  We calculate the scaling dimensions of operators with large global charge and
spin in 2+1 dimensional conformal field theories. By the state-operator
correspondence, these operators correspond to superfluids with vortices and can
be systematically studied using effective field theory. As the spin increases
from zero to the unitarity bound, the superfluid state corresponding to the
lowest dimension operator passes through three distinct regimes: (1) a single
phonon, (2) two vortices, and (3) multiple vortices. We also calculate
correlation functions with two such operators and the Noether current.
