Knowlpedia

Gauge equivalence classes of connections form an orbit space

The space of connections modulo gauge transformations is the set of orbits for the gauge group action

Let π:P→M\pi:P\to M be a , let Conn⁡(P)\operatorname{Conn}(P) be its set of , and let G(P)\mathcal G(P) be its . Pullback defines the

Conn⁡(P)×G(P)⟶Conn⁡(P),(ω,u)⟼ω⋅u:=u∗ω.\operatorname{Conn}(P)\times\mathcal G(P)\longrightarrow\operatorname{Conn}(P), \qquad (\omega,u)\longmapsto\omega\cdot u:=u^*\omega.

Its orbit relation is

ω0∼ω1⟺∃ u∈G(P) such that ω1=u∗ω0.\omega_0 \sim \omega_1 \quad\Longleftrightarrow\quad \exists\,u\in\mathcal G(P)\ \text{such that}\ \omega_1=u^*\omega_0.

Thus the

Conn(P)/G(P)\mathrm{Conn}(P)/\mathcal G(P)

is the set of gauge-equivalence classes of connections.

Local form

In a trivialization, a g:U→Gg:U\to G sends a AA to

Ag=Ad⁡g−1A+g−1dg.A^g=\operatorname{Ad}_{g^{-1}}A+g^{-1}dg.
Examples
  1. Trivial bundle: gauge action on Lie algebra valued 1-forms. If P=M×GP=M\times G is the , then a connection is represented by a g\mathfrak g-valued 11-form AA, and the gauge group identifies with C∞(M,G)C^\infty(M,G).
    A⟼Ad⁡g−1A+g−1dg.A \longmapsto \operatorname{Ad}_{g^{-1}}A+g^{-1}dg.
  2. Abelian case. For G=U(1)G=U(1) on a trivial bundle, the adjoint term is AA, so
    A⟼A+g−1dg.A \longmapsto A + g^{-1}dg.
    Locally writing g=eiθg=e^{i\theta} gives g−1dg=i dθg^{-1}dg=i\,d\theta.
  1. Flat connections. Gauge equivalence preserves the of a flat connection up to conjugation. If GG is connected, gauge classes of flat connections on the trivial bundle over S1S^1 correspond to conjugacy classes in GG.