A physical approach to the classification of indecomposable Virasoro representations from the blob algebra

TL;DR

This paper develops a physical approach linking Virasoro algebra representations to the blob algebra, enabling classification of indecomposable modules relevant for Logarithmic CFTs and analyzing related quantum spin chains.

Contribution

It extends the construction of Read and Saleur, revealing a deep relationship between Virasoro representations and the blob algebra, leading to a conjecture on classifying all indecomposable modules in Logarithmic CFTs.

Findings

Classified a wide range of Virasoro modules including projective and tilting modules.

Solved algebraic structures of quantum-group symmetric spin chains.

Analyzed boundary conditions in supersymmetric spin chains.

Abstract

In the context of Conformal Field Theory (CFT), many results can be obtained from the representation theory of the Virasoro algebra. While the interest in Logarithmic CFTs has been growing recently, the Virasoro representations corresponding to these quantum field theories remain dauntingly complicated, thus hindering our understanding of various critical phenomena. We extend in this paper the construction of Read and Saleur (2007), and uncover a deep relationship between the Virasoro algebra and a finite-dimensional algebra characterizing the properties of two-dimensional statistical models, the so-called blob algebra (a proper extension of the Temperley--Lieb algebra). This allows us to explore vast classes of Virasoro representations (projective, tilting, generalized staggered modules, etc.), and to conjecture a classification of all possible indecomposable Virasoro modules (with, in particular, L_0 Jordan cells of arbitrary rank) that may appear in a consistent physical Logarithmic CFT where Virasoro is the maximal local chiral algebra. As by-products, we solve and analyze algebraically quantum-group symmetric XXZ spin chains and sl(2|1) supersymmetric spin chains with extra spins at the boundary, together with the "mirror" spin chain introduced by Martin and Woodcock (2004).