What do the fusion rules in CFT mean when coefficients are larger than 1?

[nrqed]

Hi everyone,

This is a question about conformal field theory. Even though it is not directly about beyond the SM, it seems like an appropriate place to ask since it is an important tool in string theory.

I am confused by the fusion rules \phi_i ~ \phi_j = N^k_{ij} \phi_k
(where as usual this is meant to be relating conformal families). My confusion is due to the fact that the N coefficients may in general be different from 0 or 1. What does it mean to have a coefficient larger than 1? Books mention in passing something about having several ways to fuse the families but they never explain and then stop talking about it because the entries of N are only 0 or 1 in the minimal models. Can someone shed some light on this?

thanks!

Patrick

 

[Physics Monkey]

Imagine the fields are labelled by representations of some Lie group, e.g. you are studying some kind of sigma model with the group as a target space. Then an example of the phenomenon I think you are talking about is provided by SU(3). The tensor product of an octet with an octet contains two octets in the resulting direct sum of representations. Hence you can fuse to an octet in two different ways and so N^8_{88} = 2.

Is this what you wanted to know?

 

[nrqed]

Imagine the fields are labelled by representations of some Lie group, e.g. you are studying some kind of sigma model with the group as a target space. Then an example of the phenomenon I think you are talking about is provided by SU(3). The tensor product of an octet with an octet contains two octets in the resulting direct sum of representations. Hence you can fuse to an octet in two different ways and so N^8_{88} = 2.

Is this what you wanted to know?

Hi Physics Monkey,

Yes, this is exactly what I was talking about. I had not thought about an analogy with Lie groups. Thanks, I have to absorb this. I can understand the situation in the context of group representations as I can think about counting the number of independent "states". But in the context of CFT, I thought that the fusion rules were simply stating whether a conforma family appears or not in the OPE of two fields. In that context, what is the difference between having one way to fuse to that family instead of, say, two ways? How does one see that there are two ways of fusing into a certain conformal family when one is working out the OPE explicitly?

Thanks for your help,

Patrick