  We present here a careful study of the holographic duals of BPS surface
operators in the 6d ${\cal N}=(2,0)$ theory. Several different classes of
surface operators have been recently identified and each class has a specific
calibration form - a 3-form in $AdS_7\times S^4$ whose pullback to the M2-brane
world-volume is equal to the volume form. In all but one class, the appropriate
forms are closed, so the action of the M2-brane is easily expressed in terms of
boundary data, which is the geometry of the surface. Specifically, for surfaces
of vanishing anomaly, it is proportional to the integral of the square of the
extrinsic curvature. This can be extended to the case of surfaces with
anomalies, by taking the ratio of two surfaces with the same anomaly. This
gives a slew of new expectation values at large $N$ in this theory.
  For one specific class of surface operators, which are Lagrangian
submanifolds of ${\mathbb R}^4\subset {\mathbb R}^6$, the structure is far
richer and we find that the M2-branes are special Lagrangian submanifold of an
appropriate six-dimensional almost Calabi-Yau submanifold of $AdS_7\times S^4$.
This allows for an elegant treatment of many such examples.
