Quantum dispersion relations for the $AdS_4 \times CP^3$ GKP string

We compute the one-loop correction to the dispersion relations of the excitations of the $AdS_4 \times CP^3$ sigma model dual to ABJM theory, expanded around the cusp background. The results parallel those of N = 4 SYM. As in that case, the dispersion relations are compatible with the predictions from the Bethe ansatz for the GKP string, though showing some known discrepancies on which we comment.