  Expectation values of surface operators suffer from logarithmic divergences
reflecting a conformal anomaly. In a holographic setting, where surface
operators can be computed by a minimal surface in $AdS$, the leading
contribution to the anomaly comes from a divergence in the classical action (or
area) of the minimal surface. We study the subleading correction to it due to
quantum fluctuations of the minimal surface. In the same way that the
divergence in the area does not require a global solution but only a
near-boundary analysis, the same holds for the quantum corrections. We study
the asymptotic form of the fluctuation determinant and show how to use the heat
kernel to calculate the quantum anomaly. In the case of M2-branes describing
surface operators in the ${\cal N}=(2,0)$ theory in 6d, our calculation of the
one-loop determinant reproduces expressions for the anomaly that have been
found by less direct methods.
