  We use a mix of field theoretic and holographic techniques to elucidate
various properties of quantum entanglement entropy. In (3+1)-dimensional
conformal field theory we study the divergent terms in the entropy when the
entangling surface has a conical or a wedge singularity. In (2+1)-dimensional
field theory with a mass gap we calculate, for an arbitrary smooth entanglement
contour, the expansion of the entropy in inverse odd powers of the mass. We
show that the shape-dependent coefficients that arise are even powers of the
extrinsic curvature and its derivatives. A useful dual construction of a
(2+1)-dimensional theory, which allows us to exhibit these properties, is
provided by the CGLP background. This smooth warped throat solution of
11-dimensional supergravity describes renormalization group flow from a
conformal field theory in the UV to a gapped one in the IR. For this flow we
calculate the recently introduced renormalized entanglement entropy and confirm
that it is a monotonic function.
