The n-Category Café

It’s a really fancy way of expressing the locality and gluing properties of the path integral

And hence of replacing the path integral by something well defined: saying that a $d$-dimensional quantum field theory is a suitable functor from $d\mathrm{Cob}$ to $\mathrm{Vect}$ is specifying precisely those conditions that we would like a path integral to satisfy.

In his latest lecture M. Hopkins made that very explicit by giving a precise homotopy-theoretic way for thinking of the symbols

that appear in the path integral.

So a QFT is a functor from cobordisms to vector spaces, and in some cases we can use the path integral as a (more or less heuristic) way to construct such functors.

Perhaps overly fancy;

As with any power tool, it is overly fancy as long as applied to simple enough situations.

For QFT, there is unfortuantely only a small region in theory space lying between trivial and intractable. Systems in that region include in particular topological theories in more or less arbitrary and conformal field theories in two dimensions.

For both these cases the functorial conception of field theory has lead to insights which you can hardly even state without using the language of categories. (My favorite example is this theorem.)

My guess is that it all can be phrased in terms of (co?)simplicial sets of some sort, too – the nerve of a whatever category is always somthing simplicial as I remember it, so they’re not unrelated.

Categories internal to $C$ are the same as simplical $C$-objects, functors correspond to simplicial maps. This is established by identifying sequences of composable morphisms of length $p$ with $p$-simplices.

Similar statements hold for various flavors of $n$-categories (and $\omega$-categories), where in many cases the categorical structure is defined in terms of simplicial notions.

For instance, Mike Hopkins thinks that topological field theories should be thought of as living in the world of weak omega-categories defined in terms of complicial sets.

By geometrically realizing simplicial sets one passes from categories to topological spaces. Many things that can be said in terms of $n$-categories can and are being said in terms of homotopy theory.

You won’t find anyone applying n-categories to Yang-Mills theory yet…

Crucial aspects of S-duality in 4-dimensional Yang-Mills theory, leading to the relation to geometric Langlands, appear naturally in a categorical language.

Kapranov has written about how lifting everything by one dimension in this context (from strings to membranes, that is) requires 2-vector bundles.

Another way to understand S-duality in 4-dimensional Yang-Mills theory is to realize YM as the compactification of 6-dimensional field theory. The connection 1-form of Yang-Mills comes from a connection 2-form in the 6-dimensional theory, to be thought of as the connection on a gerbe.

For the abelian case this is very well understood. Witten explains it in his contribution Conformal field theory in four and six dimensions to the proceedings of Segal’s 60th birthday conference (Lecture Note Series 308). On slide 71 of the corresponding talk Witten mentioned the obvious guess that for nonabelian Yang-Mills S-duality can hence be understood by compactifying a nonabelian gerbe from 6 to four dimensions. (I once commented on that here.)

The jury on that is still out, though. But people are trying. Jussi Kalkkinen argued that we should patch together Yang-Mills theory that differ by S-duality transformations to get a global picture, and that the structure thus appearing is that of a nonabelian gerbe.

So, while, as yet, nobody has ever tried to say sentences like “4-dimensional Yang-Mills theory is such and such a functor”, crucial and deep properties of Yang-Mills theory find their formulation and illumination in terms of categorical language.

You won’t find anyone applying n-categories to Yang-Mills theory yet, except in 2d spacetime.

Not in the sense of what Urs and JB are discussing, certainly.

However, some may find hep-th/0312112 interesting.

[…] his work with Connes on renormalization and Hopf algebras.

Maybe interestingly, the Hopf algebra used by Connes and Kreimer is nothing but the category algebra of the multicategory of Feynman diagrams, I think #.