Lattice W-algebras and logarithmic CFTs

This paper is part of an effort to gain further understanding of 2D logarithmic conformal field theories (LCFTs) by exploring their lattice regularizations. While all work so far has dealt with the Virasoro algebra (or the product Vir⊗ Vir-bar ), the best known (although maybe not the most relevant physically) LCFTs in the continuum are characterized by a W-algebra symmetry, whose presence is powerful, but whose role as a 'symmetry' remains mysterious. We explore here the origin of this symmetry in the underlying lattice models. We consider Uqsℓ(2) XXZ spin chains for q a root of unity, and argue that the centralizer of the 'small' quantum group U-bar qsℓ(2) goes over the W-algebra in the continuum limit. We justify this identification by representation theoretic arguments, and give, in particular, lattice versions of the W-algebra generators. In the case q=i, which corresponds to symplectic fermions at central charge c=−2, we provide a full analysis of the scaling limit of the lattice Virasoro and W generators, and show in details how the corresponding continuum Virasoro and W-algebras are obtained. Striking similarities between the lattice W algebra and the Onsager algebra are observed in this case. (paper)