  We study entanglement entropy in theories with gravitational or mixed U(1)
gauge-gravitational anomalies in two, four and six dimensions. In such theories
there is an anomaly in the entanglement entropy: it depends on the choice of
reference frame in which the theory is regulated. We discuss subtleties
regarding regulators and entanglement entropies in anomalous theories. We then
study the entanglement entropy of free chiral fermions and self-dual bosons and
show that in sufficiently symmetric situations this entanglement anomaly comes
from an imbalance in the flux of modes flowing through the boundary, controlled
by familiar index theorems. In two and four dimensions we use anomalous Ward
identities to find general expressions for the transformation of the
entanglement entropy under a diffeomorphism. (In the case of a mixed anomaly
there is an alternative presentation of the theory in which the entanglement
entropy is not invariant under a U(1) gauge transformation. The free-field
manifestation of this phenomenon involves a novel kind of fermion zero mode on
a gravitational background with a twist in the normal bundle to the entangling
surface.) We also study d-dimensional anomalous systems as the boundaries of d
+ 1 dimensional gapped Hall phases. Here the full system is non-anomalous, but
the boundary anomaly manifests itself in a change in the entanglement entropy
when the boundary metric is sheared relative to the bulk.
