The String Coffee Table

The purpose of the following is to point out that the argument on pp. 29-30 and pp. 66-67 of FRS I really defines a lax functor from the theory’s quiver diagram to the suspension of the representation category of its chiral data.

This argument is as follows: Pick some vertex algebra $V$ describing the local symmetry of a class of RCFTs. Let $\mathrm{Rep}(V)$ be the (modular) category of representation of this algebra. A particular RCFT in this class (all whose members share the same local symmetries) is determined by any one boundary conditions (D-brane). Call this D-brane $N$. On the space of open string states for strings both whose ends sit on $N$, the operator product expansion defines an associative product and coproduct. This way the space of open $N-N$ string states induces a Frobenius algebra internal to $\mathrm{Rep}(V)$.

In fact, the RCFT is completely specified by $\mathrm{Rep}(V)$ together with this algebra of $N-N$ strings.

Now, $N-N$ strings can interact (in particular) with strings that stretch from the brane $N$ to some other brane, $N'$. Hence there is an object in $\mathrm{Rep}(V)$ which represents the space of $N-N'$ string states.

Using the operator product expansion once again, we find that the algebra of $N-N$ string states acts on the space of $N-N'$ string states. The latter hence forms a module for the former.

But notice, for reasons that will become important below, that we could just as well have started with the algebra of $N'-N'$-states. These would act on the space of $N-N'$ states from the other side. Hence the space of $N-N'$ states is really a bimodule, even though we may choose to forget this fact.

The upshot of this analysis is this: An RCFT with chiral data $V$ is the same as a (special, symmetric) Frobenius algebra of open $N-N$ string states internal to $\mathrm{Rep}(V)$. D-branes for this RCFT are precisely all modules for this algebra (internal to $\mathrm{Rep}(V)$).

Notice how in $\mathrm{Rep}(V)$ we may find different collections of Frobenius algebra objects and their modules. There may be several conformal field theories (and associated collections of D-branes) for a specified chiral data.

I claim that we can neatly encode the above story, which leads to a choice of Frobenius algebra and algebra modules, in terms of a choice of quiver representation, in some slightly generalized sense.

It’s just some general abstract nonsense:

For the sake of convenience, let’s forget the bialgebra and Frobenius structure for a moment, and just consider internal algebras and their modules. Let $C$ be any tensor category.

What is the neatest way to define an algebra internal to $C$? How about this one: An algebra internal to $C$ is the same as a monad in $\Sigma(C)$, which again is the same as a lax functor

Here $\Sigma(C)$ is the 2-category with a single object and one morphism per object of $C$. 1 is the category with a single morphism.

A lax functor is a functor from a 1- to a 2-category which respects units and composition only up to some coherent 2-morphisms. (Not necessarily an 2-isomorphism!) These 2-morphisms are nothing but the unit and the product of the algebra. Their coherence is the algebra’s associativity and unit law.

What is the neatest way to define a module for an inernal algebra, more precisely, to define a module which is really a bimodule? Easy: let

be the category with two objects, $N$ and $N'$, and one nontrivial morphism between these. A collection of two internal algebras in $C$ together with an internal bimodule for them is nothing but a lax functor

The $N-N$-algebra is the image of $N \overset{\mathrm{Id}}{\to} N$ under $A$, The $N'-N'$-algebra is the image of $N' \overset{\mathrm{Id}}{\to} N'$ under $A$, and so on. Now $A$ has two more coherent 2-isomorphism compared to the case before. One of them yields the left $A_{NN}$-action on the bimodule which is the image of $N \to N'$ under $A$, the other one encodes the right action.

The pattern now is clear. Consider any category $Q$ with objects $N_1, N_2, N_3, \dots$ and specified nontrivial morphisms between these (a “quiver”). A lax functor

encodes the same data as an algebra internal to $C$ for each object of $Q$, together with an internal bimodule for each nontrivial morphism of $Q$.

But it’s a bit pitiful for a functor to be just lax. It would be much nicer if it were pseudo. However, a pseudofunctor

is the same as a collection of algebras and bimodules internal to $C$, all of whose products, left and right actions are invertible morphisms. That’s an interesting special case of our lax functor, but for the most general situation that we may be intersted in it is a little too strong a condition.

But there is an obvious choice in between lax and pseudo, namely that where to every coherent 2-morphism coming from the lax functor there is one going the other way round, such that the two obvious “bubble moves” are satisfied. For lack of a better name, let me call this a “special lax functor”.

A special lax functor

is a slight generalization of the ordinary concept of a representation of the quiver $Q$. Maybe I should call it a “generalized quiver representation”.

The generalized quiver representation

is the same thing as one special Frobenius algebra internal to $C$ per object of $Q$, together with one bimodule for the Frobenius algebras per morphism of $Q$.

Hence we find that the central theorem of FRS, which says that a full RCFT is the same thing as a modular category $\mathrm{Rep}(V)$ of Moore-Seiberg data together with a special (and symmetric) Frobenius alegbra internal to $\mathrm{Rep}(V)$, can be rephrased as saying that

A background configuration of a RCFT is a generalized quiver representation with values in $\Sigma(\mathrm{Rep}(V))$.

Update: In the above I did not cleanly distinguish between the RCFT and its background configurations. (But see the pdf notes for more on that).

The point is that we want to distinguish between the RCFT with all of its admissable boundary conditions, and setups where we intentionally restrict attention to just a subcollection of these boundary conditions. The latter corresponds to choosing some “background” and “adding” some collection of D-branes to it.

Hence a lax functor $\Gamma \to \Sigma(C)$ defines a “background with D-branes” for the full RCFT which is given by any one of the boundary conditions (and all the modules of the algebra of string states on that boundary).

The interesting thing is that this allows us to define the natural notion of category of rational string backgrounds. It’s just the category of these functors. This might be just the right way to talk about the “landscape” of rational conformal field theories.