Hydrodynamics of Borromean Counterfluids

Counterflow superfluidity in a system with $N\geq 3$ components is distinctively different from the $N=2$ case. The key feature is the difference between the number ($N$) of elementary vortex excitations and the number ($N-1$) of independent branches of phonon modes, that is, the number of superfluid modes is larger than the number of ordered phase variables. We show how all the dynamical and statistical aspects of this ("Borromean") type of ordering are naturally described by effective $N$-component theory featuring compact-gauge invariance.