Editing Connection (vector bundle) (section)

== Gauge transformations ==

{{See also|Gauge group (mathematics)}}

Given two connections <math>\nabla_1, \nabla_2</math> on a vector bundle <math>E\to M</math>, it is natural to ask when they might be considered equivalent. There is a well-defined notion of an [[automorphism]] of a vector bundle <math>E\to M</math>. A section <math>u\in \Gamma(\operatorname{End}(E))</math> is an automorphism if <math>u(x)\in \operatorname{End}(E_x)</math> is invertible at every point <math>x\in M</math>. Such an automorphism is called a '''gauge transformation''' of <math>E</math>, and the group of all automorphisms is called the '''gauge group''', often denoted <math>\mathcal{G}</math> or <math>\operatorname{Aut}(E)</math>. The group of gauge transformations may be neatly characterised as the space of sections of the ''capital A adjoint bundle'' <math>\operatorname{Ad}(\mathcal{F}(E))</math> of the [[frame bundle]] of the vector bundle <math>E</math>. This is not to be confused with the ''lowercase a [[adjoint bundle]]'' <math>\operatorname{ad}(\mathcal{F}(E))</math>, which is naturally identified with <math>\operatorname{End}(E)</math> itself. The bundle <math>\operatorname{Ad} \mathcal{F}(E) </math> is the [[associated bundle]] to the principal frame bundle by the conjugation representation of <math>G=\operatorname{GL}(r)</math> on itself, <math>g\mapsto ghg^{-1}</math>, and has fibre the same general linear group <math>\operatorname{GL}(r)</math> where <math>\operatorname{rank} (E) = r</math>. Notice that despite having the same fibre as the frame bundle <math>\mathcal{F}(E)</math> and being associated to it, <math>\operatorname{Ad}(\mathcal{F}(E))</math> is not equal to the frame bundle, nor even a principal bundle itself. The gauge group may be equivalently characterised as <math>\mathcal{G} = \Gamma(\operatorname{Ad} \mathcal{F}(E)).</math>

A gauge transformation <math>u</math> of <math>E</math> acts on sections <math>s\in \Gamma(E)</math>, and therefore acts on connections by conjugation. Explicitly, if <math>\nabla</math> is a connection on <math>E</math>, then one defines <math>u\cdot \nabla</math> by
:<math>(u\cdot \nabla)_X(s) = u(\nabla_X (u^{-1}(s))</math>
for <math>s\in \Gamma(E), X\in \Gamma(TM)</math>. To check that <math>u\cdot \nabla</math> is a connection, one verifies the product rule
:<math> \begin{align} u\cdot \nabla(fs) &= u(\nabla(u^{-1}(fs)))\\&=u(\nabla(fu^{-1}(s)))\\&=u(df \otimes u^{-1}(s)) + u(f\nabla(u^{-1}(s)))\\&=df \otimes s + f u\cdot \nabla(s).\end{align}</math>
It may be checked that this defines a left [[group action]] of <math>\mathcal{G}</math> on the affine space of all connections <math>\mathcal{A}</math>.

Since <math>\mathcal{A}</math> is an affine space modelled on <math>\Omega^1(M, \operatorname{End}(E))</math>, there should exist some endomorphism-valued one-form <math>A_u\in \Omega^1(M, \operatorname{End}(E))</math> such that <math>u\cdot \nabla = \nabla + A_u</math>. Using the definition of the endomorphism connection <math>\nabla^{\operatorname{End}(E)}</math> induced by <math>\nabla</math>, it can be seen that
:<math>u\cdot \nabla = \nabla - d^{\nabla}(u) u^{-1}</math>
which is to say that <math>A_u = - d^{\nabla}(u) u^{-1}</math>.

Two connections are said to be '''gauge equivalent''' if they differ by the action of the gauge group, and the quotient space <math>\mathcal{B} = \mathcal{A}/\mathcal{G}</math> is the [[moduli space]] of all connections on <math>E</math>. In general this topological space is neither a smooth manifold or even a [[Hausdorff space]], but contains inside it the [[Yang–Mills_equations#Moduli_space_of_Yang–Mills_connections|moduli space of Yang–Mills connections]] on <math>E</math>, which is of significant interest in [[gauge theory]] and [[Yang–Mills theory|physics]].