John Baez

Lecture at the 2002 Joint Spring Meeting of the Pacific Northwest Geometry Seminar and Cascade Topology Seminar

May 12, 2002

Categorified Gauge Theory

In electromagnetism we can think of the vector potential as a 1-form A which couples to charged point particles in a very natural way - we simply integrate it over the particle's worldline to obtain a term in the action. Similarly, in string theory there naturally arises a 2-form B, the Kalb-Ramond field, which we integrate over the string worldsheet. The resulting theory of "2-form electromagnetism" is formally very similar to Maxwell's equations: in particular, we define a curvature 3-form G = dB and require that

*d*G = J
where the current J is now a 2-form.

Just as the electromagnetic vector potential should really be regarded as a connection on a U(1) bundle, the Kalb-Ramond field should really be thought of as a connection on a "U(1) gerbe". Moreover, just as U(1) bundles are classified by the 1st Cech cohomology with coefficients in the sheaf of smooth U(1)-valued functions, U(1) gerbes are classified by the 2nd Cech cohomology with coefficients in this sheaf.

Electromagnetism can be generalized to Yang-Mills theory by replacing the group U(1) by an arbitrary compact Lie group. This raises the question of whether we can similarly generalize 2-form electromagnetism to some sort of "higher-dimensional Yang-Mills theory". We show how to do this by 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.

In particular, we define a "Lie 2-group" to be a category C 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. This 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.

Just as a connection on a trivial G-bundle is the same as a Lie(G)-valued 1-form, a connection on a trivial C-2-bundle turns out to be a Lie(G)-valued 1-form together with a Lie(H)-valued 2-form. Following ideas of Breen and Messing, we give formulas defining the curvature of such a connection, which consists of a Lie(G)-valued 2-form together with a Lie(H)-valued 3-form.

We write down the obvious generalization of the Yang-Mills action for a connection on a trivial C-2-bundle, and derive the "categorified Yang-Mills equations" from this action. We also show that in certain cases these equations admit self-dual solutions in five dimensions. We conclude by sketching how nontrivial C-2-bundles can be classified by the 2nd nonabelian Cech cohomology.

Click on these to see the slides:

  1. title page

  2. electromagnetism

  3. the vector potential as U(1) connection

  4. U(1) connections mod gauge transformations

  5. classifying U(1) bundles by Cech cohomology

  6. generalizations: Yang-Mills theory

  7. generalizations: 2-form electromagnetism

  8. combining the generalizations: higher Yang-Mills theory

  9. categorification

  10. smooth categories, Lie 2-groups and Lie 2-algebras

  11. examples of smooth categories

  12. the structure of Lie 2-groups and Lie 2-algebras

  13. bundles versus 2-bundles: basic definitions

  14. bundles versus 2-bundles: connections and holonomy

  15. bundles versus 2-bundles: curvature and Bianchi identity

  16. bundles versus 2-bundles: Yang-Mills action and Yang-Mills equations

  17. self-dual solutions of the categorified Yang-Mills equations on a 5-dimensional Riemannian manifold

  18. classifying 2-bundles by nonabelian Cech cohomology

For more on this subject try these papers:

Note: the first paper corrects a mistake on slide 17. The sign conventions differ between this paper and my talk, but either one is okay.

In 2004, I gave a more advanced talk on this subject at the conference in honor of Larry Breen's 60th birthday. I learned a lot in the meanwhile, like the importance of vanishing fake curvature. So, take this earlier talk with a grain of salt!

© 2004 John Baez