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Transactions of the American Mathematical Society

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Modular invariance for conformal full field algebras


Authors: Yi-Zhi Huang and Liang Kong
Journal: Trans. Amer. Math. Soc. 362 (2010), 3027-3067
MSC (2000): Primary 17B69; Secondary 81T40
DOI: https://doi.org/10.1090/S0002-9947-09-04933-2
Published electronically: December 22, 2009
MathSciNet review: 2592945
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Abstract: Let $ V^{L}$ and $ V^{R}$ be simple vertex operator algebras satisfying certain natural uniqueness-of-vacuum, complete reducibility and cofiniteness conditions and let $ F$ be a conformal full field algebra over $ V^{L}\otimes V^{R}$. We prove that the $ q_{\tau}$- $ \overline{q_{\tau}}$-traces (natural traces involving $ q_{\tau}=e^{2\pi i\tau}$ and $ \overline{q_{\tau}}= \overline{e^{2\pi i\tau}}$) of geometrically modified genus-zero correlation functions for $ F$ are convergent in suitable regions and can be extended to doubly periodic functions with periods $ 1$ and $ \tau$. We obtain necessary and sufficient conditions for these functions to be modular invariant. In the case that $ V^{L}=V^{R}$ and $ F$ is one of those constructed by the authors in an earlier paper, we prove that all these functions are modular invariant.


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  • [BK] B. Bakalov and A. Kirillov, Jr., Lectures on tensor categories and modular functors, University Lecture Series, Vol. 21, Amer. Math. Soc., Providence, RI, 2001. MR 1797619 (2002d:18003)
  • [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. MR 757857 (86m:81097)
  • [B] R. Borcherds, Monstrous moonshine and monstrous Lie superalgebras, Invent. Math. 109 (1992), 405-444. MR 1172696 (94f:11030)
  • [FFFS] G. Felder, J. Fröhlich, J. Fuchs and C. Schweigert, Correlation functions and boundary conditions in rational conformal field theory and three-dimensional topology, Compositio Math. 131 (2002), 189-237. MR 1898435 (2003e:57051)
  • [FHL] I. Frenkel, Y.-Z. Huang and J. Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Memoirs Amer. Math. Soc. 104, 1993. MR 1142494 (94a:17007)
  • [FLM] I. Frenkel, J. Lepowsky, and A. Meurman, Vertex operator algebras and the Monster, Pure and Appl. Math., Vol. 134, Academic Press, New York, 1988. MR 996026 (90h:17026)
  • [FjFRS1] J. Fjelstad, J. Fuchs, I. Runkel and C. Schweigert, TFT construction of RCFT correlators V: Proof of modular invariance and factorisation, Theory and Applications of Categories 16 (2006), 342-433. MR 2259258 (2008a:81199)
  • [FjFRS2] I. Runkel, J. Fjelstad, J. Fuchs and C. Schweigert, Topological and conformal field theory as Frobenius algebras, in: Proceedings of the Streetfest (Canberra, July, 2005), Contemp. Math. Vol. 431, Amer. Math. Soc., Providence, RI, 2007. MR 2342831
  • [FrFRS] J. Fröhlich, J. Fuchs, I. Runkel and C. Schweigert, Correspondences of ribbon categories, Adv. Math. 199 (2006), 192-329. MR 2187404 (2007b:18007)
  • [FRS1] J. Fuchs, I. Runkel and C. Schweigert, Conformal correlation functions, Frobenius algebras and triangulations, Nucl. Phys. B624 (2002), 452-468. MR 1882479 (2002j:81215)
  • [FRS2] J. Fuchs, I. Runkel and C. Schweigert, TFT construction of RCFT correlators I: Partition functions, Nucl. Phys. B646 (2002), 353-497. MR 1940282 (2004c:81244)
  • [FRS3] J. Fuchs, I. Runkel and C. Schweigert, TFT construction of RCFT correlators II: Unoriented world sheets Nucl.Phys. B678 (2004), 511-637. MR 2026879 (2005a:81197)
  • [FRS4] J. Fuchs, I. Runkel and C. Schweigert, TFT construction of RCFT correlators III: Simple currents Nucl.Phys. B694 (2004), 277-353. MR 2076134 (2005e:81209)
  • [FRS5] J. Fuchs, I. Runkel and C. Schweigert, TFT construction of RCFT correlators, IV: Structure constants and correlation functions, Nucl. Phys. B715 (2005), 539-638. MR 2137114 (2006g:81161)
  • [H1] 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. MR 1344849 (98a:17050)
  • [H2] Y.-Z. Huang, Virasoro vertex operator algebras, (nonmeromorphic) operator product expansion and the tensor product theory, J. Alg. 182 (1996), 201-234. MR 1388864 (97h:17029)
  • [H3] Y.-Z. Huang, 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, and A. A. Voronov, Contemporary Math., Vol. 202, Amer. Math. Soc., Providence, 1997, 335-355. MR 1436926 (98a:17051)
  • [H4] Y.-Z. Huang, Genus-zero modular functors and intertwining operator algebras, Internat. J. Math. 9 (1998), 845-863. MR 1651049 (99i:17031)
  • [H5] Y.-Z. Huang, Generalized rationality and a ``Jacobi identity'' for intertwining operator algebras, Selecta Math. (N.S.), 6 (2000), 225-267. MR 1817614 (2002c:17041)
  • [H6] Y.-Z. Huang, Differential equations and intertwining operators, Comm. Contemp. Math. 7 (2005), 375-400. MR 2151865 (2006e:17037)
  • [H7] Y.-Z. Huang, Differential equations, duality and modular invariance, Comm. Contemp. Math. 7 (2005), 649-706. MR 2175093 (2007a:11057)
  • [H8] Y.-Z. Huang, Vertex operator algebras, the Verlinde conjecture and modular tensor categories, Proc. Natl. Acad. Sci. USA 102 (2005), 5352-5356. MR 2140309 (2006a:17026)
  • [H9] Y.-Z. Huang, Vertex operator algebras, fusion rules and modular transformations, in: Noncommutative Geometry and Representation Theory in Mathematical Physics, ed. J. Fuchs, J. Mickelsson, G. Rozenblioum and A. Stolin, Contemporary Math. Vol. 391, Amer. Math. Soc., Providence, 2005, 135-148. MR 2184018 (2006j:17027)
  • [H10] Y.-Z. Huang, Vertex operator algebras and the Verlinde conjecture, Commun. Contemp. Math. 10 (2008), 103-154. MR 2387861
  • [H11] Y.-Z. Huang, Rigidity and modularity of vertex tensor categories, Commun. Contemp. Math. 10 (2008), 871-911. MR 2468370
  • [HK] Y.-Z. Huang and L. Kong, Full field algebras, Comm. Math. Phys. 272 (2007), 345-396. MR 2300247 (2008g:81102)
  • [KO] A. Kapustin and D. Orlov, Vertex algebras, mirror symmetry, and D-branes: The case of complex tori, Comm. Math. Phys. 233 (2003), 79-136. MR 1957733 (2004b:14073)
  • [Kong1] L. Kong, A mathematical study of open-closed conformal field theories, Ph.D. thesis, Rutgers University, 2005.
  • [Kong2] L. Kong, Full field algebras, operads and tensor categories, Adv. Math. 213 (2007), 271-340. MR 2331245 (2008i:17036)
  • [Kont] M. Kontsevich, Rational conformal field theory and invariants of $ 3$-dimensional manifolds, preprint CPT-88/P.2189, University of Marseille, 1988.
  • [MS1] G. Moore and N. Seiberg, Polynomial equations for rational conformal field theories, Phys. Lett. B212 (1988), 451-460. MR 962600 (89m:81155)
  • [MS2] G. Moore and N. Seiberg, Classical and quantum conformal field theory, Comm. Math. Phys. 123 (1989), 177-254. MR 1002038 (90e:81216)
  • [MS3] G. Moore and N. Seiberg, Lectures on RCFT, in: Physics, geometry, and topology (Banff, AB, 1989), ed. H.C. Lee, NATO Adv. Sci. Inst. Ser. B Phys., 238, Plenum, New York, 1990, 263-361. MR 1153682 (93m:81133b)
  • [S1] G. Segal, 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, Kluwer Acad. Publ., Dordrecht, 1988, 165-171. MR 981378 (90d:58026)
  • [S2] G. Segal, Two-dimensional conformal field theories and modular functors, in: Proceedings of the IXth International Congress on Mathematical Physics, Swansea, 1988, Hilger, Bristol, 1989, 22-37. MR 1033753 (92b:81192)
  • [S3] G. Segal, 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 University Press, Cambridge, 2004, 421-577. MR 2079383 (2005h:81334)
  • [Ts] H. Tsukada, String path integral realization of vertex operator algebras, Mem. Amer. Math. Soc. 91, 1991. MR 1052556 (91m:17044)
  • [Tu] V. G. Turaev, Quantum invariants of knots and $ 3$-manifolds, de Gruyter Studies in Math., Vol. 18, Walter de Gruyter, Berlin, 1994. MR 1292673 (95k:57014)
  • [V] E. Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys. B300 (1988), 360-376. MR 954762 (89h:81238)
  • [W] E. Witten, Three-dimensional gravity reconsidered, to appear; arXiv:0706.3359.

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Additional Information

Yi-Zhi Huang
Affiliation: Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscataway, New Jersey 08854-8019
Email: yzhuang@math.rutgers.edu

Liang Kong
Affiliation: Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, D-04103, Leipzig, Germany – and – Institut Des Hautes Études Scientifiques, Le Bois-Marie, 35, Route De Chartres, F-91440 Bures-sur-Yvette, France
Address at time of publication: Institute for Advanced Study, Tsinghua University, Beijing 100084, People’s Republic of China
Email: kong.fan.liang@gmail.com

DOI: https://doi.org/10.1090/S0002-9947-09-04933-2
Received by editor(s): March 10, 2008
Published electronically: December 22, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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