On Dimensional Transmutation in 1+1D Quantum Hydrodynamics

Recently a detailed correspondence was established between, on one side, four and five-dimensional large-N supersymmetric gauge theories with $\mathcal{N}=2$ supersymmetry and adjoint matter, and, on the other side, integrable 1+1-dimensional quantum hydrodynamics. Under this correspondence the phenomenon of dimensional transmutation, familiar in asymptotically free QFTs, gets mapped to the transition from the elliptic Calogero-Moser many-body system to the closed Toda chain. In this paper we attempt to formulate the hydrodynamical counterpart of the dimensional transmutation phenomenon inspired by the identification of the periodic Intermediate Long Wave (ILW) equation as the hydrodynamical limit of the elliptic Calogero-Moser/Ruijsenaars-Schneider system. We also conjecture that the chiral flow in the vortex fluid provides the proper framework for the microscopic description of such dimensional transmutation in the 1+1d hydrodynamics. We provide a geometric description of this phenomenon in terms of the ADHM moduli space.