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Letters in Mathematical Physics

, Volume 108, Issue 12, pp 2543–2587 | Cite as

Modularity of logarithmic parafermion vertex algebras

  • Jean Auger
  • Thomas Creutzig
  • David Ridout
Article

Abstract

The parafermionic cosets \(\mathsf {C}_{k} = {\text {Com}} ( \mathsf {H} , \mathsf {L}_{k}(\mathfrak {sl}_{2}) )\) are studied for negative admissible levels k, as are certain infinite-order simple current extensions \(\mathsf {B}_{k}\) of \(\mathsf {C}_{k}\). Under the assumption that the tensor theory considerations of Huang, Lepowsky and Zhang apply to \(\mathsf {C}_{k}\), irreducible \(\mathsf {C}_{k}\)- and \(\mathsf {B}_{k}\)-modules are obtained from those of \(\mathsf {L}_{k}(\mathfrak {sl}_{2})\). Assuming the validity of a certain Verlinde-type formula likewise gives the Grothendieck fusion rules of these irreducible modules. Notably, there are only finitely many irreducible \(\mathsf {B}_{k}\)-modules. The irreducible \(\mathsf {C}_{k}\)- and \(\mathsf {B}_{k}\)-characters are computed and the latter are shown, when supplemented by pseudotraces, to carry a finite-dimensional representation of the modular group. The natural conjecture then is that the \(\mathsf {B}_{k}\) are \(C_2\)-cofinite vertex operator algebras.

Keywords

Vertex algebras Conformal field theory Modular transformations Parafermions Coset constructions 

Mathematics Subject Classification

Primary 17B69 Secondary 13A50 

Notes

Acknowledgements

We thank Shashank Kanade and Andrew Linshaw for discussions relating to the results presented here. J. A. is supported by a Doctoral Research Scholarship from the Fonds de Recherche Nature et Technologies de Québec (184131). T. C. is supported by the Natural Sciences and Engineering Research Council of Canada (RES0020460). DR’s research is supported by the Australian Research Council Discovery Project DP160101520 and the Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers CE140100049.

References

  1. 1.
    Rozansky, L., Saleur, H.: Quantum field theory for the multivariable Alexander–Conway polynomial. Nucl. Phys. B 376, 461–509 (1992)ADSCrossRefGoogle Scholar
  2. 2.
    Gurarie, V.: Logarithmic operators in conformal field theory. Nucl. Phys. B 410, 535–549 (1993). arXiv:hep-th/9303160 ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Creutzig, T., Ridout, D.: Relating the archetypes of logarithmic conformal field theory. Nucl. Phys. B 872, 348–391 (2013). arXiv:1107.2135 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Kausch, H.: Symplectic fermions. Nucl. Phys. B583, 513–541 (2000). arXiv:hep-th/0003029 ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Kausch, H.: Extended conformal algebras generated by a multiplet of primary fields. Phys. Lett. B 259, 448–455 (1991)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Gaberdiel, M., Kausch, H.: A rational logarithmic conformal field theory. Phys. Lett. B 386, 131–137 (1996). arXiv:hep-th/9606050 ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Gaberdiel, M., Kausch, H.: A local logarithmic conformal field theory. Nucl. Phys. B 538, 631–658 (1999). arXiv:hep-th/9807091 ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Ridout, D., Wood, S.: Bosonic ghosts at \(c=2\) as a logarithmic CFT. Lett. Math. Phys. 105, 279–307 (2015). arXiv:1408.4185 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Gaberdiel, M.: Fusion rules and logarithmic representations of a WZW model at fractional level. Nucl. Phys. B 618, 407–436 (2001). arXiv:hep-th/0105046 ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Lesage, F., Mathieu, P., Rasmussen, J., Saleur, H.: Logarithmic lift of the \(\widehat{su} \left(2 \right)_{-1/2}\) model. Nucl. Phys. B 686, 313–346 (2004). arXiv:hep-th/0311039 ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Ridout, D.: Fusion in fractional level \(\widehat{\mathfrak{sl}} \left(2 \right)\)-theories with \(k=-\tfrac{1}{2}\). Nucl. Phys. B 848, 216–250 (2011). arXiv:1012.2905 [hep-th]ADSCrossRefGoogle Scholar
  12. 12.
    Zhu, Y.: Modular invariance of characters of vertex operator algebras. J. Am. Math. Soc. 9, 237–302 (1996)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Creutzig, T., Gannon, T.: Logarithmic conformal field theory, log-modular tensor categories and modular forms. J. Phys., A50, 404004 (2017). arXiv:1605.04630 [math.QA]MathSciNetCrossRefGoogle Scholar
  14. 14.
    Creutzig, T., Gannon, T.: The theory of \(C_2\)-cofinite VOAs (in preparation)Google Scholar
  15. 15.
    Huang, Y-Z.: Rigidity and modularity of vertex tensor categories. Commun. Contemp. Math.10, 871–911, (2008). arXiv:math/0502533 [math.QA]MathSciNetCrossRefGoogle Scholar
  16. 16.
    Carqueville, N., Flohr, M.: Nonmeromorphic operator product expansion and \(C_2\)-cofiniteness for a family of W-algebras. J. Phys. A 39, 951–966 (2006). arXiv:math-ph/0508015 ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Adamović, D., Milas, A.: On the triplet vertex algebra \(\cal{W}\left(p\right)\). Adv. Math.217, 2664–2699 (2008). arXiv:0707.1857 [math.QA]
  18. 18.
    Abe, T.: A \(\mathbb{Z}_{2}\)-orbifold model of the symplectic fermionic vertex operator superalgebra. Math. Z.255, 755–792 (2007). arXiv:math/0503472 [math.QA]MathSciNetCrossRefGoogle Scholar
  19. 19.
    Adamović, D., Milas, A.: The \(N=1\) triplet vertex operator superalgebras. Comm. Math. Phys.288, 225–270 (2009). arXiv:0712.0379 [math.QA]
  20. 20.
    Creutzig, T., Kanade, S., Linshaw, A.: Simple current extensions beyond semi-simplicity. arXiv:1511.08754 [math.QA]
  21. 21.
    Creutzig, T., Kanade, S., Linshaw, A., Ridout, D.: Schur-Weyl duality for Heisenberg cosets. Transform. Groups (to appear) arXiv:1611.00305 [math.QA]
  22. 22.
    Zamolodchikov, A., Fateev, V.: Nonlocal (parafermion) currents in two-dimensional conformal quantum field theory and self-dual critical points in \(Z_N\)-symmetric statistical systems. Sov. Phys. JETP 62, 215–225 (1985)Google Scholar
  23. 23.
    Gepner, D.: New conformal field theories associated with Lie algebras and their partition functions. Nucl. Phys. B 290, 10–24 (1987)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Lepowsky, J., Wilson, R.: A new family of algebras underlying the Rogers–Ramanujan identities and generalizations. Proc. Natl. Acad. Sci. USA 78, 7245–7248 (1981)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Dong, C., Lepowsky, J.: Generalized Vertex Algebras and Relative Vertex Operators. Progress in Mathematics, vol. 112. Birkhäuser, Boston (1993)CrossRefGoogle Scholar
  26. 26.
    Dong, C., Wang, Q.: On \(C_2\)-cofiniteness of parafermion vertex operator algebras. J. Algebra 328, 420–431 (2011). arXiv:1005.1709 [math.QA]
  27. 27.
    Arakawa, T., Lam, C., Yamada, H.: Zhu’s algebra, \(C_2\)-algebra and \(C_2\)-cofiniteness of parafermion vertex operator algebras. Adv. Math. 264, 261–295, (2014). arXiv:1207.3909 [math.QA]
  28. 28.
    Dong, C., Ren, L.: Representations of the parafermion vertex operator algebras. arXiv:1411.6085 [math.QA]
  29. 29.
    Creutzig, T., Kanade, S., McRae R.: Tensor categories for vertex operator superalgebra extensions. arXiv:1705.05017 [math.QA]
  30. 30.
    Adamović, D., Milas, A.: Vertex operator algebras associated to modular invariant representations of \(A_1^{\left(1\right)}\). Math. Res. Lett. 2, 563–575 (1995). arXiv:q-alg/9509025
  31. 31.
    Creutzig, T., Ridout, D.: Modular data and Verlinde formulae for fractional level WZW models I. Nucl. Phys. B 865, 83–114 (2012). arXiv:1205.6513 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Creutzig, T., Ridout, D.: Modular data and Verlinde formulae for fractional level WZW models II. Nucl. Phys. B 875, 423–458 (2013). arXiv:1306.4388 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    Ridout, D., Wood, S.: Relaxed singular vectors, Jack symmetric functions and fractional level \(\widehat{\mathfrak{sl}} \left(2 \right)\) models. Nucl. Phys. B 894, 621–664 (2015). arXiv:1501.07318 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    Creutzig, T.: W-algebras for Argyres-Douglas theories. Eur. J. Math. 3, 659 (2017). arXiv:1701.05926 [hep-th]MathSciNetCrossRefGoogle Scholar
  35. 35.
    Camino, J., Ramallo, A., Sanchez de Santos, J.: Graded parafermions. Nucl. Phys. B530, 715–741 (1998). arXiv:hep-th/9805160 ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    Fortin, J.-F., Mathieu, P., Warnaar, S.: Characters of graded parafermion conformal field theory. Adv. Theor. Math. Phys. 11, 945–989 (2007). arXiv:hep-th/0602248 MathSciNetCrossRefGoogle Scholar
  37. 37.
    Ridout, D., Snadden, J., Wood, S.: An admissible level \(\widehat{\mathfrak{osp}}\left( {1}\vert {2}\right)\)-model: Modular transformations and the Verlinde formula. Lett. Math. Phys. (to appear)Google Scholar
  38. 38.
    Creutzig, T., Kanade, S., Liu, T., Ridout, D.: Admissible level \(\widehat{\mathfrak{osp}} \left({1} \vert 2 \right)\)-models (in preparation)Google Scholar
  39. 39.
    Creutzig, T., Frohlich, J., Kanade, S.: Representation theory of \(L_k(\mathfrak{osp}(1|2))\) from vertex tensor categories and Jacobi forms. Proc. Am. Math. Soc. (to appear). arXiv:1706.00242 [math.QA]
  40. 40.
    Ennes, I., Ramallo, A., Sanchez de Santos, J.: \( OSP \left({1}\vert {2} \right)\) conformal field theory. In: Trends in Theoretical Physics, volume 419 of AIP Conference Proceedings, pp. 138–150, La Plata (1997). arXiv:hep-th/9708094
  41. 41.
    Huang, Y-Z., Lepowsky, J., Zhang, L.: Logarithmic tensor product theory I–VIII. arXiv:1012.4193 [math.QA], arXiv:1012.4196 [math.QA], arXiv:1012.4197 [math.QA], arXiv:1012.4198 [math.QA], arXiv:1012.4199 [math.QA], arXiv:1012.4202 [math.QA], arXiv:1110.1929 [math.QA], arXiv:1110.1931 [math.QA]
  42. 42.
    Creutzig, T., Huang, Y-Z., Yang, J.: Braided tensor categories of admissible modules for affine Lie algebras. arXiv:1709.01865 [math.QA]
  43. 43.
    Li, H.: On abelian coset generalized vertex algebras. Commun. Contemp. Math. 3, 287–340 (2001). arXiv:math/0008062 [math.QA]MathSciNetCrossRefGoogle Scholar
  44. 44.
    Kirillov Jr, A., Ostrik, V.: On a \(q\)-analogue of the McKay correspondence and the ADE classification of \(\mathfrak{sl}_2\) conformal field theories. Adv. Math. 171, 183–227 (2002). arXiv:math/0101219 [math.QA]
  45. 45.
    Huang, Y-Z., Kirillov Jr, A., Lepowsky, J.: Braided tensor categories and extensions of vertex operator algebras. Commun. Math. Phys. 337, 1143–1159 (2015). arXiv:1406.3420 [math.QA]ADSMathSciNetCrossRefGoogle Scholar
  46. 46.
    Ridout, D., Wood, S.: The Verlinde formula in logarithmic CFT. J. Phys. Conf. Ser. 597, 012065 (2015). arXiv:1409.0670 [hep-th]CrossRefGoogle Scholar
  47. 47.
    Creutzig, T., Ridout, D.: Logarithmic conformal field theory: beyond an introduction. J. Phys. A 46, 494006 (2013). arXiv:1303.0847 [hep-th]MathSciNetCrossRefGoogle Scholar
  48. 48.
    Adamović, D.: A construction of admissible \(A_1^{\left(1\right)}\)-modules of level \(-\frac{4}{3}\). J. Pure Appl. Algebra 196, 119–134 (2005). arXiv:math/0401023 [math.QA]
  49. 49.
    Ridout, D.: \(\widehat{\mathfrak{sl}} \left(2 \right)_{-1/2}\) and the triplet model. Nucl. Phys. B 835, 314–342 (2010). arXiv:1001.3960 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  50. 50.
    Miyamoto, M.: Modular invariance of vertex operator algebras satisfying \(C_2\)-cofiniteness. Duke Math. J. 122, 51–91 (2004). arXiv:math/0209101 [math.QA]MathSciNetCrossRefGoogle Scholar
  51. 51.
    Dong, C., Li, H., Mason, G.: Vertex operator algebras associated to admissible representations of \(\widehat{sl}_2\). Comm. Math. Phys. 184, 65–93 (1997). arXiv:q-alg/9509026
  52. 52.
    Kac, V., Wakimoto, M.: Modular invariant representations of infinite-dimensional Lie algebras and superalgebras. Proc. Natl. Acad. Sci. USA 85, 4956–4960 (1988)ADSMathSciNetCrossRefGoogle Scholar
  53. 53.
    Ridout, D.: \(\widehat{\mathfrak{sl}} \left(2 \right)_{-1/2}\): a case study. Nucl. Phys. B 814, 485–521 (2009). arXiv:0810.3532 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  54. 54.
    Kawasetsu, K., Ridout, D.: Relaxed highest-weight modules I: rank \(1\) cases. arXiv:1803.01989 [math.RT]
  55. 55.
    Witten, E.: Non-abelian bosonization in two dimensions. Commun. Math. Phys. 92, 455–472 (1984)ADSCrossRefGoogle Scholar
  56. 56.
    Nagatomo, K., Tsuchiya, A.: The triplet vertex operator algebra \(W \left( p \right)\) and the restricted quantum group \(\overline{U}_q \left( sl_2 \right)\) at \(q = e^{\frac{\pi i}{p}}\). Adv. Stud. Pure Math. 61, 1–49 (2011). arXiv:0902.4607 [math.QA]
  57. 57.
    Tsuchiya, A., Wood, S.: The tensor structure on the representation category of the \(\cal{W}_p\) triplet algebra. J. Phys. A 46, 445203 (2013). arXiv:1201.0419 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  58. 58.
    Adamović, D.: Realizations of simple affine vertex algebras and their modules: the cases \(\widehat{sl(2)}\) and \(\widehat{osp(1,2)}\). arXiv:1711.11342 [math.QA]
  59. 59.
    Gaberdiel, M.: Fusion in conformal field theory as the tensor product of the symmetry algebra. Int. J. Mod. Phys. A 9, 4619–4636 (1994). arXiv:hep-th/9307183 ADSMathSciNetCrossRefGoogle Scholar
  60. 60.
    Creutzig, T., Quella, T., Schomerus, V.: Branes in the \(GL \left({1} \vert 1 \right)\) WZNW model. Nucl. Phys. B 792, 257–283 (2008). arXiv:0708.0853 [hep-th]ADSCrossRefGoogle Scholar
  61. 61.
    Creutzig, T., Ridout, D., Wood, S.: Coset constructions of logarithmic \(\left( 1,p \right)\)-models. Lett. Math. Phys.104, 553–583 (2014). arXiv:1305.2665 [math.QA]
  62. 62.
    Creutzig, T., Milas, A.: False theta functions and the Verlinde formula. Adv. Math. 262, 520–545 (2014). arXiv:1309.6037 [math.QA]MathSciNetCrossRefGoogle Scholar
  63. 63.
    Ridout, D., Wood, S.: Modular transformations and Verlinde formulae for logarithmic \(\left( p_{+}, p_{-} \right)\)-models. Nucl. Phys. B 880, 175–202 (2014). arXiv:1310.6479 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  64. 64.
    Adamović, D., Milas, A.: Some applications and constructions of intertwining operators in logarithmic conformal field theory. Contemp. Math. 695, 15–27 (2017). arXiv:1605.05561 [math.QA]
  65. 65.
    Creutzig, T., Milas, A., Rupert, M.: Logarithmic link invariants of \({\overline{U}}_{q}^{H}(\mathfrak{sl}_{2})\) and asymptotic dimensions of singlet vertex algebras (2017).  https://doi.org/10.1016/j.jpaa.2017.12.004; arXiv:1605.05634 [math.QA]MathSciNetCrossRefGoogle Scholar
  66. 66.
    Feigin, B., Semikhatov, A.: \(\cal{W}_n^{(2)}\) algebras. Nucl. Phys. B698, 409–449 (2004). arXiv:math/0401164 [math.QA]ADSMathSciNetCrossRefGoogle Scholar
  67. 67.
    Arakawa, T., Creutzig, T., Kawasetsu, K., Linshaw, A.: Orbifolds and cosets of minimal \(\cal{W}\)-algebras. Comm. Math. Phys. 355, 339–372 (2017). arXiv:1610.09348 [math.RT]
  68. 68.
    Pearce, P., Rasmussen, J., Tartaglia, E.: Logarithmic superconformal minimal models. J. Stat. Mech. 2014, P05001 (2014). arXiv:1312.6763 [hep-th]MathSciNetCrossRefGoogle Scholar
  69. 69.
    Canagasabey, M., Rasmussen, J., Ridout, D.: Fusion rules for the \(N=1\) superconformal logarithmic minimal models I: The Neveu–Schwarz sector. J. Phys. A 48, 415402 (2015). arXiv:1504.03155 [hep-th]MathSciNetCrossRefGoogle Scholar
  70. 70.
    Canagasabey, M., Ridout, D.: Fusion rules for the logarithmic \(N=1\) superconformal minimal models II: including the Ramond sector. Nucl. Phys. B 905, 132–187 (2016). arXiv:1512.05837 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  71. 71.
    Adamović, D., Milas, A.: The \(N=1\) triplet vertex operator superalgebras: twisted sector. SIGMA 4, 087 (2008). arXiv:0806.3560 [math.QA]
  72. 72.
    Inami, T., Matsuo, Y., Yamanaka, I.: Extended conformal algebras with \(N=1\) supersymmetry. Phys. Lett. B 215, 701–705 (1988)ADSMathSciNetCrossRefGoogle Scholar
  73. 73.
    Bouwknegt, P., Schoutens, K.: \(\cal{W}\) symmetry in conformal field theory. 223, 183–276 (1993). arXiv:hep-th/9210010
  74. 74.
    Arakawa, T., Creutzig, T., Linshaw, A.: Cosets of Bershadsky-Polyakov algebras and rational \(\cal{W}\)-algebras of type A. Selecta Math. New Ser. 23, 2369–2395 (2017). arXiv:1511.09143 [math.RT]
  75. 75.
    Li, H.: Symmetric invariant bilinear forms on vertex operator algebras. J. Pure Appl. Algebra 96, 279–297 (1994)MathSciNetCrossRefGoogle Scholar
  76. 76.
    Miyamoto, M.: \(C_2\)-cofiniteness of cyclic-orbifold models. Commun. Math. Phys. 335, 1279–1286 (2015). arXiv:1306.5031 [math.QA]

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Authors and Affiliations

  1. 1.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada
  2. 2.School of Mathematics and StatisticsThe University of MelbourneParkvilleAustralia

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