  We construct a gravitational-anomaly-free effective action for the coupled
system of IIA D=10 dynamical supergravity interacting with an NS5-brane. The
NS5-brane is considered as elementary in that the associated current is a
delta-function supported on its worldvolume. Our approach is based on a
Chern-kernel which encodes the singularities of the three-form field strength
near the brane in an SO(4)-invariant way and provides a solution for its
Bianchi identity in terms of a two-form potential. A dimensional reduction of
the recently constructed anomaly-free effective action for an elementary
M5-brane in D=11 is seen to reproduce our ten-dimensional action. The
Chern-kernel approach provides in particular a concrete realization of the
anomaly cancellation mechanism envisaged by Witten.
