The presence of chiral fermions in the physical Hilbert space implies
consistency conditions on the spectral action. These conditions are eq
uivalent to the absence of gauge and gravitational anomalies. Suggesti
ons for the fermionic part of the spectral action are made based on th
e supersymmetrisation of the bosonic part. (C) 1998 Published by Elsev
ier Science B.V. Ail rights reserved.