### Towards the FFRS Description of 2dCFT (B)

#### Posted by Urs Schreiber

More of our little seminar preparing us for the description of 2-dimensional conformal field theory in terms of “state sum models” internal to modular tensor categories.

Last time we had understood how to build representations of 2-dimensional cobordisms (called “2-dimensional quantum field theories”) by chopping these up into *generators*: the pair of pants, the cap, the co-pair of pants and the co-cap.

As we move up in dimension, and as we put extra structure on our cobordisms, it will become increasingly hard to find and handle a collection of such generators.

*But*: it will always be easy to chop any 2-dimensional cobordism into “**2-generators**”: little disk-shaped 2-dimensional pieces of our cobordism.

So we need to think about how to think about 2-processes that occur over such “2-generators”.

After doing, so, we rediscover the *Frobenius property* as an incarnation of the *exchange law* of 2-processes and see why we want to *triangulate* our cobordism and *color the triangulation with a Frobenius algebra*.

Secretly, this is all about expressing one 2-functor – one on the entire cobordism, say – in terms of another one # – living only over a little piece of cobordism.

This is possible if both are connected by a special ambidextrous adjunction: something weaker than an equivalence but stronger than an adjunction.

And special ambidextrous adjunctions are precisely what gives rise to special Frobenius monads.

And if we want to *glue* what we had sliced before, we need special Frobenius monads in *ribbon* categories.

And this is why we see special Frobenius algebras internal to ribbon categories decorating triangulations of surfaces.

The glue and the ribbon will appear in the next set of notes.

## Re: Towards the FFRS Description of 2dCFT (B)

I did run badly out of time with preparing this. But here are some rough notes, for the moment: