nLab KSC-theory

Idea

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

Applications

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

Related concepts

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

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:

• Edward Witten, section 5 of D-branes and K-theory, J. High Energy Phys., 1998(12):019, 1998 (arXiv:hep-th/9810188)

• Charles Doran, Stefan Mendez-Diez, Jonathan Rosenberg: T-duality For Orientifolds and Twisted KR-theory, Lett. Math. Phys. 104 11 (2014) 1333-1364 [arXiv:1306.1779, doi:10.1007/s11005-014-0715-0]