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Unitary representations of the Virasoro and super-Virasoro algebras. (English) Zbl 0588.17014
Any unitary highest weight representation of the Virasoro algebra is determined by a pair of real numbers (c,h); unitarity implies $c\ge 0$, $h\ge 0$. When $c\ge 1$ it is easy to find a corresponding representation. When $c<1$ it was shown by {\it D. Friedan}, {\it Z. Qiu} and {\it S. Shenker} [Vertex operators in mathematics and physics, Publ., Math. Sci. Res. Inst. 3, 419-449 (1985; Zbl 0559.58010)] that c belongs to an infinite discrete set and for each such c, h can take only a finite number of values. In previous papers the authors developed a method to obtain representations corresponding to some values of $c<1$. In the present paper they show that the method in fact provides all the possible unitary highest weight representations of the Virasoro algebra, thus completing its classification. By a similar method they complete the classification of the unitary highest weight representations of the super-Virasoro algebras.
Reviewer: F.Levstein

##### MSC:
 17B65 Infinite-dimensional Lie (super)algebras 17B10 Representations of Lie algebras, algebraic theory 58E05 Abstract critical point theory
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##### References:
 [1] Goddard, P., Kent, A., Olive, D.: Virasoro algebras and coset space models. Phys. Lett.152 B, 88 (1985) · Zbl 0661.17015 [2] Polyakov, A.M.: Conformal symmetry of critical fluctuations. JETP Lett.12, 381 (1970); · Zbl 0221.32004 [3] Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B241, 333 (1980) · Zbl 0661.17013 [4] Friedan, D., Qiu, Z., Shenker, S.: In: Vertex operators in mathematics and physics. Lepowsky, J. et al. (eds.). MSRI publications No. 3, Berlin, Heidelberg, New York: Springer 1984, p. 419; Conformal invariance unitarity and critical exponents in two dimensions. Phys. Rev. Lett.52, 1575 (1984) [5] Goddard, P., Olive, D.: Kac-Moody algebras, conformal symmetry and critical exponents. Nucl. Phys. B257 [FS14], 226 (1985) · Zbl 0661.17014 · doi:10.1016/0550-3213(85)90344-X [6] Friedan, D., Qiu, Z., Shenker, S.: Superconformal invariance in two dimensions and the tricritical Ising model. Phys. Lett.151 B, 37 (1985) · Zbl 0559.58010 [7] Ramond, P.: Dual theory for free fermions. Phys. Rev. D3, 2415 (1971) [8] Neveu, A., Schwarz, J.H.: Factorizable dual model of pions. Nucl. Phys. B31, 86 (1971); Quark model of dual pions. Phys. Rev. D4, 1109 (1971) · doi:10.1016/0550-3213(71)90448-2 [9] Sugawara, H.: A field theory of currents. Phys. Rev.170, 1659 (1968) · doi:10.1103/PhysRev.170.1659 [10] Sommerfield, C.: Currents as dynamical variables. Phys. Rev.176, 2019 (1968) · doi:10.1103/PhysRev.176.2019 [11] Coleman, S., Gross, D., Jackiw, R.: Fermion avatars of the Sugawara model. Phys. Rev.180, 1359 (1969); · doi:10.1103/PhysRev.180.1359 [12] Bardakci, K., Halpern, M.: New dual quark models. Phys. Rev. D3, 2493 (1971) · Zbl 1259.81043 [13] Dashen, R., Frishman, Y.: Four-fermion interactions and scale invariance. Phys. Rev. D11, 2781 (1975) [14] Kac, V.G.: Infinite dimensional Lie algebras. Boston: Birkhäuser 1983 · Zbl 0537.17001 [15] Goddard, P.: Kac-Moody algebras: representations and applications. DAMTP preprint 85/7 [16] Olive, D.: Kac-Moody algebras: an introduction for physicists. Imperial College preprint TP/84-85/14 [17] Rocha-Caridi, A.: In: Vertex operators in mathematics and physics. Lepowsky, J. et al. (eds.). MSRI publications No. 3, p. 451. Berlin, Heidelberg, New York: Springer 1984 [18] Frenkel, I.B., Kac, V.G.: Basic representations of Lie algebras and dual resonance models. Invent. Math.62, 23 (1980) · Zbl 0493.17010 · doi:10.1007/BF01391662 [19] Segal, G.: Unitary representations of some infinite dimensional groups. Commun. Math. Phys.80, 301 (1981) · Zbl 0495.22017 · doi:10.1007/BF01208274 [20] Goddard, P., Olive, D.: In: Vertex operators in mathematics and physics. Lepowsky, J. et al. (eds.). MSRI publications No. 3, p. 51. Berlin, Heidelberg, New York: Springer 1984 [21] Feigin, B.L., Fuchs, D.B.: Funct. Anal. Appl.17, 241 (1983) · Zbl 0529.17010 · doi:10.1007/BF01078118 [22] Kent, A.: Papers in preparation [23] Bardakci, K., Halpern, M.B.: New dual quark models. Phys. Rev. D3, 2493 (1971) · Zbl 1259.81043 [24] Goddard, P., Kent, A., Olive, D.: In preparation [25] Kac, V., Todorov, I.: Superconformal current algebras and their unitary representations. Commun. Math. Phys.102, 337-347 (1985) · Zbl 0599.17011 · doi:10.1007/BF01229384 [26] Altschuler, D.: University of Geneva preprint (1985) UGVA-DPT 1985/06-466