Shifted M-theory G4 quantization on a resolved Calabi–Yau fourfold

Assume Full AC. Let $T^{1,0}{\widehat{Y}}_4$ denote the holomorphic tangent bundle and $T_{\mathbb{R}}{Y}_{11}$ the real tangent bundle. Let ${\widehat{Y}}_4$ be a smooth compact connected resolved Calabi–Yau fourfold in the broad compact Calabi–Yau convention, so $K_{{\widehat{Y}}_4}$ is holomorphically trivial and $c_1(T^{1,0}{\widehat{Y}}_4)=0$. Let $Y_{11}$ be a finite-dimensional Hausdorff second-countable oriented smooth eleven-manifold without boundary, with a smooth Lorentzian metric $g$ of signature $(-,+,\dots ,+)$ and a time orientation $\tau$, a continuous choice of timelike-cone component. Supply a smooth principal ${\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^{+}(10,1)$-bundle $P\to Y_{11}$ and an equivariant two-sheeted covering $\mathrm{\Lambda }:P\to F_{g,\tau }^{+}{Y}_{11}$ of its oriented time-oriented orthonormal frame bundle. Work componentwise with path-connected components of CW homotopy type. Let $i:{\widehat{Y}}_4\hookrightarrow Y_{11}$ be an internal slice for which $i^{\ast }{T}_{\mathbb{R}}{Y}_{11}\cong (T^{1,0}{\widehat{Y}}_4{)}_{\mathbb{R}}\oplus {\epsilon }_{\mathbb{R}}^3;$ thus the external rank-three real tangent bundle is trivial along the slice. Restrict the closed curvature to the slice. Let $G_{4,CJS}$ denote the CJS-convention internal curvature and put $g_4=[G_{4,CJS}]\mathrm{/}((2\pi {)}^3{\mathrm{\ell }}_p^3)$. Define the unit-M2-coupled local potential and curvature by $C_{3,M}=T_{M2}{C}_{3,CJS}$ and $G_{4,M}=T_{M2}{G}_{4,CJS}$. Conditional on the adopted tension $T_{M2}=(2\pi {)}^{-2}{\mathrm{\ell }}_p^{-3}$ and identifying this rescaled curvature with the curvature in the Witten shift, the correctly typed CY4 conclusion is the free-period condition $g_4+\frac{1}{2}{\rho }_{{\widehat{Y}}_4}(c_2(T^{1,0}{\widehat{Y}}_4))\in \mathrm{im}\text{\hspace{0.17em}⁣}({\rho }_{{\widehat{Y}}_4}:H^4({\widehat{Y}}_4;\mathbb{Z})\to H_{\mathrm{d}\mathrm{R}}^4({\widehat{Y}}_4;\mathbb{R}))\mathrm{.}$ This is a conditional specialization of the generic postulate, not a claim of a unique integral lift or a second quantization postulate.

The Axiom of Choice

Crepant resolution and matter surface

M2 Brane Worldvolume Action

Shifted G4 quantization in M theory

M2 M5 tension normalizations

First Pontryagin class of an underlying real complex bundle

Compact Calabi–Yau manifold conventions

Naturality, normalization, and Whitney sum for Chern classes

The current M2 action [F2] has Wess–Zumino term $+T_{M2}\int C_{3,CJS}$, so the unit-coupled local potential is defined by $C_{3,M}=T_{M2}{C}_{3,CJS}$. Since this is a constant rescaling, it preserves potential patch transitions and gives $G_{4,M}=T_{M2}{G}_{4,CJS}$ on the global curvature, without assuming a global potential. The actual tension input [F2] then gives $[G_{4,M}\mathrm{/}(2\pi )]=[G_{4,CJS}]\mathrm{/}((2\pi {)}^3{\mathrm{\ell }}_p^3)=g_4$. This unit identity is conditional on identifying the physical Witten curvature with this rescaled CJS curvature; the tension supplier lacks a current Step 1 readiness record, and the action coefficient alone does not establish that physical identification.

For the internal inclusion $i$, coefficient extension and cochain pullback commute; restriction to smooth singular chains commutes; and de Rham integration is natural because ${\int }_{\sigma }{i}^{\ast }\omega ={\int }_{i\circ \sigma }\omega$. Thus pulling the free-period condition [F1] back to the slice sends its integral-image term into $\mathrm{im}{\rho }_{{\widehat{Y}}_4}$. The tangent splitting in [F4] and the CY convention [F5] give ${\rho }_{{\widehat{Y}}_4}(i^{\ast }\lambda )=\frac{1}{2}{\rho }_{{\widehat{Y}}_4}(p_1((T^{1,0}{\widehat{Y}}_4{)}_{\mathbb{R}}))=-{\rho }_{{\widehat{Y}}_4}(c_2(T^{1,0}{\widehat{Y}}_4))$, where $p_1$ is the Pontryagin class of the underlying real bundle and $c_2$ is the Chern class of the holomorphic tangent bundle; the equalities are in real de Rham cohomology and any 2-torsion in $\lambda$ is killed by coefficient extension. Substitution with the unit identity above yields the displayed free-period condition. This does not prove the physical quantization premise, a unique integral lift, or evenness of $c_2$.

The design's shorthand $g_4+c_2\mathrm{/}2\in H^4({\widehat{Y}}_4;\mathbb{Z})$ is replaced here by the current generic supplier's typed free-period statement in $\mathrm{im}\rho$; the supplier does not retain torsion or specify an integral lift. Collinucci–Savelli's separate parity argument makes the free-period shift integral for their restricted smooth global-Weierstrass subclass only. Their argument is not extended to every singular or resolved fourfold.