nLab HZR-theory

Idea

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.

Examples

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

Related concepts

chromatic homotopy theory

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

References

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 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)