Categorifying the Dijkgraaf-Witten model
Posted by John Baez
The Dijkgraaf-Witten model is a simple sort of topological quantum field theory where the only field is a gauge field, and the gauge group G is finite:
- Robbert Dijkgraaf and Edward Witten, Topological Gauge Theories and Group Cohomology, Commun. Math. Phys. 129 (1990), 393.
Martins and Porter have a new paper on how to categorify this model, replacing the group G by a categorical group, or “2-group”.
I wrote about this stuff eleven years ago in week54 of This Week’s Finds - so if you want an elementary intro to these ideas, start there.
In the simplest version of the Dijkgraaf-Witten model, the path integral is just an integral over the moduli space of principal G-bundles, using the simplest possible measure on that space. It’s nice to formulate this theory on a triangulated manifold, where we assign a group element to each edge, and require that these group elements multiply to 1 around each triangle. This formulation makes it clear that we can also “twist” the Dijkgraaf-Witten model, which in n dimensions amounts to changing the action by any element of the nth cohomology of the gauge group.
I explained this stuff in the Winter 2005 Quantum Gravity Seminar. I also discussed the generalization where G is a Lie group - this gives 3d quantum gravity. And, I explained how the path integral in such theories can be rewritten as a sum over spin foams.
For people who like higher gauge theory, it’s tempting to “categorify” the Dijkgraaf-Witten model by replacing the gauge group by a 2-group. There’s already been some work on this, going back to a paper by Yetter:
- David Yetter, TQFT’s from homotopy 2-types, Journal of Knot Theory and its Ramifications 2 (1993), 113-123.
and you can see more recent papers online:
- Hendryk Pfeiffer, Higher gauge theory and a non-Abelian generalization of 2-form electrodynamics.
- Timothy Porter and Vladimir Turaev, Formal homotopy quantum field theories, I.
- Timothy Porter and Vladimir Turaev, Formal homotopy quantum field theories, II.
Anyway, here’s a new one!
- João Faria Martins and Timothy Porter, On Yetter’s invariant and an extension of the Dijkgraaf-Witten invariant to categorical groups.
A categorical group or “crossed module” is the same as a strict 2-group. Note that this paper “twists” the categorified Dijkgraaf-Witten model using an element of the nth cohomology of the classifying space of our 2-group. So, it’s using the obvious generalization of group cohomology to 2-groups.
It would be nice to generalize this work from finite (or discrete) 2-groups to Lie 2-groups, and that’s sort of what I’m doing with Freidel and Baratin - we’re focusing on the case of the Poincaré 2-group.
Note that in all these theories, the connection or 2-connection is flat - it has to be when G is discrete, but it still is in these theories when we generalize G to a Lie group or Lie 2-group. Flat connections are a wee bit boring in physics, but good for getting TQFTs. Urs is busy working on more exciting theories that involve 2-connections or 3-connections with nontrivial curvature.
One can try to go further, replacing the group in the Dijkgraaf-Witten model by an $n$-group, but the Homotopy Hypothesis conjectures that such things are the same as pointed connected homotopy $n$-types, so at this point it’s more efficient to use simplicial techniques rather than $n$-categories to make the ideas precise. For work along these lines, try:
- Timothy Porter, Interpretations of Yetter’s notion of G-coloring: simplicial fibre bundles and non-abelian cohomology.
- Timothy Porter, TQFTs from homotopy $n$-types.
Re: Categorifying the Dijkgraaf-Witten model
Thanks, John, for inserting into my entry on Martins & Porter a link back to this entry here, which I forgot to point to.
At one point I want to get back and explain and emphasize again how really cool things happen when we replace here the finite group $G$ by our strict Fréchet Lie 2-group $\mathrm{tar} = \mathbf{B}(\Omega G \to P G)$ without the central extension (so this is still equivalent to $\mathbf{B}G$) and then realize that putting in the central extension to the strict string 2-group $(\hat \Omega G \to P G)$ amounts to introducing the “twist” here in the guise of a weak 2-functor $\mathrm{tra} : \mathbf{B}(\Omega G \to P G) \to \mathbf{B}^2 U(1) \,.$
Transgressing this setup to loop space produces the Lie-analog of Simon Willerton’s baby FHT theorem as I once described in the entry with the funny title 2-Monoid of Observables on String-G.
In particular, the representations of the loop groupoid that we transgress to are twisted equivariant vector bundles on $G$ and we make contact with non-baby FHT.
Back then I was stopped by the fact that I wasn’t sure if I could handle the smooth structure on the quotient $\Lambda(\Omega G \to P G) := \mathrm{Hom}(\mathbf{B}\mathbb{Z}, \mathbf{B}(\Omega G \to P G))/\sim \,,$ where “$/\sim$” is supposed to denote the operation of identifying isomorphic 1-morphisms.
But now I think I actually can handle this, essentially by the standard smooth space Yoga.
I should try to find the time to do that. There is something interesting lurking here, which will connect all this combinatorial and homotopy theoretic reasoning to the real thing.