A physical approach to the classification of indecomposable Virasoro representations from the blob algebra
Abstract
In the context of Conformal Field Theory (CFT), many results can be obtained from the representation theory of the Virasoro algebra. While the interest in Logarithmic CFTs has been growing recently, the Virasoro representations corresponding to these quantum field theories remain dauntingly complicated, thus hindering our understanding of various critical phenomena. We extend in this paper the construction of Read and Saleur (2007) [1,2], and uncover a deep relationship between the Virasoro algebra and a finitedimensional algebra characterizing the properties of twodimensional statistical models, the socalled blob algebra (a proper extension of the TemperleyLieb algebra). This allows us to explore vast classes of Virasoro representations (projective, tilting, generalized staggered modules, etc.), and to conjecture a classification of all possible indecomposable Virasoro modules (with, in particular, L_{0} Jordan cells of arbitrary rank) that may appear in a consistent physical Logarithmic CFT where Virasoro is the maximal local chiral algebra. As byproducts, we solve and analyze algebraically quantumgroup symmetric XXZ spin chains and sl(21) supersymmetric spin chains with extra spins at the boundary, together with the "mirror" spin chain introduced by Martin and Woodcock (2003) [3].
 Publication:

Nuclear Physics B
 Pub Date:
 August 2013
 DOI:
 10.1016/j.nuclphysb.2013.04.017
 arXiv:
 arXiv:1212.0093
 Bibcode:
 2013NuPhB.873..614G
 Keywords:

 High Energy Physics  Theory;
 Mathematical Physics;
 Mathematics  Quantum Algebra
 EPrint:
 65 pages, 19 figures. New appendix