COLLOQUIUM Alessandra Frabetti, "Non-associative renormalisation group"

by Prof. Alessandra Frabetti (ICJ, Lyon 1)

René Baire (IMB)

René Baire


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.