  We examine the one-loop partition function describing the fluctuations of the
superstring in a Schwarzschild-AdS$_5\times S^5$ background. On the bosonic
side, we demonstrate the one-loop equivalence of the Nambu-Goto action and the
Polyakov action for a general worldsheet, while on the fermionic side, we
consider the reduction of the ten-dimensional Green-Schwarz fermion action to a
two-dimensional worldsheet action. We derive the partition functions of the
worldsheets corresponding to both straight and parallel Wilson lines. We show
that the UV divergences of the functional determinants in the thermal AdS
background are canceled.
