nLab
KSC-theory

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Stable Homotopy theory

Representation theory

Contents

Idea

What is called self-conjugate K-theory of spaces XX (Anderson 64) is KR-theory on real spaces of the form X×S 0,2X \times S^{0,2}, where the second factor denotes the circle equipped with the antipodal 2\mathbb{Z}_2-action (see at real space for the notation).

Applications

In the context of type II string theory on orientifolds KSCKSC-theory is the cohomology theory in which the RR-fields of the I˜\tilde I-variant of type I superstring theory are cocycles (Witten 98, DMR 13, section 3.3.)

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

The definition of KSc theory is due to

  • D. W. Anderson, The real K-theory of classifying spaces Proc. Nat. Acad. Sci. U. S. A., 51(4):634–636, 1964.

Discussion of applications to superstring theory on orientifolds is in

Last revised on December 8, 2015 at 16:10:11. See the history of this page for a list of all contributions to it.