Sylvain Carrozza

I will present a brief review of large-N tensor models and their applications in quantum gravity. On the one hand, they provide a general platform to investigate random geometry in an arbitrary number of dimensions, in analogy with the matrix models approach to two-dimensional quantum gravity. Previously known universality classes of random geometries have been identified in this context, with continuous random trees acting as strong attractors. On the other hand, the same combinatorial structure supports a generic family of large-N quantum theories, collectively known as melonic theories.

I will review recent work on tensorial group field theories (TGFTs). The renormalization methods being developed in this context provide more and more control over their field-theoretic structures, and for models which increasingly resemble loop quantum gravity. Perhaps surprisingly, some of these models are asymptotically free and can therefore be made sense of at arbitrary values of the (abstract) scale with respect to which they are organized. They define in this sense UV complete quantum field theories.

Rank 3 tensorial group fields theories with gauge invariance condition appear to be renormalizable on dimension 3 groups such as SU(2), but also on dimension 4 groups. Building on an analogy with ordinary scalar field theories, I will generalize such models to group dimension 4 - ε, and discuss what this might teach us about the physically relevant SU(2) case.

I will start with a brief overview of tensorial group field theories with gauge invariant condition and their relation to spin foam models. The rest of the talk will be focused on the SU(2) theory in dimension 3, which is related to Euclidean 3d quantum gravity and has been proven renormalizable up to order 6 interactions. General renormalization group flow equations will be introduced, allowing in particular to understand the behavior of the relevant couplings in the neighborhood of the Gaussian fixed point.

I will recall the main motivations for considering spin foam models in their Group Field Theory (GFT) versions, which are quantum field theories defined on group manifolds. As for any other quantum field theory, a fully consistent definition of the latter must involve renormalization. I will briefly review a specific class of GFTs, called tensorial, for which progress in this direction has recently been possible. A new just-renormalizable model, in three dimensions and on the SU(2) group, will be presented.