Massive IIA Romans-mass and NS three-form flux quantization

Let $X$ be the compact oriented smooth six-manifold used as the internal space. A smooth differential three-form is a smooth section of ${\mathrm{\Lambda }}^3{T}^{\ast }X$, so $H_3\in {\mathrm{\Omega }}^3(X)$ has smooth coordinate coefficients; $F_0\in \mathbb{R}$ is a constant scalar on $X$. In local coordinates, if $H_3=\frac{1}{3!}{H}_{ijk}d{x}^i\wedge d{x}^j\wedge d{x}^k$, then $d_X{H}_3=\frac{1}{3!}{\mathrm{\partial }}_{\mathrm{\ell }}{H}_{ijk}d{x}^{\mathrm{\ell }}\wedge d{x}^i\wedge d{x}^j\wedge d{x}^k$, with repeated indices summed. The NS5-free hypothesis means that the magnetic Bianchi identity has no NS5 current: this four-form vanishes. Use the smooth singular chain complex of $X$: $C_k^{\mathrm{\infty }}(X;\mathbb{Z})$ is the free abelian group of finite integral sums of smooth singular $k$-simplices $\sigma :{\mathrm{\Delta }}^k\to X$, where each simplex extends smoothly to an open neighborhood of ${\mathrm{\Delta }}^k$ in its affine span; $\mathrm{\partial }$ is the alternating sum of face restrictions. Put $H_k^{\mathrm{\infty }}(X;\mathbb{Z}):=\ker {\mathrm{\partial }}_k\mathrm{/}\mathrm{im}{\mathrm{\partial }}_{k+1}$ and write $\mathrm{Tor}{H}_3^{\mathrm{\infty }}(X;\mathbb{Z})$ for its torsion subgroup. A free integral three-cycle ${\mathrm{\Sigma }}_3$ is a class in $H_3^{\mathrm{\infty }}(X;\mathbb{Z})\mathrm{/}\mathrm{Tor}{H}_3^{\mathrm{\infty }}(X;\mathbb{Z})$. For a representative $z={\sum }_j{n}_j{\sigma }_j$, define ${\int }_{{\mathrm{\Sigma }}_3}{H}_3:={\sum }_j{n}_j{\int }_{{\mathrm{\Delta }}^3}{\sigma }_j^{\ast }{H}_3$; Stokes' theorem and $d_X{H}_3=0$ make this period independent of the representative.

The period also descends through the torsion quotient: if a homology class $[z]$ is torsion, some integer $r>0$ and smooth chain $b$ satisfy $rz=\mathrm{\partial }b$, so $r{\int }_z{H}_3={\int }_b{d}_X{H}_3=0$ by Stokes and closedness, and therefore ${\int }_z{H}_3=0$ as a real number.

Set ${\mathrm{\ell }}_s:=\sqrt{{\alpha }^{\mathrm{\prime }}}>0$. Adopt, in the standard RR normalization of the democratic supergravity equations, $(2\pi {\mathrm{\ell }}_s)F_0\in \mathbb{Z},\frac{1}{(2\pi {\mathrm{\ell }}_s{)}^2}{\int }_{{\mathrm{\Sigma }}_3}{H}_3\in \mathbb{Z}$ for every free integral three-cycle ${\mathrm{\Sigma }}_3$. Use natural units $\mathrm{\hslash }=c=1$: ${\mathrm{\ell }}_s$ has units of length, $F_0$ has units of inverse length, the coordinate components of $H_3$ have units of inverse length, and an $H_3$ period has units of length squared. Thus both displayed normalized quantities are dimensionless. In this smooth-field formulation, $d_X{H}_3=0$ expresses the exclusion of NS5 magnetic sources; a sector with an NS5 current is outside this postulate. This is a free-period postulate; it makes no claim about torsion classes or K-theory refinements.

The DGKT canonical RR convention rescales its RR fields by $\sqrt{2}$. In the DGKT source convention, use the conversion $m_0=f_0\mathrm{/}(2\sqrt{2}\pi \sqrt{{\alpha }^{\mathrm{\prime }}})$ and hence $F_0^{\mathrm{G}\mathrm{r}\mathrm{a}\tilde{\mathrm{n}}\mathrm{a}}=\sqrt{2}{m}_0=f_0\mathrm{/}(2\pi \sqrt{{\alpha }^{\mathrm{\prime }}})$; use the source's separate $H_3$ period and cover conventions rather than identifying its canonical integers with a different RR basis.

String Tension Alpha Prime

A smooth differential $k$-form

DGKT classical smeared four-dimensional effective solution

The smooth integral chain complex and simplex convention are those of Smooth integral singular chain complex. The representative-independence of the period uses the published Stokes theorem for smooth singular chains Stokes theorem for smooth singular chains, applied to the integer chains through the coefficient inclusion $\mathbb{Z}\hookrightarrow \mathbb{R}$. These are mathematical formulation inputs, not derivations of the adopted physical quantization.