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Letters in Mathematical Physics

, Volume 104, Issue 5, pp 553–583 | Cite as

Coset Constructions of Logarithmic (1, p) Models

  • Thomas Creutzig
  • David Ridout
  • Simon Wood
Article

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.

Mathematics Subject Classification (2000)

17B68 17B69 

Keywords

logarithmic conformal field theory vertex algebras 

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany
  2. 2.Department of Theoretical Physics, Research School of Physics and Engineering, Mathematical Sciences InstituteAustralian National UniversityCanberraAustralia
  3. 3.Kavli Institute for the Physics and Mathematics of the Universe (WPI), Todai Institutes for Advanced StudyThe University of TokyoKashiwaJapan

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