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Gauge action on a trivial real principal bundle and associated line

Example

Let $M = R^{2}$ with coordinates $(x, y)$ , let $G = (R, +)$ , and let $P = M \times R$ with projection $(x, y, t) \mapsto (x, y)$ and right action $(x, y, t) s = (x, y, t + s)$ . Then $G (P) ≅ C^{\infty} (R^{2}, R)$ as groups under addition, by $u \mapsto Φ_{u}$ with $Φ_{u} (x, y, t) = (x, y, t + u (x, y))$ . A nonconstant gauge transformation can move a connection while carrying its curvature: for $H = span {X = \partial_{x}, Y = \partial_{y} - x \partial_{t}},$ the curvature is $K_{H} (X, Y) = \partial_{t}$ . The transformation $u (x, y) = x$ sends $X$ to $X + \partial_{t}$ and $Y$ to $Y$ , and the pushed connection has the same vertical curvature value.

For the smooth representation $ρ (s) = e^{s}$ of $(R, +)$ on $R$ , the associated line bundle $E_{ρ} = P \times_{ρ} R$ is trivial, with global vector-bundle coordinates $[(x, y, t), v] \mapsto (x, y, e^{t} v)$ . In these coordinates the induced map of $Φ_{u}$ is ${\overline{Φ}}_{u} (x, y, v) = (x, y, e^{u (x, y)} v) .$ Thus the gauge group acts linearly on the smooth sections of this associated line bundle, and the nonconstant choice $u (x, y) = x$ multiplies a section by $e^{x}$ .

Facts & Assumptions

Given: ${A C}_{ω}$ , $M = R^{2}$ , $G = (R, +)$ , $P = M \times R$ , its product projection, and standard right action.

[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 is inherited here from the declared smooth tangent-bundle convention.

[F1]

Smooth principal $G$ -bundle: a smooth principal bundle has a smooth projection and right action, together with equivariant smooth local product charts.

[F2]

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

[F3]

Gauge group of a smooth principal bundle: gauge transformations are smooth principal-bundle diffeomorphisms covering the identity of $M$ .

[F4]

Gauge transformations act on principal connections and carry curvature: pushforward by a gauge transformation preserves a principal connection and carries its curvature by the differential of that gauge transformation.

[F5]

Associated bundles: the associated line is the quotient with relation $[p s, v] = [p, e^{s} v]$ .

[F6]

Associated vector bundles are well-defined: principal local sections give smooth associated vector-bundle charts.

[F7]

Gauge transformations induce associated bundle automorphisms: $[p, v] \mapsto [Φ (p), v]$ is the induced smooth bundle automorphism and gives a linear action on smooth sections.

[L1]

The Lie bracket of smooth vector fields: $[A, B] f = A (B f) - B (A f)$ .

Verification

1.1F1given

The product projection and right-translation action are smooth, the identity product chart is equivariant, and each fibre is the free transitive right $(R, +)$ -torsor; therefore [F1] makes $P$ a smooth principal bundle.

2.1F3givenconstructstep 1.1

A gauge transformation has the form $Φ (x, y, t) = (x, y, τ (x, y, t))$ ; equivariance gives $τ (x, y, t + s) = τ (x, y, t) + s$ , so $u (x, y) : = τ (x, y, 0)$ is smooth and $τ = t + u$ . Conversely each smooth $u$ gives $Φ_{u}$ with inverse $Φ_{- u}$ , and $Φ_{u} \circ Φ_{v} = Φ_{u + v}$ , proving the asserted group isomorphism.

2.2F5F6givenstep 1.1

Define $ρ (s) = e^{s}$ , a smooth homomorphism to ${GL}_{1} (R)$ . By [F5] the map $[(x, y, t), v] \mapsto (x, y, e^{t} v)$ is well-defined, with inverse $(x, y, w) \mapsto [(x, y, 0), w]$ ; it is the global associated chart supplied by [F6].

2.3A1F2L1givenstep 1.1

The vertical bundle is spanned by $\partial_{t}$ , while $X, Y$ project to $\partial_{x}, \partial_{y}$ ; their span complements the vertical direction and is invariant under $t$ -translations, so [F2] makes $H$ a principal connection. By [L1], $[X, Y] = [\partial_{x}, \partial_{y} - x \partial_{t}] = - \partial_{t}$ , hence $K_{H} (X, Y) = \partial_{t}$ .

3.1F7step 2.1step 2.2algebra

By [F7], $Φ_{u}$ induces an associated bundle automorphism. In the coordinates of step 2.2 it sends $[(x, y, t), v]$ to $[(x, y, t + u (x, y)), v]$ , hence $(x, y, v) \mapsto (x, y, e^{u (x, y)} v)$ ; the section $s_{0} (x, y) = [(x, y, 0), 1]$ is sent to $e^{u} s_{0}$ .

3.2F2F4L1step 2.1step 2.3

For $u (x, y) = x$ , $d Φ_{u}$ sends $X$ to $X + \partial_{t}$ and $Y$ to $Y$ . The pushed connection is principal by [F4], and [L1] gives $[X + \partial_{t}, Y] = - \partial_{t}$ , so $K_{Φ_{u *} H} (X + \partial_{t}, Y) = \partial_{t} = Φ_{u *} K_{H} (X, Y)$ . The horizontal subspace changes, so this nonconstant gauge map moves the connection while carrying its curvature.

4.1F7step 2.1step 2.2step 2.3step 3.1algebra∎

The zero gauge function gives the identity on the connection, curvature, associated bundle, and its sections. A constant $u = c$ has identity differential and leaves $H$ and its curvature fixed, while its associated bundle map is scalar multiplication by $e^{c}$ in the coordinates of step 2.2.

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Sources

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)
Jack Smith, Differential Geometry, Cambridge Part III Michaelmas 2022 lecture notes taken by Leonard Tomczak (standard reference, not scraped)
Karl-Hermann Neeb, Differential Topology of Fiber Bundles (lecture notes, 2024) (standard reference, not scraped)