Lattice fusion rules and logarithmic operator product expansions

TL;DR

This paper rigorously analyzes lattice fusion rules in logarithmic conformal field theories using Temperley-Lieb modules, revealing how indecomposability manifests in lattice models and relates to operator-product expansions in quantum fields.

Contribution

It provides a rigorous calculation of lattice fusion rules for Temperley-Lieb modules, connecting lattice indecomposability with field theory operator-product expansions.

Findings

Explicit fusion rules for TL modules are derived.

Indecomposability mechanisms are compared between lattice models and field theories.

Physical interpretation of lattice fusion rules in terms of operator-product expansions.

Abstract

The interest in Logarithmic Conformal Field Theories (LCFTs) has been growing over the last few years thanks to recent developments coming from various approaches. A particularly fruitful point of view consists in considering lattice models as regularizations for such quantum field theories. The indecomposability then encountered in the representation theory of the corresponding finite-dimensional associative algebras exactly mimics the Virasoro indecomposable modules expected to arise in the continuum limit. In this paper, we study in detail the so-called Temperley-Lieb (TL) fusion functor introduced in physics by Read and Saleur [Nucl. Phys. B 777, 316 (2007)]. Using quantum group results, we provide rigorous calculations of the fusion of various TL modules. Our results are illustrated by many explicit examples relevant for physics. We discuss how indecomposability arises in the "lattice" fusion and compare the mechanisms involved with similar observations in the corresponding field theory. We also discuss the physical meaning of our lattice fusion rules in terms of indecomposable operator-product expansions of quantum fields.