Type IIA String Theory and tmf with Level Structure

Abstract

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 $MStrin{g}^h$ in such a way that it orients $tm{f}_1(3)$; we extend Devalapurkar's result, showing that $MStrin{g}^h$ orients $tm{f}_1(n)$. We compute the homotopy groups of $MStrin{g}^h$ in the dimensions relevant for physical applications, and apply them to anomaly cancellation applications for certain compactifications of type IIA string theory.

56 pages, v2 has include a new section, 2.2