ratiufest/Header

Speaker: Luc Haine (Louvain-la-Neuve)

Title: A centerless representation of the Virasoro algebra associated with the unitary circular ensemble

Abstract: We consider the 2-dimensional Toda lattice tau functions deforming the probabilities that a randomly chosen matrix from the unitary group, for the Haar measure, has no eigenvalues within an arc of the unit circle. We show that these tau functions satisfy a centerless Virasoro algebra of constraints, with a boundary part in the sense of Adler, Shiota and van Moerbeke. As an application, we obtain a new derivation of a differential equation due to Tracy and Widom, satisfied by these probabilities, linking it to the Painlevé VI equation. The Virasoro operators precisely describe the master symmetries of the Ablowitz-Ladik hierarchy on the tau functions of the hierarchy. This raises interesting questions in relation with the bi-hamiltonian structure of this hierarchy.