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

We consider the 2-dimensional Toda lattice tau functions $τ_n(t,s;η,θ)$ deforming the probabilities $τ_n(η,θ)$ that a randomly chosen matrix from the unitary group U(n), for the Haar measure, has no eigenvalues within an arc $(η,θ)$ 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 Painleve VI equation.