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 al