On non-planar ABJM anomalous dimensions from M2 branes in AdS$_{4}\times S^{7}/\mathbb{Z}_{k}$

Planar parts of conformal dimensions of primary operators in $U_k(N) \times U_{-k}(N)$ ABJM theory are controlled by integrability. Strong coupling asymptotics of planar dimensions of operators with large spins can be found from the energy of semiclassical strings in AdS$_{4}\times$CP$^3$ but computing non-planar corrections requires understanding higher genus string corrections. As was pointed out in arXiv:2408.10070, there is an alternative way to find the non-planar corrections by quantizing M2 branes in AdS$_{4}\times S^7/\mathbb{Z}_{k}$ which are wrapped around the 11d circle of radius $1/k= \lambda/N$ and generalize spinning strings in AdS$_4\times$CP$^3$. Computing the 1-loop correction to the energy of M2 brane that corresponds to the long folded string with large spin $S$ in AdS$_4$ allowed to obtain a prediction for the large $\lambda$ limit of non-planar corrections to the cusp anomalous dimension. Similar predictions were found for non-planar dimensions of operators dual to M2 branes that generalize the ''short'' and ''long'' circular strings with two equal spins $J_1=J_2$ in CP$^3$. Here we consider two more non-trivial examples of 1-loop M2 brane computations that correspond to: (i) long folded string with large spin $S$ in AdS$_4$ and orbital momentum $J$ in CP$^3$ whose energy determines the generalized cusp anomalous dimension, and (ii) circular string with spin $S$ in AdS$_4$ and spin $J$ in CP$^3$. We find the leading terms of the expansion of the corresponding 1-loop M2 brane energies in $1/k$. We also discuss similar semiclassical 1-loop M2 brane computation in flat 11d background and comment on possible relation to higher genus corrections to energies in 10d string theory.