One Loop Effects in Various Dimensions and D-Branes

We calculate some one loop corrections to the effective action of theories in $d$ dimensions that arise on the dimensional reduction of a Weyl fermion in $D$ dimensions. The terms that we are interested in are of a topological nature. Special attention is given to the effective actions of the super Yang Mills theories that arise on dimensional reduction of the N=1 theory in six dimensions or on the dimensional reduction of the N=1 theory in ten dimensions. In the latter case we suggest an interpretation of the quantum effect as a coupling of the gauge field on the brane to a relative background gauge field.