$E_8$ gauge theory, and a derivation of $K$-theory from $M$-theory

The partition function of Ramond-Ramond p-form fields in Type IIA supergravity on a ten-manifold X contains subtle phase factors that are associated with T-duality, self-duality, and the relation of the RR fields to K-theory.The analogous partition function of M-theory on X x S 1 contains subtle phases that are similarly associated with E% gauge theory.We analyze the detailed phase factors on the two sides and show that they agree, thereby testing M-theory/Type IIA duality as well as the Jf-theory formalism in an interesting way.We also show that certain .D-brane states wrapped on nontrivial homology cycles are actually unstable, that (-1) FL symmetry in Type IIA superstring theory depends in general on a cancellation between a fermion anomaly and an anomaly of RR fields, and that Type IIA superstring theory with no wrapped branes is well-defined only on a spacetime with W*? = 0.