The genuinely $\mathbb{Z}_2$-equivariant cohomology version of ordinary cohomology, taking into account the $\mathbb{Z}_2$-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 $\mathbb{Z}_2$-equivariant super line 2-bundle
chromatic level | complex oriented cohomology theory | E-∞ ring/A-∞ ring | real oriented cohomology theory |
---|---|---|---|
0 | ordinary cohomology | Eilenberg-MacLane spectrum $H \mathbb{Z}$ | HZR-theory |
0th Morava K-theory | $K(0)$ | ||
1 | complex K-theory | complex K-theory spectrum $KU$ | KR-theory |
first Morava K-theory | $K(1)$ | ||
first Morava E-theory | $E(1)$ | ||
2 | elliptic cohomology | elliptic spectrum $Ell_E$ | |
second Morava K-theory | $K(2)$ | ||
second Morava E-theory | $E(2)$ | ||
algebraic K-theory of KU | $K(KU)$ | ||
3 …10 | K3 cohomology | K3 spectrum | |
$n$ | $n$th Morava K-theory | $K(n)$ | |
$n$th Morava E-theory | $E(n)$ | BPR-theory | |
$n+1$ | algebraic K-theory applied to chrom. level $n$ | $K(E_n)$ (red-shift conjecture) | |
$\infty$ | complex cobordism cohomology | MU | MR-theory |
A detailed model (“Jandl gerbes”) for differential $HZR$-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 August 2007, Volume 274, Issue 1, pp 31-64 (arXiv:hep-th/0512283)
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 (arXiv:1608.06466)