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 = Jwhere 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:
For more on this subject try these papers:
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!