W-Extended Logarithmic Minimal Models

TL;DR

This paper investigates the structure and fusion rules of W-extended logarithmic minimal models, revealing their indecomposable representations, explicit characters, and the algebraic closure of their fusion processes.

Contribution

It provides a detailed classification of W-indecomposable representations, explicit character formulas, and a complete fusion algebra for the models, extending understanding of their symmetry structure.

Findings

Identified 6pp'-2p-2p' W-indecomposable representations with ranks 1, 2, and 3.

Derived explicit W-extended characters and their decompositions into irreducible characters.

Established a closed fusion algebra without an identity element for p>1.

Abstract

We consider the continuum scaling limit of the infinite series of Yang-Baxter integrable logarithmic minimal models LM(p,p') as `rational' logarithmic conformal field theories with extended W symmetry. The representation content is found to consist of 6pp'-2p-2p' W-indecomposable representations of which 2p+2p'-2 are of rank 1, 4pp'-2p-2p' are of rank 2, while the remaining 2(p-1)(p'-1) are of rank 3. We identify these representations with suitable limits of Yang-Baxter integrable boundary conditions on the lattice. The W-indecomposable rank-1 representations are all W-irreducible while we present a conjecture for the embedding patterns of the W-indecomposable rank-2 and -3 representations. The associated W-extended characters are all given explicitly and decompose as finite non-negative sums of W-irreducible characters. The latter correspond to W-irreducible subfactors and we find that there are 2pp'+(p-1)(p'-1)/2 of them. We present fermionic character expressions for some of the rank-2 and all of the rank-3 W-indecomposable representations. To distinguish between inequivalent W-indecomposable representations of identical characters, we introduce `refined' characters carrying information also about the Jordan-cell content of a representation. Using a lattice implementation of fusion on a strip, we study the fusion rules for the W-indecomposable representations and find that they generate a closed fusion algebra, albeit one without identity for p>1. We present the complete set of fusion rules and interpret the closure of this fusion algebra as confirmation of the proposed extended symmetry. Finally, 2pp' of the W-indecomposable representations are in fact W-projective representations and they generate a closed fusion subalgebra.