Notes on relativistic superfluidity and gauge/string duality

We consider selected topics of relativistic superfluidity within gauge/string duality. Non-relativistically, the only conservation law relevant to the hydrodynamic approximation is the energy-momentum conservation. Relativistically, one has to introduce an extra conserved U(1) current and an extra three-dimensional scalar field which is condensed. Finding out a proper U(1) symmetry becomes a crucial point. We emphasize that in dual models there do arise extra U(1) symmetries associated with wrapping of the strings around (extra) compact directions in Euclidean space-time. Moreover, if the geometry associated with an extra compact dimension is cigar-like then the corresponding U(1) symmetry could well be spontaneously broken. The emerging Goldstone particle survives in the hydrodynamic limit. A specific point is that the chemical potential conjugated with the corresponding U(1) charge is vanishing. Within the standard approach the vanishing chemical potential implies no superfluidity. We argue that an exotic liquid, introduced recently in the literature, with vanishing energy density and non-vanishing pressure represents a viable description of the liquid associated with 3d Goldstone particles in Euclidean space-time. Since it lives on the stretched membrane, it describes energy-momentum transport in the deep infrared. We discuss briefly possible applications to the quark-gluon plasma.