Formula omitted]-extended logarithmic minimal models

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 6 p p ′ − 2 p − 2 p ′ W -indecomposable representations of which 2 p + 2 p ′ − 2 are of rank 1, 4 p p ′ − 2 p − 2 p ′ 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 2 p p ′ + ( 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, 2 p p ′ of the W -indecomposable representations are in fact W -projective representations and they generate a closed fusion subalgebra.