  We study the one-loop effective action of certain classical type IIA string
configurations in $AdS_4\times \mathbb{CP}^3$. These configurations are dual to
Wilson loops in the $\mathcal{N}= 6\:$ $U(N)_k \times U(N)_{-k}$ Chern-Simons
theory coupled to matter whose expectation values are known via supersymmetric
localization. We compute the one-loop effective actions using two methods:
perturbative heat kernel techniques and full $\zeta$-function regularization.
We find that the result of the perturbative heat kernel method matches the
field theory prediction at the appropriate order while the $\zeta$-function
approach seems to lead to a disagreement. We explore various issues that might
be responsible for this state of affairs.
