THE GEOMETRY OF VECTOR BUNDLES AND

AN INTRODUCTION TO GAUGE THEORY

LECTURE 6

Professor Steven Bradlow Class Notes From Math 433

University of Illinois at Urbana-Champaign

February 4, 1998

Recall that given a rank n vector bundle, we were able to construct a principal GL(n, ℝ) bundle. Furthermore, given a principal G−bundle, P, and a representation ρ : G → GL(V), we constructed a vector bundle, E = P ×_{G} V = P ×_{ρ} V. Furthermore, if V is of rank n, then

E = (∐ U_{α} × ℝ^{n})/{g_{αβ}}

where the g_{αβ} : U_{α} → U_{β} → ρ(G) ⊆ GL(n, ℝ) are the transition functions. In general, we say that the structure group of a bundle can be reduced if its structure group G is a subset of GL(n, ℝ). In general, a vector bundle can be reduced from GL(n, ℝ) to some subgroup, G, if and only if we can find a system of local trivializations such that all transition functions take their values in G.

Local Description of a Section

Suppose that a vector bundle,π : E → B, is described as

E = (∐ U_{α} × ℝ^{n})/{g_{αβ}}

and that σ : B → E is a section. Let ψ_{α} : E_{|U_{α}} → U_{α} × ℝ^{n} be the trivializing maps. We had that the following diagram was commutative:

E|U_{α}
[figure: σ; ψ_{α}]
U_{α}  \overline{}_{σ_{α}=ψ_{α}∘σ} U_{α} × ℝ^{n}

Use the fiber preserving properties of the maps involved to check that σ_{α}(b) = (b, s_{α}(b)) where s_{α} : U_{α} → ℝ^{n}. Now, σ determines a collection of local sections,

{σ : U_{α} → ℝ^{n}}

one for each trivializing open set.

The natural question to ask here is if given a collection of local sections, {σ_{α} : U_{α} → ℝ^{n}}, one for each trivializing open set, when does the collection determine a global section, σ : B → E with σ_{|U_{α}} = σ_{α}? On U_{α} ∩ U_{β} = U_{αβ}, we have the following commutative diagram:

Typeset by AMS-TEX

[figure: U_{αβ} × ℝ^{n}; ψ_{β}; E|U_{αβ}; ψ_{α}; U_{αβ} × ℝ^{n}]
[figure: Id; σ; Id]
[figure: U_{αβ} × ℝ^{n}; σ_{β}; U_{αβ}; σ_{α}; U_{αβ} × ℝ^{n}]

Writing g_{βα} = ψ_{β}ψ_{α}^{−1}, we see that s_{β} = g_{βα}s_{α}. Hence, given a collection of local sections, {σ_{α}(b) = (b, s_{α}(b))} such that s_{β}(b) = g_{βα}(b)s_{α}(b), then they define a global section by defining σ(b) = ψ_{α}^{−1} ∘ σ_{α}(b).

Example 1. Set s_{α}(b) = 0 for all b ∈ U_{α}. This defines the local zero section and the collection {s_{α}} will glue together to define the global zero section.

Observe that the same analysis of sections for vector bundles applies to sections of principal G−bundles. In other words, given a principal G−bundle, described by

P = (∐ U_{α} × G)/{g_{αβ}},

the description of a section is the same as above, but with ℝ^{n} replaced by G. Now, in the relation, s_{β} = g_{βα}s_{α}, the multiplication on the right hand side is group multiplication. Conversely, using a section, say σ, the transition functions can be written as g_{βα} = s_{β}s_{α}^{−1}. The significance of this will become clear shortly.

A Return to Bundle Maps

Suppose that we have a bundle map,

E  \overline{}^{h} 𝔼′

[figure: π; π′]

B

with

\begin{aligned}E & = (∐ U_{α} × ℝ^{n})/\{g_{αβ}\} \\ E′ & = (∐ U_{α} × ℝ^{m})/\{g′_{αβ}\} \end{aligned}

Then, we have the following commutative diagram specified by the trivializations for E and E′ respectively,

E|U_{α} \underline{}^{h}  E′|U_{α}
[figure: ψ_{α}; ψ′_{α}]
U_{α} × ℝ^{n} \underline{}_{ψ′_{α}∘h∘ψ_{α}^{−1}=h_{α}} U_{α} × ℝ^{m}

Exercise 1. Show that we can regard h_{α} : U_{α} → Hom(ℝ^{n}, ℝ^{m}). That is, we can think of h as an element of Mat_{n,m}(ℝ), real n by m matrices.

The exercise says that given h, we obtain a set, {h_{α} : U_{α} → Hom(ℝ^{n}, ℝ^{m})}. How are h_{α} and h_{β} related? Using the transition functions from E and E′ respectively, we require that the following diagram commute:

ℝ^{n} \underline{}^{h_{α}}  ℝ^{m}
[figure: g_{βα}; g′_{βα}]
ℝ^{n}  \overline{}_{h_{β}} ℝ^{m}

Consider the special case of E = E′. Then h is a bundle endomorphism and

h_{α} : U_{α} → End(ℝ^{n}, ℝ^{n}) ≅ Mat_{n,n}(ℝ)

are such that h_{β}g_{βα} = g_{βα}h_{α}. If h is, in fact, a bundle automorphism then,

h_{β} = g_{βα}h_{α}g_{βα}^{−1} = g_{βα}h_{α}g_{αβ}

273 Altgeld Hall, 1409 W. Green Street, Urbana, IL 61801 E-mail address: bradlow@math.uiuc.edu