M2 matter-curve zero modes for smooth algebraic flux backgrounds

Assume Full AC. Fix a smooth compact resolved flat sectioned elliptic Calabi–Yau fourfold and a chosen weight component whose matter surface $S_R$ is a smooth compact complex surface fibred by the selected rational curve over a smooth compact matter curve $q:S_R\to C_R$. Require $q$ to be a proper smooth submersion with closed oriented real two-dimensional fibers, with the orientation of the selected weight fixed, and write $i:S_R\hookrightarrow {\widehat{Y}}_4$. All integral cohomology below is singular cohomology. For a smooth compact complex curve $C$, let $j_C:H^2(C;\mathbb{Z})\to H_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}^2(C;\mathbb{R})$ be coefficient change: on singular cochains it is composition with $\mathbb{Z}\hookrightarrow \mathbb{R}$. Let $J_C:H_{\mathrm{d}\mathrm{R}}^2(C;\mathbb{R})\to H_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}^2(C;\mathbb{R})$ be the de Rham comparison, and define ${\rho }_C=J_C^{-1}\circ j_C$. For a holomorphic line bundle $L$, use the integral first-Chern normalization $c_1(L):=e_{\mathrm{T}\mathrm{h}}(L_{\mathbb{R}})$, with the underlying real plane oriented by complex multiplication. Define its degree by $\mathrm{deg}L:=⟨c_1(L),[C]⟩\in \mathbb{Z}$, where $[C]$ is the complex-oriented integral fundamental class, summed over connected components. Write $K_C=T^{\ast 1,0}C$ for the holomorphic cotangent line: its local frame is $dz$, with the holomorphic nonzero derivative as transition factor. These conventions and their local well-definedness checks are specified below.

For an algebraic gauge-background flux in the stated geometric regime, form the real fiber-integrated class $\alpha =q_{\ast }(i^{\ast }[g_4])\in H_{\mathrm{d}\mathrm{R}}^2(C_R;\mathbb{R})$. Assume a chosen integral lift $c\in H^2(C_R;\mathbb{Z})$ with ${\rho }_{C_R}(c)=\alpha$, and a supplied holomorphic line bundle ${\mathcal{L}}_R$ with $c_1({\mathcal{L}}_R)=c$. The pushforward is de Rham integration along these oriented compact fibers. Orient the total surface by the complex orientation of the base followed by the selected fiber orientation; the integral uses this same weight convention. Its defining evaluation identity and the degree convention above give $\mathrm{deg}({\mathcal{L}}_R)={\int }_{S_R}{g}_4\mathrm{.}$ The real class alone neither guarantees this integral lift nor determines the holomorphic or flat part of the line bundle; these are additional data, not consequences of shifted quantization. Choose a spin line ${\mathcal{S}}_R$ with ${\mathcal{S}}_R^{\otimes 2}\cong K_{C_R}$. For any holomorphic line $E$ on $C$, let ${\mathcal{O}}_C(E)$ be its sheaf of holomorphic sections, let $H^q(C,E):=H^q(C,{\mathcal{O}}_C(E))$ be derived sheaf cohomology, and write $h^q(C,E):={\dim }_{\mathbb{C}}{H}^q(C,E)$. The groups for $q=0,1$ are finite-dimensional for every such $E$ by the exact general-line Dolbeault and sheaf comparisons below; in particular this applies to the supplied $E={\mathcal{L}}_R\otimes {\mathcal{S}}_R$, without a divisor representation being assumed. The net four-dimensional chiral multiplicity is the index ${\chi }_R=h^0(C_R,{\mathcal{L}}_R\otimes {\mathcal{S}}_R)-h^1(C_R,{\mathcal{L}}_R\otimes {\mathcal{S}}_R)\mathrm{.}$ This is an adopted physical zero-mode dictionary, supported by the later smooth-curve derivation in Weigand's lecture notes. The bosonic M2 Wess–Zumino term alone does not prove it. The convention does not count vectorlike pairs and makes no claim for singular curves or general non-algebraic cohomology classes.

The Axiom of Choice

Crepant resolution and matter surface

M2 Brane Worldvolume Action

Codimension-two matter and codimension-three Yukawa dictionary

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

M-theory/F-theory elliptic-fiber limit

Singular cochain complex with coefficients

Singular cohomology with coefficients

The de Rham theorem

Thom-defined Euler class of an oriented vector bundle

Fundamental class of a compact oriented manifold

Kronecker evaluation pairing

Top de Rham integration equals fundamental-class evaluation

Holomorphic line bundles and meromorphic sections on a Riemann surface

Sheaf cohomology as right derived global sections

Hermitian metric and $L^2$ pairing on a compact Riemann surface

Dolbeault cohomology of a compact riemann surface is finite dimensional

Cech--Dolbeault comparison for holomorphic line bundles on a compact Riemann surface

Coefficient change commutes with coboundary, since $\delta (j\circ \phi )=j\circ \phi \circ \mathrm{\partial }=j\circ \delta \phi$. It therefore descends to $j_C$, and [F6] gives the inverse comparison defining ${\rho }_C$. For the supplied $c_1({\mathcal{L}}_R)=c$, coefficient change commutes with evaluation on the integral fundamental cycle, and [F7] gives $\mathrm{deg}{\mathcal{L}}_R=⟨j_C(c),[C_R{]}_{\mathbb{R}}⟩={\int }_{C_R}{\rho }_{C_R}(c)={\int }_{C_R}{q}_{\ast }(i^{\ast }[g_4])={\int }_{S_R}{g}_4\mathrm{.}$ The last equality uses the stated base-before-fiber orientation and defining integration identity. Changing a closed-form representative changes the integral by the integral of an exact form on a closed manifold, hence by zero. For the cotangent line, the chain rule makes the $dz$ transition factors satisfy the cocycle rule; they are holomorphic and nonzero, so [F8] applies. Tensor products of the supplied holomorphic line frames again have holomorphic nonzero transitions, giving the actual line ${\mathcal{L}}_R\otimes {\mathcal{S}}_R$.

For each connected component of $C_R$, choose the metrics whose existence is proved in [F9]. Apply its arbitrary-line Dolbeault finiteness and global sheaf comparison to $E=({\mathcal{L}}_R\otimes {\mathcal{S}}_R){\mathrm{\mid }}_C$. This gives finite $H^0$ and $H^1$ for this actual supplied line, rather than only for $\mathcal{O}(D)$. No connectedness restriction has been added: a compact smooth curve has finitely many connected components, because components are open in a locally connected manifold and compactness gives a finite subcover. Sheaves on a finite disjoint union are exactly tuples of sheaves on its components, by restriction and gluing; global sections are their finite product. This equivalence and exactness of finite products apply to injective resolutions in [F8], so derived global sections, and hence $H^0,H^1$, are the finite products of the component groups. They are finite-dimensional, including the empty case. The comparison is with the fixed sheaf groups, so the auxiliary metric choices impose no restriction on the given holomorphic line and do not affect $h^q$. This proves only the mathematical meaning and finiteness of the adopted index; its physical zero-mode interpretation remains the Postulate.