## Final Activity Report Summary - REN-NCFT (Renormalisation of non-commutative quantum field theory)

Nowadays, the main goal of fundamental high energy physics is to unify the two pillars of our description of the world, namely general relativity and quantum mechanics. Several propositions have been made to achieve this target: string theory, loop quantum gravity, non-commutative geometry etc. It is believed, at least by part of the physics community, that space-time, at intermediate scales between the current energy scale reached by particles colliders and the Planck scale, may be modelled by a non-commutative geometry. It is then a natural challenge to try to define mathematically coherent models on such spaces. This was exactly the main objective of this project. By model here, we mean a quantum field theory and by mathematically coherent, a renormalisable one. In the early thirties, it was hoped that the non-commutativity of space could regularise the unavoidable divergences of quantum field theory. It was naively thought that a non-commutative field theory (NCFT) would be simply free of divergences. Only recently, it was discovered that this is not the case but that a non-commutative space may nevertheless lead to a certain kind a regularisation. The first renormalisable non trivial NCFT was indeed found to be asymptotically safe. The project focused on defining new examples of renormalisable non-commutative models, testing their regularisation properties and studying some general mathematical aspects of such models.

The first renormalisable NCFT was obtained by considering a commutative Lagrangian and modifying its propagator. Such a process is now called a vulcanisation. Our work essentially revealed the complexity of such a procedure. We proved the perturbative renormalisability of another NCFT, discovering that the needed modification is far more complicated that we expected. Moreover we found that another model, thought to be properly vulcanised, is actually not. In particular, it does not regularise the commutative renormalisation group flow.

At a different level, we studied some of the mathematical aspects of such non-commutative models. Concretely, we derived the combinatorial properties of the graph invariants which enter the Feynman amplitudes of these theories.

As a conclusion, our work points to the need of a more general approach to the understanding of the vulcanisation. This approach should be more physically inspired but also rely on a more abstract mathematical foundation.

The first renormalisable NCFT was obtained by considering a commutative Lagrangian and modifying its propagator. Such a process is now called a vulcanisation. Our work essentially revealed the complexity of such a procedure. We proved the perturbative renormalisability of another NCFT, discovering that the needed modification is far more complicated that we expected. Moreover we found that another model, thought to be properly vulcanised, is actually not. In particular, it does not regularise the commutative renormalisation group flow.

At a different level, we studied some of the mathematical aspects of such non-commutative models. Concretely, we derived the combinatorial properties of the graph invariants which enter the Feynman amplitudes of these theories.

As a conclusion, our work points to the need of a more general approach to the understanding of the vulcanisation. This approach should be more physically inspired but also rely on a more abstract mathematical foundation.