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.

40pp, v2: Acknowledgments added, v3: typos fixed and few explanations added, for a version in JHEP

References in corpus (4)

• Associative-algebraic approach to logarithmic conformal field theories
• Enlarged symmetry algebras of spin chains, loop models, and S-matrices
• The periodic sl(2|1) alternating spin chain and its continuum limit as a bulk Logarithmic Conformal Field Theory at c=0
• On the correspondence between boundary and bulk lattice models and (logarithmic) conformal field theories

Cited by in corpus (12)

• Geometric Algebra and Algebraic Geometry of Loop and Potts Models
• Spaces of states of the two-dimensional O(n) and Potts models
• Fusion in the periodic Temperley-Lieb algebra and connectivity operators of loop models
• Four-point geometrical correlation functions in the two-dimensional $Q$-state Potts model: connections with the RSOS models
• On the computation of fusion over the affine Temperley-Lieb algebra
• The action of the Virasoro algebra in quantum spin chains. I. The non-rational case
• From combinatorial maps to correlation functions in loop models
• Bootstrap approach to geometrical four-point functions in the two-dimensional critical $Q$-state Potts model: A study of the $s$-channel spectra
• Realizing the braided Temperley-Lieb-Jones C*-tensor categories as Hilbert C*-modules
• Fusion of irreducible modules in the periodic Temperley--Lieb algebra
• On correlation functions in models related to the Temperley-Lieb algebra
• Uncoiled affine Temperley-Lieb algebras and their Wenzl-Jones projectors