Type IIA String Theory and tmf with Level Structure

We look at a new stringʰ tangential structure first introduced by Devalapurkar and relate it to the W₇=0 condition of Diaconescu-Moore-Witten for type IIA string theory and M-theory. We show that a stringʰ structure on the target space automatically satisfies the W₇=0 condition, and we explain when the W₇=0 condition lifts to a stringʰ structure. Devalapurkar initially constructed MTStringʰ in such a way that it orients tmf₁(3); we~extend Devalapurkar's result by showing that MTStringʰ orients tmf₁(n). We compute the homotopy groups of MTStringʰ in dimensions relevant to physical applications and apply them to anomaly cancellation for certain compactifications of type IIA string theory.