# Gauge Theory and Gravity

### Andrew Waldron and Derek Wise

#### Fall 2008

In the Fall 2008 quarter, we're giving a series of lectures on gauge theories, especially with a view toward understanding gravity as a gauge theory. Andrew is giving lectures on "constrained systems", while Derek is lecturing on "gauge theories" in the sense of principal bundles. This course is one of the activities for the VIGRE Research Focus Group on Quantum Geometry.

Here are the course notes:

Andrew's Lectures:

Derek's Lectures:

• The Real Scalar Field – Basic example of a gauge theory: the "N-component real scalar field"; Promoting global symmetries to local 'gauge' symmetries; Bundles; Scalar field as a section of a trivial vector bundle.

• Scalar Field in Bundle Language – Generalizing the N-component real scalar field to the case of a nontrivial bundle.

• Principal BundlesG-bundles and principal G-bundles; The frame bundle; Getting a principal bundle from any "bundle of gadgets" using "generalized frames".
(Actually, this lecture was delivered by Eric Babson, in my absence; these are the notes I gave Eric for the lecture, not notes from his actual lecture. Eric gave some examples, including the principal bundles corresponding to the two nontrivial vector bundle examples in Lecture 2.)

• Associated Bundles – Reconstructing a "bundle of gadgets" from its principal bundle of generalized frames; Examples of associated bundles; Vector bundles from representations; Gauge transformations as sections of an associated bundle of groups.

• Connections and Holonomy – Ehresmann connections on a principal G-bundle; Horizontal lifts and parallel translation; Holonomy; Wilson loops.

• Covariant Derivatives – Using a principal bundle connection for parallel transport in any associated bundle; Covariant derivatives in associated vector bundles; Connection and covariant derivatives in local coordinates.

• BF Theory and 3d Gravity – Introduction to BF theory, with 3d gravity as a special case.

(If you have a UCD Math account, you can also get these notes via the svn by navigating to wally/quantum_geometry)