Type II string theory and modularity

This paper, in a sense, completes a series of three papers. In the previoustwo hep-th/0404013, hep-th/0410293, we have explored the possibility ofrefining the K-theory partition function in type II string theories usingelliptic cohomology. In the present paper, we make that more concrete bydefining a fully quantized free field theory based on elliptic cohomology of10-dimensional spacetime. Moreover, we describe a concrete scenario how this isrelated to compactification of F-theory on an elliptic curve leading to IIA andIIB theories. We propose an interpretation of the elliptic curve in the contextof elliptic cohomology. We discuss the possibility of orbifolding of theelliptic curves and derive certain properties of F-theory. We propose a link ofthis to type IIB modularity, the structure of the topological Lagrangian ofM-theory, and Witten's index of loop space Dirac operators.Comment: 27 pages; clarifications added and typos corrected; to appear in JHE