Coset Constructions of Logarithmic (1,p)-Models

TL;DR

This paper constructs and analyzes coset realizations of logarithmic (1,p)-models, including singlet and triplet algebras, revealing their structure via kernel subalgebras and explicit character decompositions.

Contribution

It introduces coset constructions for (1,p) logarithmic models, connecting them to W^(2)_n algebras and providing explicit realizations for p=2,3, with conjectures for p>5.

Findings

B_p algebras are homomorphic images of W^(2)_{p-1} for p ≤ 5

Triplet algebra W(p) is realized as a coset inside the kernel of screening operators

Explicit coset character decompositions are provided for p=2 and 3

Abstract

One of the best understood families of logarithmic conformal field theories is that consisting 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 M(p) and the triplet algebras W(p), as well as the ubiquitous symplectic fermions theory. In this work, these algebras are realized 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 affine sl(n)_k, generalising the Bershadsky-Polyakov algebra W^(2)_3. Inspired by work of Adamovic for p=3, vertex algebras 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 B_p is a homomorphic image 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 W(p) is then realised as a coset inside the full kernel of the screening operator, while the singlet algebra M(p) is similarly realised inside B_p. As an application, and to illustrate these results, the coset character decompositions are explicitly worked out for p=2 and 3.