The M-theory/Type II partition functions and generalized cohomology

Diaconescu, Moore and Witten (DMW) have matched the M-theory partition function, described by an E8 gauge theory, with the K-theoretic partition function of type IIA string theory. In this talk we report on recent work with Igor Kriz and with Varghese Mathai on the corresponding situation obtained by including the NSNS fluxes and cancelling the DMW anomaly. This leads naturally to a refinement of the K-theoretic description to twisted K-theory and to elliptic cohomology. We then briefly discuss the compatibility with S-duality in type IIB string theory, as well as the possiblity of having a generalized cohomology theory describing the nongravitational field(s) of M-theory.