Curvature-corrected M2 Gauss law

Given: The stated finite-dimensional Hausdorff second-countable smooth compact oriented boundaryless spin eight-manifold $X$ and its componentwise path-connected CW-type condition, the specified integral orientation and fundamental class, the adopted AC and ${\mathrm{A}\mathrm{C}}_{\omega }$ background, the smooth global closed normalized curvature form, the signed localized charge, and the adopted integrated field-law premise.

The R4 CJS carrier gives $2{\kappa }_{11}^2=(2\pi {)}^8{\mathrm{\ell }}_p^9$ in the source-free local convention (Eleven-Dimensional CJS Supergravity Bosonic Action).

The M2 action carrier adopts $S=-T_{\mathrm{M}2}{\int }_{{\mathrm{\Sigma }}_3}\sqrt{-\det g}+T_{\mathrm{M}2}{\int }_{{\mathrm{\Sigma }}_3}{X}^{\ast }{C}_3$ with the $+C_3$ Wess–Zumino sign and an independent positive tension (M2 Brane Worldvolume Action). This fixes the local membrane coupling used to define $C_{3,M}$; it does not by itself reconcile the source conventions in the integrated curvature equation.

The shifted-flux postulate gives the free-period condition and explicitly leaves torsion/differential refinements unresolved (Shifted G4 quantization in M theory).

The normalization theorem proves $2{\kappa }_{11}^2{T}_{\mathrm{M}2}=(2\pi {)}^6{\mathrm{\ell }}_p^6$ by exact multiplication (M2 M5 tension product identity).

Pontryagin classes are integral characteristic classes and the cup product multiplies their singular cohomology classes (Pontryagin classes by complexification, Singular cohomology ring).

The specified integral orientation determines $[X{]}_{\mathbb{Z}}$ (Fundamental class of a compact oriented manifold).

The evaluation pairing on singular cohomology and homology is defined on quotient classes and independent of representatives; its ring-coefficient version uses rational cochains and rational chains (Kronecker evaluation pairing, The kronecker pairing is independent of cocycle and cycle representatives, Singular cohomology with coefficients, The singular chain complex and singular homology).

The denominator $48$ defines a rational scalar, and the singular characteristic pairing in [F7] is integer-valued before division (The rationals as equivalence classes of pairs of integers).

A smooth differential $k$-form is a smooth section of the $k$th exterior-power bundle of the cotangent bundle (A smooth differential $k$-form).

The Pontryagin-class definition assumes full AC and, for a real rank-$n$ bundle, defines $p_i(E)=(-1{)}^i{c}_{2i}(E_{\mathbb{C}})$, with $p_i(E)=0$ when $2i>n$. Its base is path-connected CW (or CW type); since $TX$ has rank $8$, $p_2(TX)=c_4(T{X}_{\mathbb{C}})$ is defined componentwise under the stated hypotheses (Pontryagin classes by complexification).

The wedge of smooth forms is their pointwise exterior product, and its smoothness is supplied by the wedge-product definition and its well-definedness result (The wedge product of differential forms).

A form has compact support when the closure of its nonzero locus is compact; every smooth form on compact $X$ therefore has compact support (Compact support of a differential form).

Under ${\mathrm{A}\mathrm{C}}_{\omega }$, the integral of a compactly supported top form on an oriented smooth manifold is defined by a chart-subordinate partition-of-unity sum with only finitely many nonzero terms near the compact support; independence of the choices is supplied by the integration theorem (Integral of a compactly supported top form).

The fluxed Gauss law, localized source sign, and one-loop coefficient are adopted physical premises; their crosswalk is not derived from [F1]–[F4].

Full AC is an adopted mathematical background axiom used by the Pontryagin-class definition. Applied to the range of an at-most-countable family, it gives the stated ${\mathrm{A}\mathrm{C}}_{\omega }$ hypothesis of [F13] (The Axiom of Choice, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).