gauge group

Let us start with two vector bundlesMathworldPlanetmath E and F over a space B




The first objective is to show how to create a bundle called Hom(E,F). There are two different ways to do this. The first way is to observe that since for vector spaces V,W that Hom(V,W)WV (here just meaning the module of homomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath from V to W) thus if we take the bundle Hom(E,F):=FE then we have that since the fibers (FE)b=FbEb we have what we want.

Another way of looking at the creation of the bundle Hom(E,F) is to look at representations and what we would ideally like our bundle to look like. If we have that the bundle that represents F is given by taking the principle Gl(m,) bundle


and a trivial representation then the bundle afforded by the trivial representation is simply F. The same thing is true with the bundle E. We then have that if we look at the structure group of our proposed new bundle it should be Gl(m,)×Gl(n,). The fibers of our proposed new bundle definitely should be Hom(n,m) thus we have that the representation we are looking for should take something in our structure group and something in the fiber and give us something new in the fiber. The proposed representation is ρ(A,B)(U)=AUB-1. Then looking at the bundle associated to the representation of ρ gives us that if




we can similarly define


The group of sectionsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of Aut(E) is called the gauge group. It is a group since we have that if (f,f) is a section of Aut(E) and (τ,τ) is also a section of Aut(E) then we have a group operationMathworldPlanetmath given by composition:


The fact that these bundle mapsMathworldPlanetmath are isomorphismsMathworldPlanetmathPlanetmath of bundles gives the existence of an inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. The reason for this is that for each section (f,f), EbEf(b) since fb is a vector space isomorphism. Thus we now have that if we look at the bundle map given by taking (f-1,(f)-1)) (where f-1 means (fb)-1 for each bB). Associativity is clear by composition of functions being associative and the identity map acts as the identity elementMathworldPlanetmath.

Title gauge group
Canonical name GaugeGroup
Date of creation 2013-03-22 17:02:34
Last modified on 2013-03-22 17:02:34
Owner sjm1979 (13837)
Last modified by sjm1979 (13837)
Numerical id 9
Author sjm1979 (13837)
Entry type Definition
Classification msc 55R25
Defines bundle of homomorphisms
Defines bundle of endomorphisms
Defines automorphism bundle