  Using $\zeta$-function regularization, we study the one-loop effective action
of fundamental strings in $AdS_5\times S^5$ dual to the latitude
$\frac{1}{4}$-BPS Wilson loop in $\mathcal{N}=4$ Super-Yang-Mills theory. To
avoid certain ambiguities inherent to string theory on curved backgrounds we
subtract the effective action of the holographic $\frac{1}{2}$-BPS Wilson loop.
We find agreement with the expected field theory result at first order in the
small latitude angle expansion but discrepancies at higher order.
