Speakers: Alessandra Frabetti (ICJ, Lyon 1)

It is well known that quantum mechanics motivates the extension from usual to non-commutative geometry. Quantum field theory requires extra renormalisation steps, and recent developments show that its unification with gravity leads to non-associative geometry.

These geometrical extensions apply to the coordinate ring of the relevant spaces, bundles, differential forms and gauge groups. Most of the results concern non-associative deformations based on quasi-Hopf algebras (Drinfel’d, Strominger, Szabo), or higher structures $L_infini$ and $A_infini$ (Stasheff, Zwiebach). Few results rely on « loops », that is, non-associative groups (Moufang, Malcev, Sabinin, Loginov).

I will show that a proalgebraic « renormalisation loop » is indeed hidden in standard Dyson’s renormalisation formulas, related to renormalisation Hopf algebras and their non-commutative versions, and I will give a hint to the relationship with higher structures.

https://indico.math.cnrs.fr/event/9346/