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Heterotic Calabi–Yau bundle chiral index

Physical scope

Conditional index deduction for an actual paired gauge-adjoint matter sector in a smooth projective heterotic bundle phase, at leading tree level in the two-derivative linearized massless-mode approximation as g_str=exp(Phi_0) tends to zero and alpha-prime/L^2 tends to zero for string-frame g_L=L^2 g_0 at fixed smooth shape and holomorphic bundle. No finite-parameter error bound or persistence after corrections is claimed. It predicts the signed net index, not vector-like pairs, the full spectrum, experimental data or vacuum existence.

Statement

Assume the Axiom of Choice. Let $X$ be a smooth, connected, irreducible projective complex threefold with specified trivialization $K_{X} ≅ O_{X}$ and complex orientation. For a locally free holomorphic bundle $E$ on $X$ , write $c_{i}^{t o p} (E) \in H^{2 i} (X; Z)$ for the topological Chern class of its underlying complex bundle, and write $c_{i}^{C H} (E) \in A^{i} (X)$ for its algebraic Chow Chern class. Let $A^{*} (X)_{Q} = A^{*} (X) \otimes Q$ and let $\deg_{X} : A^{3} (X)_{Q} \to Q$ be the degree of the resulting zero-cycle class, extended rationally; this algebraic degree is distinct in definition from evaluation of a topological cohomology class on $[X]$ . Let $V_{R}$ be a rank- $r > 0$ locally free holomorphic bundle in an actual paired gauge-adjoint matter sector of the adopted post-cmp-heterotic-cohomology-spectrum-dictionary: $R$ and $\overline{R}$ are distinct conjugate representations of the unbroken gauge centralizer, and $V_{R}$ comes from the full structure-group multiplicity representation paired with $R$ in the adjoint branching. Use the dictionary only at leading tree level in its two-derivative, linearized massless-mode approximation. In natural units fix a smooth internal shape metric $g_{0}$ , complex structure and holomorphic bundle, use the string-frame metrics $g_{L} = L^{2} g_{0}$ and constant dimensionless dilaton $Φ_{0}$ , and take the joint formal limit $g_{s t r} = e^{Φ_{0}} \to 0$ , $α^{'} / L^{2} \to 0$ , where $[L] = l e n g t h$ and $[α^{'}] = {l e n g t h}^{2}$ . No finite-parameter error bound or persistence of unprotected massless states after corrections is asserted. Assume $c_{1}^{t o p} (T_{X}) = c_{1}^{t o p} (V_{R}) = 0, c_{1}^{C H} (T_{X}) = c_{1}^{C H} (V_{R}) = 0 in A^{1} (X)_{Q},$ and coherent cohomology $H^{0} (X, V_{R}) = H^{3} (X, V_{R}) = 0$ , where $V_{R}$ is the coherent sheaf of holomorphic sections of $V_{R}$ . By the Dolbeault resolution comparison in the Proof, put $h^{q} (X, V_{R}) = \dim_{C} H_{\overset{ˉ}{\partial}}^{q} (X, V_{R}) = \dim_{C} H^{q} (X, V_{R})$ , and use the adopted chirality convention $N_{R} - N_{\overline{R}} = h^{1} (X, V_{R}) - h^{2} (X, V_{R})$ . Then $N_{R} - N_{\overline{R}} = - χ (X, V_{R}) = - \deg_{X} ⁣ ([{ch}^{C H} (V_{R}) {td}^{C H} (T_{X})]_{3}) = - \frac{1}{2} \deg_{X} (c_{3}^{C H} (V_{R})) .$ Replacing $V_{R}$ by its dual exchanges the representation assignment and reverses this signed index, with the same endpoint vanishings transferred by Serre duality. This index does not determine vector-like pairs or the full massless spectrum.

Statement references

Perturbative heterotic Calabi–Yau bundle compactification data

Chern classes from the projective-bundle relation

Chern classes of a vector bundle on a smooth scheme

E-valued Dolbeault operators and their L2 structure

Heterotic massless matter and bundle-cohomology dictionary

The acyclic-resolution theorem for right derived functors

The local Dolbeault lemma on nested polydiscs

Sheaves of smooth-function modules are cohomologically acyclic

Grothendieck-Riemann-Roch for projective morphisms

The Chern character and the Todd class

Serre duality for locally free sheaves on a smooth projective variety

Facts & Assumptions

Given: The Axiom of Choice; a smooth connected irreducible projective complex threefold $X$ with $K_{X} ≅ O_{X}$ ; the specified leading-order joint-limit matter sector; the locally free holomorphic bundle $V_{R}$ and its coherent sheaf $V_{R}$ ; the topological and Chow Chern classes typed in the Statement; and the stated endpoint and Chow-class vanishing hypotheses.

[F9]

Projective threefold coherent comparison and holomorphic bundle algebraization algebraizes the given holomorphic bundle on the actual arbitrary smooth projective threefold, uniquely up to isomorphism, and canonically identifies analytic and algebraic coherent cohomology in every degree and identifies duals. In the algebraic expressions below, $V_{R}$ denotes this finite locally free algebraization; its analytification is the original holomorphic section sheaf. Thus the Chow Chern classes and the input to GRR have a defined algebraic carrier. No projective-space hypothesis on $X$ is imposed.

[F1]

Grothendieck-Riemann-Roch for projective morphisms applies to $X \to Spec C$ and $[V_{R}] \in K_{0} (X)$ ; it identifies $χ (X, V_{R})$ with $\deg_{X} ([{ch}^{C H} (V_{R}) {td}^{C H} (T_{X})]_{3})$ in the rational Chow ring.

[F2]

The Chern character and the Todd class defines the Chern character and Todd class in $A^{*} (X)_{Q}$ and gives ${ch}_{1}^{C H} = c_{1}^{C H}$ , ${ch}_{2}^{C H} = \frac{1}{2} ((c_{1}^{C H})^{2} - 2 c_{2}^{C H})$ , ${ch}_{3}^{C H} = \frac{1}{6} ((c_{1}^{C H})^{3} - 3 c_{1}^{C H} c_{2}^{C H} + 3 c_{3}^{C H})$ , and ${td}^{C H} = 1 + \frac{1}{2} c_{1}^{C H} + \frac{1}{12} ((c_{1}^{C H})^{2} + c_{2}^{C H}) + \frac{1}{24} c_{1}^{C H} c_{2}^{C H} + \dots$ .

[F3]

Euler characteristic of a coherent sheaf defines $χ (X, V_{R}) = \sum_{q = 0}^{3} (- 1)^{q} \dim_{C} H^{q} (X, V_{R})$ , with finite-dimensional terms on this projective variety.

[F4]

Heterotic massless matter and bundle-cohomology dictionary adopts $h^{1} (X, V_{R})$ and $h^{2} (X, V_{R})$ as multiplicities in the paired distinct conjugate unbroken-gauge representations $R$ and $\overline{R}$ , where $V_{R}$ is the associated bundle for the corresponding full structure-group multiplicity representation in the actual gauge-adjoint branching. This applies only in the 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. These physical conditions are hypotheses of this theorem, including when using the dual paired sector.

[F5]

Associated holomorphic bundle in a heterotic representation defines $V_{R}$ as the locally free holomorphic vector bundle in the chosen representation.

[F6]

Chern classes from the projective-bundle relation defines $c_{i}^{t o p}$ in integral singular cohomology, while Chern classes of a vector bundle on a smooth scheme defines $c_{i}^{C H}$ in the algebraic Chow ring of a smooth scheme; these are distinct typed classes here.

[F7]

E-valued Dolbeault operators and their L2 structure defines the global $V_{R}$ -valued Dolbeault complex. In a holomorphic frame the local Dolbeault lemma The local Dolbeault lemma on nested polydiscs applies componentwise, and each sheaf of smooth $V_{R}$ -valued forms is acyclic by Sheaves of smooth-function modules are cohomologically acyclic. Thus the Dolbeault complex is an acyclic resolution of the original holomorphic section sheaf, identified with $V_{R}^{a n}$ by [F9]. Applying The acyclic-resolution theorem for right derived functors to the global-sections functor, with the injective-resolution data defining sheaf cohomology and all successive kernel sheaves in its domain, identifies its global cohomology canonically with coherent sheaf cohomology. Its DC hypothesis is covered by the assumed AC.

[F8]

Serre duality for locally free sheaves on a smooth projective variety gives the perfect pairings $H^{q} (X, V_{R}) \times H^{3 - q} (X, V_{R}^{\lor} \otimes K_{X}) \to C$ in the stated smooth projective hypotheses.

[A1]

The Axiom of Choice is assumed as an ambient mathematical hypothesis and is inherited by the exact imported GRR, coherent cohomology, and Serre-duality carriers. Its use is mathematical; no physical measurement or experimental premise enters this deduction.

[A2]

The topological equalities $c_{1}^{t o p} (V_{R}) = c_{1}^{t o p} (T_{X}) = 0$ , the rational Chow equalities $c_{1}^{C H} (V_{R}) = c_{1}^{C H} (T_{X}) = 0$ , and $H^{0} (X, V_{R}) = H^{3} (X, V_{R}) = 0$ are explicit hypotheses. The topological vanishings alone are not used as substitutes for the Chow-class vanishings; neither is inferred merely from the phrase “Calabi–Yau.”

Proof

Given: The projective threefold, bundle and hypotheses above, with [F1]–[F9] and [A1]–[A2].

1.1givenF5F7F9algebra

First use [F9] to choose the finite locally free algebraization of the given holomorphic $V_{R}$ ; all algebraic Chow and GRR notation refers to this algebraization. On each holomorphic coordinate polydisc with a holomorphic frame of rank $r$ , the sheaf complex $0 \to O (V_{R}) \to A^{0, 0} (V_{R}) \overset{{\overset{ˉ}{\partial}}_{V_{R}}}{\to} \dots \overset{{\overset{ˉ}{\partial}}_{V_{R}}}{\to} A^{0, 3} (V_{R}) \to 0$ is the direct sum of $r$ scalar Dolbeault resolutions in its coefficient functions: at degree zero its kernel consists of holomorphic sections, and at every positive degree local closedness is exact by [F7] and the componentwise theorem The local Dolbeault lemma on nested polydiscs. Each $A^{0, q} (V_{R})$ is a sheaf of modules over the sheaf of smooth real functions, so it is acyclic for global sections by [F7]. The acyclic resolution theorem The acyclic-resolution theorem for right derived functors in [F7] therefore identifies the cohomology of its global-section complex, which is $H_{\overset{ˉ}{\partial}}^{q} (X, V_{R})$ by the Definition, with $H^{q} (X^{a n}, V_{R}^{a n})$ . Combining this with [F9] gives the Dolbeault-to-algebraic-coherent comparison used below; it applies here because $X$ is a smooth projective complex threefold and $V_{R}$ is holomorphic.

2.1givenF3F4A2step 1.1algebra

By [F3] and [A2], $χ (X, V_{R}) = h^{0} - h^{1} + h^{2} - h^{3} = - h^{1} (X, V_{R}) + h^{2} (X, V_{R})$ . The adopted assignment [F4] defines $N_{R} - N_{\overline{R}} = h^{1} - h^{2}$ , so $N_{R} - N_{\overline{R}} = - χ (X, V_{R})$ .

3.1givenF1F5F9A1step 2.1

The algebraic sheaf $V_{R}$ is finite locally free by [F9], and its cohomology is the analytic cohomology used in step 2.1. Apply [F1] to the projective structure morphism $f : X \to Spec C$ ; its source and target are nonsingular irreducible quasi-projective schemes, and AC is [A1]. The Hirzebruch–Riemann–Roch clause gives $χ (X, V_{R}) = \deg_{X} ([{ch}^{C H} (V_{R}) {td}^{C H} (T_{X})]_{3})$ . Combining this with step 2.1 gives the first equality in the Formula.

4.1givenF2A2step 3.1algebra

In the rational Chow ring, [F2] gives ${ch}_{1}^{C H} (V_{R}) = c_{1}^{C H} (V_{R}) = 0$ , ${ch}_{2}^{C H} (V_{R}) = - c_{2}^{C H} (V_{R})$ , and ${ch}_{3}^{C H} (V_{R}) = \frac{1}{2} c_{3}^{C H} (V_{R})$ under [A2]. The same formulas and $c_{1}^{C H} (T_{X}) = 0$ give ${td}_{1}^{C H} (T_{X}) = 0$ and ${td}_{3}^{C H} (T_{X}) = 0$ . Therefore the codimension-three part of the product in step 3.1 is exactly ${ch}_{3}^{C H} (V_{R}) = \frac{1}{2} c_{3}^{C H} (V_{R})$ : all other degree-three summands contain either ${ch}_{1}^{C H} (V_{R})$ , ${td}_{1}^{C H} (T_{X})$ , or ${td}_{3}^{C H} (T_{X})$ . Taking the Chow degree yields the second Formula equality.

5.1F2F4F8F9A2step 1.1algebra∎

The dual compatibility and cohomology comparisons of [F9] transport the algebraic Serre pairing [F8] to the original analytic bundles. This pairing, the specified trivialization $K_{X} ≅ O_{X}$ , and the definition of $V_{R}^{\lor}$ give $H^{q} (X, V_{R}^{\lor}) ≅ H^{3 - q} (X, V_{R})^{*}$ for every $q$ . Thus the assumed endpoint vanishings transfer to $H^{0} (X, V_{R}^{\lor}) = H^{3} (X, V_{R}^{\lor}) = 0$ , and $h^{1} (X, V_{R}^{\lor}) = h^{2} (X, V_{R})$ while $h^{2} (X, V_{R}^{\lor}) = h^{1} (X, V_{R})$ ; the adopted dictionary therefore reverses the signed multiplicity. Also the algebraic Chern roots of $V_{R}^{\lor}$ are $- x_{i}$ when those of $V_{R}$ are $x_{i}$ , so $c_{3}^{C H} (V_{R}^{\lor}) = (- 1)^{3} c_{3}^{C H} (V_{R}) = - c_{3}^{C H} (V_{R})$ by the splitting definition in [F2]. Hence the GRR expression itself changes sign as well.

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