On the algebraic K theory of the massive D8 and M9-branes

In this paper we review some basic relations of algebraic K theory and we f ormulate them in the language of D-branes. Then we study the relation betwe en the D8-branes wrapped on an orientable, compact manifold W in a massive Type IIA, supergravity background and the M9-branes wrapped on a compact ma nifold Z in a massive d = 11 supergravity background from the K-theoretic p oint of view. By interpreting the D8-brane charges as elements of K-0(C(W)) and the (inequivalent classes of) spaces of gauge fields on the M9-branes as the elements of K-0(C(Z) x ((k) over bar*) G) where G is a one-dimension al compact group, a connection between charges and gauge fields is argued t o exists. This connection could be realized as a composition map between th e corresponding algebraic K theory groups.