  This paper, in a sense, completes a series of three papers. In the previous
two hep-th/0404013, hep-th/0410293, we have explored the possibility of
refining the K-theory partition function in type II string theories using
elliptic cohomology. In the present paper, we make that more concrete by
defining a fully quantized free field theory based on elliptic cohomology of
10-dimensional spacetime. Moreover, we describe a concrete scenario how this is
related to compactification of F-theory on an elliptic curve leading to IIA and
IIB theories. We propose an interpretation of the elliptic curve in the context
of elliptic cohomology. We discuss the possibility of orbifolding of the
elliptic curves and derive certain properties of F-theory. We propose a link of
this to type IIB modularity, the structure of the topological Lagrangian of
M-theory, and Witten's index of loop space Dirac operators.
