nLab
KO-theory

Contents

Contents

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.

Propertis

Homotopy groups

The stable homotopy groups of KO

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

are:

n=n = 8k+08k + 08k+18k + 18k+28k + 28k+38k + 38k+48k + 48k+58k + 58k+68k + 68k+78k + 7
π n(KO)=\pi_n(KO) = \mathbb{Z}/2\mathbb{Z}/2/2\mathbb{Z}/200\mathbb{Z}000000

With Bott periodicity 8.

cohomology theories of string theory fields on orientifolds

string theoryB-fieldBB-field moduliRR-field
bosonic stringline 2-bundleordinary cohomology H 3H\mathbb{Z}^3
type II superstringsuper line 2-bundlePic(KU)// 2Pic(KU)//\mathbb{Z}_2KR-theory KR KR^\bullet
type IIA superstringsuper line 2-bundleBGL 1(KU)B GL_1(KU)KU-theory KU 1KU^1
type IIB superstringsuper line 2-bundleBGL 1(KU)B GL_1(KU)KU-theory KU 0KU^0
type I superstringsuper line 2-bundlePic(KU)// 2Pic(KU)//\mathbb{Z}_2KO-theory KOKO
type I˜\tilde I superstringsuper line 2-bundlePic(KU)// 2Pic(KU)//\mathbb{Z}_2KSC-theory KSCKSC

References

General

The differential K-theory for KO is discussed in

The full twisted differential orthogonal K-theory is discussed in

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

and was then re-amplified in

With further developments in

Discussion of orbi-orienti-folds using equivariant KO-theory is in

An elaborate proposal for the correct flavour of real equivariant K-theory needed for orientifolds is sketched in

Last revised on April 7, 2021 at 12:34:50. See the history of this page for a list of all contributions to it.