Loop groups, Kaluza-Klein reduction and M-theory

We show that the data of a principal G-bundle over a principal circle bundle is equivalent to that of a d (LG) over cap =(def) U(1) x LG-bundle over the base of the circle bundle. We apply this to the Kaluza-Klein reduction of M-theory to IIA and show that certain generalized characteristic classes of the loop group bundle encode the Bianchi identities of the antisymmetric tensor fields of IIA supergravity. We further show that the low dimensional characteristic classes of the central extension of the loop group encode the Bianchi identities of massive IIA, thereby adding support to the conjectures of [ 1].