Logarithmic Conformal Field Theory: a Lattice Approach

TL;DR

This paper reviews how lattice models with non semi-simple algebras like Temperley-Lieb can be used to understand the structure and couplings in logarithmic conformal field theories, especially in bulk cases.

Contribution

It summarizes recent progress in connecting lattice algebraic structures to bulk LCFT properties, extending previous boundary-focused studies.

Findings

Lattice models reflect continuum LCFT indecomposable modules

Fusion rules and couplings can be experimentally measured from lattice models

Progress has been made in understanding bulk LCFT structures

Abstract

Logarithmic Conformal Field Theories (LCFT) play a key role, for instance, in the description of critical geometrical problems (percolation, self avoiding walks, etc.), or of critical points in several classes of disordered systems (transition between plateaus in the integer and spin quantum Hall effects). Much progress in their understanding has been obtained by studying algebraic features of their lattice regularizations. For reasons which are not entirely understood, the non semi-simple associative algebras underlying these lattice models - such as the Temperley-Lieb algebra or the blob algebra - indeed exhibit, in finite size, properties that are in full correspondence with those of their continuum limits. This applies to the structure of indecomposable modules, but also to fusion rules, and provides an `experimental' way of measuring couplings, such as the `number b' quantifying the logarithmic coupling of the stress energy tensor with its partner. Most results obtained so far have concerned boundary LCFTs, and the associated indecomposability in the chiral sector. While the bulk case is considerably more involved (mixing in general left and right moving sectors), progress has also been made in this direction recently, uncovering fascinating structures. This article provides a short general review of our work in this area.