Quantum Gravity Seminar - Winter 2002

John Baez

In Winter 2002 I began some lectures on categorified gauge theory. These lectures start with a quick sketch of a very big subject: weak n-categories. Then they focus on certain special cases: groups, groupoids, and 2-groups. Then they sketch how 2-groups can be used to describe a generalization of gauge theory which describes not just the parallel transport of point particles, but also of 1-dimensional extended objects - strings, if you will, or spin network edges.

The most familiar example is 2-form electromagnetism. This is a modified version of Maxwell's equations in which the connection 1-form is replaced by a 2-form, sometimes called the Kalb-Ramond field. Just as you can integrate the A field over the worldline of a particle to get a term in the action which describes its coupling to the electromagnetic field, you can integrate the B field over the worldsheet of a string to get a term in the action which describes its coupling to the Kalb-Ramond field! But the really interesting thing is that you can think of the Kalb-Ramond field as a connection on a "principal 2-bundle" where the relevant gauge "2-group" has only one object, but a U(1)'s worth of morphisms.

To get a general overview of these ideas, you should start by peeking at the slides of this talk:

There are also a lot of links to papers here.

The course notes below were taken by Alissa Crans, who has kindly scanned them in and made PDF files from them:

Eventually I will try to give descriptive titles for the notes from each class, but right now I only have them listed by date. I should also create lower-resolution files that are quicker to download. Of course, if anyone else wants to do this, I'll be eternally grateful, and my thanks will appear here.

If you want to dig deeper into this subject, try the Spring 2002 notes!


© 2002 John Baez and Alissa Crans