Type IIA String Theory and tmf with Level Structure

We look at a new string$^h$ tangential structure first introduced by Devalapurkar and relate it to the $W_7=0$ condition of Diaconescu-Moore-Witten for type IIA string theory and M-theory. We show that a string$^h$ structure on the target space automatically satisfies the $W_7=0$ condition and we also explain when the $W_7=0$ condition gives rise to a string$^h$ structure. Devalapurkar initially constructed $MString^h$ in such a way that it orients $tmf_1(3)$; we extend Devalapurkar's result, showing that $MString^h$ orients $tmf_1(n)$. We compute the homotopy groups of $MString^h$ in the dimensions relevant for physical applications, and apply them to anomaly cancellation applications for certain compactifications of type IIA string theory.