nLab KO-theory

Idea

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.

Properties

Homotopy groups

The stable homotopy groups of KO

$$\pi_n(KO) \,\simeq\, \widetilde {KO}^0( S^n )$$

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.

Related concepts

• KO-dimension

• quaternionic oriented cohomology theory

• KR-theory, KU-theory, KQ-theory

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 $KSC$

References

General

Generalization of the Atiyah-Jänich theorem (to any degree and) to KO-theory:

• Michael F. Atiyah, Isadore M. Singer: Index theory for skew-adjoint Fredholm operators, Publications Mathématiques de l’IHÉS 37 (1969) 5-26 [doi:10.1007/BF02684885, numdam:PMIHES_1969__37__5_0, pdf]

• Max Karoubi: Espaces Classifiants en K-Théorie, Trans. Amer. Math. Soc. 147 (1970) 75-115 [doi:10.2307/1995218, jstor:1995218]

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:

• Daniel Grady, Hisham Sati, Twisted differential KO-theory [arXiv:1905.09085]

For D-brane charge

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

• Witten 98, section 5

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

• Varghese Mathai, Michael Murray, Daniel Stevenson, Type I D-branes in an H-flux and twisted KO-theory, JHEP 0311 (2003) 053 (arXiv:hep-th/0310164)

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

• Jacques Distler, Dan Freed, Greg Moore, Orientifold Précis in: Hisham Sati, Urs Schreiber (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory Proceedings of Symposia in Pure Mathematics, AMS (2011) (arXiv:0906.0795, slides)