Why are fractional charges of orientifolds compatible with Dirac quantization?

TL;DR

This work explains why fractional D$p$-charges of orientifold planes with $p\le4$ do not violate Dirac quantization: the worldvolume fermion anomalies are encoded by the $\eta$ invariant in the Dai-Freed framework, producing a precise shift in flux quantization that cancels the naive phase mismatch. By computing $\eta(\mathcal{D}_{X})$ for real projective spaces $X=\mathbb{RP}^{q+2}$ via an equivariant APS approach, the authors show that $\eta(\mathcal{D}_{XP})=\pm 2^{p-5}$ and that the RR-phase $\exp(2\pi i\,\mathsf{q}_p)$ is canceled by $\exp(-2\pi i\,\eta)$, ensuring a well-defined partition function for wrapped D$q$-branes. The analysis is placed in a broader context by linking it to cobordism classifications of interacting fermionic SPT phases and by elaborating how extensions of spin groups (e.g., $\mathsf{spin}[k]$) organize the anomalies across all O$p$-planes, with detailed discussions for $O4$ through $O0$ and RP$^n$ spaces. An appendix clarifies that the type IIB duality group is the $\mathsf{pin}^+$ version of the double cover of $GL(2,\mathbb{Z})$, connecting nonperturbative dualities to these topological structures.

Abstract

Orientifold $p$-planes with $p\le4$ have fractional D$p$-charges, and therefore appear inconsistent with Dirac quantization with respect to D$(6-p)$-branes. We explain in detail how this issue is resolved by taking into account the anomaly of the worldvolume fermions using the $η$ invariants. We also point out relationships to the classification of interacting fermionic symmetry protected topological phases. In an appendix, we point out that the duality group of type IIB string theory is the pin+ version of the double cover of $GL(2,\mathbb{Z})$.

Figures (2)

• Figure 1: To compare the phases of the partition functions of D-branes wrapped around $Y_1$ and $Y_2$, we use a manifold $X$ satisfying $\partial X=Y_1 \sqcup \overline{Y_2}$. There can be multiple such choices.
• Figure 2: The $\eta$ invariant of a freely-acting quotient $X=\tilde{X}/G$ can often be computed by considering the equivariant APS index theorem on $\tilde{X}=\partial \tilde{W}$ such that the fixed points $p$ of $G$ action on $\tilde{W}$ are isolated.