  We show that supercocycles on super $L_\infty$-algebras capture, at the
rational level, the twisted cohomological charge structure of the fields of
M-theory and of type IIA string theory. We show that rational 4-sphere-valued
supercocycles for M-branes in M-theory descend to supercocycles in type IIA
string theory with coefficients in the free loop space of the 4-sphere, to
yield the Ramond-Ramond fields in the rational image of twisted K-theory, with
the twist given by the B-field. In particular, we derive the M2/M5
$\leftrightarrow$ F1/Dp/NS5 correspondence via dimensional reduction of
sphere-valued super-$L_\infty$-cocycles.
