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 i n D dimensions. The terms that we are interested in are of a topological na ture. 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 d imensions. In the latter case we suggest an interpretation of the quantum e ffect as a coupling of the gauge field on the brane to a relative backgroun d gauge field.