A new tangential structure for type IIA string theory

The Green-Schwarz anomaly cancellation condition says that the target space of heterotic string theory must come with a string structure for the theory to be consistent. In this talk we discuss a new tangential structure called string^h, first introduced by Devalapurkar, as a spin^c analogue of string. Approximating the correct tangential structure of string theory has many implications, primarily is the Swampland program via the cobordism conjecture of Mcnamara and Vafa. We will show that the spectrum of string^h has the notable property that it orients tmf_1(n), just like how the spectrum of string orients tmf, by the work of Ando-Hopkins-Rezk. Finally we will show that the anomaly condition of the partition function of M-theory, studied by Diaconescu-Moore-Witten is implied by a string^h structure on the target space of type IIA, in parallel to the Green-Schwarz anomaly for heterotic string theory, and discuss applications for anomaly cancellation.