Fusion rules and modular transformations in 2D conformal field theory. (English) Zbl 1180.81120

Summary: We study conformal field theories with a finite number of primary fields with respect to some chiral algebra. It is shown that the fusion rules are completely determined by the behavior of the characters under the modular group. We illustrate with some examples that conversely the modular properties of the characters can be derived from the fusion rules. We propose how these results can be used to find restrictions on the values of the central charge and conformal dimensions.


81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
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[1] Belavin, A.A.; Polyakov, A.M.; Zamolodchikov, A.B., Nucl. phys., B241, 333, (1984)
[2] Friedan, D.; Shenker, S.H., Nucl. phys., B281, 509, (1987)
[3] Cardy, J.L., Nucl. phys., B270 [FS16], 186, (1986)
[4] Capelli, A.; Itzykson, C.; Zuber, J.-B.; Capelli, A.; Itzykson, C.; Zuber, J.-B.; Gepner, D., Nucl. phys., Comm. math. phys., Nucl. phys., B287, 111, (1987)
[5] D. Friedan and S. H. Shenker, unpublished;
[6] J. Harvey, G. Moore and C. Vafa, Quasicrystalline compactification, Harvard/Princeton-preprint HUTP-87/A072/IASSNS/HEP-87/48/PUPT-1068;
[7] E. Martinec, appendix in ref. [16]
[8] R. Dijkgraaf, E. Verlinde and H. Verlinde, c = 1 conformal field theories on Riemann surfaces, Utrecht preprint THU-87/17, Comm. Math. Phys., to be published
[9] R. Dijkgraaf, E. Verlinde and H. Verlinde, Conformal field theory at c = 1, of the 1987 Cargèse Summer School on Nonperturbative quantum field theory, to be published · Zbl 0649.32019
[10] Bershadsky, M.; Knizhnik, V.; Teitelman, M.; Friedan, D.; Qiu, Z.; Shenker, S.H., Phys. lett., Phys. lett., 151B, 21, (1985)
[11] Knizhnik, V.; Zamolodchikov, A.B., Nucl. phys., B247, 83, (1984)
[12] Gepner, D.; Witten, E., Nucl. phys., B278, 493, (1986)
[13] Fateev, V.A.; Zamolodchikov, A.B.; Fateev, V.A.; Zamolodchikov, A.B.; Gepner, D.; Qiu, Z., Jetp, Jetp, Nucl. phys., B285, 423, (1987)
[14] Eguchi, T.; Ooguri, H., Nucl. phys., B282, 308, (1987)
[15] Vafa, C., Phys. lett., 199B, 191, (1987)
[16] Friedan, D.; Qiu, Z.; Shenker, S.H., Phys. rev. lett., 52, 1575, (1984)
[17] in Vertex operators in mathematics and physics, ed. J. Lepowski et al. p. 419
[18] Goddard, P.; Kent, A.; Olive, D.; Goddard, P.; Kent, A.; Olive, D., Phys. lett., Comm. math. phys., 103, 105, (1986)
[19] Fateev, V.A.; Zamolodchikov, A.B., Nucl. phys., B280 [FS18], 644, (1987)
[20] F. A. Bais, P. Bouwknegt, K. Schoutens and M. Surridge, Coset construction for extended Virasoro algebras, Amsterdam/Utrecht-preprint ITFA 87-18/THU 87-21
[21] J. Bagger, D. Nemeschansky and S. Yankielowicz, Virasoro algebras with central charge c > 1, Harvard preprint HUTP-87/A073
[22] D. Kastor, E. Martinec and Z. Qiu, Current algebra and conformal discrete series, E. Fermi Inst. preprint EFI-87-58
[23] M. R. Douglas, G/H conformal field theory, Caltech preprint CALT-68-1453
[24] Dixon, L.; Harvey, J.; Vafa, C.; Witten, E., Nucl. phys., B274, 285, (1986)
[25] R. Dijkgraaf, C. Vafa, E. Verlinde and H. Verlinde, in preparation
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