GKP free-period quantization for Type IIB three-form flux

Adopt the GKP/Dirac normalization on the selected smooth covering space $X$ of the Type IIB O3/O7 Calabi–Yau orientifold. Assume $F_3,H_3\in {\mathrm{\Omega }}^3(X;\mathbb{R})$ are globally defined smooth closed forms, so $d_X{F}_3=d_X{H}_3=0$ and their periods depend only on the represented homology classes. This is the sector with no NS5/D5 magnetic source currents; a model with such currents requires separately stated sourced Bianchi identities and is outside this postulate. On each chosen local potential patch $U\subseteq X$ on which the potentials are defined, they satisfy $F_3{\mathrm{\mid }}_U=d{C}_2$ and $H_3{\mathrm{\mid }}_U=d{B}_2$; global potentials are not required. A smooth integral three-cycle $z$ is a finite integer sum of smooth maps from the standard three-simplex to $X$ whose alternating face boundary is zero. For every such $z$, require $\frac{1}{(2\pi {)}^2{\alpha }^{\mathrm{\prime }}}{\int }_z{F}_3\in \mathbb{Z},\frac{1}{(2\pi {)}^2{\alpha }^{\mathrm{\prime }}}{\int }_z{H}_3\in \mathbb{Z}\mathrm{.}$ Equivalently, $\frac{1}{2\pi {\alpha }^{\mathrm{\prime }}}{\int }_z{F}_3\in 2\pi \mathbb{Z}$ and likewise for $H_3$. Closedness and Stokes' theorem make these periods invariant under adding a boundary to $z$; a torsion cycle has zero period because some positive integer multiple is a boundary. Thus the nontrivial period lattice is on the free homology quotient. Since ${\alpha }^{\mathrm{\prime }}$ has dimension length squared, the periods have dimension length squared and their normalized values are dimensionless. This is an adopted physical Dirac-quantization input, not a theorem of de Rham cohomology. It does not assert that ordinary integral cohomology detects torsion RR data, that the global RR field is classified by ordinary differential cohomology, that twisted K-theory refinements or Freed–Witten shifts vanish, or that quotient/orientifold multiplicities can be omitted.

Type IIB Calabi–Yau orientifold flux background conventions

Cheeger–Simons differential character

Differential twist and twisted differential K-theory

Giddings–Kachru–Polchinski, Eq. (2.25)

Graña, §2.2

The differential-character and twisted differential K-theory definitions are formulation context only. This postulate does not use either definition to derive its period lattice or to identify torsion, flat holonomy, or a universal RR charge class. Its physical range is limited to the stated GKP free-period sector.