Lattice fusion rules and logarithmic operator product expansions

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 [N. Read and H. Saleur, Associative-algebraic approach to logarithmic conformal field theories, Nucl. Phys. B 777 (2007) 316]. Using quantum group results, we provide rigorous calculations of the fusion of various TL modules at roots of unity cases. 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.