nLab KO-theory

Redirected from "KO".
Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Algebraic topology

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.

Properties

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 0KU^0
type IIB superstringsuper line 2-bundleBGL 1(KU)B GL_1(KU)KU-theory KU 1KU^1
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

On the differential K-theory for KO:

On the full twisted differential orthogonal K-theory:

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 June 5, 2024 at 12:51:18. See the history of this page for a list of all contributions to it.