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

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