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We show that the exact partition function of U(N) six-dimensional gaugetheory with eight supercharges on C^2 x S^2 provides the quantization of theintegrable system of hydrodynamic type known as gl(N) periodic IntermediateLong Wave (ILW). We characterize this system as the hydrodynamic limit ofelliptic Calogero-Moser integrable system. We compute the Bethe equations fromthe effective gauged linear sigma model on S^2 with target space the ADHMinstanton moduli space, whose mirror computes the Yang-Yang function of gl(N)ILW. The quantum Hamiltonians are given by the local chiral ring observables ofthe six-dimensional gauge theory. As particular cases, these provide the gl(N)Benjamin-Ono and Korteweg-de Vries quantum Hamiltonians. In the fourdimensional limit, we identify the local chiral ring observables with theconserved charges of Heisenberg plus W_N algebrae, thus providing a gaugetheoretical proof of AGT correspondence.

Six-dimensional supersymmetric gauge theories, quantum cohomology of instanton moduli spaces and gl(N) Quantum Intermediate Long Wave Hydrodynamics