Worldvolume and target space anomalies in the D=10 super--fivebrane sigma--model

The fields of the conjectured ``heterotic" super--fivebrane sigma--model in ten dimensions are made out of a well known gravitational sector, the $X$ and the $\vartheta$, and of a still unknown heterotic sector which should be coupled to the Yang--Mills fields. We compute the one--loop $d=6$ worldvolume and $D=10$ target space Lorentz--anomalies which arise from the gravitational sector of the heterotic super--fivebrane sigma--model, using a method which we developed previously for the Green--Schwarz heterotic superstring. These anomalies turn out to carry an overall coefficient which is $1/2$ of that required by the string/fivebrane duality conjecture. As a consequence the worldvolume anomaly vanishes if the heterotic fields consist of 16 (rather than 32) complex Weyl fermions on the worldvolume. This implies that the string/fivebrane duality conjecture can not be based on a ``heterotic" super--fivebrane sigma--model with only fermions in the heterotic sector. Possible implications of this result are discussed.