Perturbative computation of string one-loop corrections to Wilson loop minimal surfaces in $AdS_5 \times S^5$

We revisit the computation of the 1-loop string correction to the "latitude" minimal surface in $AdS_5 \times S^5$ representing 1/4 BPS Wilson loop in planar $\cal N$=4 SYM theory previously addressed in arXiv:1512.00841 and arXiv:1601.04708. We resolve the problem of matching with the subleading term in the strong coupling expansion of the exact gauge theory result (derived previously from localization) using a different method to compute determinants of 2d string fluctuation operators. We apply perturbation theory in a small parameter (angle of the latitude) corresponding to an expansion near the $AdS_2$ minimal surface representing 1/2 BPS circular Wilson loop. This allows us to compute the corrections to the heat kernels and zeta-functions of the operators in terms of the known heat kernels on $AdS_2$. We apply the same method also to two other examples of Wilson loop surfaces: generalized cusp and $k$-wound circle.