The Candelas–Horowitz–Strominger–Witten four-generation quotient

Given: The connected Fermat quintic, the specified $G$-action and projective quotient, an $E_8\times E_8$ standard embedding, a fixed $27$-representation convention, the physical postulates, and the following explicit conditional inputs.

Free CHSW action and Euler characteristic of its quintic quotient proves that the displayed action is free, $Y$ is smooth connected projective with $K_Y$ trivial, and $\chi (Y)=-8$. Its Statement additionally supplies a finite surjective quotient map $\pi :Y_0\to Y$, an affine cover $\mathrm{Spec}B$ with invariant affine inverse images $\mathrm{Spec}A$ and $B=A^G$, and closed-point fibers equal to $G$-orbits. These are the exact quotient properties used in step 1.2; algebraic finite étaleness and degree are reconstructed there.

Chern classes from the projective-bundle relation defines topological Chern classes, so $c_1^{\mathrm{t}\mathrm{o}\mathrm{p}}(T_Y)=c_1^{\mathrm{t}\mathrm{o}\mathrm{p}}(\det T_Y)$. Since $\det T_Y=K_Y^{\vee }$, First Chern class of tensor, dual, and conjugate lines gives $c_1^{\mathrm{t}\mathrm{o}\mathrm{p}}(\det T_Y)=-c_1^{\mathrm{t}\mathrm{o}\mathrm{p}}(K_Y)=0$.

The Euler number of the tangent bundle is the Euler characteristic gives $⟨e(T{Y}_{\mathbb{R}}),[Y]⟩=\chi (Y)$. The compact connected smooth $Y$ admits a finite triangulation by Finite smooth triangulation of compact corners, relative to admissible boundary data, hence is a path-connected CW complex; its tangent bundle is numerable using a finite trivializing cover and a subordinate smooth partition of unity. Therefore Top Chern class equals Euler class of the underlying real bundle gives $c_3^{\mathrm{t}\mathrm{o}\mathrm{p}}(T_Y)=e(T{Y}_{\mathbb{R}})$, so $⟨c_3^{\mathrm{t}\mathrm{o}\mathrm{p}}(T_Y),[Y]⟩=\chi (Y)$ in the complex orientation.

SU(3)-fixed chiral spinor lines in six dimensions: with a supplied spin structure and exact lifted holonomy $SU(3)$, each six-dimensional chiral spinor bundle has one complex line of parallel sections, and either ten-dimensional heterotic Majorana–Weyl chirality has one four-real-dimensional four-dimensional Majorana module in its holonomy-fixed subspace.

Spin structure and parallel spinors from Calabi–Yau data constructs the canonical ${\mathrm{Spin}}^c(6)$ lift of the unitary frame bundle, with determinant $K_Y^{-1}$, and reduces it by the unit parallel section proportional to ${\mathrm{\Omega }}^{-1}$ to a Spin lift. Its complex spinor bundle is ${\mathrm{\Lambda }}^{0,\ast }{T}_Y^{\ast }$ and the degree-zero vacuum is a nonzero parallel spinor. Its group lift restricted to determinant-one frames is the standard $SU(3)$ lift, acting trivially on that vacuum.

Ricci-flat Kähler tangent connection is Hermitian–Yang–Mills proves that, for a supplied Ricci-flat Kähler metric with fundamental form $\omega$, the tangent Chern connection satisfies $F^{0,2}=0$ and ${\omega }^2\wedge F=0$.

Tangent-bundle end cohomology vanishing under exact SU(3) holonomy gives $H^0(Y,T_Y)=H^3(Y,T_Y)=0$ when the metric has exact $SU(3)$ holonomy.

Heterotic Calabi–Yau bundle chiral index gives $N_R-N_{\bar{R}}=-\frac{1}{2}{\mathrm{deg}}_Y(c_3^{\mathrm{C}\mathrm{H}}(T_Y))$ for the declared cohomological chirality convention, provided its endpoint, topological Chern, Chow Chern, AC, actual adjoint-sector and joint-limit leading-order hypotheses hold. Its integral is the degree of an algebraic Chow zero-cycle, not evaluation of a topological class.

Perturbative heterotic Calabi–Yau bundle compactification data requires an actual HYM connection, a source configuration satisfying the chosen local Bianchi identity, and the declared perturbative, trace, and warping assumptions. Heterotic Type I Ten Dimensional N1 Supergravity Action selects the heterotic branch; Heterotic Green Schwarz Bianchi And Anomaly Cancellation supplies only the local source-free Bianchi formula in its own trace convention.

Heterotic massless matter and bundle-cohomology dictionary adopts the cohomology-to-massless-matter assignment for actual paired adjoint sectors in its joint weak-coupling, fixed-shape large-radius limit, at leading tree-level, two-derivative, linearized massless-mode order, with no finite-parameter error estimate or persistence claim. The branching in [A4] supplies precisely the pair $(\mathbf{3},\mathbf{27})$ and its conjugate; the bundle representation $\mathbf{3}$ and the gauge representation $\mathbf{27}$ are distinct. This physical dictionary is not a mathematical theorem.

A Ricci-flat Kähler metric in the chosen class is supplied by Ricci-flat metric in every Kähler class when c₁=0 using the locally proved Calabi theorem. The example still fixes such a metric as background data; metric existence does not establish its exact tangent holonomy or physical trace normalization. The canonical Spin lift is chosen using the descended volume form in [F1] and [F10].

Assume exact tangent $SU(3)$ holonomy for [F3]. Fix the nowhere-zero descended holomorphic volume form $\mathrm{\Omega }$ supplied by [F1], and choose exactly the ${\mathrm{\Omega }}^{-1}$-induced canonical Spin lift of [F10], rather than an arbitrary spin structure. Full lifted holonomy means all spin parallel transports around piecewise-smooth loops at one base point. Verification step 3.2 proves that this chosen lift has full lifted holonomy equal to the standard $SU(3)$ subgroup of $\mathrm{Spin}(6)$ required by [F9]. The quotient calculation does not prove exact tangent holonomy, and that hypothesis is not inferred merely from $K_Y\cong {\mathcal{O}}_Y$.

The standard embedding uses the tangent connection with $H_3=0$. The heterotic gauge trace is normalized so that the embedded curvature form and the tangent term in the adopted Bianchi identity agree. The current Bianchi/action carriers deliberately do not prove a trace conversion, so this matching remains an explicit physical convention needed to instantiate the source-free branch.

Fix a homomorphism $\iota :SU(3)\to E_8$ whose differential is the selected standard maximal ${\mathfrak{s}\mathfrak{u}}_3\subset {\mathfrak{e}}_8$ with centralizer ${\mathfrak{e}}_6$ and adjoint branching as stated in the Example. The gauge connection is the image under $\iota$ of the tangent connection; the associated $27$ sector is assigned to $T_Y$ for the sign convention, and the conjugate sector to $T_Y^{\ast }$. Replacing it by $T_Y^{\ast }$ exchanges $27$ and $\overline{27}$ and reverses the sign.

Adopt only the joint-limit approximation of [F6]: $g_{\mathrm{s}\mathrm{t}\mathrm{r}}=e^{{\mathrm{\Phi }}_0}\to 0$ and ${\alpha }^{\mathrm{\prime }}\mathrm{/}{L}^2\to 0$ for string-frame $g_L=L^2{g}_0$ at fixed smooth Ricci-flat shape, complex structure and holomorphic tangent bundle. Here ${\mathrm{\Phi }}_0$ and $g_{\mathrm{s}\mathrm{t}\mathrm{r}}$ are dimensionless, $L$ has length dimension and ${\alpha }^{\mathrm{\prime }}$ has length-squared dimension. Constant positive metric rescaling preserves the Levi-Civita connection, its holonomy and Ricci-flatness; it multiplies ${\omega }^2\wedge F$ by a positive scalar, so the HYM condition remains valid. The index interpretation below is only the leading-order one and makes no corrected finite-parameter spectrum claim.

Chern classes of a vector bundle on a smooth scheme defines algebraic Chern classes in $A^{\ast }(X)$ and Operational Chern classes and the Whitney formula supplies their exact-sequence Whitney formula and injective splitting principle.

Chow groups of projective space gives ${\mathrm{deg}}_{{\mathbb{P}}^4}(h^4\cap [{\mathbb{P}}^4])=1$ for $h=c_1^{\mathrm{C}\mathrm{H}}({\mathcal{O}}_{{\mathbb{P}}^4}(1))$. Intersection with an invertible sheaf and the first Chern class identifies a hypersurface cut by a section of $\mathcal{O}(5)$ with its first-Chern divisor class, and Flat pullback of cycles and of rational equivalence defines flat pullback on Chow groups and its generic-length multiplicities.

Over an algebraically closed field, every maximal ideal is an evaluation ideal identifies maximal ideals of a complex polynomial ring with complex points; hence a proper ideal in a finitely generated complex algebra has a complex zero. Chinese remainder theorem for pairwise comaximal ideals gives simultaneous residues for pairwise comaximal ideals. The diagonal ideal modulo its square is Omega identifies relative differentials with the diagonal ideal modulo its square. Under AC, Finite étale algebras have finite locally free underlying modules says that a module-finite algebra is finite étale precisely when its underlying module is finitely presented and flat and its relative differentials vanish. Step 1.2 applies these local proved tools to the invariant affine quotient charts constructed in [F1], and reconstructs finite étaleness and degree $25$ without importing a free-action torsor theorem.

Naturality of the Chow ring and the projection formula gives naturality of algebraic Chern classes under pullback and ${\pi }_{\ast }({\pi }^{\ast }\alpha \cdot \beta )=\alpha \cdot {\pi }_{\ast }\beta$ for a proper map of smooth projective varieties.