Tensor Field Theories: Renormalization and Random Geometry

Abstract

This thesis focuses on renormalization of quantum field theories. Its first part considers three tensor models in three dimensions, a Fermionic quartic with tensors of rank-3 and two Bosonic sextic, of ranks 3 and 5. We rely upon the large-$N$ melonic expansion of tensor models. For the first model, invariant under $U(N)^3$, we obtain the RG flow of the two melonic couplings and the vacuum phase diagram, from a reformulation with a diagonalizable matrix intermediate field. The discrete chiral symmetry breaks spontaneously and we compare with the three-dimensional Gross-Neveu model. Beyond the massless $U(N)^3$ symmetric phase, we also observe a massive phase of same symmetry and another where the symmetry breaks into $U(N^2)\times U(N/2)\times U(N/2)$. A matrix model invariant under $U(N)\times U(N^2)$, with close properties, is also studied. For the other models, with symmetry groups $U(N)^3$ and $O(N)^5$, a non-melonic coupling (the "wheel") with an optimal scaling in $N$ drives us to a generalized melonic expansion. The kinetic terms are taken of short- and long-range, and we analyze perturbatively, at large-$N$, the RG flows of the sextic couplings up to four loops. Only the rank-3 model displays non-trivial fixed points (two real Wilson-Fisher-like in the short-range case and a line of fixed points in the other). We finally obtain the real conformal dimensions of the primary bilinear operators. In the second part, we establish the first results of perturbative multi-scale renormalization for a quartic scalar field on critical Galton-Watson trees, with a long-range kinetic term. At criticality, an emergent infinite spine provides a space of effective dimension $4/3$ on which to compute averaged correlation fonctions. This approach formalizes the notion of a QFT on a random geometry. We use known probabilistic bounds on the heat-kernel on a random graph reviewed in detail.