Integrable Boundary Conditions and W-Extended Fusion in the Logarithmic Minimal Models LM(1,p)

TL;DR

This paper explores the boundary conditions and fusion rules in logarithmic minimal models LM(1,p), connecting lattice models with extended W symmetry and confirming the consistency of the fusion algebra through lattice approaches.

Contribution

It identifies specific boundary conditions as limits of integrable lattice models and derives fusion rules for extended W representations in LM(1,p).

Findings

Identified 4p-2 boundary conditions matching W-representations.

Derived fusion rules consistent with previous theoretical results.

Confirmed closure of the fusion algebra on a finite set of representations.

Abstract

We consider the logarithmic minimal models LM(1,p) as `rational' logarithmic conformal field theories with extended W symmetry. To make contact with the extended picture starting from the lattice, we identify 4p-2 boundary conditions as specific limits of integrable boundary conditions of the underlying Yang-Baxter integrable lattice models. Specifically, we identify 2p integrable boundary conditions to match the 2p known irreducible W-representations. These 2p extended representations naturally decompose into infinite sums of the irreducible Virasoro representations (r,s). A further 2p-2 reducible yet indecomposable W-representations of rank 2 are generated by fusion and these decompose as infinite sums of indecomposable rank-2 Virasoro representations. The fusion rules in the extended picture are deduced from the known fusion rules for the Virasoro representations of LM(1,p) and are found to be in agreement with previous works. The closure of the fusion algebra on a finite number of representations in the extended picture is remarkable confirmation of the consistency of the lattice approach.