### Bulk Fields and Induced Bimodules

#### Posted by Urs Schreiber

As I mentioned recently, Fjelstad, Fuchs, Runkel & Schweigert know how to describe 2-dimensional (rational) conformal field theory in terms of tangle diagrams in modular tensor categories $C_2$.

There are various hints # that one can understand this formalism from the point of view of 2-transport # with values in $C_2$.

Notice how this is rather analogous to *principal* 2-transport #, with values in a 2-group $G_2$.

Instead of a 2-group, $C_2$ is just a 2-*monoid*. But the look and feel of both is rather similar: being modular tensor, $C_2$ in particular has left and right duals for all objects.

Many aspects of the diagrams drawn onto the worldsheet in the FRS formalism can be understood # from locally trivializing a 2-transport

which sends pieces of worldsheet to morphisms in $C_2$.

One aspect of this is however a little troubling: $F$ only sees the 2-dimensional parameter space. However, FRS show that *bulk* field insertions in 2D CFT are represented diagrammatically by insertion points on the worldsheet at which ribbons emanate *perpendicular* to the worldsheet, into a third dimension.

In fact, this leads to a big story where the entire 2-dimensional field theory is described as the boundary part of a 3-dimensional topological field theory, generalizing the old observation by Witten on the relation between Chern-Simons theory and the Wess-Zumino model. I used to be puzzled about how to capture this 3-dimensional aspect of 2-dimensional CFT in terms of 2-dimensional transport.

But, remember, I also used to be puzzled about how to describe non fake-flat # principal 2-transport. There, the solution is # to pass from a 2-functor

with values in the the 2-group $G_2$ to the pseudo functor (a 3-functor, really!)

with values in the 3-group of automorphisms of $G_2$.

Given what I said so far, there is really one question one should ask:

*What happens if we consider weak 2-functors that send pieces of worldsheet not to a modular tensor category $C_2$, but to the 3-monoid*

of endomorphisms of $C_2$?

To get started, let’s just look at what happens at these bulk field insertions, first.

On the worldsheet, we usually have edges running which are labelled by a (special symmetric Frobenius) algebra object

internal to $C_2$. More generally, the labels take values in bimodules

for (special symmetric Frobenius) algebra objects internal to $C_2$.

At a bulk field insertion, we have a ribbon labelled by a simple object $U_i$ of $C_2$ coming down from above the worldsheet, and another ribbon, labelled by another simple object $U_j$, running below the worldsheet. Where they both meet with a bimodule ribbon $N$ on the worldsheet, we rewire them using a morphism

of induced bimodules in $C_2$.

This means in particular that we regard the object $N \otimes U_i$ again as an $A$-bimodule. $A$ acts from the right by first passig *beneath* $U_i$ (using the braiding in $C_2$) and then acting on $N$ as before.

Similarly, $A$ acts from the left on $U_j \otimes N$ by first passing *above* $U_j$, and then acting on $N'$ as before.

Clearly, the braidings involved here reflect the 3-dimensional situation which I tried to describe.

Using the fact that the braiding of $C_2$ allows us to mimic the 3-dimensional structure (after all, a braided monoidal category is nothing but a 3-category with a single object, in disguise), I was able to essentially describe this situation in terms of locally trivialized 2-transport with values in $\Sigma(C_2)$. See the diagram on p. 7 here.

A problem with this description is that there is no sense in which the $U_i$- and the $U_j$-ribbons really run off in a perpendicular direction. Everything gets sort of projected onto the worldsheet as long as we just work in $\Sigma(C_2)$.

So I asked myself: what happens if we consider bimodule homomorphisms not in $\Sigma(C_2)$ but in $\mathrm{End}(\Sigma(C_2))$?

Unless I made a mistake, the answer seems to be that we indeed get this way what is needed here. I describe some of the details in

For every object $R$ of $C_2$, we get an inner endomorphism of $C_2$ (by an op-lax 2-functor) by conjugation

That’s how the 1-morphisms of $\Sigma(C_2)$ sit inside $\mathrm{End}(\Sigma(C_2))$.

The crucial new aspect is that a 2-morphism between two such 1-morphisms is now no longer just

but, since it now comes to us as a pseudonatural transformation of the above action by conjugation, is in fact a square

Recall that for the principal 2-transport, it was the freedom contained in the vertical morphisms of these squares which gave rise to the fake curvature #. Here, as is now quite manifest, these vertical morphisms play exactly the role that the objects $U_i$ and $U_j$ play in the bimodule homomorphism

mentioned above.

Moreover, we now also have 3-morphisms between these 2-morphisms. Since these come from modifications of pseudonatural transformations, it is not hard to see that they indeed describe “ribbons” attached to $u$ and $v$ running off into a third dimension.

Looking at the details, one finds that a morphism of bimodules in $\mathrm{End}(\Sigma(C_2))$ (using “inner endomorphisms” only, in a way I make precise in my notes), is in fact a morphism

of induced bimodules in $C_2$, with $U_i$ and $U_j$ running off into the third dimension the way they should.