Associative-algebraic approach to logarithmic conformal field theories

By N. Read, H. Saleur. 2007

141 citations recorded by the source corpus.

Abstract We set up a strategy for studying large families of logarithmic conformal field theories by using the enlarged symmetries and non-semisimple associative algebras appearing in their lattice regularizations (as discussed in a companion paper [N. Read, H. Saleur, Enlarged symmetry algebras of spin chains, loop models, and S -matrices, cond-mat/0701259 ]). Here we work out in detail two examples of theories derived as the continuum limit of XXZ spin-1/2 chains, which are related to spin chains with supersymmetry algebras gl ( n | n ) and gl ( n + 1 | n ) , respectively, with open (or free) boundary conditions in all cases. These theories can also be viewed as vertex models, or as loop models. Their continuum limits are boundary conformal field theories (CFTs) with central charge c = − 2 and c = 0 respectively, and in the loop interpretation they describe dense polymers and the boundaries of critical percolation clusters, respectively. We also discuss the case of dilute (critical) polymers as another boundary CFT with c = 0 . Within the supersymmetric formulations, these boundary CFTs describe the fixed points of certain nonlinear sigma models that have a supercoset space as the target manifold, and of Landau–Ginzburg field theories. The submodule structures of indecomposable representations of the Virasoro algebra appearing in the boundary CFT, representing local fields, are derived from the lattice. A central result is the derivation of the fusion rules for these fields.

Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/fa6f3b50111ee7aa49c22942c0f67eaa208204b1

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Citations in the corpus

Logarithmic conformal field theory: a lattice approach (2013, methodology): “Finally, we will show in the next section how the idea of studying Virasoro modules by taking scaling limits of the spin chains can be extended in order to compute fusion rules using an induction procedure [50, 54, 65].”
Logarithmic Tensor Category Theory for Generalized Modules for a Conformal Vertex Algebra, I: Introduction and Strongly Graded Algebras and their Generalized Modules (2010, background): “Results have also been obtained for the vertex operator subalgebras of the algebras W(p, q) generated by the Virasoro algebra (see [16, 24, 44, 103, 115, 124, 126] and [50]).”
Logarithmic conformal field theory: beyond an introduction (2013, background): “In particular, we work almost entirely in the continuum, expecting that the reader is familiar with rational conformal field theory as described (for example) in [14], eschewing approaches based on statistical lattice models and conjectured scaling limits (see [15–18]).”