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 . Write

Conn(P)\mathrm{Conn}(P)

for the set (indeed an affine space) of all on PP.

Let G(P)\mathcal G(P) be the of PP, i.e. the group of covering the identity on MM.

By , there is a well-defined left action

G(P)×Conn(P)⟶Conn(P),(u,ω)⟼u∗ω.\mathcal G(P)\times \mathrm{Conn}(P)\longrightarrow \mathrm{Conn}(P),\qquad (u,\omega)\longmapsto u^*\omega.

Because this is a genuine group action, it determines an equivalence relation on Conn(P)\mathrm{Conn}(P):

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.

Corollary (orbit space of connections modulo gauge)

The quotient set

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

is therefore a well-defined orbit space: its elements are precisely the gauge equivalence classes of connections, i.e. the orbits of the action u⋅ω:=u∗ωu\cdot \omega := u^*\omega.

In local data, this action is the familiar gauge transformation law for connection 1-forms: on a chart, a g:U→Gg:U\to G sends a AA to AgA^g, as in .

Examples
  1. Trivial bundle: gauge action on Lie algebra valued 1-forms. If P=M×GP=M\times G is the , then specifying a connection is equivalent to specifying a g\mathfrak g-valued 1-form A∈Ω1(M;g)A\in\Omega^1(M;\mathfrak g). The gauge group identifies with C∞(M,G)C^\infty(M,G), and the action is
    A⟼g−1Ag+g−1dg,A \longmapsto g^{-1}Ag + g^{-1}dg,
    exactly the transformation described by .
  1. Abelian case: U(1)U(1) connections differ by exact 1-forms on a trivial bundle. For G=U(1)G=U(1) on a trivial bundle, the adjoint term g−1Agg^{-1}Ag is just AA, so the gauge action becomes
    A⟼A+g−1dg.A \longmapsto A + g^{-1}dg.
    Writing g=eiθg=e^{i\theta} locally, one has g−1dg=i dθg^{-1}dg = i\,d\theta. Thus gauge-equivalent connections differ by an exact 1-form, and the orbit space records precisely the ambiguity coming from adding exact forms.
  1. Flat connections and holonomy data. For a , gauge equivalence preserves the up to conjugation. On M=S1M=S^1, gauge classes of flat connections on the trivial bundle correspond to conjugacy classes in GG via the holonomy element around the loop (constructed as in ).