The genuinely -equivariant cohomology version of ordinary cohomology, taking into account the -action on the coefficients. A real-oriented cohomology theory. In degree 3 it serves as a twist for KR-theory.
The B-field over orientifold background of the bosonic string is a cocycle in (twisted) HZR-theory. More generally for the type II superstring it is a genuinely -equivariant super line 2-bundle
A detailed model (“Jandl gerbes”) for differential -theory in degree 3 (“orientifold B-fields”) is in
Urs Schreiber, Christoph Schweigert, Konrad Waldorf, Unoriented WZW models and Holonomy of Bundle Gerbes, Communications in Mathematical Physics 274 1 (2007) 31-64 [arXiv:hep-th/0512283, doi:10.1007/s00220-007-0271-x]
Krzysztof Gawedzki, Rafal R. Suszek, Konrad Waldorf, Bundle Gerbes for Orientifold Sigma Models, Adv. Theor. Math. Phys. 15(3), 621-688 (2011) (arXiv:0809.5125)
Pedram Hekmati, Michael Murray, Richard Szabo, Raymond Vozzo, Real bundle gerbes, orientifolds and twisted KR-homology, Advances in Theoretical and Mathematical Physics Volume 23 (2019) Number (arXiv:1608.06466, doi:10.4310/ATMP.2019.v23.n8.a5)
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