  In the presence of background Neveu-Schwarz flux, the description of the
Ramond-Ramond fields of type IIB string theory using twisted K-theory is not
compatible with S-duality. We argue that other possible variants of twisted
K-theory would still not resolve this issue. We propose instead a possible path
to a solution using elliptic cohomology. We also discuss T-duality relation of
this to a previous proposal for IIA theory, and higher-dimensional limits. In
the process, we obtain some other results which may be interesting on their
own. In particular, we prove a conjecture of Witten that the 11-dimensional
spin cobordism group vanishes on K(Z,6), which eliminates a potential new
theta-angle in type IIB string theory.
