Zeta-function Regularization of Holographic Wilson Loops

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.