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.

💡 Research Summary

This paper investigates a newly introduced tangential structure, called “string‑h”, and establishes its deep connections with both the anomaly‑cancellation condition in type IIA string theory and with topological modular forms (tmf) equipped with level structures. The motivation stems from the classic Green‑Schwarz mechanism, Killingback’s refinement of the heterotic anomaly condition, and Witten’s discovery that a string structure on a manifold yields a modular form via the Witten genus. In the context of M‑theory and type IIA compactifications, Diaconescu‑Moore‑Witten identified a crucial consistency requirement: the seventh integral Stiefel‑Whitney class (W_{7}(TM)) must vanish. This paper shows that the existence of a string‑h structure on the tangent bundle automatically provides a canonical trivialization of (W_{7}), and conversely, for manifolds of dimension ≤ 8 (and closed manifolds of dimension 9) any trivialization of (W_{7}) lifts to a string‑h structure. Thus, in the dimensions relevant for ten‑dimensional physics, the two notions are essentially equivalent.

Four equivalent definitions of a string‑h structure are presented. The primary definition (following Devalapurkar) requires a trivialization of the ku‑theoretic Bockstein (\square_{ku}(\lambda_{c}(V))) for a spin(^{c}) vector bundle (V). Equivalent formulations involve a class (c^{2}{ku}(V)\in ku^{4}(X)) whose image under the Postnikov truncation equals (\lambda{c}(V)), a twisted string structure over the universal complex bundle ((BU,S)), and a virtual‑bundle description. Theorem 2.23 proves the equivalence of all four, while Propositions 2.33, 2.37, 2.38 describe how string‑h structures interact with ordinary string, spin, and complex structures. Low‑degree homotopy groups of the classifying space (BString^{h}) are computed (Lemma 2.46), providing the groundwork for later spectral constructions.

The paper then explores the relationship between string‑h structures and spin(^{c}) structures on free loop spaces. Unlike ordinary string structures, which induce spin structures on (LM), a string‑h structure does not induce any spin(^{c}) structure on the loop space, even after allowing for the “level” ambiguity described by Huang‑Han‑Duan. Theorem 3.14 constructs an explicit closed string‑h manifold whose loop space fails to admit a spin(^{c}) structure at any level, highlighting a key distinction between the two notions.

A central homotopical result is the construction of an (E_{\infty})‑ring map \