Combining an abelianized Yang-Mills field with its dual in 6D N=(1,0) supergravity yields a class in twisted K-theory, with the B-field as twist.
Douglas, On the twisted K -homology of simple Lie groups , Topology 45 (2006), 955-988, [ arXiv:math.AT/0402082 https://arxiv.org/abs/math.AT/0402082]
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Six-dimensional gauge theories and (twisted) generalized cohomology
Combining an abelianized Yang-Mills field with its dual in 6D N=(1,0) supergravity yields a class in twisted K-theory, with the B-field as twist.