A fusion for the periodic Temperley-Lieb algebra and its continuum limit

TL;DR

This paper extends the concept of fusion from boundary to bulk conformal field theories using the affine Temperley-Lieb algebra, establishing fusion rules that relate to the continuum limit but differ from standard non-chiral CFT fusion.

Contribution

It introduces a topological fusion method for affine Temperley-Lieb algebra modules and derives fusion rules with a novel continuum interpretation.

Findings

Fusion rules for ATL modules established using Frobenius reciprocity.

Fusion interpretation involves glueing right movers of one field to left movers of another.

Results differ from traditional non-chiral CFT fusion, highlighting a new perspective.

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.