To begin, we review the definition of geometric quantizations of Sobolev spaces of classical fields and Fock spaces as defined by the Segal-Bargmann construction usingfractional Gaussian fields. First, we show that the classical periodic Benjamin-Ono equation is a Liouville integrable Hamiltonian system with respect to the Gardner-Faddeev-Zakharov bracket and describe the classicalconserved density in the dispersionless limit \bar e → 0. Second, we give Nazarov-Sklyanin's integrable geometric quantization of this system inFock space with dimensionless quantization parameter hbar and describe the spectrum explicitly after a non-trivial change of variables \bar e = e1 + e2, \hbar = - e1 e2 invariant under e2 \leftrightarrow e1. Finally,we identify the result with the spectrum of commuting Chern classes of the tautological bundle in equivariant cohomology of Hilbert schemes (the abelian case of Nekrasov's BPS/CFT Correspondence for pure N=2 SUSY Yang-Mills on R4 in the (e2,e1)-Omega background), in which \bar e is the deformation parameter of Maulik-Okounkov's Yangian and \hbar is the handle-gluing element.