Mathematical physics

After an introduction to the notion of Leonard pairs, I explain their different uses. Then, I provide the definition of the associated Heun operator and how it allows us to simplify the computation of the quantum entanglement entropy. Finally, I show, in the simplest example, how the Bethe ansatz can be used to diagonalize the Heun operator.

In this talk, I will present the construction of the Yangian of a cotangent Lie algebra from the geometry of the equivariant affine Grassmannian. Furthermore, I will discuss how this quantum group can be dynamically twisted to a quantum groupoid over a neighborhood in the moduli space of G-bundles over a compact Riemann surface. These constructions are motivated by relations between a certain holomorphic-topological 4d gauge field theory and the geometric Langlands correspondence. Representations of the Yangian are pertubative line operators of said theory, while the dynamical twist of the Yangian controls the action of these operators via Hecke modifications in this setting. This talk is based on the two joint works arXiv:2405.19906 and arXiv:2411.05068 with Wenjun Niu.