a trajectory starting at (u * 2 , u * 4 ) in the UV to also start moving when reaching small enough values of the scale k. This is a manifestation of the non-autonomous nature of the exact flow, as also recently observed in [33]. A second consequence of the presence of the non-trivial fixed point is a focusing of the trajectories in the u 2 direction towards the infrared.…”

“…It represents a good opportunity for further applications of the FRG to GFTs, which would be undeniably valuable if they were available in 4d quantum gravity models. The FRG has recently been applied to matrix models and TGFTs without gauge invariance condition [33,49,50], hence we think that the D = 3 model of the present paper is the natural next step in this research direction, and it could provide a UV completion of otherwise diverging renormalization group trajectories. We also proposed a speculative idea towards a more physical application, would the existence of this fixed point be confirmed: its presence might favor dynamically generated ultra-local theories in the infrared, hence possibly evading the question of the gravitational interpretation of the bare Laplace-type propagator.…”

“…The sign ≈ indicates that the short-time asymptotics of the heat-kernel is invoked to evaluate the right-hand-sides. It is only in this limit, justified by our assumption (15), that we will obtain an autonomous system of flow equations (see [33] for a general perspective on the non-autonomous character of the renormalization group flow in GFTs). Solving for the wave-function normalization Z k in (23) and (22) we obtain:…”

“…Thanks to recent breakthroughs in the closely related field of tensor models [18][19][20], the usual quantum field theory tools are currently being generalized to GFTs which increasingly resemble LQG. These developments include first and foremost perturbative renormalization [21][22][23][24][25][26][27][28], but also constructive results [29][30][31], and generalizations of the usual functional renormalization group (FRG) methods [32,33].…”

The Tensor Track V: Holographic Tensors

“…Their renormalization group flow in the tensor theory space [68] can be investigated via functional equations and truncations to search numerically for non-trivial fixed points [69].…”