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

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Gauge transformations act on principal connections and carry curvature

Statement

Assume ${A C}_{ω}$ . Let $P \to M$ be a smooth right principal $G$ -bundle with a principal connection $H$ , and let $Φ \in G (P)$ . Then $Φ_{*} H : = d Φ (H)$ is a principal connection. If $K_{H}$ and $K_{Φ_{*} H}$ are the vertical-valued curvatures from Principal connections and vertical curvature, then for $X, Y \in T_{p} P$ , $K_{Φ_{*} H} (d Φ_{p} X, d Φ_{p} Y) = d Φ_{p} (K_{H} (X, Y)) .$ Composition makes this a left action of $G (P)$ on the set of principal connections.

Facts & Assumptions

Given: ${A C}_{ω}$ , a smooth right principal $G$ -bundle $π : P \to M$ , a principal connection $H$ , and a smooth gauge transformation $Φ : P \to P$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ asserts that every countable family of nonempty sets has a choice function; it supplies the declared smooth tangent-bundle convention used by the connection definition.

[F1]

Principal connections and vertical curvature: a principal connection is a smooth right-invariant complement $T P = V P \oplus H$ , and its curvature is $K_{H} (X, Y) = - v ([h X, h Y])$ .

[F2]

Gauge group of a smooth principal bundle: a gauge transformation is a smooth diffeomorphism $Φ$ satisfying $π \circ Φ = π$ and $Φ (p g) = Φ (p) g$ .

[L1]

The chain rule for differentials of smooth maps: for smooth $F, G$ , $d (G \circ F)_{p} = d G_{F (p)} \circ d F_{p}$ .

[L2]

Diffeomorphism pushforward preserves Lie brackets: a diffeomorphism satisfies $F_{*} [X, Y] = [F_{*} X, F_{*} Y]$ .

Proof

1.1A1F1F2L1

Under [A1], the smooth tangent-bundle convention used by [F1] is in force. Since $π \circ Φ = π$ , [L1] gives $d π_{Φ (p)} \circ d Φ_{p} = d π_{p}$ ; thus $d Φ_{p} (V_{p} P) \subseteq V_{Φ (p)} P$ , and applying the same identity to $Φ^{- 1}$ gives equality.

2.1F1F2L1step 1.1

Set $H^{'} = d Φ (H)$ . The isomorphism $d Φ_{p}$ maps vertical and horizontal complements to complements, and differentiating $Φ \circ R_{g} = R_{g} \circ Φ$ by [L1] shows right invariance; hence [F1] makes $H^{'}$ a principal connection.

3.1F1L2step 2.1

At $q = Φ (p)$ , the projections satisfy $h^{'} \circ d Φ = d Φ \circ h$ and $v^{'} \circ d Φ = d Φ \circ v$ . By [L2] pushforward preserves brackets, so the curvature formula [F1] gives $K_{H^{'}} (d Φ X, d Φ Y) = - v^{'} ([d Φ (h X), d Φ (h Y)]) = - d Φ (v ([h X, h Y])) = d Φ (K_{H} (X, Y))$ .

4.1F2step 2.1algebra∎

Pushforward composition gives $(Φ \circ Ψ)_{*} H = Φ_{*} (Ψ_{*} H)$ and the identity pushes $H$ to itself, so $Φ \cdot H : = Φ_{*} H$ is a left group action.

Depends on

Used by

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

Dependency tree · two levels

23 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

Matthijs Vákár, Principal Bundles and Gauge Theories (standard reference, not scraped)
Jack Smith, Differential Geometry, Cambridge Part III Michaelmas 2022 lecture notes taken by Leonard Tomczak (standard reference, not scraped)