Shifted G4 quantization in M theory

Let $Y$ be a finite-dimensional, Hausdorff, second-countable oriented smooth eleven-manifold without boundary, equipped with a smooth Lorentzian metric $g$ of signature $(-,+,\dots ,+)$ and a time orientation $\tau$: a continuous choice of one component of the timelike cone $\{v\in T_{y}Y:g_y(v,v)<0\}$ at every $y\in Y$. Supply a spin structure in the convention of the eleven-dimensional spin supplier: a smooth principal ${\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^{+}(10,1)$-bundle $P\to Y$ and an equivariant two-sheeted covering $\mathrm{\Lambda }:P\to F_{g,\tau }^{+}Y$ over ${\mathrm{id}}_Y$, where $F_{g,\tau }^{+}Y$ is the bundle of oriented, time-oriented orthonormal frames. Work componentwise, with each connected component path connected and of CW type. Let $G_4\in {\mathrm{\Omega }}^4(Y;\mathbb{R})$ be a smooth closed real four-form in the source's unit-membrane-coupling normalization. This postulate specifies only curvature-level data: the closed form, the integral spin class below, and their de Rham period condition. No global C-field object, descent data, holonomy, or differential-cohomology lift is assumed or constructed; any local three-form potentials are additional data and are not used in the condition. Write $H^4(Y;\mathbb{Z})$ for singular cohomology and let ${\rho }_Y:H^4(Y;\mathbb{Z})\to H_{\mathrm{d}\mathrm{R}}^4(Y;\mathbb{R})$ be the coefficient-to-period map constructed from the standard cohomology comparisons. Adopt the source-reported spin class $\lambda \in H^4(Y;\mathbb{Z})$ satisfying $2\lambda =p_1(TY)$ as an additional physical topological input. Then adopt the free-period condition ${\left[\frac{G_4}{2\pi }\right]}_{\mathrm{d}\mathrm{R}}-\frac{1}{2}{\rho }_Y(\lambda )\in \mathrm{im}\text{\hspace{0.17em}⁣}({\rho }_Y:H^4(Y;\mathbb{Z})\to H_{\mathrm{d}\mathrm{R}}^4(Y;\mathbb{R}))\mathrm{.}$ Here unit-membrane-coupling normalization means the following local convention in $\mathrm{\hslash }=c=1$ units. Whenever an additional smooth local potential $c_3\in {\mathrm{\Omega }}^3(U;\mathbb{R})$ on an open $U\subseteq Y$ is supplied with $d{c}_3=G_4{\mathrm{\mid }}_U$, its electric membrane factor is $\exp (i{\int }_{{\mathrm{\Sigma }}_3}{X}^{\ast }{c}_3)$ for a smooth map $X:{\mathrm{\Sigma }}_3\to U$ from a compact oriented smooth three-manifold without boundary. The coefficient in this exponent is $+1$ and the integral is dimensionless; in length-valued coordinates the components of $c_3$ and $G_4$ have dimensions length${}^{-3}$ and length${}^{-4}$. This specifies a local factor only, not a globally single-valued membrane amplitude independently of the fermion factor.

This is the typed de Rham form of the source's shifted integral-period statement. It is an anomaly/quantization postulate, not a theorem of ordinary cohomology and not a consequence of the local CJS field equations. The CJS carrier instead defines $F_4=d{C}_3$ with dimensionless components of $C_3$; it does not identify $F_4$ with the normalized $G_4$. If an additional constant electric coupling $q>0$ of dimension length${}^{-3}$ is supplied through a local membrane term $+q{\int }_{{\mathrm{\Sigma }}_3}{X}^{\ast }{C}_3$, the corresponding conversion is $c_3=q{C}_3$ and $G_4{\mathrm{\mid }}_U=q{F}_4{\mathrm{\mid }}_U$. No value of $q$, or identification with a membrane tension, follows from the source-free CJS action or is needed for the displayed condition on $G_4$. A globally defined CJS potential yields an exact rescaled curvature and alone cannot represent a nonzero real flux class. The displayed condition forgets torsion and does not specify a global C-field or a differential-cohomology refinement.

Witten, §2.2

The condition controls the image in real de Rham cohomology only. Applications with torsion, boundaries, localized sources, or singular spacetime require additional relative or differential-cohomology data.