Advertisement

Realizations of Simple Affine Vertex Algebras and Their Modules: The Cases \({\widehat{sl(2)}}\) and \({\widehat{osp(1,2)}}\)

  • Dražen AdamovićEmail author
Article
  • 9 Downloads

Abstract

We study the embeddings of the simple admissible affine vertex algebras \({V_k(sl(2))}\) and \({V_k({osp}(1,2))}\), \({k \notin {\mathbb Z}_{\ge 0}}\), into the tensor product of rational Virasoro and N = 1 Neveu–Schwarz vertex algebra with lattice vertex algebras. By using these realizations we construct a family of weight, logarithmic, and Whittaker \({\widehat{sl(2)}}\) and \({\widehat{osp(1,2)}}\)-modules. As an application, we construct all irreducible degenerate Whittaker modules for \({V_k(sl(2))}\).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The author is partially supported by the Croatian Science Foundation under the Project 2634 and by the QuantiXLie Centre of Excellence, a Project cofinanced by the Croatian Government and European Union through the European Regional Development Fund—the Competitiveness and Cohesion Operational Programme (Grant KK.01.1.1.01.0004).We would like to thank T. Creutzig, A.Milas, G. Radobolja, D. Ridout, and S. Wood for valuable discussions. Finally, we thank the referee for a careful reading of the paper and helpful comments.

References

  1. 1.
    Adamović D.: Rationality of Neveu–Schwarz vertex operator superalgebras. Int. Math. Res. Not. IMRN 17, 865–874 (1997)CrossRefzbMATHGoogle Scholar
  2. 2.
    Adamović D.: Representations of the N =  2 superconformal vertex algebra. Int. Math. Res. Not. IMRN 2, 61–79 (1999)CrossRefzbMATHGoogle Scholar
  3. 3.
    Adamović D.: Regularity of certain vertex operator superalgebras. Contemp. Math. 343, 1–16 (2004)CrossRefzbMATHGoogle Scholar
  4. 4.
    Adamović D.: A construction of admissible A 1 (1)-modules of level − 4/3. J. Pure Appl. Algebra 196, 119–134 (2005)CrossRefzbMATHGoogle Scholar
  5. 5.
    Adamović D.: Lie superalgebras and irreducibility of certain A 1(1)-modules at the critical level. Commun. Math. Phys. 270, 141–161 (2007)ADSCrossRefzbMATHGoogle Scholar
  6. 6.
    Adamović D.: A realization of certain modules for the N = 4 superconformal algebra and the affine Lie algebra A 2 (1). Transform. Groups 21(2), 299–327 (2016)CrossRefzbMATHGoogle Scholar
  7. 7.
    Adamović D.: A note on the affine vertex algebra associated to \({{\mathfrak{g}\mathfrak{l} (1|1)}}\) at the critical level and its generalizations. Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 21(532), 75–87 (2017) arXiv:1706.09143 zbMATHGoogle Scholar
  8. 8.
    Adamović, D.: On Whittaker modules for \({\widehat{osp}(1, 2) }\). In preparationGoogle Scholar
  9. 9.
    Adamović D., Lü R., Zhao K.: Whittaker modules for the affine Lie algebra A 1 (1). Adv. Math. 289, 438–479 (2016)CrossRefzbMATHGoogle Scholar
  10. 10.
    Adamović D., Milas A.: Vertex operator algebras associated to the modular invariant representations for A 1(1). Math. Res. Lett. 2, 563–575 (1995)CrossRefzbMATHGoogle Scholar
  11. 11.
    Adamović D., Milas A.: The N = 1 triplet vertex operator superalgebras. Commun. Math. Phys. 288, 225–270 (2009)ADSCrossRefzbMATHGoogle Scholar
  12. 12.
    Adamović D., Milas A.: The N = 1 triplet vertex operator superalgebras: twisted sector, Symmetry. Integr. Geom. Methods Appl. (SIGMA) 087, 24 (2008)zbMATHGoogle Scholar
  13. 13.
    Adamović D., Milas A.: Lattice construction of logarithmic modules for certain vertex algebras. Sel. Math. (N.S.) 15(4), 535–561 (2009)CrossRefzbMATHGoogle Scholar
  14. 14.
    Adamović D., Milas A.: On W-algebras associated to (2, p) minimal models and their representations. Int. Math. Res. Not. 2010 20, 3896–3934 (2010)zbMATHGoogle Scholar
  15. 15.
    Adamović D., Milas A.: An explicit realization of logarithmic modules for the vertex operator algebra W p,p'. J. Math. Phys. 073511, 16 (2012)zbMATHGoogle Scholar
  16. 16.
    Adamović, D., Milas, A.: Vertex operator superalgebras and LCFT. J. Phys. A Math. Theoret. 46–49, 494005. Special Issue on Logarithmic conformal field theory (2013)Google Scholar
  17. 17.
    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 CrossRefzbMATHGoogle Scholar
  18. 18.
    Adamović D., Kac V.G., Möseneder Frajria P., Papi P., Perše O.: Conformal embeddings of affine vertex algebras in minimal W-algebras I: structural results. J. Algebra 500, 117–152 (2018)  https://doi.org/10.1016/j.jalgebra.2016.12.005 CrossRefzbMATHGoogle Scholar
  19. 19.
    Adamović D., Kac V.G., Möseneder Frajria P., Papi P., Perše O.: Conformal embeddings of affine vertex algebras in minimal W-algebras II: decompositions. Jpn. Jo. Math. 12(2), 261–315 (2017)CrossRefzbMATHGoogle Scholar
  20. 20.
    Adamović, D., Pedić, V.: On fusion rules and intertwining operators for the Weyl vertex algebra (to appear)Google Scholar
  21. 21.
    Adamović, D., Radobolja, G.: Self-dual and logarithmic representations of the twisted Heisenberg–Virasoro algebra at level zero. Commun. Contemp. Math. (to appear). arXiv:1703.00531
  22. 22.
    Arakawa T.: W-algebras at the critical level. Contemp. Math 565, 1–14 (2012)CrossRefzbMATHGoogle Scholar
  23. 23.
    Arakawa T.: Two-sided BGG resolutions of admissible representations. Represent. Theory 18, 183–222 (2014)CrossRefzbMATHGoogle Scholar
  24. 24.
    Arakawa, T.: Rationality of admissible affine vertex algebras in the category \({{\mathcal{O}}}\). Duke Math. J 165(1), 67–93 (2016) arXiv:1207.4857
  25. 25.
    Arakawa T., Futorny V., Ramirez L.E.: Weight representations of admissible affine vertex algebras. Commun. Math. Phys. 353(3), 1151–1178 (2017)ADSCrossRefzbMATHGoogle Scholar
  26. 26.
    Auger J., Creutzig T., Ridout D.: Modularity of logarithmic parafermion vertex algebras. Lett. Math. Phys. 108(12), 2543–2587 (2018) arXiv:1704.05168 ADSCrossRefzbMATHGoogle Scholar
  27. 27.
    Berman S., Dong C., Tan S.: Representations of a class of lattice type vertex algebras. J. Pure Appl. Algebra 176, 27–47 (2002)CrossRefzbMATHGoogle Scholar
  28. 28.
    Creutzig T., Milas A.: False theta functions and the Verlinde formula. Adv. Math. 262, 520–545 (2014)CrossRefzbMATHGoogle Scholar
  29. 29.
    Creutzig, T., Gannon, T.: Logarithmic conformal field theory, log-modular tensor categories and modular forms. J. Phys. A 50(40):404004, 37 pp. arXiv:1605.04630 (2017)
  30. 30.
    Creutzig T., Huang Y.Z., Yang J.: Braided tensor categories of admissible modules for affine Lie algebras. Commun. Math. Phys. 362(3), 827–854 (2018)ADSCrossRefzbMATHGoogle Scholar
  31. 31.
    Creutzig T., Linshaw A.: Cosets of affine vertex algebras inside larger structures. J. Algebra Vol. 517(1), 396–438 (2019) arXiv:1407.8512v4 CrossRefzbMATHGoogle 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)ADSCrossRefzbMATHGoogle Scholar
  33. 33.
    Dong C., Lepowsky J.: Generalized Vertex Algebras and Relative Vertex Operators. Birkhäuser, Boston (1993)CrossRefzbMATHGoogle Scholar
  34. 34.
    Frenkel E.: Lectures on Wakimoto modules, opers and the center at the critical level. Adv. Math. 195, 297–404 (2005)CrossRefzbMATHGoogle Scholar
  35. 35.
    Eicher, C.: Relaxed highest weight modules from \({\mathcal{D}}\)-modules on the Kashiwara flag scheme. arXiv:1607.06342
  36. 36.
    Ennes I. P., Ramallo A. V., Sanchezde Santos J.M.: On the free field realization of the osp(1,2) current algebra. Phys. Lett. B 389, 485–493 (1996) arXiv:hep-th/9606180 ADSCrossRefGoogle Scholar
  37. 37.
    Feingold A.J., Frenkel I.B.: Classical affine algebras. Adv. Math. 56, 117–172 (1985)CrossRefzbMATHGoogle Scholar
  38. 38.
    Fjelstad, J., Fuchs, J., Hwang, S., Semikhatov, AM., Tipunin, IY.: Logarithmic conformal field theories via logarithmic deformations. Nucl. Phys. B 633 (2002)Google Scholar
  39. 39.
    Feigin B.L., Semikhatov A.M., Tipunin I.Yu.: Equivalence between chain categories of representations of affine sl(2) and N =  2 superconformal algebras. J. Math. Phys. 39, 3865–3905 (1998)ADSCrossRefzbMATHGoogle Scholar
  40. 40.
    Blondeau-Fournier, O., Mathieu, P., Ridout, D., Wood, S.: Superconformal minimal models and admissible Jack polynomials. Adv. Math. 314, 71–123 (2016)Google Scholar
  41. 41.
    Frenkel I.B., Zhu Y.: Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66, 12–168 (1992)CrossRefzbMATHGoogle Scholar
  42. 42.
    Gaberdiel M.: Fusion rules and logarithmic representations of a WZW model at fractional level. Nucl. Phys. B 618, 407–436 (2001)ADSCrossRefzbMATHGoogle Scholar
  43. 43.
    Huang, Y. -Z., Lepowsky, J., Zhang, L.: Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, Parts I–VIII, arXiv:1012.4193 arXiv:1012.4196 arXiv:1012.4197 arXiv:1012.4198 arXiv:1012.4199 arXiv:1110.1929 arXiv:1110.1931; Part I published in Conformal Field Theories and Tensor Categories, pp. 169–248. Springer, Berlin (2014)
  44. 44.
    Kac V.G., Wakimoto M.: Quantum reduction and representation theory of superconformal algebras. Adv. Math. 185, 400–458 (2004)CrossRefzbMATHGoogle Scholar
  45. 45.
    Kawasetsu, K., Ridout, D.: Relaxed highest-weight modules I: rank 1 cases. Commun. Math. Phys. (to appear). arXiv:1803.01989
  46. 46.
    Iohara, K, Koga, Y: Representation theory of the Virasoro algebra. Springer Monographs in Mathematics, Springer, London (2011)Google Scholar
  47. 47.
    Lashkevich, M.Y.: Superconformal 2D minimal models and an unusual coset constructions. Modern Phys. Lett. A 851–860, arXiv:hep-th/9301093 (1993)
  48. 48.
    Li H.: Symmetric invariant bilinear forms on vertex operator algebras. J. Pure Appl. Algebra 96, 279–297 (1994)CrossRefzbMATHGoogle Scholar
  49. 49.
    Li H.: The phyisical superselection principle in vertex operator algebra theory. J. Algebra 196, 436–457 (1997)CrossRefzbMATHGoogle Scholar
  50. 50.
    Lam, C., Yamauchi, H.: 3-dimensional Griess algebras and Miyamoto involutions. arXiv:1604.04470
  51. 51.
    Lesage F., Mathieu P., Rasmussen J., Saleur H.: Logarithmic lift of the su(2)−1/2 model. Nucl. Phys. B 686, 313–346 (2004) arXiv:hep-th/0311039 ADSCrossRefzbMATHGoogle Scholar
  52. 52.
    Milas, A.: Weak modules and logarithmic intertwining operators for vertex operator algebras. In: Berman, S., Fendley, P., Huang, Y.-Z., Misra, K., Parshall, B. (eds.) Recent Developments in Infinite-Dimensional Lie Algebras and Conformal Field Theory. Contemp. Math., Vol. 297, pp. 201–225. American Mathematical Society, Providence (2002)Google Scholar
  53. 53.
    Milas A.: Characters, supercharacters and Weber modular functions. J. Reine Angew. Math. (Crelle’s J.) 608, 35–64 (2007)zbMATHGoogle Scholar
  54. 54.
    Miyamoto M.: Modular invariance of vertex operator algebra satisfying C 2-cofiniteness. Duke Math. J. 122, 51–91 (2004)CrossRefzbMATHGoogle Scholar
  55. 55.
    Ridout D.: sl(2) −1/2 and the Triplet Model. Nucl. Phys. B 835, 314–342 (2010)ADSCrossRefzbMATHGoogle Scholar
  56. 56.
    Ridout D.: Fusion in fractional level sl(2)-theories with k = − 1/2. Nucl. Phys. B 848, 216–250 (2011)ADSCrossRefzbMATHGoogle Scholar
  57. 57.
    Ridout D., Snadden J., Wood S.: An admissible level \({\widehat{osp}(1,2)}\)-model: modular transformations and the Verlinde formula. Lett. Math. Phys. 108, 11, 2363–2423 (2018)zbMATHGoogle Scholar
  58. 58.
    Ridout D., Wood S.: From Jack polynomials to minimal model spectra. J. Phys. A 48, 045201 (2015)ADSCrossRefzbMATHGoogle Scholar
  59. 59.
    Ridout D., Wood S.: Relaxed singular vectors, Jack symmetric functions and fractional level sl(2). Models Nucl. Phys. B 894, 621–664 (2015)ADSCrossRefzbMATHGoogle Scholar
  60. 60.
    Sato, R.: Modular invariant representations of the N = 2 superconformal algebra. Int. Math. Res. Not. (to appear). arXiv:1706.04882
  61. 61.
    Semikhatov A.: The MFF singular vectors in topological conformal theories. Modern Phys. Lett. A 09(20), 1867–1896 (1994)ADSCrossRefzbMATHGoogle Scholar
  62. 62.
    Semikhatov, A.: Inverting the Hamiltonian reduction in string theory. arXiv:hep-th/9410109
  63. 63.
    Tsuchiya A., Kanie Y.: Fock space representations of the Virasoro algebra—intertwining operators. Publ. Res. Inst. Math. Sci 22, 259–327 (1986)CrossRefzbMATHGoogle Scholar
  64. 64.
    Wakimoto M.: Fock representations of affine Lie algebra A 1 (1). Commun. Math. Phys. 104, 605–609 (1986)ADSCrossRefzbMATHGoogle Scholar
  65. 65.
    Wang W.: Rationality of Virasoro vertex operator algebras. Duke Math. J./Int. Math. Res. Not. 71(1), 97–211 (1993)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceUniversity of ZagrebZagrebCroatia

Personalised recommendations