Knowlpedia

Convention: local curvature is F = dA + A wedge A

The sign and bracket convention relating a local connection 1 form to its local curvature 2 form.

A principal connection can be described locally by a Lie algebra–valued 1-form, and its curvature is then expressed by a standard structure equation. Different sign conventions appear in the literature; this knowl fixes the convention used here.

Let π:P→M\pi:P\to M be a and let ω∈Ω1(P;g)\omega\in\Omega^1(P;\mathfrak{g}) be a . Let Ω∈Ω2(P;g)\Omega\in\Omega^2(P;\mathfrak{g}) be its curvature (see ).

Fix a local section s:U→Ps:U\to P over an open set U⊆MU\subseteq M. Define:

  • the local connection 1-form on UU by
    A:=s∗ω∈Ω1(U;g),A := s^*\omega \in \Omega^1(U;\mathfrak{g}),
    as in ;
  • the local curvature 2-form on UU by
    F:=s∗Ω∈Ω2(U;g),F := s^*\Omega \in \Omega^2(U;\mathfrak{g}),
    as in .
Convention

We use the local curvature formula

F=dA+A∧A.F = dA + A\wedge A.

This is the convention recorded in .

Here dd is the , and A∧AA\wedge A is the g\mathfrak{g}-valued 2-form defined by

(A∧A)(X,Y):=[A(X),A(Y)],(A\wedge A)(X,Y) := [A(X),A(Y)],

using the on g\mathfrak{g}. (This is a standard way to combine the on forms with the bracket on the Lie algebra.)

With this convention, the global structure equation on PP can be written in the parallel form

Ω=dω+ω∧ω,\Omega = d\omega + \omega\wedge\omega,

where ω∧ω\omega\wedge\omega uses the same bracketed wedge construction.

Examples
  1. Abelian structure group If GG is abelian (for instance U(1)U(1)), then [ ⋅,⋅ ]=0[\,\cdot,\cdot\,]=0 on g\mathfrak{g}, so A∧A=0A\wedge A=0 and the formula reduces to
    F=dA.F=dA.
  1. Pure gauge gives zero curvature On a trivial bundle over UU, a pure gauge local connection can be written as
    A=g−1dgA = g^{-1}dg
    for a smooth map g:U→Gg:U\to G (compare ). With the convention above, one obtains
    F=dA+A∧A=0,F = dA + A\wedge A = 0,
    which is the local manifestation of flatness.
  1. Non-abelian contribution from A wedge A For a non-abelian GG (for example SU(2)SU(2)), even if the components of AA have constant coefficients in a coordinate chart, the term A∧AA\wedge A can be nonzero because it depends on the Lie bracket. This is the origin of genuinely non-linear curvature effects in gauge theory.