On the Algebraic K-theory of The Massive D8 and M9 Branes

We study the relation between 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 manifold $Z$ in a massive d=11 supergravity background from the K-theoretic point of view. By speculating on the use of the dimensional reduction to relate the two theories in different dimensions and 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)\times_{\bar{k}^*}G} a connection between charges and gauge fields is argued to exist. This connection is realized as a map between the corresponding algebraic K-theory groups.