group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
The generalized cohomology theory represented by the KO-spectrum, hence the “orthogonal” version of complex K-theory.
This is supposed to be the generalized cohomology theory which measures D-brane charge in type I string theory/on orientifold planes.
The stable homotopy groups of KO
are:
$n =$ | $8k + 0$ | $8k + 1$ | $8k + 2$ | $8k + 3$ | $8k + 4$ | $8k + 5$ | $8k + 6$ | $8k + 7$ |
---|---|---|---|---|---|---|---|---|
$\pi_n(KO) =$ | $\mathbb{Z}$ | $\mathbb{Z}/2$ | $\mathbb{Z}/2$ | $0$ | $\mathbb{Z}$ | $0$ | $0$ | $0$ |
With Bott periodicity 8.
cohomology theories of string theory fields on orientifolds
string theory | B-field | $B$-field moduli | RR-field |
---|---|---|---|
bosonic string | line 2-bundle | ordinary cohomology $H\mathbb{Z}^3$ | |
type II superstring | super line 2-bundle | $Pic(KU)//\mathbb{Z}_2$ | KR-theory $KR^\bullet$ |
type IIA superstring | super line 2-bundle | $B GL_1(KU)$ | KU-theory $KU^0$ |
type IIB superstring | super line 2-bundle | $B GL_1(KU)$ | KU-theory $KU^1$ |
type I superstring | super line 2-bundle | $Pic(KU)//\mathbb{Z}_2$ | KO-theory $KO$ |
type $\tilde I$ superstring | super line 2-bundle | $Pic(KU)//\mathbb{Z}_2$ | KSC-theory $KSC$ |
On the differential K-theory for KO:
Daniel Grady, Hisham Sati, Differential KO-theory: Constructions, computations, and applications, Advances in Mathematics Volume 384, 25 June 2021, 107671 (arXiv:1809.07059, doi:10.1016/j.aim.2021.107671)
Kiyonori Gomi, Mayuko Yamashita, Differential KO-theory via gradations and mass terms (arXiv:2111.01377)
On the full twisted differential orthogonal K-theory:
The original observation that D-brane charge for orientifolds should be in KR-theory, hence in KO-theory right on the O-planes, is due to
and was then re-amplified in
Sergei Gukov, K-Theory, Reality, and Orientifolds, Commun.Math.Phys. 210 (2000) 621-639 (arXiv:hep-th/9901042)
Oren Bergman, E. Gimon, Shigeki Sugimoto, Orientifolds, RR Torsion, and K-theory, JHEP 0105:047, 2001 (arXiv:hep-th/0103183)
With further developments in
Discussion of orbi-orienti-folds using equivariant KO-theory is in
N. Quiroz, Bogdan Stefanski, Dirichlet Branes on Orientifolds, Phys.Rev. D66 (2002) 026002 (arXiv:hep-th/0110041)
Volker Braun, Bogdan Stefanski, Orientifolds and K-theory (arXiv:hep-th/0206158)
H. Garcia-Compean, W. Herrera-Suarez, B. A. Itza-Ortiz, O. Loaiza-Brito, D-Branes in Orientifolds and Orbifolds and Kasparov KK-Theory, JHEP 0812:007, 2008 (arXiv:0809.4238)
An elaborate proposal for the correct flavour of real equivariant K-theory needed for orientifolds is sketched in
Last revised on June 5, 2024 at 12:51:18. See the history of this page for a list of all contributions to it.