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Virasoro algebras and coset space models. (English) Zbl 0661.17015
This article is an extension of a previous one [P. Goddard and D. Olive, Nucl. Phys. B 257, No.2, 226-252 (1985; see the preceding review Zbl 0661.17014)] in which the authors have used the affine Kac- Moody algebras to construct representations of the Virasoro algebra. By considering the coset spaces by dividing out a subalgebra $${\mathfrak h}$$ of $${\mathfrak g}$$, where $${\mathfrak g}$$ is the Lie algebra on which the Kac-Moody algebra is modelled, one obtains again the Virasoro algebra but with different central charge. Furthermore, representations of the Kac-Moody algebra on positive-definite Hilbert spaces obeying a unitarity property automatically yield in this way unitary representations of the Virasoro algebra. In particular, choosing for the coset the quaternionic projective space $${\mathbb{H}}P^{n-1}$$, a connection is found with the results of D. Friedan, Z. Qiu and S. Shenker [Vertex operators in mathematics and physics, Publ., Math. Sci. Res. Inst. 3, 419-449 (1985; Zbl 0559.58010)]. The representations thus constructed can be made explicit by using the “quark model” picture, realizing the Kac- Moody generators in terms of Fermi fields arising from a realization of either the Neveu-Schwarz or Ramond algebra. Similar constructions in the case of supersymmetry are mentioned.

##### MSC:
 17B65 Infinite-dimensional Lie (super)algebras 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations 81S05 Commutation relations and statistics as related to quantum mechanics (general) 81T60 Supersymmetric field theories in quantum mechanics
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##### References:
 [1] Belavin, A.; Polyakov, A.; Zamolodchikov, A., Nucl. phys., B241, 333, (1984) [2] D. Friedan, Z. Qiu and S. Shenker, Chicago preprint EFI 83-60, to be published in: Proc. MSRI Workshop on Vertex operators, ed. J. Lepowsky (Springer, Berlin); [3] P. Goddard and D. Olive, DAMTP preprint 84/16. [4] Neveu, A.; Schwarz, J.H.; Ramond, P., Nucl. phys., Phys. rev., Phys. rev., D3, 2415, (1971) [5] P. Goddard, A. Kent and D. Olive, in preparation.
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