Communications in Mathematical Physics

, Volume 272, Issue 2, pp 345–396 | Cite as

Full Field Algebras

Article

Abstract

We introduce a notion of full field algebra which is essentially an algebraic formulation of the notion of genus-zero full conformal field theory. For any vertex operator algebras VL and VR, \({V^{L}\otimes V^{R}}\) is naturally a full field algebra and we introduce a notion of full field algebra over \({V^{L}\otimes V^{R}}\) . We study the structure of full field algebras over \({V^{L}\otimes V^{R}}\) using modules and intertwining operators for VL and VR. For a simple vertex operator algebra V satisfying certain natural finiteness and reductivity conditions needed for the Verlinde conjecture to hold, we construct a bilinear form on the space of intertwining operators for V and prove the nondegeneracy and other basic properties of this form. The proof of the nondegenracy of the bilinear form depends not only on the theory of intertwining operator algebras but also on the modular invariance for intertwining operator algebras through the use of the results obtained in the proof of the Verlinde conjecture by the first author. Using this nondegenerate bilinear form, we construct a full field algebra over \({V\otimes V}\) and an invariant bilinear form on this algebra.

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References

  1. ABD.
    Abe T., Buhl G. and Dong C. (2004). Rationality, regularity and C 2-cofiniteness. Trans. Amer. Math. Soc. 356(8): 3391–3402 MATHCrossRefMathSciNetGoogle Scholar
  2. BK.
    Bakalov, B., Kirillov, A., Jr.: Lectures on tensor categories and modular functors, University Lecture Series, Vol. 21, Providence, RI: Amer. Math. Soc., 2001Google Scholar
  3. BPZ.
    Belavin A.A., Polyakov A.M. and Zamolodchikov A.B. (1984). Infinite conformal symmetries in two-dimensional quantum field theory. Nucl. Phys. B241: 333–380 CrossRefADSMathSciNetGoogle Scholar
  4. Bi.
    Birman J.S. (1974). Braids, links and mapping class groups Annals of Mathematics Studies, Vol 82. Princeton University Press, Princeton, NJ Google Scholar
  5. Bo.
    Borcherds R.E. (1986). Vertex algebras, Kac-Moody algebras and the Monster. Proc. Natl. Acad. Sci. USA 83: 3068–3071 CrossRefADSMathSciNetGoogle Scholar
  6. DMZ.
    Dong, C., Mason, G., Zhu, Y.: Discrete series of the Virasoro algebra and the moonshine module. In: Algebraic Groups and Their Generalizations: Quantum and infinite-dimensional Methods, Proc. 1991 Amer. Math. Soc. Summer Research Institute, ed. by W. J. Haboush, B. J. Parshall, Proc. Symp. Pure Math. 56, Part 2, Providence, RI: Amer. Math. Soc., 1994, pp. 295–316 (1991)Google Scholar
  7. FFFS.
    Felder G., Fröhlich J., Fuchs J. and Schweigert C. (2002). Correlation functions and boundary conditions in rational conformal field theory and three-dimensional topology. Compositio. Math. 131: 189–237 MATHCrossRefMathSciNetGoogle Scholar
  8. FHL.
    Frenkel I.B., Huang Y.-Z. and Lepowsky J. (1993). On axiomatic approaches to vertex operator algebras and modules. Memoirs Amer. Math. Soc. 104: 593 MathSciNetGoogle Scholar
  9. FLM.
    Frenkel I.B., Lepowsky J. and Meurman A. (1988). Vertex operator algebras and the Monster Pure and Appl Math, Vol. 134. Academic Press, NewYork Google Scholar
  10. FFRS.
    Fjelstad J., Fuchs J., Runkel I. and Schweigert C. (2006). TFT construction of RCFT correlators V: Proof of modular invariance and factorisation. Theory and Appl. of Categories 16: 342–433 MATHMathSciNetGoogle Scholar
  11. FRS1.
    Fuchs J., Runkel I. and Schweigert C. (2002). Conformal correlation functions, Frobenius algebras and triangulations. Nucl. Phys. B624: 452–468 CrossRefADSMathSciNetGoogle Scholar
  12. FRS2.
    Fuchs J., Runkel I. and Schweigert C. (2002). TFT construction of RCFT correlators I: Partition functions. Nucl. Phys. B646: 353–497 CrossRefADSMathSciNetGoogle Scholar
  13. FRS3.
    Fuchs J., Runkel I. and Schweigert C. (2005). TFT construction of RCFT correlators, IV: Structure constants and correlation functions. Nucl. Phys. B715: 539–638 CrossRefADSMathSciNetGoogle Scholar
  14. H1.
    Huang Y.-Z. (1995). A theory of tensor products for module categories for a vertex operator algebra, IV, J. Pure. Appl. Alg. 100: 173–216 MATHCrossRefGoogle Scholar
  15. H2.
    Huang Y.-Z. (1996). Virasoro vertex operator algebras, (nonmeromorphic) operator product expansion and the tensor product theory. J. Alg. 182: 201–234 MATHCrossRefGoogle Scholar
  16. H3.
    Huang, Y.-Z.: Intertwining operator algebras, genus-zero modular functors and genus-zero conformal field theories. In: Operads: Proceedings of Renaissance Conferences, ed. J.-L. Loday, J. Stasheff, A. A. Voronov, Contemporary Math., Vol. 202, Providence, RI: Amer. Math. Soc., pp. 335–355, 1997.Google Scholar
  17. H4.
    Huang Y.-Z. (1998). Genus-zero modular functors and intertwining operator algebras. Internat, J Math. 9: 845–863 MATHCrossRefADSMathSciNetGoogle Scholar
  18. H5.
    Huang Y.-Z. (2000). Generalized rationality and a “Jacobi identity” for intertwining operator algebras. Selecta Math. (N.S.) 6: 225–267 MATHCrossRefMathSciNetGoogle Scholar
  19. H6.
    Huang Y.-Z. (2005). Vertex operator algebras, the Verlinde conjecture and modular tensor categories. Proc. Natl. Acad. Sci. USA 102: 5352–5356 CrossRefADSMathSciNetGoogle Scholar
  20. H7.
    Huang Y.-Z. (2005). Differential equations and intertwining operators. Comm. Contemp. Math. 7: 375–400 MATHCrossRefGoogle Scholar
  21. H8.
    Huang Y.-Z. (2005). Differential equations, duality and modular invariance. Comm. Contemp. Math. 7: 649–706 MATHCrossRefGoogle Scholar
  22. H9.
    Huang, Y.-Z.: Vertex operator algebras and the Verlinde conjecture. Comm. Contemp. Math., to appear; http://arxiv.org/list/math.QA/0406291, 2004Google Scholar
  23. H10.
    Huang, Y.-Z.: Rigidity and modularity of vertex tensor categories. Comm. Contemp. Math., to appear; http://arxiv.org/list/math.QA/0502533, 2005Google Scholar
  24. KO.
    Kapustin A. and Orlov D. (2003). Vertex algebras, mirror symmetry and D-branes: The case of complex tori. Commun. Math. Phys. 233: 79–136 MATHCrossRefADSMathSciNetGoogle Scholar
  25. K.
    Kong, L.: A mathematical study of open-closed conformal field theories. Ph.D. thesis, Rutgers University, 2005Google Scholar
  26. L.
    Li H.S. (1999). Some finiteness properties of regular vertex operator algebras. J. Algebra, 212: 495–514 MATHCrossRefMathSciNetGoogle Scholar
  27. MS1.
    Moore G. and Seiberg N. (1988). Polynomial equations for rational conformal field theories. Phys. Lett. B212: 451–460 ADSMathSciNetGoogle Scholar
  28. MS2.
    Moore G. and Seiberg N. (1989). Classical and quantum conformal field theory. Commun. Math. Phys. 123: 177–254 MATHCrossRefADSMathSciNetGoogle Scholar
  29. MS3.
    Moore G. and Seiberg N. (1989). Naturality in conformal field theory,. Nucl. Phys. B313: 16–40 CrossRefADSMathSciNetGoogle Scholar
  30. R1.
    Rosellen, M.: OPE-algebras, Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 2002, Bonner Mathematische Schriften [Bonn Mathematical Publications], Vol. 352, Universität Bonn, Mathematisches Institut, Bonn, 2002Google Scholar
  31. R2.
    Rosellen M. (2005). OPE-algebras and their modules. Int. Math. Res. Not. 2005: 433–447 MATHCrossRefMathSciNetGoogle Scholar
  32. S1.
    Segal, G.: The definition of conformal field theory. In: Differential geometrical methods in theoretical physics (Como, 1987), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 250, Dordrecht: Kluwer Acad. Publ., pp. 165–171, 1988Google Scholar
  33. S2.
    Segal, G.: Two-dimensional conformal field theories and modular functors. In: Proceedings of the IXth International Congress on Mathematical Physics, Swansea, 1988, Bristol: Hilger, pp. 22–37, 1989Google Scholar
  34. S3.
    Segal, G.: The definition of conformal field theory, preprint, 1988; also in: Topology, geometry and quantum field theory, ed. U. Tillmann, London Math. Soc. Lect. Note Ser., Vol. 308, Cambridge: Cambridge University Press, pp. 421–457, 2004Google Scholar
  35. Ts.
    Tsukada H. (1991). String path integral realization of vertex operator algebras. Mem. Amer. Math. Soc. 91(444): vi+138 MathSciNetGoogle Scholar
  36. Tu.
    Turaev V.G. (1994). Quantum invariants of knots and 3-manifolds. de Gruyter Studies in Math., Vol. 18. Walter de Gruyter, BerlinGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of MathematicsRutgers UniversityPiscatawayUSA
  2. 2.Max Planck Institute for Mathematics in the SciencesLeipzigGermany
  3. 3.Institut Des Hautes Études ScientifiquesBures-sur-YvetteFrance

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