Six-dimensional gauge theories and (twisted) generalized cohomology

We consider the global aspects of the 6-dimensional $\mathcal{N}=(1, 0)$ theory arising from the coupling of the vector multiplet to the tensor multiplet. We show that the Yang-Mills field and its dual, when both are abelianized, combine to define a class in twisted cohomology with the twist arising from the class of the $B$-field, in a duality-symmetric manner. We then show that this lifts naturally to a class in twisted (differential) K-theory. Alternatively, viewing the B-field in both $\mathcal{N}=(1,0)$ and $\mathcal{N}=(2,0)$ theories, not as a twist but as an invertible element, leads to a description within untwisted chromatic level two generalized cohomology theories, including forms of elliptic cohomology and Morava K-theory.