Boson-Fermion Correspondence of Type B and Twisted Vertex Algebras

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 36)


The boson-fermion correspondence of type A is an isomorphism between two super vertex algebras (and so has singularities in the operator product expansions only at z = w). The boson-fermion correspondence of type B plays similarly important role in many areas, including representation theory, integrable systems, random matrix theory and random processes. But the vertex operators describing it have singularities in their operator product expansions at both z = w and \(z = -w\), and thus need a more general notion than that of a super vertex algebra. In this paper we present such a notion: the concept of a twisted vertex algebra, which generalizes the concept of super vertex algebra. The two sides of the correspondence of type B constitute two examples of twisted vertex algebras. The boson-fermion correspondence of type B is thus an isomorphism between two twisted vertex algebras.


Vertex Operator Operator Product Expansion Random Matrix Theory Vertex Algebra High Weight Representation 
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.


  1. 1.
    Anguelova, Iana I.: Twisted vertex algebras, bicharacter construction and boson-fermion correspondences. (arxiv:1206.4026[math-ph]), preprintGoogle Scholar
  2. 2.
    Bakalov, B., Kac, V.: Twisted modules over lattice vertex algebras. In: Lie Theory and Its Applications in Physics V, pp. 3–26. World Scietific Publishing, River Edge, NJ (2004)Google Scholar
  3. 3.
    Bakalov, B., Kac, V.: Generalized vertex algebras. In: Doebner, H.-D., Dobrev, V.K. (eds.) Proceedings of the 6-th International Workshop “Lie Theory and Its Applications in Physics” (LT-6), Varna, Bulgaria, pp. 3–25. Heron Press (2006)Google Scholar
  4. 4.
    Borcherds, Richard E.: Vertex algebras, Kac-Moody algebras, and the Monster. Proc. Natl. Acad. Sci. USA 83(10), 3068–3071 (1986)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Transformation groups for soliton equations. IV. A new hierarchy of soliton equations of KP-type. Phys. D 4(3), 343–365 (1981/82)Google Scholar
  6. 6.
    Dong, C., Lepowsky, J.: Generalized Vertex Algebras and Relative Vertex Operators, vol. 112 of Progress in Mathematics. Birkhäuser Boston, Boston, MA (1993)Google Scholar
  7. 7.
    Frenkel, E., Reshetikhin, N.: Towards deformed chiral algebras. In: Proceedings of the Quantum Group Symposium at the XXIth International Colloquium on Group Theoretical Methods in Physics, Goslar 1996, pp. 6023–+ (1997)Google Scholar
  8. 8.
    Frenkel, I.B.: Two constructions of affine Lie algebra representations and boson-fermion correspondence in quantum field theory. J. Funct. Anal. 44(3), 259–327 (1981)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Frenkel, I.B., Lepowsky, J., Meurman, A.: Vertex Operator Algebras and the Monster, vol. 134 of Pure and Applied Mathematics. Academic Press, Boston, MA (1988)Google Scholar
  10. 10.
    Frenkel, I.B., Huang, Y-Z., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Mem. Am. Math. Soc. 104(494), viii+64 (1993)Google Scholar
  11. 11.
    Kac, V.: Vertex algebras for beginners, vol. 10 of University Lecture Series, 2nd edn. American Mathematical Society, Providence, RI (1998)Google Scholar
  12. 12.
    Kac, V., Raina, A.K.: Bombay lectures on highest weight representations of infinite-dimensional Lie algebras, vol. 2 of Advanced Series in Mathematical Physics. World Scientific Publishing, Teaneck, NJ (1987)Google Scholar
  13. 13.
    Li, H.S.: Symmetric invariant bilinear forms on vertex operator algebras. J. Pure Appl. Algebra 96(3), 279–297 (1994)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Lepowsky, J., Li, H.: Introduction to vertex operator algebras and their representations, vol. 227 of Progress in Mathematics. Birkhäuser Boston, Boston, MA (2004)Google Scholar
  15. 15.
    Littlewood, D.E.: A University Algebra: An Introduction to Classic and Modern Algebra. Dover, New York (1970)MATHGoogle Scholar
  16. 16.
    Miwa, T., Jimbo, M., Date, E.: Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  17. 17.
    van de Leur, J.W., Orlov, A. Yu.: Random turn walk on a half line with creation of particles at the origin. Phys. Lett. A 373(31), 2675–2681 (2009)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    You, Y.: Polynomial solutions of the BKP hierarchy and projective representations of symmetric groups. In: Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988), vol. 7 of Adv. Ser. Math. Phys., pp. 449–464. World Scientific Publishing, Teaneck, NJ (1989)Google Scholar

Copyright information

© Springer Japan 2013

Authors and Affiliations

  1. 1.Math DepartmentCollege of CharlestonCharlestonUSA

Personalised recommendations