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

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.

Bootstrap approach to geometrical four-point functions in the two-dimensional critical Q-state Potts model: a study of the s-channel spectra

“…Remark 3. The quotient W 0,q 2 is but one example of representations that appear more generally in T a N when q is still generic, but z takes particular values [26,28]. Indeed, the standard module W j ,z has a non-zero homomorphism to W j,z ,…”

“…For the reader's convenience we summarise here some of the notations used in this paper. They are, as far as possible, the same as those used in [27,28,[37][38][39][40][41][42].…”

“…The key question we want to address in this paper is thus to determine the set S of values of ∆,∆, which we will tackle by a combination of algebraic and numerical methods. In its crudest form, the latter involves the brute force numerical determination of a (very) large number of terms appearing in the right-hand side of (28).…”

“…The four-point function (125) involves two conformal blocks, corresponding to the fusion channels σσ ∼ 1 and σσ ∼ . Expanding the function G as in (28) gives then…”

“…This corresponds to sending 1 − z → 1 1−z , and thus to changing the sign of P abab − P abba , as required from the geometrical interpretation. This does not mean that only odd spins appear in the expression (28), because in this case there are large degeneracies (in fact, only a few of the terms correspond to primary fields).…”