Abstract :
We consider the continuum scaling limit of the infinite series of Yang–Baxter integrable logarithmic minimal models image as ‘rational’ logarithmic conformal field theories with extended image symmetry. The representation content is found to consist of image image-indecomposable representations of which image are of rank 1, image are of rank 2, while the remaining image are of rank 3. We identify these representations with suitable limits of Yang–Baxter integrable boundary conditions on the lattice. The image-indecomposable rank-1 representations are all image-irreducible while we present a conjecture for the embedding patterns of the image-indecomposable rank-2 and -3 representations. The associated image-extended characters are all given explicitly and decompose as finite non-negative sums of image-irreducible characters. The latter correspond to image-irreducible subfactors and we find that there are image of them. We present fermionic character expressions for some of the rank-2 and all of the rank-3 image-indecomposable representations. To distinguish between inequivalent image-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 image-indecomposable representations and find that they generate a closed fusion algebra, albeit one without identity for image. We present the complete set of fusion rules and interpret the closure of this fusion algebra as confirmation of the proposed extended symmetry. Finally, image of the image-indecomposable representations are in fact image-projective representations and they generate a closed fusion subalgebra.
