John Baez

Lecture at Arithmetic, Geometry and Topology,
a conference in honor of Larry Breen's 60th birthday.

December 15, 2004

Categorified Gauge Theory

The problem of defining higher gauge fields is an old one. Ordinarily, gauge field are used in physics to describe the parallel transport of point particles. Higher gauge fields would similarly describe the parallel transport of higher-dimensional extended objects.

Not surprisingly, interest in this problem has been reawakened by string theory, since 1-dimensional strings and higher-dimensional "branes" play a crucial role here. Indeed, a higher analogue of the electromagnetic field called the "Kalb-Ramond field" arises naturally in string theory. It describes the phase acquired by a string as it moves through spacetime. Electromagnetism is described locally by a 1-form, which we integrate over the worldline of a charged particle to compute the phase it acquires as it moves. Similarly, the Kalb-Ramond field is described locally by a 2-form, which we integrate over the worldsheet of a string.

However, mathematical physicists know that globally, the electromagnetic field is described by a connection on a U(1) bundle, which would be nontrivial in the presence of magnetic monopoles. Similarly, the Kalb-Ramond field is described globally by a connection on a "U(1) gerbe".

In traditional physics, forces other than electromagnetism are described by replacing the group U(1) by other Lie groups G and treating these forces as connection on G-bundles. This raises the question of whether we can similarly see connections on U(1) gerbes as a special case of higher gauge fields involving more general Lie groups - particularly nonabelian  groups.

Breen and Messing have raised the possibility that we can, by using their theory of "connections on nonabelian gerbes". This has been taken up by the physicists Aschieri and Jurco, who have applied it to string theory. However, none of this work includes a theory of parallel transport for these connections.

Toby Bartels, Alissa Crans, Aaron Lauda, Urs Schreiber and I have developed such a theory by systematically categorifying  the concepts of smooth manifold, Lie group and Lie algebra, and setting up a theory of bundles, connections and curvature in this new context.

Categorification is the process of replacing sets by categories. Ordinary gauge theory already uses categories to describe parallel transport, since there is a a category  where the objects are points of spacetime and the morphisms are paths. Higher gauge theory goes further and uses a 2-category  where the objects are points, the morphisms are paths and the 2-morphisms are "paths of paths" - since a path of paths is a mathematical way of describing the surface traced out by the motion of a string. In a sneaky self-referential move, the concept of 2-category is most efficiently obtained by categorifying the concept of "category" itself. So, higher gauge theory is naturally tied to categorification!

In particular, if we categorify the concept of "Lie group" we get the concept of "Lie 2-group". The simplest version of this idea - a so-called "strict" Lie 2-group - is a category G where the set of objects and the set of morphisms are Lie groups, and source, target, identity and composition maps are homomorphisms of Lie groups. A strict Lie 2-group turns out to be the same as a "Lie crossed module": a pair of Lie groups G and H with a homomorphism t: H → G and an action of G on H satisfying the equations in the usual definition of crossed module.

Using this idea we can then go ahead and categorify the concept of bundle, getting the concept of a "2-bundle". Just as a bundle has some gauge group G, a 2-bundle has a "gauge 2-group", which we take here to be any strict Lie 2-group G.

Just as a connection on a trivial G-bundle is the same as a Lie(G)-valued 1-form, a 2-connection on a trivial G-2-bundle turns out to be a Lie(G)-valued 1-form A together with a Lie(H)-valued 2-form B. We show that a 2-connection gives well-behaved parallel transport along both curves and surfaces if a certain quantity called the "fake curvature" vanishes. This quantity was introduced by Breen and Messing. It equals

dA + A2 - dt(B)

When this is zero, we can define nonabelian parallel transport over surfaces in a way that doesn't depend on the parametrization of the surface! This was always the sticking point in attempts at higher gauge theory.

Click on this to see the transparencies of my talk:

For more on this subject, try these papers:

Also, Urs Schreiber and I will soon be coming out with a paper which contains many of the results mentioned in this talk. You can see the latest draft here:

In 2002, I gave a preliminary talk on this subject at a joint meeting of the Pacific Northwest Geometry Seminar and the Cascade Topology Seminar. I've learned a lot since then, like the importance of vanishing fake curvature. So, take this earlier talk with a grain of salt!

In 2005, I gave a related talk on Lie 2-algebras and Lie 2-groups at the U. C. Irvine Algebra Seminar.

© 2004 John Baez