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September 27, 2006

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 2C_2.

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

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

Instead of a 2-group, C 2C_2 is just a 2-monoid. But the look and feel of both is rather similar: being modular tensor, C 2C_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

(1)F:P 2Σ(C 2) F : P_2 \to \Sigma(C_2)

which sends pieces of worldsheet to morphisms in C 2C_2.

One aspect of this is however a little troubling: FF 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

(2)tra:P 2(X)Σ(G 2) \mathrm{tra} : P_2(X) \to \Sigma(G_2)

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

(3)tra:P 2(X)Aut(Σ(G 2)) \mathrm{tra} : P_2(X) \to \mathrm{Aut}(\Sigma(G_2))

with values in the 3-group of automorphisms of G 2G_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 2C_2, but to the 3-monoid

(4)End(Σ(C 2)) \mathrm{End}(\Sigma(C_2))

of endomorphisms of C 2C_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

(1)A A

internal to C 2C_2. More generally, the labels take values in bimodules

(2)N N

for (special symmetric Frobenius) algebra objects internal to C 2C_2.

At a bulk field insertion, we have a ribbon labelled by a simple object U iU_i of C 2C_2 coming down from above the worldsheet, and another ribbon, labelled by another simple object U jU_j, running below the worldsheet. Where they both meet with a bimodule ribbon NN on the worldsheet, we rewire them using a morphism

(3)NU i f U jN \array{ N \otimes U_i \\ \;\; \downarrow f \\ U_j \otimes N' }

of induced bimodules in C 2C_2.

This means in particular that we regard the object NU iN \otimes U_i again as an AA-bimodule. AA acts from the right by first passig beneath U iU_i (using the braiding in C 2C_2) and then acting on NN as before.

Similarly, AA acts from the left on U jNU_j \otimes N by first passing above U jU_j, and then acting on NN' 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 2C_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 Σ(C 2)\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 iU_i- and the U jU_j-ribbons really run off in a perpendicular direction. Everything gets sort of projected onto the worldsheet as long as we just work in Σ(C 2)\Sigma(C_2).

So I asked myself: what happens if we consider bimodule homomorphisms not in Σ(C 2)\Sigma(C_2) but in End(Σ(C 2))\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

End(Σ(C 2))\mathrm{End}(\Sigma(C_2))-2-Transport.

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

(4)(U)(R *UR). (\bullet \stackrel{U}{\to} \bullet) \mapsto (\bullet\stackrel{R^*}{\to} \bullet \stackrel{U}{\to} \bullet \stackrel{R}{\to} \bullet) \,.

That’s how the 1-morphisms of Σ(C 2)\Sigma(C_2) sit inside End(Σ(C 2))\mathrm{End}(\Sigma(C_2)).

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

(5)R R, \array{ \bullet \stackrel{R}{\to} \bullet \\ \Downarrow \\ \bullet \stackrel{R'}{\to} \bullet} \,,

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

(6) R v u R . \array{ \bullet &\stackrel{R}{\to}& \bullet \\ v\downarrow\;\;&\Downarrow & \;\; \downarrow u \\ \bullet &\stackrel{R'}{\to}& \bullet } \,.

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 iU_i and U jU_j play in the bimodule homomorphism

(7)NU i f U jN \array{ N \otimes U_i \\ \;\; \downarrow f \\ U_j \otimes N' }

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 uu and vv running off into a third dimension.

Looking at the details, one finds that a morphism of bimodules in End(Σ(C 2))\mathrm{End}(\Sigma(C_2)) (using “inner endomorphisms” only, in a way I make precise in my notes), is in fact a morphism

(8)NU i f U jN \array{ N \otimes U_i \\ \;\; \downarrow f \\ U_j \otimes N' }

of induced bimodules in C 2C_2, with U iU_i and U jU_j running off into the third dimension the way they should.

Posted at September 27, 2006 2:45 PM UTC

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