PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Gauge transformations induce associated bundle automorphisms

Statement

Let $G$ be a finite-dimensional real Lie group, let $P \to M$ be a smooth right principal $G$ -bundle, let $V$ be a finite-dimensional real vector space, and let $ρ : G \to GL (V)$ be a smooth representation, where $GL (V)$ is the group of invertible linear maps $V \to V$ . Write $G (P)$ for the gauge group of Gauge group of a smooth principal bundle. Put $E_{ρ} : = P \times_{ρ} V$ for the associated smooth vector bundle of Associated bundles and Associated vector bundles are well-defined. For every $Φ \in G (P)$ define $\overline{Φ} : E_{ρ} \to E_{ρ}, [p, v] ⟼ [Φ (p), v] .$ Here a smooth vector-bundle automorphism means a smooth fibrewise-linear diffeomorphism covering ${id}_{M}$ . The map $\overline{Φ}$ is such an automorphism, and $\overline{Φ \circ Ψ} = \overline{Φ} \circ \overline{Ψ}, \overline{{id}_{P}} = {id}_{E_{ρ}} .$ Thus $Φ \mapsto \overline{Φ}$ is a group homomorphism to the group of smooth vector-bundle automorphisms of $E_{ρ}$ . If $Γ^{\infty} (E_{ρ})$ denotes the smooth sections as in Smooth sections, local sections, and support, then $Φ \cdot s : = \overline{Φ} \circ s$ is a linear left action of $G (P)$ on $Γ^{\infty} (E_{ρ})$ . No injectivity or surjectivity of either induced action is asserted.

The square below records the quotient descent, where $q : P \times V \to E_{ρ}$ is the orbit projection. It commutes because $Φ$ is $G$ -equivariant; the pointwise verification appears in step 1.1.

Facts & Assumptions

Given: A smooth right principal $G$ -bundle $π : P \to M$ , a smooth finite-dimensional representation $ρ : G \to GL (V)$ , its associated smooth vector bundle $E_{ρ}$ , and a gauge transformation $Φ \in G (P)$ .

[F1]

Gauge group of a smooth principal bundle: $Φ$ is a smooth diffeomorphism with $π \circ Φ = π$ and $Φ (p g) = Φ (p) g$ ; these maps form a group under composition.

[L1]

Smooth principal $G$ -bundle: over a principal chart, each $p \in P$ has a unique smooth group coordinate $g$ in $p = s (π (p)) g$ .

[F2]

Associated bundles: $E_{ρ}$ is the quotient with relation $[p g, v] = [p, ρ (g) v]$ .

[F3]

Associated vector bundles are well-defined: a smooth principal section gives an associated vector-bundle chart $[s (x), v] \leftrightarrow (x, v)$ , and these charts define the smooth vector-bundle structure.

[F4]

Vector bundle maps over a smooth base map: a smooth vector-bundle map over $f$ is smooth, covers $f$ , and is linear on each fibre.

[F5]

Smooth sections, local sections, and support: a smooth section $s$ is a smooth map with bundle projection composed with $s$ equal to the identity.

[F6]

Smooth sections form a module over smooth functions: smooth sections are closed under pointwise addition and multiplication by smooth functions, hence form a real vector space.

[F7]

Identity maps and composites of smooth maps are smooth: composites of smooth maps are smooth.

Proof

1.1F1F2given

If $[p g, v] = [p, ρ (g) v]$ , then $[Φ (p g), v] = [Φ (p) g, v] = [Φ (p), ρ (g) v]$ by [F1] and [F2]; hence the formula is representative-independent, and $π (Φ (p)) = π (p)$ shows that it covers ${id}_{M}$ . The displayed square commutes on each $(p, v)$ .

2.1F1F3F4F7L1step 1.1

Let $s : U \to P$ be any local section from a smooth principal chart. The unique $a_{s} : U \to G$ with $Φ (s (x)) = s (x) a_{s} (x)$ exists by [L1] and is smooth, since it is the group coordinate of $Φ \circ s$ . In the associated chart from [F3], $\overline{Φ} (x, v) = (x, ρ (a_{s} (x)) v)$ , which is smooth and fibrewise linear; the same calculation for $Φ^{- 1}$ proves that $\overline{Φ^{- 1}}$ is smooth.

3.1F1step 2.1algebra

For every $[p, v] \in E_{ρ}$ , $\overline{Φ} (\overline{Ψ} ([p, v])) = [Φ (Ψ (p)), v] = \overline{Φ \circ Ψ} ([p, v])$ , and $\overline{{id}_{P}} = {id}_{E_{ρ}}$ . Taking $Ψ = Φ^{- 1}$ shows that $\overline{Φ^{- 1}}$ is the inverse of $\overline{Φ}$ ; thus $\overline{Φ}$ is a smooth bundle automorphism and the assignment preserves composition.

4.1F1F5F6F7step 2.1step 3.1∎

For $s \in Γ^{\infty} (E_{ρ})$ , $\overline{Φ} \circ s$ is smooth by [F7] and step 2.1, and it is a section because $\overline{Φ}$ covers ${id}_{M}$ . Fibrewise linearity makes this assignment linear on the real vector space [F6] of smooth sections; [F1] and step 3.1 give identity and composition, so these maps form a linear left action of $G (P)$ .

Depends on

Used by

Gauge action on a trivial real principal bundle and associated line Example

Dependency tree · two levels

36 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Karl-Hermann Neeb, Differential Topology of Fiber Bundles (lecture notes, 2024) (standard reference, not scraped)
D. Husemöller, M. Joachim, B. Jurčo, and M. Schottenloher, Basic Bundle Theory and K-Cohomology Invariants (standard reference, not scraped)
Matthijs Vákár, Principal Bundles and Gauge Theories (standard reference, not scraped)