Coset Constructions of Logarithmic (1, p) Models

• Published in: Letters in mathematical physics Vol. 104; no. 5; pp. 553 - 583
• Main Authors: Creutzig, Thomas, Ridout, David, Wood, Simon
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.05.2014
• Subjects: Complex Systems; Geometry; Group Theory and Generalizations; Mathematical and Computational Physics; Physics; Physics and Astronomy; Theoretical; 17B68; 17B69; vertex algebras; logarithmic conformal field theory
• ISSN: 0377-9017; 1573-0530
• DOI: 10.1007/s11005-014-0680-7

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 M ( p ) and the triplet algebras W ( p ) , as well as the ubiquitous symplectic fermions theory. In this work, these algebras are realised through a coset construction. The W n ( 2 ) algebra of level k was introduced by Feigin and Semikhatov as a (conjectured) quantum hamiltonian reduction of sl ^ ( n ) k , generalising the Bershadsky–Polyakov algebra W 3 ( 2 ) . Inspired by work of Adamović 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 quotient of W p - 1 ( 2 ) 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.