Journées de topologie quantique

lundi 13 nov. 2023, 11:00 → 16:30 Europe/Paris

Bâtiment Mirande Aile A Salle 318 - étage 3 (Institut de Mathématiques de Bourgogne)

The “journées de topologie quantique” (quantum topology days) are a regular event aiming at bringing together, in a somewhat informal setting, people thinking about quantum topology in a broad sense. As the name suggest this is a one day event, with three talks and lots of time for discussions, taking place either in Paris or in Dijon.

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Program

• 11h00 - Ramanujan Santharoubane: On the kernel of SO(3)-Witten-Reshetikhin-Turaev quantum representations

Given a surface S, SO(3)-Witten-Reshetikhin-Turaev TQFT's provide a sequence, indexed by odd integers, of complex finite dimensional representations of the mapping class group of S called quantum representations. One major and difficult problem is to find the kernels of these quantum representations. For prime index, Gilmer and Masbaum proved that the representation preserves a lattice over the ring of integers of a cyclotomic field. This allows to approximate each quantum representation by homomorphisms from the mapping class group in to finite groups. In this talk, we will see that certain of these approximations can be completely understood and are related to the Johnson filtration of the mapping class group. As a consequence we can find some non trivial 'upper bound' for the kernels of quantum representations. This is a joint work with Renaud Detcherry.

• 14h - Anna-Katharina Hirmer: Generalised Kitaev models from Hopf monoids: topological invariance and examples

Quantum double models were introduced by Kitaev to obtain a realistic model for a topological quantum computer. They are based on a directed ribbon graph and a finite-dimensional semisimple Hopf algebra. The ground state of these models is a topological invariant of a surface, i.e. only depends on the homeomorphism class of the oriented surface but not the ribbon graph. Meusburger and Voß generalised part of the construction from Hopf algebras to pivotal Hopf monoids in symmetric monoidal categories. We explain the construction of the ground state for involutive Hopf monoids and show that it is topological invariant. We explicitly describe this construction for Hopf monoids in Set, Top, Cat and SSet.

• 15h30 - Emmanuel Graff: The Homotopy Braid Group is Torsion-Free

V. Lin, in the 'Kourkova notebook', questions the existence of a non-trivial epimorphism of the braid group onto a non-abelian torsion-free group. The homotopy braid group studied by Goldsmith in 1974 an known to be nilpotent, naturally appears as a potential candidate. In 2001, Humphries showed that this homotopy braid group is torsion-free for less than 6 strands. In this presentation, we will see a new approach based on the broader concept of welded braids, as well as algebraic techniques. This will allow us to demonstrate that the homotopy braid group is torsion-free for any number of strands.