M-theory, type IIA superstrings, and elliptic cohomology

The topological part of the M-theory partition function was shown by Witten to be encoded in the index of an E₈ bundle in eleven dimensions. This partition function is, however, not automatically anomalyfree. We observe here that the vanishing W₇ = 0 of the Diaconescu-Moore-Witten anomaly in IIA and compactified M-theory partition function is equivalent to orientability of spacetime with respect to (complex-oriented) elliptic cohomology. Motivated by this, we define an elliptic cohomology correction to the IIA partition function, and propose its relationship to interaction between 2- and 5-branes in the M-theory limit.