Notes on Spinning Operators in Fermionic CFT

The Gross-Neveu model defines a unitary CFT of interacting fermions in $2<d<4$ which has perturbative descriptions in the $1/N$ expansion and in the epsilon-expansion near two and four dimensions. In each of these descriptions, the CFT has an infinite tower of nearly conserved currents of all spins. We determine the structure of the non-conservation equations both at large $N$ and in the epsilon-expansion, and use it to find the leading order anomalous dimensions of the broken currents. Similarly, we use the fact that the CFT spectrum includes a nearly free fermion to fix the leading anomalous dimensions of a few scalar composite operators. We also compute the scaling dimensions of double-trace spinning operators in the large $N$ expansion, which correspond to interaction energies of two-particle states in the AdS dual higher-spin theory. We first derive these anomalous dimensions by a direct Feynman diagram calculation, and then show that the result can be exactly reproduced by analytic bootstrap methods, provided the sum over the tower of weakly broken higher-spin currents is suitably regularized. Finally, we apply the analytic bootstrap approach to derive the anomalous dimensions of the double-trace spinning operators in the 3d bosonic and fermion vector models coupled to Chern-Simons theory, to leading order in $1/N$ but exactly in the `t Hooft coupling.