Tensor field theory is the quantum field theoretic counterpart of tensor models. One may "enhance" certain interactions which are not of conventional melonic type so that they contribute to the dominant amplitudes, which consequently may drive us away from the branched polymer phase characterized by the usual melonic limit of tensor models. Therefore, such enhanced tensor field theories are of interest for the random geometric approach to quantum gravity. We consider two types of enhanced models + and × with order-d tensor fields ϕ : (U(1)^D)^d → ℂ and with the enhanced quartic interactions of the form p^2aϕ⁴ reminiscent of derivative couplings expressed in momentum space. Scrutinising the degree of divergence via multiscale renormalization analysis, we study their renormalizability at all orders of perturbation. We furthermore compute the beta functions of the couplings to understand their renormalization group flow behaviour. At all orders of perturbation, both models have a constant wave function renormalisation, therefore no anomalous dimension. Despite such a peculiar behaviour, both models acquire nontrivial radiative corrections for the coupling constants. In particular, we observe in some of the coupling constants linear behaviour in the log of momentum.

Random Tensor Network (RTNs) are random quantum states associated to decorated graphs, which provide a computable platform to investigate generic entanglement properties of quantum many-body systems. More precisely, a global state is obtained by stitching together local pieces of data: to each edge is associated a bipartite entangled state, to each vertex an independent random tensor, and those are glued together following the combinatorics of the graph. The entanglement structure of a RTN can be understood analytically in some detail, and is found to reproduce key expected features of quantum gravity states in the context of holography. This is due to the fact that the computation of the Rényi-n entropy of some subregion can be reduced to the evaluation of the partition function of a classical ’spin’ model on the network (where the ’spin’ associated to each vertex is an element of the symmetric group S_n). In a RTN, the tensor associated to a given vertex is usually averaged over the whole unitary group of the corresponding Hilbert space, with respect to the Haar measure. In this talk, I will investigate what happens when one averages over the much smaller subgroup of Local Unitary (LU) transformations. As we will discuss, this situation can be analysed with the help of Weingarten calculus and colored diagrammatics. Interestingly, it allows for richer entanglement structures which can be mapped to suitably modified classical spin models.

Feynman integrals are complicated objects and it is generally hard to evaluate them analytically. However, if their inherent mathematical structure is fully put to use, these integrals turn out to be remarkably well-suited for numerical evaluation. Feynman integrals with up to 30 propagators can be integrated quickly. I will illustrate how these tropical geometric structures can be employed, explain the key algorithmic step, tropical sampling, in detail and show first empirical results on the large loop behaviour of the beta function in D = 4 ϕ⁴ based on numerical computations up to 15 loops.

We apply the Functional Renormalization Group analysis to Tensor Field Theory (TFT) endowed with both local and nonlocal degrees of freedom and in the cyclic "melonic" truncation. For simplicity, we concentrate on the so-called local potential approximation without inspecting the flow of the wave function renormalization. A notion of effective dimension d_eff = d+(r−1)/ζ is identified from the dimension of our configuration space ℝ^d×G^r where G is a compact Lie group and ζ is one of our theory parameters. The compact dimensions vanish along the flow yielding, in the IR limit, d_eff = d. This positively allows phase transition in TFT as soon as d > 2. Due to the richness of the TFT model, we examine the phase structure of sundry limiting situations.

The perturbative non-renormalizability of the Einstein-Hilbert action is often taken as a hint that a quantum field theory of gravity should be non-perturbative, and asymptotically safe quantum gravity is often viewed as an example. In this talk, I will present indications that asymptotically safe quantum gravity with matter is instead near perturbative and I will discuss implications, both for the control of approximations as well as for the relation of Euclidean to Lorentzian signature settings.

A single scalar with higher-derivative kinetic term and derivative quartic interactions is a toy model for higher derivative gravity. The Functional Renormalization Group shows that this theory is asymptotically free both in the UV and in the IR. The physical implications of this result are clarified in part by by computing the scattering amplitudes.