Letters in Mathematical Physics

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

Coset Constructions of Logarithmic (1, p) Models

  • Thomas CreutzigEmail author
  • David Ridout
  • Simon Wood


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 


logarithmic conformal field theory vertex algebras 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adamović D.: Classification of irreducible modules of certain subalgebras of free boson vertex algebra. J. Algebra 270, 115132 (2003)Google Scholar
  2. 2.
    Adamović D.: A construction of admissible \({A^{(1)}_1}\) -modules of level −4/3. J. Pure Appl. Algebra 196, 119134 (2005)Google Scholar
  3. 3.
    Adamović, D.: On realization of certain admissible \({A^{(1)}_1}\)-modules. In: Functional analysis IX, Various Publ. Ser. (Aarhus), vol. 48Google Scholar
  4. 4.
    Adamović D., Milas A.: Lattice construction of logarithmic modules for certain vertex algebras. Selecta Math. (N.S.) 15, 535561 (2009)Google Scholar
  5. 5.
    Adamović D., Milas A.: On W-algebras associated to (2,p) minimal models and their representations. Int. Math. Res. Not. 2010, 3896 (2010)zbMATHGoogle Scholar
  6. 6.
    Adamović D., Milas A.: Vertex operator algebras associated to modular invariant representations of \({A_1^{(1)}}\). Math. Res. Lett. 2, 563 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Adamović D., Milas A.: On the triplet vertex algebra W(p). Adv. Math. 217, 2664 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Bershadsky M.: Conformal field theories via Hamiltonian reduction. Comm. Math. Phys. 139, 71 (1991)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Creutzig, T., Gao, P., Linshaw, A.R.: A commutant realization of \({W^{(2)}_n}\) at critical level. Int. Math. Res. Not. (2012). doi: 10.1093/imrn/rns229
  10. 10.
    Creutzig T., Linshaw A.R.: A commutant realization of Odake’s algebra. Transform. Groups 18(3), 615–637 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Creutzig T., Ridout D.: Modular data and Verlinde formulae for fractional level WZW models I. Nucl. Phys. B 865, 83 (2012)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Creutzig T., Ridout D.: W-algebras extending \({\widehat{{\rm gl}}}\) (1|1). Springer Proc. Math. Stat. 36, 349 (2013)Google Scholar
  13. 13.
    Creutzig T., Ridout D.: Relating the archetypes of logarithmic conformal field theory. Nucl. Phys. B 872, 348 (2013)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Creutzig T., Ridout D.: Modular data and Verlinde formulae for fractional level WZW models II. Nucl. Phys. B 875, 423 (2013)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Creutzig, T., Ridout, D.: Logarithmic conformal field theory: beyond an introduction. J. Phys. A 46, 494006 (2013)Google Scholar
  16. 16.
    Creutzig T., Rønne P.B.: The GL(1|1)-symplectic fermion correspondence. Nucl. Phys. B 815, 95 (2009)ADSzbMATHCrossRefGoogle Scholar
  17. 17.
    Creutzig T., Schomerus V.: Boundary correlators in supergroup WZNW models. Nucl. Phys. B 807, 471 (2009)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Dong C.: Vertex algebras associated with even lattices. J. Algebra 161, 245 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Dong, C., Mason, G.: Coset constructions and dual pairs for vertex operator algebra. arXiv:9904155 [math.QA]Google Scholar
  20. 20.
    Feigin B.L., Gainutdinov A.M., Semikhatov A.M., Yu I.l: Tipunin, logarithmic extensions of minimal models: characters and modular transformations. Nucl. Phys. B 757, 303 (2006)ADSzbMATHCrossRefGoogle Scholar
  21. 21.
    Fuchs J., Hwang S., Semikhatov A.M., Tipunin I.Y.: Nonsemisimple fusion algebras and the Verlinde formula. Comm. Math. Phys. 247, 713 (2004)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Feigin B.L., Semikhatov A.M.: W\({_{2}^{(n)}}\)-algebras. Nucl. Phys. B 698, 409 (2004)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Feigin B., Semikhatov A., Sirota V., Yu Tipunin I.: Resolutions and characters of irreducible representations of the N =  2 superconformal algebra. Nucl. Phys. B 536, 617 (1998)ADSCrossRefGoogle Scholar
  24. 24.
    Gaberdiel M.R.: Fusion rules and logarithmic representations of a WZW model at fractional level. Nucl. Phys. B 618, 407 (2001)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Gaberdiel M.R., Kausch H.G.: A local logarithmic conformal field theory. Nucl. Phys. B 538, 631 (1999)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Gaberdiel M.R., Runkel I.: From boundary to bulk in logarithmic CFT. J. Phys. A 41, 075402 (2008)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Kausch H.G.: Extended conformal algebras generated by a multiplet of primary fields. Phys. Lett. B 259(4), 448 (1991)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Kac V., Roan S.S., Wakimoto M.: Quantum reduction for affine superalgebras. Commun. Math. Phys. 241(2-3), 307–342 (2003)ADSzbMATHMathSciNetGoogle Scholar
  29. 29.
    Linshaw A.: Invariant chiral differential operators and the W3 algebra. J. Pure Appl. Algebra 213, 632 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Lian B., Linshaw A.: Howe pairs in the theory of vertex algebras. J. Algebra 317, 111 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Nagatomo, K., Tsuchiya, A.: The triplet vertex operator algebra W(p) and the restricted quantum group \({\bar{U}_q(sl_2)}\) at qe π i/p. In: Advanced studies in pure mathematics. Exploring new structures andnatural constructions in mathematical physics. American MathematicalSociety, vol. 61 (2011)Google Scholar
  32. 32.
    Polyakov A.M.: Gauge transformations and diffeomorphisms. Int. J. Mod. Phys. A 5, 833 (1990)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    Ridout D.: Fusion in fractional level \({\widehat{{\rm sl}}(2)}\) -theories with k =  −1/2. Nucl. Phys. B 848, 216 (2011)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Ridout D.: \({\widehat{{\rm sl}}(2)_{-1/2}}\) and the triplet model. Nucl. Phys. B 835, 314 (2010)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Ridout D.: \({\widehat{{\rm sl}}(2)_{-1/2}}\) : a case study. Nucl. Phys. B 814, 485 (2009)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Schomerus V., Saleur H.: The GL(1|1) WZW model: from supergeometry to logarithmic CFT. Nucl. Phys. B 734, 221 (2006)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    Tsuchiya, A., Wood, S.: The tensor structure on the representation category of the W p triplet algebra. arXiv:1201.0419 [hep-th]Google Scholar
  38. 38.
    Tsuchiya, A., Wood, S.: On the extended W-algebra of type \({\mathfrak{sl}_2}\) at positive rational level.arXiv:1302.6435 [math.QA]Google Scholar

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

Personalised recommendations