Abstract
The equivalent of fusion in boundary conformal field theory (CFT) can be realized quite simply in the context of lattice models by essentially glueing two open spin chains. This has led to many developments, in particular in the context of chiral logarithmic CFT.We consider in this paper a possible generalization of the idea to the case of bulk conformal field theory. This is of course considerably more difficult, since there is no obvious way of merging two closed spin chains into a big one. In an earlier paper, two of us had proposed a “topological” way of performing this operation in the case of models based on the affine Temperley-Lieb (ATL) algebra, by exploiting the associated braid group representation and skein relations. In the present work, we establish — using, in particular, Frobenius reciprocity — the resulting fusion rules for standard modules of ATL in the generic as well as partially degenerate cases. These fusion rules have a simple interpretation in the continuum limit. However, unlike in the chiral case this interpretation does not match the usual fusion in non-chiral CFTs. Rather, it corresponds to the glueing of the right moving component of one conformal field with the left moving component of the other.
Highlights
The study of relations between lattice models and conformal field theories (CFT) has a long history, and stems from several different motivations
Two of us had proposed a “topological” way of performing this operation in the case of models based on the affine Temperley-Lieb (ATL) algebra, by exploiting the associated braid group representation and skein relations
While we believe that our definition of fusion and the corresponding results for finitedimensional ATL modules are interesting in their own right, it is clear that we have not obtained what one would like to call a “lattice version” of bulk fusion
Summary
The study of relations between lattice models and conformal field theories (CFT) has a long history, and stems from several different motivations. It has proven extremely valuable to study in detail the corresponding lattice regularizations, and to infer from these results about the continuum limit Crucial in this approach has been the construction of a lattice equivalent of conformal fusion in the chiral case. This glueing is not what one would normally call fusion in non-chiral CFT.
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