Orientifolds, RR Torsion, and K-theory

TL;DR

The paper addresses how RR fluxes in orientifold backgrounds are classified and quantized, arguing that K-theory, not integral cohomology, provides the correct framework to capture fractional charges, flux correlations, and identifications of orientifold variants.By employing real K-theory (KR, KH) and the Atiyah-Hirzebruch spectral sequence, the authors compute RR-field groups, compare them to cohomology, and show how higher-differential obstructions and extension problems produce physical effects such as half-integer shifts and equivalences between variants.Key results include the identification of conditions under which O$p^-$ and O$p^+$ variants are equivalent in K-theory (notably for lower p), the obstruction of certain RR torsions in specific backgrounds, and a detailed analysis of the orientifold 6-plane demonstrating parity-dependent correlations between $G_0$ and $G_2$.Overall, the work provides a K-theoretic explanation for previously puzzling RR-charge phenomena in orientifolds and outlines open questions about the absolute charges and the full structure of differentials in the AHSS.

Abstract

We analyze the role of RR fluxes in orientifold backgrounds from the point of view of K-theory, and demonstrate some physical implications of describing these fluxes in K-theory rather than cohomology. In particular, we show that certain fractional shifts in RR charge quantization due to discrete RR fluxes are naturally explained in K-theory. We also show that some orientifold backgrounds, which are considered distinct in the cohomology classification, become equivalent in the K-theory description, while others become unphysical.

Figures (4)

• Figure 1: Brane realization of (a) $H$ torsion, (b) $G_{6-p}$ torsion.
• Figure 2: Deforming the intersecting brane to a wrapped brane.
• Figure 3: T-duality relates (a) a configuration of two O$p^+$-planes at opposite points on a circle to a single O$(p+1)^+$-plane wrapping the circle, and (b) a configuration of one O$p^+$ and one $\widetilde{\hbox{O}{p}}^ {\raisebox{-4.5pt}{${+}$}}$ to a wrapped $\widetilde{\hbox{O}{(p+1)}}^ {\raisebox{-4.5pt}{${+}$}}$. The two configurations are equivalent for $p\leq 2$.
• Figure 4: Consistent $SL(3,\mathbb Z)$-inequivalent configurations of O6-planes: (a) eight O$6^-$ planes (rank 16), (b) six O$6^-$ planes and two O$6^+$ planes (rank 8), (c)+(d) two inequivalent configurations of four O$6^-$ planes and four O$6^+$ planes (rank 0).