Knowlpedia

Local connection 1-form

A Lie algebra valued 1-form on an open set obtained by pulling back a principal connection along a local section

Let π:P→M\pi:P\to M be a with Lie algebra g\mathfrak g, and let ω∈Ω1(P;g)\omega\in\Omega^1(P;\mathfrak g) be a representing a .

Given a local section s:U→Ps:U\to P over an open set U⊆MU\subseteq M, the local connection 1-form (also called the gauge potential in that trivialization) is the g\mathfrak g-valued 1-form

A:=s∗ω∈Ω1(U;g).A:=s^*\omega \in \Omega^1(U;\mathfrak g).

This is the local representative of the global connection in the trivialization determined by ss (compare ).

Change of section (local gauge transformation)

If s′(x)=s(x)⋅g(x)s'(x)=s(x)\cdot g(x) for a smooth map g:U→Gg:U\to G (a ), then the pulled-back forms satisfy the standard transformation law

A′=g−1Ag+g−1dg,A' = g^{-1} A g + g^{-1}dg,

as recorded in .

On overlaps

If {si:Ui→P}\{s_i:U_i\to P\} is a system of local sections and gij:Ui∩Uj→Gg_{ij}:U_i\cap U_j\to G are the associated transition functions defined by sj=si gijs_j=s_i\,g_{ij} (see ), then on Ui∩UjU_i\cap U_j the local forms obey

Aj=gij−1Aigij+gij−1dgij.A_j = g_{ij}^{-1} A_i g_{ij} + g_{ij}^{-1}dg_{ij}.

The corresponding local curvature 2-form is

F:=dA+A∧A,F:=dA + A\wedge A,

as in , and it transforms by conjugation (see ).

Examples
  1. Trivial principal bundle. If P≅M×GP\cong M\times G is trivial, choosing the global section s(x)=(x,e)s(x)=(x,e) identifies a principal connection with a single global g\mathfrak g-valued 1-form AA on MM. The corresponds to A=0A=0, while a connection has the form A=g−1dgA=g^{-1}dg.
  1. Dirac monopole on the Hopf bundle. For the as a principal U(1)U(1)-bundle, the is described by two local connection 1-forms ANA_N and ASA_S on the northern and southern charts of S2S^2, related on the overlap by a U(1)U(1)-valued gauge function.
  1. Levi-Civita connection in a local frame. For a Riemannian manifold, the can be viewed as a principal connection on the . Choosing a local orthonormal frame gives a matrix-valued local connection 1-form whose entries are the usual connection coefficients; its curvature is the matrix of curvature 2-forms in that frame (see ).