DefinitionDefinition: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-11-05 (gpt-6.1-sol)

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Associated holomorphic bundle in a heterotic representation

Physical scope

Purely mathematical chart construction of a holomorphic associated vector bundle; no heterotic spectrum or stability theorem is asserted.

Definition

Let $n \geq 0$ and let $X$ be a complex $n$ -manifold, meaning a Hausdorff, second-countable space with an atlas of charts to open subsets of $C^{n}$ whose transition maps are biholomorphic. Let $(U_{i})_{i \in I}$ be an open cover of $X$ . A complex Lie group $H$ is a finite-dimensional complex manifold equipped with a group structure for which multiplication and inversion are holomorphic. Let $g_{i j} : U_{i} \cap U_{j} \to H$ be holomorphic maps, for all $i, j \in I$ , satisfying $g_{i i} = e$ , $g_{i j} g_{j k} = g_{i k}$ on triple intersections, and $g_{j i} = g_{i j}^{- 1}$ . Let $W$ be a complex vector space of finite dimension $r$ , and let $ρ : H \to {G L}_{C} (W)$ be a holomorphic group homomorphism, where ${G L}_{C} (W)$ is the group of invertible complex-linear maps $W \to W$ ; this is a finite-dimensional holomorphic representation. The associated holomorphic vector bundle in representation $ρ$ is the rank- $r$ bundle obtained by gluing the local products $U_{i} \times W$ with transition maps from the $j$ -coordinates to the $i$ -coordinates given by $(x, w_{j}) \mapsto (x, ρ (g_{i j} (x)) w_{j})$ . Its local sections are holomorphic maps $s_{i} : U_{i} \to W$ satisfying $s_{i} = ρ (g_{i j}) s_{j}$ on overlaps. In a heterotic application, $H$ is the declared structure group of the gauge bundle and $ρ$ is the chosen representation sector; the mathematical definition does not choose an embedding into $E_{8} \times E_{8}$ or $Spin (32) / Z_{2}$ .

Definition references

Perturbative heterotic Calabi–Yau bundle compactification data

Facts & Assumptions

Given: The complex manifold $X$ , the holomorphic transition cocycle $g_{i j}$ , the holomorphic representation $ρ$ , and the finite-dimensional representation space $W$ .

[A1]

The cocycle is part of the supplied bundle atlas; this definition makes no claim that an arbitrary collection of maps satisfying only some cocycle equations defines a global principal bundle.

Proof

Given: The data in the Definition and [A1].

1.1givenA1algebra

Each map $ρ (g_{i j})$ is holomorphic and invertible because $g_{i j}$ is holomorphic and $ρ$ is a holomorphic group homomorphism into $G L (W)$ . On a triple overlap, $ρ (g_{i j}) ρ (g_{j k}) = ρ (g_{i j} g_{j k}) = ρ (g_{i k})$ , and the diagonal and inverse identities follow from $g_{i i} = e$ and $g_{j i} = g_{i j}^{- 1}$ . Thus the stated gluing maps satisfy the vector-bundle cocycle identities.

2.1step 1.1∎

The cocycle identities make the local products $U_{i} \times W$ into a rank- $r$ bundle atlas, and their holomorphic transition maps make its total-space charts holomorphic. The compatibility equation for $s_{i}$ is exactly the equality of the local representatives under this gluing. Therefore the construction is a well-defined holomorphic vector bundle with the stated local sections.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Angel M. Uranga, String Theory: A Graduate Course, Lecture 9: Calabi–Yau compactification and Heterotic String phenomenology