• affine algebras
• Kac-Moody algebras
• representation theory
• space of Opers
• vertex algebras

The goal of the thesis is the description of the center of the completed enveloping algebra U_k(g) associated to the affine Kac-Moody algebra g_k as the level k, which is a complex number, varies. We focus on the critical level k = - 1/2 for which this center turns out to be incredibly large. The Lie algebra g_k carries a natural action of the group of automorphisms of the formal disc, our goal is actually to describe the center not only algebraically but also geometrically, taking into account the action of this group.

The proof is carried out exploiting the formalism of vertex operator algebras, which arises quite naturally in this context. We associate to g_k a vertex algebra V_k(g) which is closely related to the completed enveloping algebra U_k(g). The connection between V_k(g) and U_k(g) allows us to derive the description of the center of U_k(g) from the description of the center V_k(g).

We prove, using tools coming from representation theory and algebraic geometry, that the center of the vertex algebra at the critical level is isomorphic to the algebra of functions on the space of Opers on the formal disc D, in an equivariant way with respect of the action of the group of automorphisms of the disc. Finally, this description allow us to prove that the center of the completed enveloping algebra is isomorphisc to the algebra of functions on the space of Opers on the pointed formal disc D^*.