Selecta Mathematica

, 1:699 | Cite as

A theory of tensor products for module categories for a vertex operator algebra, I

  • Y. -Z. Huang
  • J. Lepowsky
Article

Abstract

This is the first part in a series of papers developing a tensor product theory for modules for a vertex operator algebra. The goal of this theory is to construct a “vertex tensor category” structure on the category of modules for a suitable vertex operator algebra. The notion of vertex tensor category is essentially a “complex analogue” of the notion of symmetric tensor category, and in fact a vertex tensor category produces a braided tensor category in a natural way. The theory applies in particular to many familiar “rational” vertex operator algebras, including those associated with WZNW models, minimal models and the moonshine module. In this paper (Part I), we introduce the notions ofP(z)- andQ(z)-tensor product, whereP(z) andQ(z) are two special elements of the moduli space of spheres with punctures and local coordinates, and we present the fundamental properties and constructions ofQ(z)-tensor products.

References

  1. [BPZ]
    A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov.Infinite conformal symmetries in two-dimensional quantum field theory. Nucl. Phys.,B241 (1984), 333–380.CrossRefMathSciNetGoogle Scholar
  2. [B1]
    R. E. Borcherds.Vertex algebras, Kac-Moody algebras, and the Monster. Proc. Natl. Acad. Sci. USA83 (1986), 3068–3071.CrossRefMathSciNetGoogle Scholar
  3. [B2]
    R. E. Borcherds.Monstrous moonshine and monstrous Lie superalgebras. Invent. Math.109 (1992), 405–444.MATHCrossRefMathSciNetGoogle Scholar
  4. [CN]
    J. H. Conway and S. P. Norton.Monstrous moonshine. Bull. London Math. Soc.,11 (1979), 308–339.MATHCrossRefMathSciNetGoogle Scholar
  5. [Do]
    C. Dong.Representations of the moonshine module vertex operator algebra. in: Mathematical Aspects of Conformal and Topological Field Theories and Quantum Groups, Proc. Joint Summer Research Conference, Mount Holyoke, 1992, ed. P. Sally, M. Flato, J. Lepowsky, N. Reshetikhin and G. Zuckerman, Contemporary Math., Vol. 175, Amer. Math. Soc., Providence, 1994, 27–36.Google Scholar
  6. [DL]
    C. Dong and J. Lepowsky.Generalized Vertex Algebras and Relative Vertex Operators. Progress in Math., Vol.112 Birkhäuser, Boston, 1993.Google Scholar
  7. [Dr1]
    V. Drinfeld.On quasi-cocommutative Hopf algebras. Algebra and Analysis1 (1989), 30–46.MathSciNetGoogle Scholar
  8. [Dr2]
    V. Drinfeld.On quasitriangular quasi-Hopf algebras and a certain group closely related to Gal. Algebra and Analysis2 (1990), 149–181.MathSciNetGoogle Scholar
  9. [F]
    M. Finkelberg.Fusion categories. Ph.D. thesis, Harvard University, 1993.Google Scholar
  10. [FHL]
    I. B. Frenkel, Y.-Z. Huang and J. Lepowsky.On axiomatic approaches to vertex operator algebras and modules. preprint, 1989; Memoirs Amer. Math. Soc.104, 1993.Google Scholar
  11. [FLM1]
    I. B. Frenkel, J. Lepowsky and A. Meurman.A natural representation of the Fischer-Griess Monster with the modular function J as character. Proc. Natl. Acad. Sci. USA81 (1984), 3256–3260.MATHCrossRefMathSciNetGoogle Scholar
  12. [FLM2]
    I. B. Frenkel, J. Lepowsky and A. Meurman.Vertex Operator Algebras and the Monster, Pure and Appl. Math., Vol. 134, Academic Press, Boston, 1988.Google Scholar
  13. [FZ]
    I. B. Frenkel and Y. Zhu.Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J.66 (1992), 123–168.MATHCrossRefMathSciNetGoogle Scholar
  14. [FS]
    D. Friedan and S. Shenker.The analytic geometry of two-dimensional conformal field theory. Nucl. Phys.B281 (1987), 509–545.CrossRefMathSciNetGoogle Scholar
  15. [G]
    M. Gaberdiel.Fusion in conformal field theory as the tensor product of the symmetry algebra. preprint, 1993.Google Scholar
  16. [H1]
    Y.-Z. Huang.On the geometric interpretation of vertex operator algebras. Ph.D. thesis, Rutgers University, 1990.Google Scholar
  17. [H2]
    Y.-Z. Huang.Geometric interpretation of vertex operator algebras. Proc. Natl. Acad. Sci. USA88 (1991), 9964–9968.MATHCrossRefGoogle Scholar
  18. [H3]
    Y.-Z. Huang.Applications of the geometric interpretation of vertex operator algebras. in: Proc. 20th International Conference on Differential Geometric Methods in Theoretical Physics, New York, 1991, ed. S. Catto and A. Rocha, World Scientific, Singapore, 1992, Vol. 1, 333–343.Google Scholar
  19. [H4]
    Y.-Z. Huang.Vertex operator algebras and conformal field theory. Intl. J. Mod. Phys.A7 (1992), 2109–2151.CrossRefGoogle Scholar
  20. [H5]
    Y.-Z. Huang.Two-dimensional conformal geometry and vertex operator algebras. Birkhäuser, Boston, to appear.Google Scholar
  21. [H6]
    Y.-Z. Huang.A theory of tensor products for module categories for a vertex operator algebra, IV. J. Pure Appl. Alg.100 (1995), 173–216.MATHCrossRefGoogle Scholar
  22. [H7]
    Y.-Z. Huang.A nonmeromorphic extension of the moonshine module vertex operator algebra. in: Moonshine, the Monster and Related Topics, Proc. Joint Summer Research Conference, Mount Holyoke, 1994, ed. C. Dong and G. Mason, Contemporary Math., Amer. Math. Soc., Providence, 1995, 123–148.Google Scholar
  23. [H8]
    Y.-Z. Huang.Virasoro vertex operator algebras, the (nonmeromorphic) operator product expansion and the tensor product theory. J. Alg., to appear.Google Scholar
  24. [HL1]
    Y.-Z. Huang and J. Lepowsky.Toward a theory of tensor products for representations of a vertex operator algebra. in: Proc. 20th International Conference on Differential Geometric Methods in Theoretical Physics, New York, 1991, ed. S. Catto and A. Rocha, World Scientific, Singapore, 1992, Vol. 1, 344–354.Google Scholar
  25. [HL2]
    Y.-Z. Huang and J. Lepowsky.Vertex operator algebras and operads. in: The Gelfand Mathematical Seminars, 1990–1992, ed. L. Corwin, I. Gelfand and J. Lepowsky, Birkhäuser, Boston, 1993, 145–161.Google Scholar
  26. [HL3]
    Y.-Z. Huang and J. Lepowsky.Operadic formulation of the notion of vertex operator algebra. in: Mathematical Aspects of Conformal and Topological Field Theories and Quantum Groups, Proc. Joint Summer Research Conference, Mount Holyoke, 1992, ed. P. Sally, M. Flato, J. Lepowsky, N. Reshetikhin and G. Zuckerman, Contemporary Math., Vol. 175, Amer. Math. Soc., Providence, 1994, 131–148.Google Scholar
  27. [HL4]
    Y.-Z. Huang and J. Lepowsky.A theory of tensor products for module categories for a vertex operator algebra, II. Selecta Mathematics, New Series,1 (1995), 757–786.MATHCrossRefMathSciNetGoogle Scholar
  28. [HL5]
    Y.-Z. Huang and J. Lepowsky.Tensor products of modules for a vertex operator algebra and vertex tensor categories. in: Lie Theory and Geometry, in Honor of Bertram Kostant, ed. J.-L. Brylinski, R. Brylinski, V. Guillemin and V. Kac, Progress in Math., Vol. 123, Birkhäuser, Boston, 1994, 349–383.Google Scholar
  29. [HL6]
    Y.-Z. Huang and J. Lepowsky.A theory of tensor products for module categories for a vertex operator algebra, III. J. Pure Appl. Alg.,100 (1995), 141–171.MATHCrossRefMathSciNetGoogle Scholar
  30. [J]
    V. F. R. Jones.Hecke algebra representations of braid groups and link polynomials. Ann. Math.,126 (1987), 335–388.CrossRefGoogle Scholar
  31. [JS]
    F. A. Joyal and R. Street.Braided monoidal categories. Macquarie Mathematics Reports, Macquarie University, Australia, 1986.Google Scholar
  32. [KL1]
    D. Kazhdan and G. Lusztig.Affine Lie algebras and quantum groups. International Mathematics Research Notices (in Duke Math. J.)2 (1991), 21–29.CrossRefMathSciNetGoogle Scholar
  33. [KL2]
    D. Kazhdan and G. Lusztig.Tensor structures arising from affine Lie algebras, I. J. Amer. Math. Soc.,6 (1993), 905–947.MATHCrossRefMathSciNetGoogle Scholar
  34. [KL3]
    D. Kazhdan and G. Lusztig.Tensor structures arising from affine Lie algebras, II. J. Amer. Math. Soc.,6 (1993), 949–1011.CrossRefMathSciNetGoogle Scholar
  35. [KL4]
    D. Kazhdan and G. Lusztig.Tensor structures arising from affine Lie algebras, III. J. Amer. Math. Soc.,7 (1994), 335–381.MATHCrossRefMathSciNetGoogle Scholar
  36. [KL5]
    D. Kazhdan and G. Lusztig.Tensor structures arising from affine Lie algebras, IV. J. Amer. Math. Soc.,7 (1994), 383–453.MATHCrossRefMathSciNetGoogle Scholar
  37. [KZ]
    V. G. Knizhnik and A. B. Zamolodchikov.Current algebra and Wess-Zumino models in two dimensions. Nucl. Phys.,B247 (1984), 83–103.CrossRefMathSciNetGoogle Scholar
  38. [K1]
    T. Kohno.Linear representations of braid groups and classical Yang-Baxter equations. in: Braids, Santa Cruz, 1986, Contemporary Math.,78 (1988), 339–363.MathSciNetGoogle Scholar
  39. [K2]
    T. Kohno.Monodromy representations of braid groups and Yang-Baxter equations. Ann. Inst. Fourier37 (1987), 139–160.MATHMathSciNetGoogle Scholar
  40. [Le]
    J. Lepowsky.Remarks on vertex operator algebras and moonshine. in: Proc. 20th International Conference on Differential Geometric Methods in Theoretical Physics, New York, 1991, ed. S. Catto and A. Rocha, World Scientific, Singapore, 1992, Vol. 1, 362–370.Google Scholar
  41. [Li]
    H. Li.Representation theory and the tensor product theory for vertex operator algebras. Ph.D. Thesis, Rutgers University, 1994.Google Scholar
  42. [M]
    J. P. May.The geometry of iterated loop spaces. Lecture Notes in Mathematics, No. 271, Springer-Verlag, 1972.Google Scholar
  43. [MS]
    G. Moore and N. Seiberg.Classical and quantum conformal field theory. Comm. Math. Phys.,123 (1989), 177–254.MATHCrossRefMathSciNetGoogle Scholar
  44. [RT]
    N. Reshetikhin and V. G. Turaev.Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math.,103 (1991), 547–597.MATHCrossRefMathSciNetGoogle Scholar
  45. [SV]
    V. V. Schechtman and A. N. Varchenko.Arrangements of hyperplanes and Lie algebra homology. Invent. Math.,106 (1991), 139–194.MATHCrossRefMathSciNetGoogle Scholar
  46. [Se]
    G. Segal.The definition of conformal field theory. preprint, 1988.Google Scholar
  47. [St1]
    J.D. Stasheff.Homotopy associativity of H-spaces, I. Trans. Amer. Math. Soc.,108 (1963), 275–292.CrossRefMathSciNetGoogle Scholar
  48. [St2]
    J.D. Stasheff.Homotopy associativity of H-spaces, II. Trans. Amer. Math. Soc.108 (1963), 293–312.CrossRefMathSciNetGoogle Scholar
  49. [TK]
    A. Tsuchiya and Y. Kanie.Vertex operators in conformal field theory on1 and monodromy representations of braid group. in: Conformal Field Theory and Solvable Lattice Models, Advanced Studies in Pure Math., Vol. 16, Kinokuniya Company Ltd., Tokyo, 1988, 297–372.Google Scholar
  50. [TUY]
    A. Tsuchiya, K. Ueno and Y. Yamada.Conformal field theory on universal family of stable curves with gauge symmetries. in: Advanced Studies in Pure Math., Vol. 19, Kinokuniya Company Ltd., Tokyo, 1989, 459–565.Google Scholar
  51. [Va]
    A. N. Varchenko.Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups. Advanced Series in Mathematical Physics, Vol. 21, World Scientific, Singapore, to appear.Google Scholar
  52. [Ve]
    E. Verlinde.Fusion rules and modular transformations in 2D conformal field theory. Nucl. Phys.,B300 (1988), 360–376.CrossRefMathSciNetGoogle Scholar
  53. [W1]
    E. Witten.Non-abelian bosonization in two dimensions. Comm. Math. Phys.92 (1984), 455–472.MATHCrossRefMathSciNetGoogle Scholar
  54. [W2]
    E. Witten.Quantum field theory and the Jones polynomial. Comm. Math. Phys.,121 (1989), 351–399.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag 1995

Authors and Affiliations

  • Y. -Z. Huang
    • 1
  • J. Lepowsky
    • 2
  1. 1.Department of MathematicsUniversity of PennsylvaniaPhiladelphiaUSA
  2. 2.Department of MathematicsRutgers UniversityNew BrunswickUSA

Personalised recommendations