Satisfying Freed--Witten quantization on smooth divisors in Calabi--Yau orientifolds

Abstract

Freed--Witten quantization conditions on several divisors wrapped by D-branes must be satisfied with one bulk $B$-field. On a smooth Calabi--Yau threefold $X$ with a holomorphic O3/O7 involution $σ$, we show that the common background $B_{2,+}^{\star}=\tfrac12[R]$, where $R$ is the total O7 divisor, permits Freed--Witten period quantization on the full two-cycle lattice of \emph{every smooth connected divisor}, modulo torsion. The worldvolume fluxes are chosen individually and obey the orientifold relation between each divisor and its image. The central step concerns setwise invariant divisors $D$: we prove that the bulk Freed--Witten class $η_D^\star=\tfrac12([R]+[D])|_D$ has integer periods on every $σ$-even integral two-cycle class, including inherited and autochthonous classes. Its periods on purely $σ$-odd classes vanish, so any remaining non-integer periods occur only on classes that are neither $σ$-even nor $σ$-odd. We show how to choose $σ$-odd worldvolume fluxes on these invariant divisors that make all periods integral. For exchanged divisor pairs, quantizing the flux on one member automatically quantizes its orientifold image. On invariant divisors with $h^{2,0}_-(D)=0$, the constructed flux is of type $(1,1)$ and admits a primitive representative; this includes every smooth rigid invariant divisor. We also study zero worldvolume flux on invariant divisors. For $h^{1,1}_-(X)\leq1$, we construct common bulk backgrounds satisfying every inherited intersection period on all such divisors. A binary compatibility test determines whether a common shift exists for larger $h^{1,1}_-(X)$. Non-inherited period tests then determine whether the resulting backgrounds allow zero flux on the full worldvolume lattices.