On the CFT operator spectrum at largeg lobal charge

We calculate the anomalous dimensions of operators with large global charge J in certain strongly coupled conformal field theories in three dimensions, such as the O(2) model and the supersymmetric fixed point with a single chiral super field and a W = Phi(3) superpotential. Working in a 1/J expansion, we find that the large-J sector of both examples is controlled by a conformally invariant effective Lagrangian for a Goldstone boson of the global symmetry. For both these theories, we find that the lowest state with charge J is always a scalar operator whose dimension Delta(J) satisfies the sum rule J(2)Delta(J) - (J(2)/2 + J/4 + 3/16) Delta(J-1) - (J(2)/2 - J/4 + 3/16)Delta(J+1) = 0: 04067, up to corrections that vanish at large J. The spectrum of low-lying excited states is also calculable explcitly: for example, the second-lowest primary operator has spin two and dimension Delta(J) + root 3. In the supersymmetric case, the dimensions of all half-integer-spin operators lie above the dimensions of the integer-spin operators by a gap of order J(+1/2). The propagation speeds of the Goldstone waves and heavy fermions are 1/root 2 and +/- 1/2 times the speed of light, respectively. These values, including the negative one, are necessary for the consistent realization of the superconformal symmetry at large J.