  Diaconescu, Moore and Witten proved that the partition function of type IIA
string theory coincides (to the extent checked) with the partition function of
M-theory. One of us (Kriz) and Sati proposed in a previous paper a refinement
of the IIA partition function using elliptic cohomology and conjectured that it
coincides with a partition function coming from F-theory. In this paper, we
define the geometric term of the F-theoretical effective action on type IIA
compactifications. In the special case when the first Pontrjagin class of
spacetime vanishes, we also prove a version of the Kriz-Sati conjecture by
extending the arguments of Diaconescu-Moore-Witten. We also briefly discuss why
even this special case contains interesting examples.
