Gauge Theory and Gravity

Andrew Waldron and Derek Wise

Fall 2008

Here are the course notes:

Andrew's Lectures:

• Classical Mechanics Primer – Manifolds; Vector fields; Symplectic manifolds and what they're good for.

• More Mechanics – Hamiltonian flows and hamiltonian vector fields; Tautological 1-form on the cotangent bundle; The symplectic form; Geodesic equations and Christoffel symbols.

• Hamiltonian Flows and Actions – Proof that Hamiltonian flow for the energy function gives geodesic motion; Poisson brackets; The action principle.

• The Quantum Particle – Parameterized particle and its quantization; Operator ordering ambiguities, and diffeomorphism invariance a principle for resolving them; removing the parameterization dependence.

• The Supersymmetric Quantum Particle – Adding supersymmetry to the particle action; Symmetries of the action and their conserved charges.

• (Lecture #6)

• (Lecture #7)

• BRST & Lie Algebra Cohomology Mostly....

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 Bundles – G-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)