Lattice W-algebras and logarithmic CFTs

TL;DR

This paper investigates the lattice origins of W-algebra symmetries in 2D logarithmic conformal field theories by analyzing quantum spin chains and their continuum limits, revealing the algebraic structures underlying these models.

Contribution

It establishes a connection between lattice models and continuum W-algebras in LCFTs, providing lattice versions of W-algebra generators and detailed continuum limit analysis.

Findings

Identification of the lattice W-algebra as the centralizer of the small quantum group.

Explicit lattice constructions of W-algebra generators.

Continuum limit analysis showing the emergence of W-algebras from lattice models.

Abstract

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 of left and right Virasoro), 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 difficult to understand physically. We explore here the origin of this symmetry in the underlying lattice models. We consider U_q sl(2) XXZ spin chains for q a root of unity, and argue that the centralizer of the "small" quantum group 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.