Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Orientifolds and K-theory

V. Braun, B. Stefanski

Abstract

Recently it has been shown that D-branes in orientifolds are not always described by equivariant Real K-theory. In this paper we define a previously unstudied twisted version of equivariant Real K-theory which gives the D-brane spectrum for such orientifolds. We find that equivariant Real K-theory can be twisted by elements of a generalised group cohomology. This cohomology classifies all orientifolds just as group cohomology classifies all orbifolds. As an example we consider the $Ω\times\I_4$ orientifolds. We completely determine the equivariant orthogonal K-theory $KO_{\Zop_2}(\R^{p,q})$ and analyse the twisted versions. Agreement is found between K-theory and Boundary Confromal Field Theory (BCFT) results for both integrally- and torsion-charged D-branes.

Orientifolds and K-theory

Abstract

Recently it has been shown that D-branes in orientifolds are not always described by equivariant Real K-theory. In this paper we define a previously unstudied twisted version of equivariant Real K-theory which gives the D-brane spectrum for such orientifolds. We find that equivariant Real K-theory can be twisted by elements of a generalised group cohomology. This cohomology classifies all orientifolds just as group cohomology classifies all orbifolds. As an example we consider the orientifolds. We completely determine the equivariant orthogonal K-theory and analyse the twisted versions. Agreement is found between K-theory and Boundary Confromal Field Theory (BCFT) results for both integrally- and torsion-charged D-branes.

Paper Structure

This paper contains 13 sections, 4 theorems, 87 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Computation of $KO_{{\sf Z\!\!Z}_2}$
  3. Twisting in equivariant K--theory
  4. Twisting for ordinary orbifolds
  5. Twisted equivariant complex K--theory
  6. Defining $KR_{{\sf Z\!\!Z}_2\times{\sf Z\!\!Z}_2}^{[D_8]}$
  7. Classification of Orientifolds
  8. Physical interpretation
  9. Conclusion and Outlook
  10. Clifford algebras and $K^{p,q}$
  11. Understanding $H^2({\sf Z\!\!Z}_2\times {\sf Z\!\!Z}_2,\widetilde{U}_1)$
  12. The D-brane spectrum and K--groups for $\Omega\times{\cal I}_4$ orientifolds
  13. Cohomology for general abelian groups

Key Result

Theorem 1

Let $G$ be a compact Lie group, $V$ a $G$ vector space with a positive definite form (i.e. the generated Clifford algebra is ${\mathop{\mathbf{C}}\nolimits_{}^{}}{(V)}$ with all $\gamma_i^2=+1$) and $X$ a Real $G$ space. Then

Theorems & Definitions (7)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Theorem 3: Künneth theorem
  • Theorem 4
  • proof