The String Coffee Table

(R)CFT on more general 2-Categories

Eric wrote:

Reminds of this old s.p.r. thread: NCG/SUSY/*-Product

Right. That was also about the Moyal star. The Moyal star is neat, interesting, deserves our attention, can be used for lots of explicit constructions and what not - but nevertheless I would rather forget the Moyal star for a moment! :-)

Let me try to rephrase what I was hoping to get at.

What is 2-dimensional conformal field theory? For the special case of rational conformal field theory we have the following answer ($\to$):

A full rational conformal field theory (RCFT) (where “full RCFT” means a consistent assignment of $n$-point correlators to all kinds of Riemann surfaces) is specified by two pieces of data.

1) There is the complex-analytic content of the RCFT. This is specified by the local symmetries of the RCFT, encoded in the operator algebra of the Virasoro current and possibly other symmetry currents (like affine current algebras in WZW models).

This “chiral” component of the RCFT gives us, for every Riemann surface, a vector space of potential correlation functions (called “conformal blocks”), namely a vector space of all those would-be correlation functions that are consistent with the local symmetries of the RCFT (with the chiral Ward identities).

In order to get a full CFT, we need to pick from these potential correlation functions the actual correlation functions, such that our choice is consistent with the sewing and cutting of Riemann surfaces. This choice is additional information not contained in the Virasoro and other chiral operators.

2) Hence there is also topological content to an RCFT. This turns out to be encoded in a (special, symmetric) Frobenius algebra object $A$ in the representation category of the chiral algebra (namely the algebra of open string states for any one boundary condition in our RCFT). In practice this means, that the way to pick a consistent set of correlation functions from the spaces of conformal blocks obtained in 1) is to choose a dual triangulation for every Riemann surface and, roughly, interpret it as a “flow chart” for some algebraic computation using that algebra $A$. The result of that computation is a vector, and interpreted as a vector in the space of conformal blocks, this yields the correlation function.

There are many aspects of this that one could ponder. In the present context I am interested in the following observation:

The above construction has nicely abstracted away from any details of an RCFT to the crucial structure.

full RCFT = chiral data + internal Frobenius algebra

Given this alone, we can assign consistent $n$-point correlators to all Riemann surfaces.

We don’t even need to know that our chiral data is obtained from Virasoro currents, all we need is that it behaves like chiral data. Technically, all we need is a modular tensor category $C$ and and a certain algebra object in $C$.

My question is: Can we also abstract away from the nature of Riemann surfaces? I.e., can we use the data given by a symmetric special Frobenius algebra in a modular tensor category and assign vectors to something else than ordinary Riemann surfaces, such that some analog of the sewing constraints holds??

If that “something else” is a Moyal-deformed Riemann surface that’s fine. But since I don’t see how restricting attention to technical details of Moyal star NCG helps to adresss the general question, I would rather like to ignore this for a moment!

Here is an observation close to your heart, Eric. Let me sketch how one could imagine performing the generalization that I am talking about to a setup where Riemann surfaces are replaced by Diamond complexes.

As you may have seen, I am in the process ($\to$) of showing that the above mentioned “flow chart computation” on dual triangulations of Riemann surfaces is secretly the result of applying a locally trivialized 2-functor to our Riemann surface. This way of looking at things has the advantage that one sees what structure of the Riemann surface we really need: what we need is some 2-category whose composition of 2-morphisms behaves like gluing of little pieces of Riemann surface!

It is easy to construct such a 2-category $P_2$ for instance from any 2-dimensional diamond graph. (We have discussed similar - if not the same - things before.)

Let the objects of $P_2$ be the points of the diamond complex, let the 1-morphisms be the edges and those freely generated by composing these, and let the 2-morphisms be the diamonds in between four edges, and all those obtained by freely composing these. Finally, allow for a means to identify edges in order to obtain topologically nontrivially situations (as sketched in section 1.3 of these notes).

There is nothing much which can stop you from applying the whole construction of RCFTs as outline above to a situation where instead of 2-categories of faces in Riemann surfaces we use such 2-categories obtained from diamond complexes.

In fact, somebody should seriously think about this, the diamond complex setup might even allow to carry through the full program without any serious modification. After all, the light-cone structure implicit in the diamond structure defines a conformal structure. Furthermore, restriction to diamonds may not be a restriction at all as far as Riemann surfaces go. We know ($\to$) that every Riemann surface may be decomposed into a collection of conformal rectangles with appropriately identified boundaries.

In any case, from my point of view the interesting question is this:

How much can we generalize the 2-category of 2-paths in Riemann surfaces and still be able to define (R)CFTs on it?

I would not be surprised if it turns out that there is a 2-category whose 2-morphisms are to be regarded as Moyal-deformed disks and that, with due care, we can define an RCFT 2-transport on this 2-category. But fiddling around with the details of the Moyal star may not be the best way to see if and how this can work. Instead, I believe we would first need to understand which general properties of a 2-category we need in order to be able to define an (R)CFT on it.

P.S. Since I thought it might be worthwhile to move this from the comment section to a place where it is more visible, I have turned the above comments into a new entry ($\to$).