  We obtain the ratio of semiclassical partition functions for the extension
under mixed flux of the minimal surfaces subtending a circumference and a line
in Euclidean $AdS_{3}\times S^{3}\times T^{4}$. We reduce the problem to the
computation of a set of functional determinants. If the Ramond-Ramond flux does
not vanish, we find that the contribution of the $B$-field is comprised in the
conformal anomaly. In this case, we successively apply the Gel'fand-Yaglom
method and the Abel-Plana formula to the flat-measure determinants. To cancel
the resultant infrared divergences, we shift the regularization of the sum over
half-integers depending on whether it corresponds to massive or massless
fermionic modes. We show that the result is compatible with the zeta-function
regularization approach. In the limit of pure Neveu-Schwarz-Neveu-Schwarz flux
we argue that the computation trivializes. We extend the reasoning to other
surfaces with the same behavior in this regime.
