The Level One Zhu Algebra for the Heisenberg Vertex Operator Algebra

  • Katrina BarronEmail author
  • Nathan Vander Werf
  • Jinwei Yang
Part of the Springer INdAM Series book series (SINDAMS, volume 37)


The level one Zhu algebra for the Heisenberg vertex operator algebra is calculated, and implications for the use of Zhu algebras of higher level for vertex operator algebras are discussed. In particular, we show the Heisenberg vertex operator algebra gives an example of when the level one Zhu algebra, and in fact all its higher level Zhu algebras, do not provide new indecomposable non simple modules for the vertex operator algebra beyond those detected by the level zero Zhu algebra.


Vertex operator algebra Heisenberg algebra Conformal field theory 



The authors thank Darlayne Addabbo and Kiyo Nagatomo for reading a draft of this paper and making comments, suggestions, and corrections. The first author is the recipient of a Simons Foundation Collaboration Grant 282095, and greatly appreciates their support.


  1. 1.
    Adamović, D., Milas, A.: On the triplet vertex algebra \(W(p)\). Adv. Math. 217, 2664–2699 (2008)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Adamovic, D., Milas, A.: The structure of Zhu’s algebras for certain \(W\)-algebras. Adv. Math. 227, 2425–2456 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Barron, K., Vander Werf, N., Yang, J.: Higher level Zhu algebras and modules for vertex operator algebras. J. Pure Appl. Alg. 223, 3295–3317 (2019)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Barron, K., Vander Werf, N., Yang, J.: The level one Zhu algebra for the vertex operator algebra associated to the Virasoro algebra and implications. In: Krauel, M., Tuite, M., Yamskulna, G. (eds.) Vertex Operator Algebras, Number Theory, and Related Topics (To appear). Contemporary Mathematics. American Mathematical Society, ProvidenceGoogle Scholar
  5. 5.
    Dong, C., Li, H., Mason, G.: Vertex operator algebras and associative algebras. J. Alg. 206, 67–98 (1998)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Frenkel, I., Zhu, Y.-C.: Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66, 123–168 (1992)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Lepowsky, J., Li, H.: Introduction to Vertex Operator Algebras and Their Representations, Progress in Math, vol. 227. Birkhäuser, Boston (2003)Google Scholar
  8. 8.
    Milas, A.: Logarithmic intertwining operators and vertex operators. Commun. in Math. Phys. 277, 497–529 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Miyamoto, M.: Modular invariance of vertex operator algebras satisfying \(C_2\)-cofiniteness. Duke Math. J. 122, 51–91 (2004)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Nagatomo, K., Tsuchiya, A.: The triplet vertex operator algebra \(\cal{W}(p)\) and the restricted quantum group at root of unity. Exploring New Structures and Natural Constructions in Mathematical Physics. Advanced Studies in Pure Mathematics, vol. 61, p. 149. Mathematical Society of Japan, Tokyo (2011)Google Scholar
  11. 11.
    Tsuchiya, A., Wood, S.: The tensor structure on the representation category of the \(\cal{W}_p\) triplet algebra. J. Phys. A: Math. Theor. 46, 445203 (2013)CrossRefGoogle Scholar
  12. 12.
    Zhu, Y.-C.: Modular invariance of characters of vertex operator algebras. J. Amer. Math. Soc. 9, 237–302 (1996)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Katrina Barron
    • 1
    Email author
  • Nathan Vander Werf
    • 2
  • Jinwei Yang
    • 3
  1. 1.Department of MathematicsUniversity of Notre DameNotre DameUSA
  2. 2.Department of Mathematics and StatisticsUniversity of NebraskaKearneyUSA
  3. 3.Department of MathematicsUniversity of AlbertaEdmontonCanada

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