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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

Neveu-Schwarz 5-branes in type-IIA supergravity and gravitational anomalies