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 V L and V R, \({V^{L}\otimes V^{R}}\) 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 V L and V R. 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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Abe T., Buhl G. and Dong C. (2004). Rationality, regularity and C 2-cofiniteness. Trans. Amer. Math. Soc. 356(8): 3391–3402
Bakalov, B., Kirillov, A., Jr.: Lectures on tensor categories and modular functors, University Lecture Series, Vol. 21, Providence, RI: Amer. Math. Soc., 2001
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
Birman J.S. (1974). Braids, links and mapping class groups Annals of Mathematics Studies, Vol 82. Princeton University Press, Princeton, NJ
Borcherds R.E. (1986). Vertex algebras, Kac-Moody algebras and the Monster. Proc. Natl. Acad. Sci. USA 83: 3068–3071
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)
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
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
Frenkel I.B., Lepowsky J. and Meurman A. (1988). Vertex operator algebras and the Monster Pure and Appl Math, Vol. 134. Academic Press, NewYork
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
Fuchs J., Runkel I. and Schweigert C. (2002). Conformal correlation functions, Frobenius algebras and triangulations. Nucl. Phys. B624: 452–468
Fuchs J., Runkel I. and Schweigert C. (2002). TFT construction of RCFT correlators I: Partition functions. Nucl. Phys. B646: 353–497
Fuchs J., Runkel I. and Schweigert C. (2005). TFT construction of RCFT correlators, IV: Structure constants and correlation functions. Nucl. Phys. B715: 539–638
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
Huang Y.-Z. (1996). Virasoro vertex operator algebras, (nonmeromorphic) operator product expansion and the tensor product theory. J. Alg. 182: 201–234
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.
Huang Y.-Z. (1998). Genus-zero modular functors and intertwining operator algebras. Internat, J Math. 9: 845–863
Huang Y.-Z. (2000). Generalized rationality and a “Jacobi identity” for intertwining operator algebras. Selecta Math. (N.S.) 6: 225–267
Huang Y.-Z. (2005). Vertex operator algebras, the Verlinde conjecture and modular tensor categories. Proc. Natl. Acad. Sci. USA 102: 5352–5356
Huang Y.-Z. (2005). Differential equations and intertwining operators. Comm. Contemp. Math. 7: 375–400
Huang Y.-Z. (2005). Differential equations, duality and modular invariance. Comm. Contemp. Math. 7: 649–706
Huang, Y.-Z.: Vertex operator algebras and the Verlinde conjecture. Comm. Contemp. Math., to appear; http://arxiv.org/list/math.QA/0406291, 2004
Huang, Y.-Z.: Rigidity and modularity of vertex tensor categories. Comm. Contemp. Math., to appear; http://arxiv.org/list/math.QA/0502533, 2005
Kapustin A. and Orlov D. (2003). Vertex algebras, mirror symmetry and D-branes: The case of complex tori. Commun. Math. Phys. 233: 79–136
Kong, L.: A mathematical study of open-closed conformal field theories. Ph.D. thesis, Rutgers University, 2005
Li H.S. (1999). Some finiteness properties of regular vertex operator algebras. J. Algebra, 212: 495–514
Moore G. and Seiberg N. (1988). Polynomial equations for rational conformal field theories. Phys. Lett. B212: 451–460
Moore G. and Seiberg N. (1989). Classical and quantum conformal field theory. Commun. Math. Phys. 123: 177–254
Moore G. and Seiberg N. (1989). Naturality in conformal field theory,. Nucl. Phys. B313: 16–40
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, 2002
Rosellen M. (2005). OPE-algebras and their modules. Int. Math. Res. Not. 2005: 433–447
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, 1988
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, 1989
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, 2004
Tsukada H. (1991). String path integral realization of vertex operator algebras. Mem. Amer. Math. Soc. 91(444): vi+138
Turaev V.G. (1994). Quantum invariants of knots and 3-manifolds. de Gruyter Studies in Math., Vol. 18. Walter de Gruyter, Berlin
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Communicated by L. Takhtajan
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Huang, YZ., Kong, L. Full Field Algebras. Commun. Math. Phys. 272, 345–396 (2007). https://doi.org/10.1007/s00220-007-0224-4
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DOI: https://doi.org/10.1007/s00220-007-0224-4