The genuinely 2\mathbb{Z}_2-equivariant cohomology version of ordinary cohomology, taking into account the 2\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 2\mathbb{Z}_2-equivariant super line 2-bundle

chromatic homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum HH \mathbb{Z}HZR-theory
0th Morava K-theoryK(0)K(0)
1complex K-theorycomplex K-theory spectrum KUKUKR-theory
first Morava K-theoryK(1)K(1)
first Morava E-theoryE(1)E(1)
2elliptic cohomologyelliptic spectrum Ell EEll_E
second Morava K-theoryK(2)K(2)
second Morava E-theoryE(2)E(2)
algebraic K-theory of KUK(KU)K(KU)
3 …10K3 cohomologyK3 spectrum
nnnnth Morava K-theoryK(n)K(n)
nnth Morava E-theoryE(n)E(n)BPR-theory
n+1n+1algebraic K-theory applied to chrom. level nnK(E n)K(E_n) (red-shift conjecture)
\inftycomplex cobordism cohomologyMUMR-theory


A detailed model (“Jandl gerbes”) for differential HZRHZR-theory in degree 3 (“orientifold B-fields”) is in

Revised on January 3, 2017 11:55:06 by Urs Schreiber (