Finite-Size Corrections of the mathbb{CP}³ Giant Magnons: the L\"{u}scher terms

We compute classical and first quantum finite-size corrections to the recently found giant magnon solutions in two different subspaces of $\mathbb{CP}^3$. We use the L\"{u}scher approach on the recently proposed exact S-matrix for $\mathcal{N}=6$ superconformal Chern-Simons theory. We compare our results with the string and algebraic curve computations and find agreement, thus providing a non-trivial test for the new $AdS_4/CFT_3$ correspondence within an integrability framework.