Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 100, 44 pages      arXiv:2411.07299      https://doi.org/10.3842/SIGMA.2026.100

Type IIA String Theory and tmf with Level Structure

Arun Debray a and Matthew Yu b
a) Department of Mathematics, University of Kentucky, 719 Patterson Office Tower, Lexington, KY 40506-0027, USA
b) Mathematical Institute, University of Oxford, Woodstock Road, Oxford, UK

Received March 04, 2026, in final form September 20, 2026; Published online October 19, 2026

Abstract
We look at a new stringh 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 stringh structure on the target space automatically satisfies the $W_7=0$ condition, and we explain when the $W_7=0$ condition lifts to a stringh structure. Devalapurkar initially constructed ${\rm MTString}^h$ in such a way that it orients ${\rm tmf}_1(3)$; we extend Devalapurkar's result by showing that ${\rm MTString}^h$ orients ${\rm tmf}_1(n)$. We compute the homotopy groups of ${\rm MTString}^h$ in dimensions relevant to physical applications and apply them to anomaly cancellation for certain compactifications of type IIA string theory.

Key words: string structures; topological modular forms; level structures; anomaly cancellation; type IIA string theory.

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