Editing Lecture 7 - Monday Week 4 (section)

== Lecture 7 ==

Recap from last week:

def: principal bundle: Let <math display="inline">G</math> be a lie group and <math display="inline">M</math> be a manfiold, then a principal <math display="inline">G</math>-bundle is a fibre bundle where all fibres are right <math display="inline">G</math>-torsors,

i.e. <math display="inline">\exists</math> right action of <math display="inline">G</math> on <math display="inline">P</math> that reserves fibres, and is free and transitive on all fibres.

As dibre bundles, principal bundles are locally trivial, i.e. <math display="inline">\exists</math> <math display="inline">U_\alpha</math> open cove of <math display="inline">M</math> such that [DIAGRAM OMITTED HERE] commutes.

We can costruct <math display="inline">P</math> from local trivialisation through glueing by transition functions, <math display="block">\rho_{\beta\alpha} \rightarrow G</math> <math display="inline">\rho_{\beta\alpha}</math> act by multiplication on left!

For <math display="inline">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 display="inline">X = G</math>, and <math display="inline">G</math> acts on <math display="inline">X</math> on the right.

Questions: what are the <math display="inline">G</math>-equivariant maps <math display="inline">X \rightarrow X</math>, i.e. <math display="inline">\phi: X \rightarrow X</math> such that <math display="inline">phi(x \cdot g) = \phi(x) \cdot g</math>?

Answer: <math display="inline">\{\phi\} = G</math>, where <math display="inline">\Phi: G \rightarrow \{G-\text{equiv } \phi:X \rightarrow X \}</math> given by <math display="inline">g \mapsto \Phi_g:X \rightarrow X</math> which is given by <math display="inline">x \mapsto g \cdot x</math>.

<math display="inline">\Phi</math> is an isomorphism.<br />
Prop: there is a <math display="inline">1:1</math> correspondence <math display="block">G-\text{principal bundles} \leftrightarrow \text{free and proper (right)}G-\text{actions}.</math>

Example: For topological space <math display="inline">M</math> and <math display="inline">\tilde{M} \rightarrow</math> it’s universal cover, we can consider the fundamenta group a principal bundle <math display="inline">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 display="inline">S^1 \circlearrowright S^3 \rightarrow S^2</math> is a principal bundle. In this case, the circle <math display="inline">S^2</math> is our lie group. It is abelian so left and right action are the same. This is a non-trivial bundle.<br />
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.<br />
Remark about terminoligy: Given a <math display="inline">G-</math> principal bundle <math display="inline">G \rightarrow P \rightarrow M</math>, we can look at the group [DIAGRAM OMITTED HERE]. This is a group. If dim<math display="inline">M</math>, dim<math display="inline">G</math> both <math display="inline">\geq 1</math>, then it is an <math display="inline">\infty-</math>dimensional Lie group. '''Terminoligy''': <math display="inline">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.<br />
Definition: If <math display="inline">E</math> is a rank <math display="inline">r</math> vector bundle (over <math display="inline">k = \mathbb{R}</math> or <math display="inline">\mathbb{C}</math>), then associated with it is the frame bundle of <math display="inline">E</math>, denoted <math display="inline">\text{Fr}(E)</math>, which as a set is a <math display="inline">\text{GL}(r, k)</math> principal bundle over <math display="inline">M</math>. It is the collection of all ordered bases <math display="inline">(v_1, ..., v_n)</math> in a fibre of <math display="inline">E</math>. We have a natural action of <math display="inline">\text{GL}(r, k)</math> on <math display="inline">\text{Fr}(E)</math> were <math display="inline">(v_1, ..., v_n)\cdot A</math> is given by linear combinations of the columns of <math display="inline">A \in \text{GL}(r, k)</math>.<br />


<span id="associated-fibre-bundles"></span>
=== Associated fibre bundles ===

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

e.g. <math display="inline">G = \text{GL}(r, k)</math>, <math display="inline">G \circlearrowright k^2</math>, with principal bundle <math display="inline">P</math>, gives fibre bundle <math display="inline">P_{k^r} \rightarrow M</math> rank <math display="inline">r</math> vector bundle.