Hydrodynamic integrable systems are characterized by an infinite number of conserved quantities and can be described in terms of integrable partial differential equations. I will focus on the periodic Intermediate Long Wave (ILW) system, both at the classical and quantum level. The quantum system has not been solved yet, if not in a particular limit (the Benjamin-Ono limit) which is related to the AGT correspondence. In this talk I will show how a particular two dimensional N=(2,2) gauge theory on S^2 can be used to determine the spectrum of the quantum ILW system via Bethe Ansatz equations, as well as the norm of the eigenstates. In addition the partition function of this theory computes genus zero Gromov-Witten invariants for the ADHM instanton moduli space, thus relating quantum cohomology to quantum hydrodynamics.