Coset Constructions of Logarithmic (1, p) Models

Abstract

One of the best understood families of logarithmic onformal field theories consists of the (1, p) models (p = 2, 3, . . .) of central charge c 1, p =1 − 6(p − 1)2/p. This family includes the theories corresponding to the singlet algebras $${\mathcal{M}(p)}$$ and the triplet algebras $${\mathcal{W}(p)}$$ , as well as the ubiquitous symplectic fermions theory. In this work, these algebras are realised through a coset construction. The $${W^{(2)}_n}$$ algebra of level k was introduced by Feigin and Semikhatov as a (conjectured) quantum hamiltonian reduction of $${\widehat{\mathfrak{sl}}(n)_k}$$ , generalising the Bershadsky–Polyakov algebra $${W^{(2)}_3}$$ . Inspired by work of Adamović for p = 3, vertex algebras $${\mathcal{B}_p}$$ are constructed as subalgebras of the kernel of certain screening charges acting on a rank 2 lattice vertex algebra of indefinite signature. It is shown that for p≤5, the algebra $${\mathcal{B}_p}$$ is a quotient of $${W^{(2)}_{p-1}}$$ at level −(p − 1)2/p and that the known part of the operator product algebra of the latter algebra is consistent with this holding for p> 5 as well. The triplet algebra $${\mathcal{W}(p)}$$ is then realised as a coset inside the full kernel of the screening operator, while the singlet algebra $${\mathcal{M}(p)}$$ is similarly realised inside $${\mathcal{B}_p}$$ . As an application, and to illustrate these results, the coset character decompositions are explicitly worked out for p = 2 and 3.