### (R)CFT on more general 2-Categories

#### Posted by Urs Schreiber

I am grateful for the comments to the recent entry “Generalized Worldsheets?”, but I feel they indicate that I did not manage to get my main point across. I now tried again, providing more details, in the latest comment to that entry ($\to $), which, for that reason, I thought I should lift to blog top-level. It is reproduced below.

*What IS 2-dimensional conformal field theory, really?*

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!

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** ($\to $).

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.

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 outlined 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.

## Re: (R)CFT on more general 2-Categories

Hi Urs, interesting topic in my opinion, but I’m mostly confused about what it precisely means so I’ll scatter some comments and questions. They may be stupid, however, owing to my present confusion.

First though, let me just say that I find the idea interesting partly because it looks like an alternative to the deformation of RCFT that one usually considers, i.e. something like relaxing the conditions on the category C, and also consequently generalising the modular functor. I would expect in that case, though, that the domain category can still be thought of as some category of decorated surfaces. At least that’s the basic idea I have of deforming RCFT to the non-rational case.

Anyway, as I understand your idea, it would mean that one should generalise the modular functor such that the domain category might possibly be thought of as C-decorated somethings, where C is still modular, but the somethings are no longer surfaces. Do you agree with this?

What confuses me about this is that I have no clue what one would mean by “sewing” of things which are not geometrical. In the RCFT case it is possible to define a RCFT quite abstractly, in purely categorical language, such that, essentially, the only reference to the geometrical notion of sewing is in the axioms of a modular functor. I would certainly guess that this could easily be generalised in many ways which would have absolutely nothing to do with CFT if one does not have some good characterisation of sewing that can be applied to more general categories. So, this is my main confusion, how does one characterise sewing abstractly? Maybe this is in fact what you’re trying to say with your example, that it’s contained in the structure of some 2-category, or 2-functor?

You’ll have to forgive me for still not having grasped the 2-cat stuff you’re using, I know I should by now…

I hope I managed to communicate my confusion well enough, let me know otherwise, cause I’d be very greatful if you could clear some of my confusions regarding these matters

Cheers!

Jens