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 using fractional 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 classical conserved density in the dispersionless limit $\bar e \to 0$. Second, we give Nazarov-Sklyanin’s integrable geometric quantization of this system in Fock space with dimensionless quantization parameter hbar and describe the spectrum explicitly after a non-trivial change of variables $\bar e = e_1 + e_2$, $\hbar = - e_1 e_2$ invariant under $e_2 \leftrightarrow e_1$. 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 $R^4$ in the $(e_2,e_1)$-Omega background), in which $\bar e$ is the deformation parameter of Maulik-Okounkov’s Yangian and $\hbar$ is the handle-gluing element.