  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.
