Operator dimensions from moduli

We consider the operator spectrum of a three-dimensional N = 2 superconformal field theory with a moduli space of one complex dimension, such as the fixed point theory with three chiral superfields X, Y, Z and a superpotential W = XYZ. By using the existence of an effective theory on each branch of moduli space, we calculate the anomalous dimensions of certain low-lying operators carrying large R-charge J. While the lowest primary operator is a BPS scalar primary, the second-lowest scalar primary is in a semi-short representation, with dimension exactly J + 1, a fact that cannot be seen directly from the XYZ Lagrangian. The third-lowest scalar primary lies in a long multiplet with dimension J + 2 - c(-3) J(-3) + O(J(-4)), where c(-3) is an unknown positive coefficient. The coefficient c(-3) is proportional to the leading superconformal interaction term in the effective theory on moduli space. The positivity of c(-3) does riot follow from supersymmetry, but rather from unitarity of moduli scattering and the absence of superluminal signal propagation in the effective dynamics of the complex modulus. We also prove a general lemma, that scalar semi-short representations form a module over the chiral ring in a natural way, by ordinary multiplication of local operators. Combined with the existence of scalar semi-short states at large J, this proves the existence of scalar semi-short states at all values of J. Thus the combination of N = 2 superconformal symmetry with the large-J expansion is more powerful than the sum of its parts.