Abstract
We consider the decomposition of the conformal blocks under the conformal embeddings. The case\(\widehat{gl}\left( {lr} \right)_1 \supset \widehat{sl}\left( l \right)_r \times \widehat{sl}\left( r \right)_l \times \hat a\) (â is an affine extension of the abelian subalgebra of the central elements ofgl(lr)) is studied in detal. The reciprocal decompositions of\(\widehat{gl}\left( {lr} \right)_1 \)-modules induce a pairing between the spaces of conformal blocks of\(\widehat{sl}\left( l \right)_r \) and\(\widehat{sl}\left( r \right)_l \) Wess-Zumino-Witten models on the Riemann sphere. The completeness of the pairing is shown. Hence it defines aduality between two spaces.
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[ABI] Altschuler, D., Bauer, M., Itzykson, C.: Commun. Math. Phys.132, 349 (1990)
[ABS] Altschuler, D., Bauer, M., Saleur, H.: J. Phys. A. Math. Gen.32, L789 (1990)
[BB] Bais, F.A., Bouwknegt, P.: Nucl. Phys. B279, 561 (1987)
[BETZ] Bais, F.A., Englert, F., Taormina, A., Zizzi, P.: Nucl. Phys. B279, 529 (1987)
[BR] Bazhanov, V.V., Reshetikhin, N.: J. Phys. A. Math. Gen.23, 1477 (1990)
[F] Frenkel, I.B.: In: Lect. Note in Math. vol. 933, p. 71. Berlin, Heidelberg, New York: Springer 1982
[Fu] Fuchs, J.: Z. Phys. C35, 89 (1987)
[FD] Fuchs, J., van Driel, P.: J. Math. Phys.31, 1770 (1990)
[FMS] Friedan, D., Martinec, E., Shenkar, S.: Nucl. Phys. B271, 93 (1986)
[FS] Friedan, D., Shenker, S.: Nucl. Phys. B281, 509 (1987)
[G] Gepner, D.: Nucl. Phys. B252, 481 (1985)
[GO] Goddard, P., Olive, D.: Nucl. Phys. B257, [FS14], 226 (1985)
[GNO] Goddard, P., Nahm, W., Olive, D.: Phys. Lett. B160, 111 (1985)
[GR] Gonzales, D., Redlich, A.N.: Phys. Lett. B147, 150 (1984); Nucl. Phys. B256, 621 (1985)
[GWe] Goodman, F.M., Wenzl, H.: Adv. Math.82, 244 (1990)
[GWi] Gepner, D., Witten, E.: Nucl. Phys. B278, 493 (1986)
[H] Hasegawa, K.: Publ. RIMS25, 741 (1989)
[JM] Jimbo, M., Miwa, T.: Adv. Stud. Pure Math.6, 17 (1985)
[JMO] Jimbo, M., Miwa, T., Okado, M.: Lett. Math. Phys.14, 123 (1987); Nucl. Phys.300, [FS22], 74 (1988)
[KN] Kuiba, A., Nakanishi, T.: In: Proc. of International Colloquium on Modern Quantum Field Theory. Bombay, Jan. 1990. Singapore: World Scientific 1991
[KNS] Kuniba, A., Nakanishi, T., Suzuki, J.: Nucl. Phys. B356, 750 (1991)
[KNTY] Kawamoto, N., Namikawa, Y., Tsuchiya, A., Yamada, Y.: Commun. Math. Phys.116, 247 (1988)
[KP] Kac, V.G., Peterson, D.: Proc. Natl. Acad. Sci. USA78, 3308 (1981)
[KT] Kanie, Y., Tsuchiya, A.: In preparation
[KZ] Knizhnik, V., Zamolodchikov, A.B.: Nucl. Phys. B247, 83 (1984)
[MS] Moore, G., Seiberg, N.: Commun. Math. Phys.123, 177 (1989)
[NS] Naculich, S.G., Schnitzer, H.J.: Phys. Lett. B244, 235 (1990)
[NRS] Naculich, S.G., Riggs, H.A., Schnitzer, H.J.: Phys. Lett. B246, 417 (1990)
[P] Polyakov, A.M.: Gauge fields and strings. Chur: Harwood Academic 1987
[SA] Saleur, H., Altschler, D.: Nucl. Phys. B354, 579 (1991)
[SW] Schellekens, A.N., Warner, N.P.: Phys. Rev. D34, 3092 (1987)
[TK] Tsuchiya, A., Kanie, Y.: Adv. Stud. Pure Math.16, 297 (1988)
[TUY] Tsuchiya, A., Ueno, K., Yamada, Y.: Adv. Stud. Pure Math.19, 459 (1989)
[Wa] Walton, M.A.: Nucl. Phys. B322, 775 (1989)
[We] Wenzl, H.: Inv. Math.92, 349 (1988)
[Wi] Witten, E.: Commun. Math. Phys.92, 455 (1984)
[Wi2] Witten, E.: Commun. Math. Phys.113, 529 (1988)
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Communicated by H. Araki
Dedicated to Professor Masahisa Adachi on his 60th birthday
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Nakanishi, T., Tsuchiya, A. Level-rank duality of WZW models in conformal field theory. Commun.Math. Phys. 144, 351–372 (1992). https://doi.org/10.1007/BF02101097
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DOI: https://doi.org/10.1007/BF02101097