Lecture 7

Recap from last week:

def: principal bundle: Let [math]\displaystyle{ G }[/math] be a lie group and [math]\displaystyle{ M }[/math] be a manfiold, then a principal [math]\displaystyle{ G }[/math]-bundle is a fibre bundle where all fibres are right [math]\displaystyle{ G }[/math]-torsors,

i.e. [math]\displaystyle{ \exists }[/math] right action of [math]\displaystyle{ G }[/math] on [math]\displaystyle{ P }[/math] that reserves fibres, and is free and transitive on all fibres.

As dibre bundles, principal bundles are locally trivial, i.e. [math]\displaystyle{ \exists }[/math] [math]\displaystyle{ U_\alpha }[/math] open cove of [math]\displaystyle{ M }[/math] such that [DIAGRAM OMITTED HERE] commutes.

We can costruct [math]\displaystyle{ P }[/math] from local trivialisation through glueing by transition functions, [math]\displaystyle{ \rho_{\beta\alpha} \rightarrow G }[/math] [math]\displaystyle{ \rho_{\beta\alpha} }[/math] act by multiplication on left!

For [math]\displaystyle{ m \in U_\alpha \cap U_\beta, \quad (m, g) \in U_\alpha \times G, \quad (m, g) \sim (m, \rho_{\beta\alpha} g) \in U\beta \times G }[/math]

Lemma: [math]\displaystyle{ X = G }[/math], and [math]\displaystyle{ G }[/math] acts on [math]\displaystyle{ X }[/math] on the right.

Questions: what are the [math]\displaystyle{ G }[/math]-equivariant maps [math]\displaystyle{ X \rightarrow X }[/math], i.e. [math]\displaystyle{ \phi: X \rightarrow X }[/math] such that [math]\displaystyle{ phi(x \cdot g) = \phi(x) \cdot g }[/math]?

Answer: [math]\displaystyle{ \{\phi\} = G }[/math], where [math]\displaystyle{ \Phi: G \rightarrow \{G-\text{equiv } \phi:X \rightarrow X \} }[/math] given by [math]\displaystyle{ g \mapsto \Phi_g:X \rightarrow X }[/math] which is given by [math]\displaystyle{ x \mapsto g \cdot x }[/math].

[math]\displaystyle{ \Phi }[/math] is an isomorphism.
Prop: there is a [math]\displaystyle{ 1:1 }[/math] correspondence [math]\displaystyle{ G-\text{principal bundles} \leftrightarrow \text{free and proper (right)}G-\text{actions}. }[/math]

Example: For topological space [math]\displaystyle{ M }[/math] and [math]\displaystyle{ \tilde{M} \rightarrow }[/math] it’s universal cover, we can consider the fundamenta group a principal bundle [math]\displaystyle{ a \pi_1(M) }[/math] where we treat the fundamental group as a discrete group, i.e. a 0-dimensional lie group, one that has the discrete topology.

Example: The hopf fibration [math]\displaystyle{ S^1 \circlearrowright S^3 \rightarrow S^2 }[/math] is a principal bundle. In this case, the circle [math]\displaystyle{ S^2 }[/math] is our lie group. It is abelian so left and right action are the same. This is a non-trivial bundle.
Remark: A principal bundle is not a bundle of groups! It is a bundle of torsors. Bundles of groups are also examples of fibre bundles, but this is a different notion.
Remark about terminoligy: Given a [math]\displaystyle{ G- }[/math] principal bundle [math]\displaystyle{ G \rightarrow P \rightarrow M }[/math], we can look at the group [DIAGRAM OMITTED HERE]. This is a group. If dim[math]\displaystyle{ M }[/math], dim[math]\displaystyle{ G }[/math] both [math]\displaystyle{ \geq 1 }[/math], then it is an [math]\displaystyle{ \infty- }[/math]dimensional Lie group. Terminoligy: [math]\displaystyle{ g(P), \text{Aut}(P) }[/math], this is clarified in [DIAGRAM OMITTED HERE]. We will avoid the term “Gauge group” for clarity.

The existance of a global section of a principal bundle corresponds to a global trivialisation.
Definition: If [math]\displaystyle{ E }[/math] is a rank [math]\displaystyle{ r }[/math] vector bundle (over [math]\displaystyle{ k = \mathbb{R} }[/math] or [math]\displaystyle{ \mathbb{C} }[/math]), then associated with it is the frame bundle of [math]\displaystyle{ E }[/math], denoted [math]\displaystyle{ \text{Fr}(E) }[/math], which as a set is a [math]\displaystyle{ \text{GL}(r, k) }[/math] principal bundle over [math]\displaystyle{ M }[/math]. It is the collection of all ordered bases [math]\displaystyle{ (v_1, ..., v_n) }[/math] in a fibre of [math]\displaystyle{ E }[/math]. We have a natural action of [math]\displaystyle{ \text{GL}(r, k) }[/math] on [math]\displaystyle{ \text{Fr}(E) }[/math] were [math]\displaystyle{ (v_1, ..., v_n)\cdot A }[/math] is given by linear combinations of the columns of [math]\displaystyle{ A \in \text{GL}(r, k) }[/math].

Associated fibre bundles

Suppose we have a [math]\displaystyle{ G- }[/math]principal bundle, and left action [math]\displaystyle{ G \circlearrowright F }[/math] at least smooth, possible preserving extra structure. Then [math]\displaystyle{ P_F = {P \times F} / G }[/math] (transform them both into either left or right actions). This is equivalent to [math]\displaystyle{ P \times F / \sim }[/math] where [math]\displaystyle{ (p, gf) \sim (pg, f) }[/math].
prop: The action [math]\displaystyle{ G \circlearrowright P \times F }[/math] is always free and proper, so the quotient above always exists.
Have [math]\displaystyle{ P\times F / G = P_F \rightarrow M }[/math], so [math]\displaystyle{ P_F }[/math] is a left fibre-bundle over [math]\displaystyle{ M }[/math].

e.g. [math]\displaystyle{ G = \text{GL}(r, k) }[/math], [math]\displaystyle{ G \circlearrowright k^2 }[/math], with principal bundle [math]\displaystyle{ P }[/math], gives fibre bundle [math]\displaystyle{ P_{k^r} \rightarrow M }[/math] rank [math]\displaystyle{ r }[/math] vector bundle.