Full field algebras. (English) Zbl 1153.17012
Authors’ 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 nondegeneracy 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.
Reviewer: Martin Schlichenmaier (Luxembourg)
MSC:
| 17B69 | Vertex operators; vertex operator algebras and related structures |
| 81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
References:
| [1] | Abe, T.; Buhl, G.; Dong, C., Rationality, regularity and C_2-cofiniteness, Trans. Amer. Math. Soc., 356, 8, 3391-3402 (2004) · Zbl 1070.17011 · doi:10.1090/S0002-9947-03-03413-5 |
| [2] | Bakalov, B., Kirillov, A., Jr.: Lectures on tensor categories and modular functors, University Lecture Series, Vol. 21, Providence, RI: Amer. Math. Soc., 2001 · Zbl 0965.18002 |
| [3] | Belavin, A. A.; Polyakov, A. M.; Zamolodchikov, A. B., Infinite conformal symmetries in two-dimensional quantum field theory, Nucl. Phys., B241, 333-380 (1984) · Zbl 0661.17013 · doi:10.1016/0550-3213(84)90052-X |
| [4] | Birman, J. S., Braids, links, and mapping class groups Annals of Mathematics Studies, Vol 82 (1974), Princeton, NJ: Princeton University Press, Princeton, NJ |
| [5] | Borcherds, R. E., Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Natl. Acad. Sci. USA, 83, 3068-3071 (1986) · Zbl 0613.17012 · doi:10.1073/pnas.83.10.3068 |
| [6] | 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) · Zbl 0813.17019 |
| [7] | Felder, G.; Fröhlich, J.; Fuchs, J.; Schweigert, C., Correlation functions and boundary conditions in rational conformal field theory and three-dimensional topology, Compositio. Math., 131, 189-237 (2002) · Zbl 1002.81045 · doi:10.1023/A:1014903315415 |
| [8] | Frenkel, I. B.; Huang, Y.-Z.; Lepowsky, J., On axiomatic approaches to vertex operator algebras and modules, Memoirs Amer. Math. Soc., 104, 593 (1993) · Zbl 0789.17022 |
| [9] | Frenkel, I. B.; Lepowsky, J.; Meurman, A., Vertex operator algebras and the Monster Pure and Appl Math, Vol. 134 (1988), NewYork: Academic Press, NewYork · Zbl 0674.17001 |
| [10] | Fjelstad, J.; Fuchs, J.; Runkel, I.; Schweigert, C., TFT construction of RCFT correlators V: Proof of modular invariance and factorisation, Theory and Appl. of Categories, 16, 342-433 (2006) · Zbl 1151.81038 |
| [11] | Fuchs, J.; Runkel, I.; Schweigert, C., Conformal correlation functions, Frobenius algebras and triangulations, Nucl. Phys., B624, 452-468 (2002) · Zbl 0985.81113 · doi:10.1016/S0550-3213(01)00638-1 |
| [12] | Fuchs, J.; Runkel, I.; Schweigert, C., TFT construction of RCFT correlators I: Partition functions, Nucl. Phys., B646, 353-497 (2002) · Zbl 0999.81079 · doi:10.1016/S0550-3213(02)00744-7 |
| [13] | Fuchs, J.; Runkel, I.; Schweigert, C., TFT construction of RCFT correlators, IV: Structure constants and correlation functions, Nucl. Phys., B715, 539-638 (2005) · Zbl 1151.81384 · doi:10.1016/j.nuclphysb.2005.03.018 |
| [14] | Huang, Y.-Z., A theory of tensor products for module categories for a vertex operator algebra, IV, J, Pure. Appl. Alg., 100, 173-216 (1995) · Zbl 0841.17015 · doi:10.1016/0022-4049(95)00050-7 |
| [15] | Huang, Y.-Z., Virasoro vertex operator algebras, (nonmeromorphic) operator product expansion and the tensor product theory, J. Alg., 182, 201-234 (1996) · Zbl 0862.17022 · doi:10.1006/jabr.1996.0168 |
| [16] | 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. · Zbl 0871.17022 |
| [17] | Huang, Y.-Z., Genus-zero modular functors and intertwining operator algebras, Internat, J Math., 9, 845-863 (1998) · Zbl 0918.17022 · doi:10.1142/S0129167X9800035X |
| [18] | Huang, Y.-Z., Generalized rationality and a “Jacobi identity” for intertwining operator algebras, Selecta Math. (N.S.), 6, 225-267 (2000) · Zbl 1013.17026 · doi:10.1007/PL00001389 |
| [19] | Huang, Y.-Z., Vertex operator algebras, the Verlinde conjecture and modular tensor categories, Proc. Natl. Acad. Sci. USA, 102, 5352-5356 (2005) · Zbl 1112.17029 · doi:10.1073/pnas.0409901102 |
| [20] | Huang, Y.-Z., Differential equations and intertwining operators, Comm. Contemp. Math., 7, 375-400 (2005) · Zbl 1070.17012 · doi:10.1142/S0219199705001799 |
| [21] | Huang, Y.-Z., Differential equations, duality and modular invariance, Comm. Contemp. Math., 7, 649-706 (2005) · Zbl 1124.11022 · doi:10.1142/S021919970500191X |
| [22] | Huang, Y.-Z.: Vertex operator algebras and the Verlinde conjecture. Comm. Contemp. Math., to appear; http://arxiv.org/list/math.QA/0406291, 2004 |
| [23] | Huang, Y.-Z.: Rigidity and modularity of vertex tensor categories. Comm. Contemp. Math., to appear; http://arxiv.org/list/math.QA/0502533, 2005 |
| [24] | Kapustin, A.; Orlov, D., Vertex algebras, mirror symmetry, and D-branes: The case of complex tori, Commun. Math. Phys., 233, 79-136 (2003) · Zbl 1051.17017 · doi:10.1007/s00220-002-0755-7 |
| [25] | Kong, L.: A mathematical study of open-closed conformal field theories. Ph.D. thesis, Rutgers University, 2005 |
| [26] | Li, H. S., Some finiteness properties of regular vertex operator algebras, J. Algebra,, 212, 495-514 (1999) · Zbl 0953.17017 · doi:10.1006/jabr.1998.7654 |
| [27] | Moore, G.; Seiberg, N., Polynomial equations for rational conformal field theories, Phys. Lett., B212, 451-460 (1988) |
| [28] | Moore, G.; Seiberg, N., Classical and quantum conformal field theory, Commun. Math. Phys., 123, 177-254 (1989) · Zbl 0694.53074 · doi:10.1007/BF01238857 |
| [29] | Moore, G.; Seiberg, N., Naturality in conformal field theory,, Nucl. Phys., B313, 16-40 (1989) · doi:10.1016/0550-3213(89)90511-7 |
| [30] | 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 · Zbl 1009.81063 |
| [31] | Rosellen, M., OPE-algebras and their modules, Int. Math. Res. Not., 2005, 433-447 (2005) · Zbl 1072.81027 · doi:10.1155/IMRN.2005.433 |
| [32] | 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 · Zbl 0657.53060 |
| [33] | 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 |
| [34] | 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 |
| [35] | Tsukada, H., String path integral realization of vertex operator algebras, Mem. Amer. Math. Soc., 91, 444, vi+138 (1991) · Zbl 0749.17029 |
| [36] | Turaev, V. G., Quantum invariants of knots and 3-manifolds. de Gruyter Studies in Math., Vol. 18 (1994), Berlin: Walter de Gruyter, Berlin · Zbl 0812.57003 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
