Quantum hydrodynamics from large- n supersymmetric gauge theories.

We study the connection between periodic finite-difference Intermediate Long Wave ( $$\Delta \hbox {ILW}$$ ) hydrodynamical systems and integrable many-body models of Calogero and Ruijsenaars-type. The former describe quantum cohomology and quantum K-theory of the ADHM moduli space of Abelian instantons, while the latter arise in the instanton counting of four- and five-dimensional supersymmetric gauge theories with eight supercharges in the presence of defects. Using string theory dualities, we provide correspondences between hydrodynamical and many-body integrable systems. In particular, we match the energy spectra on both sides. [ABSTRACT FROM AUTHOR]