Communications in Mathematical Physics

, Volume 288, Issue 1, pp 225–270 | Cite as

The N = 1 Triplet Vertex Operator Superalgebras

Article
First Online:
Received:
Accepted:

Abstract

We introduce a newfamily of C 2-cofinite N = 1 vertex operator superalgebras \({\mathcal{SW}(m)}\), m ≥ 1, which are natural super analogs of the triplet vertex algebra family \({\mathcal{W}(p)}\), p ≥ 2, important in logarithmic conformal field theory. We classify irreducible \({\mathcal{SW}(m)}\)-modules and discuss logarithmic modules. We also compute bosonic and fermionic formulas of irreducible \({\mathcal{SW}(m)}\) characters. Finally, we contemplate possible connections between the category of \({\mathcal{SW}(m)}\)-modules and the category of modules for the quantum group \({U^{small}_q(sl_2)}\), \({q = e^{\frac{2 \pi i}{2m+1}}}\), by focusing primarily on properties of characters and the Zhu’s algebra \({A(\mathcal{SW}(m))}\). This paper is a continuation of our paper Adv. Math. 217, no.6, 2664–2699 (2008).

Keywords

Vertex Operator Singular Vector Irreducible Character Fusion Rule Vertex Operator Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ab.
    Abe T.: A \({\mathbb{Z}_{2}}\)-orbifold model of the symplectic fermionic vertex operator superalgebra. Math. Z. 255, 755–792 (2007)MATHCrossRefMathSciNetGoogle Scholar
  2. A1.
    Adamović D.: Rationality of Neveu-Schwarz vertex operator superalgebras. Internat. Math. Res. Notices 17, 865–874 (1997)CrossRefGoogle Scholar
  3. A2.
    Adamović D.: Representations of the vertex algebra \({\mathcal{W}_{1+\infty}}\) with a negative integer central charge. Comm. Algebra 29(7), 3153–3166 (2001)MATHCrossRefMathSciNetGoogle Scholar
  4. A3.
    Adamović D.: Classification of irreducible modules of certain subalgebras of free boson vertex algebra. J. Algebra 270, 115–132 (2003)MATHCrossRefMathSciNetGoogle Scholar
  5. AM1.
    Adamović D., Milas A.: Logarithmic intertwining operators and \({\mathcal{W}(2, 2p - 1)}\) -algebras. J. Math. Physics 48, 073503 (2007)CrossRefADSGoogle Scholar
  6. AM2.
    Adamović D., Milas A.: On the triplet vertex algebra \({\mathcal{W}(p)}\) . Adv. Math. 217, 2664–2699 (2008)MATHMathSciNetGoogle Scholar
  7. AM3.
    Adamović D., Milas A.: The N = 1 triplet vertex operator superalgebras: twisted sector. SIGMA 4(87), 1–24 (2008)Google Scholar
  8. Ar.
    Arakawa T.: Representation theory of superconformal algebras and the Kac-Roan-Wakimoto conjecture. Duke Math. J. 130, 435–478 (2005)MATHCrossRefMathSciNetGoogle Scholar
  9. BS.
    Bouwknegt P., Schoutens K.: \({\mathcal{W}}\) –symmetry in Conformal Field Theory. Phys. Rept. 223, 183–276 (1993)CrossRefADSMathSciNetGoogle Scholar
  10. CF.
    Carqueville N., Flohr M.: Nonmeromorphic operator product expansion and C 2-cofiniteness for a family of \({\mathcal{W}}\)-algebras. Phys. A: Math. Gen. 39, 951–966 (2006)MATHCrossRefADSMathSciNetGoogle Scholar
  11. D.
    Dong C.: Vertex algebras associated with even lattices. J. Algebra 160, 245–65 (1993)CrossRefGoogle Scholar
  12. DL.
    Dong C., Lepowsky J.: Generalized vertex algebras and relative vertex operators. Birkhäuser, Boston (1993)MATHGoogle Scholar
  13. DJ.
    Dong, C., Jiang, C.: Rationality of vertex operator algebras. http://arxiv.org/abs/math/0607679.v1[math.QA], 2006
  14. DK.
    De Sole A., Kac V.: Finite vs. affine W-algebras. Japanese J. Math. 1, 137–261 (2006)MATHCrossRefGoogle Scholar
  15. EFHHNV.
    Eholzer W., Flohr M., Honecker A., Hubel R., Nahm W., Vernhagen R.: Representations of \({\mathcal{W}}\) –algebras with two generators and new rational models. Nucl. Phys. B 383, 249–288 (1992)CrossRefADSGoogle Scholar
  16. FFR.
    Feingold, A.J., Frenkel, I.B., Ries, J.: Spinor Construction of Vertex Operator Algebras, Triality, and E 8(1). Cont. Math. 121, Providence, RI: Amer. Math. Soc., 1991Google Scholar
  17. FRW.
    Feingold, A.J., Ries, J., Weiner, M.: Spinor construction of the \({c=\frac{1}{2}}\) minimal model. In: Moonshine, The Monster and related topics, Contemporary Math. 193, Chongying Dong, Geoffrey Mason, eds., Providence, RI: Amer. Math. Soc., 1995, pp. 45–92Google Scholar
  18. FF.
    Feigin, B., Fuchs, D.B.: Representations of the Virasoro algebra. In: Representations of infinite-dimensional Lie groups and Lie algebras. New York: Gordon andd Breach, 1989Google Scholar
  19. FFT.
    Feigin, B., Feigin, E., Tipunin, I.: Fermionic formulas for (1, p) logarithmic model characters in \({\Phi_{2,1}}\) quasiparticle realisation. http://arxiv.org/abs/0704.2464.v4[hepth], 2007
  20. FGST1.
    Feigin B.L., Gaĭnutdinov A.M., Semikhatov A.M., Yu Tipunin I.: Modular group representations and fusion in logarithmic conformal field theories and in the quantum group center. Commun. Math. Phys. 265, 47–93 (2006)MATHCrossRefADSGoogle Scholar
  21. FGST2.
    Feigin B.L., Gaĭnutdinov A.M., Semikhatov A.M., Yu Tipunin I.: The Kazhdan-Lusztig correspondence for the representation category of the triplet W-algebra in logorithmic conformal field theories. Theor. Math. Phys. 148, 1210–1235 (2006)CrossRefGoogle Scholar
  22. FHL.
    Frenkel, I.B., Huang, Y.-Z., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104, 1993Google Scholar
  23. Fi.
    Finkelberg, M.: Am equivalence of fusion categories. Geom. Funct. Anal. 249–267 (1996)Google Scholar
  24. F1.
    Flohr M.: On modular invariant partition functions of conformal field theories with logarithmic operators. Internat. J. Modern Phys. A 11, 4147–4172 (1996)MATHCrossRefADSMathSciNetGoogle Scholar
  25. F2.
    Flohr M.: Bits and pieces in logarithmic conformal field theory. Proceedings of the School and Workshop on Logarithmic Conformal Field Theory and its Applications (Tehran 2001(Internat. J. Mod. Phys. A 18), 4497–4591 (2003)Google Scholar
  26. FGK.
    Flohr M., Grabow C., Koehn M.: Fermionic formulas for the characters of c p,1 logarithmic field theory. Nucl. phys. B 768, 263–276 (2007)MATHCrossRefADSMathSciNetGoogle Scholar
  27. FB.
    Frenkel, E., Ben-Zvi, D.: Vertex algebras and algebraic curves, Mathematical Surveys and Monographs; no. 88, Providence, RI: Amer. Math. Soc., 2001Google Scholar
  28. FLM.
    Frenkel I.B., Lepowsky J., Meurman A.: Vertex Operator Algebras and the Monster, Pure and Applied Math, Vol. 134. Academic Press, New York (1988)Google Scholar
  29. FZ.
    Frenkel I.B., Zhu Y.: Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66, 123–168 (1992)MATHCrossRefMathSciNetGoogle Scholar
  30. FHST.
    Fuchs J., Hwang S., Semikhatov A.M., Tipunin I.Yu.: Nonsemisimple fusion algebras and the Verlinde formula. Commun. Math. Phys. 247(3), 713–742 (2004)MATHCrossRefADSMathSciNetGoogle Scholar
  31. GK1.
    Gaberdiel M., Kausch H.G.: A rational logarithmic conformal field theory. Phys. Lett B 386, 131–137 (1996)CrossRefADSMathSciNetGoogle Scholar
  32. GK2.
    Gaberdiel M., Kausch H.G.: A local logarithmic conformal field theory. Nucl. Phys. B 538, 631–658 (1999)MATHCrossRefADSMathSciNetGoogle Scholar
  33. GL.
    Gao Y., Li H.: Generalized vertex algebras generated by parafermion-like vertex operators. J. Algebra. 240, 771–807 (2001)MATHCrossRefMathSciNetGoogle Scholar
  34. HK.
    Heluani, R., Kac, V.: SUSY lattice vertex algebras. http://arxiv.org/abs/0710.1587.v1[math.QA], 2007
  35. H.
    Honecker A.: Automorphisms of \({\mathcal{W}}\) algebras and extended rational conformal field theories. Nucl. Phys. B 400, 574–596 (1993)MATHCrossRefADSMathSciNetGoogle Scholar
  36. HLZ.
    Huang, Y-Z., Lepowsky, J., Zhang, L.: Logarithmic tensor product theory for generalized modules for a conformal vertex algebra. http://arxiv.org/abs/0710.2687.v3[math.QA], 2007
  37. HM.
    Huang Y.-Z., Milas A.: Intertwining operator superalgebras and vertex tensor categories for superconformal algebras, I. Commun. Contemp. Math. 4, 327–355 (2002)MATHCrossRefMathSciNetGoogle Scholar
  38. IK1.
    Iohara K., Koga Y.: Fusion algebras for N =1 superconformal field theories through coinvariants, II, N = 1 super-Virasoro-symmetry. J. Lie Theory 11, 305–337 (2001)MATHMathSciNetGoogle Scholar
  39. IK2.
    Iohara K., Koga Y.: Representation theory of the Neveu-Schwarz and Ramond algebras II: Fock modules. Ann. Inst. Fourier. Grenoble 53(6), 1755–1818 (2003)MATHMathSciNetGoogle Scholar
  40. K.
    Kac, V.G.: Vertex Algebras for Beginners, University Lecture Series, Second Edition. Providence, RI: Amer. Math. Soc., Vol. 10, 1998Google Scholar
  41. KWn.
    Kac V.G., Wang W.: Vertex operator superalgebras and their representations. Contemp. Math. 175, 161–191 (1994)MathSciNetGoogle Scholar
  42. KWak.
    Kac V., Wakimoto M.: Quantum Reduction and Representation Theory of Superconformal Algebras. Adv. Math. 185, 400–458 (2004)MATHCrossRefMathSciNetGoogle Scholar
  43. Ker.
    Kerler T.: Mapping class group actions on quantum doubles. Commun. Math. Phys. 168(2), 353–388 (1995)MATHCrossRefADSMathSciNetGoogle Scholar
  44. La.
    Lachowska A.: On the center of the small quantum group. J. Algebra 262, 313–331 (2003)MATHCrossRefMathSciNetGoogle Scholar
  45. LL.
    Lepowsky, J., Li, H.: Introduction to Vertex Operator Algebras and Their Representations, Progress in Mathematics Vol. 227. Boston: Birkhäuser, 2003Google Scholar
  46. Li.
    Li H.: Local systems of vertex operators, vertex superalgebras and modules. J. Pure Appl. Algebra 109, 143–195 (1996)MATHCrossRefMathSciNetGoogle Scholar
  47. MS.
    Mavromatos N., Szabo R.: The Neveu-Schwarz and Ramond algebras of logarithmic superconformal field theory. JHEP 0301, 041 (2003)CrossRefADSMathSciNetGoogle Scholar
  48. MR.
    Meurman A., Rocha-Caridi A.: Highest weight representations of the Neveu-Schwarz and Ramond algebras. Commun. Math. Phys. 107, 263–294 (1986)MATHCrossRefADSMathSciNetGoogle Scholar
  49. M1.
    Milas A.: Fusion rings for degenerate minimal models. J. Algebra 254(2), 300–335 (2002)MATHCrossRefMathSciNetGoogle Scholar
  50. M2.
    Milas, A.: Weak modules and logarithmic intertwining operators for vertex operator algebras. In: Recent developments in infinite-dimensional Lie algebras and conformal field theory (Charlottesville, VA, 2000), Contemp. Math. 297, Providence, RI: Amer. Math. Soc. 2002, pp. 201–225Google Scholar
  51. M3.
    Milas A.: Logarithmic intertwining operators and vertex operators. Commun. Math. Phys. 277, 497–529 (2008)MATHCrossRefADSMathSciNetGoogle Scholar
  52. M4.
    Milas A.: Characters, Supercharacters and Weber modular functions. Crelle’s Journal 608, 35–64 (2007)MATHMathSciNetGoogle Scholar
  53. MY.
    Mimachi K., Yamada Y.: Singular vectors of the Virasoro algebra in terms of Jack symmetric polynomials. Commun. Math. Phys. 174, 447–455 (1995)MATHCrossRefADSMathSciNetGoogle Scholar
  54. Miy.
    Miyamoto M.: Modular invariance of vertex operator algebras satisfying C 2-cofiniteness. Duke Math. J. 122, 51–91 (2004)MATHCrossRefMathSciNetGoogle Scholar
  55. Se.
    Semikhatov A.: Factorizable ribbon quantum groups in logarithmic conformal field theories. Theo. Math. Phys. 154, 433–453 (2008)MATHCrossRefMathSciNetGoogle Scholar
  56. Wa.
    Ole Warnaar S.: Proof of the Flohr-Grabow-Koehn conjecture for characters of logarithmic field theory. J. Phys. A: Math. Theor. 40, 12243–12254 (2007)MATHCrossRefGoogle Scholar
  57. W.
    Wang W.: Rationality of Virasoro vertex operator algebras. Internat. Math. Res. Notices 71(1), 197–211 (1993)CrossRefGoogle Scholar
  58. Z.
    Zhu Y.-C.: Modular invariance of characters of vertex operator algebras. J. Amer. Math. Soc. 9, 237–302 (1996)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ZagrebZagrebCroatia
  2. 2.Department of Mathematics and StatisticsUniversity at Albany (SUNY)AlbanyUSA

Personalised recommendations