The n-Category Café

So recall that in the Heisenberg picture of $n$-dimensional quantum field theory we do not propagate states (vectors) over $n$-dimensional cobordisms (as we would in the Schrödinger picture), but instead pull back the algebras of endomorphisms of these spaces along that propagation map and then assign algebras of local observables to subsets of our cobordisms.

We assume that the inductive limit over this net of algebras exists, that’s our algebra of observables $$A \,.$$

Instead of remembering that this comes from an algebra of endomorphisms of certain spaces of states in the Schrödinger picture, we want to allow ourselves to postulate suitable algebras of local observables in the first place and then reconstruct the spaces they are represented on.

If we just remember the total thing, $A$, this means that we are looking for the category $$\mathrm{Mod}_A$$ of its modules. In fact, since we want to think of these as arising from representions of the local algebras mentioned above, we restrict attention to “local” modules, sitting in $$\mathrm{Mod}^{\mathrm{loc}}_A \,.$$ See the last entry for more details on “local” modules.

As indicated there, this locality requirement makes $$\mathrm{Mod}^{\mathrm{loc}}_A$$ into a monoidal category. For $n$-dimensional quantum field theory with $n \gt 2$ this monoidal structure will even be symmetric monoidal.

For $n=2$, however, there will be a nontrivial braiding.

This can be traced back to the fact that the braiding on these modules arises from literally translating the supports of the local modules around each other. In dimensions greater than 2 this is always possible in a way that neigher support passes through the future light cone of the other. Only in 2-dimensions are the supports bound to cross each other lightcones as they change places.

The general mechanism behind this is that one which gives exotic statistics to $n$-particles in $(n+1)$-dimensions.

In the context of quantum field theory, we are concerned with 1-particles. (Well, these 1-particles may be quanta of fields propagating on the $n$-dimensional worldvolume of an $n$-particle. But never mind for the moment.)

So let’s look at the case $(n=2)$. If the 2-dimensional quantum field theory that we are talking about happens to be a rational conformal 2-dimensional quantum field theory, then $A$ will be a certain von Neumann algebra factor and then our category of local modules, which we now call $$\mathcal{C} := \mathrm{Mod}^\mathrm{loc}_A$$ is not just braided monoidal, but has lots of nice properties that make it a modular tensor category.

Now, it turns out that correlators in this theory (which are essentially the propagation morphism that would be part of the definition had we started with the Schrödinger picture and defined the 2-dimensonal CFT following Segal as a projective functor on 2-dimensional conformal cobordisms) are given by a 2-vector transport with values in $\mathcal{C}$-linear 2-vector spaces $$2\mathrm{Vect}_{\mathcal{C}} := \mathrm{Mod}_{\mathcal{C}} \,.$$

But now recall that $\mathcal{C}$ itself came to us as the category of modules of the algebra of observables. This means that $$2\mathrm{Vect}_{\mathcal{C}} := \mathrm{Mod}_{\mathcal{C}} = \mathrm{Mod}_{\mathrm{Mod}_A} \,.$$

Moreover, according to the FFRS theorem, the dynamics of this 2-dimensional conformal theory is best understood in terms of the kinematics of a 3-dimensional topological quantum field theory. This should be governed by the 3-vector space $$3\mathrm{Vect}_{2\mathrm{Vect}_{\mathcal{C}}} := \mathrm{Mod}_{\mathrm{Mod}_{\mathrm{Mod}_A}} \,,$$ which we can form since $\mathrm{Mod}_{\mathrm{Mod}_A}$ is still monoidal, using the fact that $\mathrm{Mod}_A$ is braided monoidal.

But then it stops. Unless we make further simplifying assumptions, like that we have not just a conformal but even a topological 2-dimensional theory to start with, the 3-category $$\mathrm{Mod}_{\mathrm{Mod}_{\mathrm{Mod}_A}}$$ won’t have a monoidal structure anymore. So we end up with a finite hierarchy of 1-, 2- and 3-particles, points, strings and membranes.

If you know about the relation of all these things to string theory, you’ll see that the fact that this hierarchy breaks off at $n=3$ is essentially the statement that there is a fundamental string (a 2-particle) which may be understood in terms of a fundamental membrane (a 3-particle), but that’s it. While there are $n$-particles for $n\gt 3$ in this game, they are not “fundamental” in some sense.

Anyway, here we see an abstract aspect of this phenomenon. Somehow it is the periodic table of $n$-categories which is at work in the background.

But what happens if we do assume that our 2-dimensional quantum field theory that we started with already was topological itself?

In that case, there are no nontrivial local observables, and we just get $$A = \mathbb{C} \,.$$

This algebra is of course particularly simple. It is commutative.

Now we can form $$2\mathrm{Vect}_{\mathbb{C}} := \mathrm{Mod}_{\mathcal{C}} = \mathrm{Mod}_{\mathrm{Mod}_\mathbb{C}} = \mathrm{Mod}_{\mathrm{Vect}_\mathbb{C}} \,.$$

Then the above description of 2d CFT in terms of 2-vector transport reduces essentially to the Fukuma-Hosono-Kawai state sum model. of the topological string.

In this case, we can go on forming higher and higher $n$-vector spaces $$n\mathrm{Vect}_\mathbb{C} = \mathrm{Mod}_{ \mathrm{Mod}_{ \mathrm{Mod}_{ \mathrm{Mod}_{ \mathrm{Mod}_{ \mathrm{Mod}_{ \mathrm{Mod}_{ \cdots_{ \mathrm{Mod}_\mathbb{C} }}}}}}}} \,.$$

I wonder if this can be used to understand the fact that in topological string theory the notion of $n$-vector spaces that occur are the Baez-Crans $\infty$-vector spaces over $\mathbb{C}$, also known as module $\infty$-categories for the discrete $\infty$-category $\mathrm{Disc}(\mathbb{C})$ also known as the category of complexes of vector spaces, or rather it’s derived version.