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Elliptic curves play a natural and important role in elliptic cohomology. Inearlier work with I. Kriz, thes elliptic curves were interpreted physically intwo ways: as corresponding to the intersection of M2 and M5 in the context of(the reduction of M-theory to) type IIA and as the elliptic fiber leading toF-theory for type IIB. In this paper we elaborate on the physical setting forvarious generalized cohomology theories, including elliptic cohomology, and wenote that the above two seemingly unrelated descriptions can be unified usingSen's picture of the orientifold limit of F-theory compactification on K3,which unifies the Seiberg-Witten curve with the F-theory curve, and throughwhich we naturally explain the constancy of the modulus that emerges fromelliptic cohomology. This also clarifies the orbifolding performed in theprevious work and justifies the appearance of the w_4 condition in the ellipticrefinement of the mod 2 part of the partition function. We comment on thecohomology theory needed for the case when the modular parameter varies in thebase of the elliptic fibration.Comment: 23 pages, typos corrected, minor clarification

The elliptic curves in gauge theory, string theory, and cohomology