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.

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Creutzig, T., Ridout, D. & Wood, S. Coset Constructions of Logarithmic (1, p) Models. Lett Math Phys 104, 553–583 (2014). https://doi.org/10.1007/s11005-014-0680-7

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Mathematics Subject Classification (2000)

  • 17B68
  • 17B69

Keywords

  • logarithmic conformal field theory
  • vertex algebras