Abstract
The coadjoint orbits of the Virasoro group, which have been investigated by Lazutkin and Pankratova and by Segal, should according to the Kirillov-Kostant theory be related to the unitary representations of the Virasoro group. In this paper, the classification of orbits is reconsidered, with an explicit description of the possible centralizers of coadjoint orbits. The possible orbits are diff(S 1) itself, diff(S 1)/S 1, and diff(S 1)/SL (n)(2,R), withSL (n)(2,R) a certain discrete series of embeddings ofSL(2,R) in diff(S 1), and diffS 1/T, whereT may be any of certain rather special one parameter subgroups of diffS 1. An attempt is made to clarify the relation between orbits and representations. It appears that quantization of diffS 1/S 1 is related to unitary representations with nondegenerate Kac determinant (unitary Verma modules), while quantization of diffS 1/SL (n)(2,R) is seemingly related to unitary representations with null vectors in leveln. A better understanding of how to quantize the relevant orbits might lead to a better geometrical understanding of Virasoro representation theory. In the process of investigating Virasoro coadjoint orbits, we observe the existence of left invariant symplectic structures on the Virasoro group manifold. As is described in an appendix, these give rise to Lie bialgebra structures with the Virasoro algebra as the underlying Lie algebra.
Similar content being viewed by others
References
Pressley, A., Segal, G.: Loop groups. Oxford: Clarendon Press 1986
Lazutkin, V.F., Pankratova, T.F.: Funkts. Anal. Prilozh.9 (1975)
Segal, G.: Unitary representations of some infinite dimensional groups. Commun. Math. Phys.80, 307 (1981)
Bowick, M.J., Rajeev, S.G.: String theory as the holomorphic geometry of loop space. The holomorphic geometry of closed bosonic string theory and diffS 1/S 1, MIT preprints, November 1986 and February, 1987
Kirillov, A.A., Yurzev, D.B.: Kahler geometry of the infinite dimensional homogeneous manifoldM = diff+ S 1/RotS 1. Preprint (1987)
Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry of critical fluctuations in two dimensions. J. Stat. Phys.34, 763 (1984); Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys.B 241, 333 (1984)
Gervais, J.-L., Neveu, A.: Dual string spectrum in Polyakov's quantization (II). Mode separation. Nucl. Phys.B 209, 125 (1982). New quantum treatment of Liouville field theory.B 224, 329 (1983)
Friedan, D., Qiu, Z., Shenker, S.: Conformal invariance, unitarity, and critical exponents in two dimensions. Phys. Rev. Lett.52, 1575 (1984) and in: Vertex operators in mathematics and physics. Lepowsky, J. et al. (eds.). Berlin, Heidelberg, New York: Springer 1984
Gervais, J.L.: Infinite family of polynomial functions of the Virasoro generators with vanishing poisson brackets. Phys. Lett.160 B, 277 (1985); Transport matrices associated with the Virasoro algebra. Phys. Lett.160 B, 281 (1985)
Thorn, C.B.: Computing the Kac determinant using dual model techniques, and more about the no-ghost theorem. Nucl. Phys. B248, 551 (1984)
Kirillov, A.A.: Elements of the theory of representations. Berlin, Heidelberg, New York: Springer 1975, Chap. 15
Crnkovic, C., Witten, E.: Covariant description of canonical formalism in geometrical theories. Princeton preprint (1986), to appear in the Newton Tercentenary Volume (Hawking, S., Israel, W. (eds.))
Atiyah, M.F., Singer, I.: Ann. Math.87, 484, 586 (1968)
Atiyah, M.F., Segal, G.B.: Ann. Math.87, 531 (1968)
Witten, E.: Fermion quantum numbers in Kaluza-Klein theory. In: The proceedings of the 1983 Shelter Island conference on quantum field theory and the foundations of physics. Khuri, N. et al. (eds.). MIT Press 1985
Vogan, D.: Lecture at the symposium on the mathematical heritage of Hermann Weyl. Durham, North Carolina, May, 1987
Brill, D., Deser, S.: Variational methods and positive energy in general relativity. Ann. Phys. (NY)50, 548 (1968)
Polyakov, A.M.: Quantum geometry of Bosonic strings. Phys. Lett.103B, 207 (1981)
Kac, V.G.: Proceedings of the international congress of mathematicians. Helsinki, 1978. Lecture Notes in Physics, Vol. 94, p. 441. Berlin, Heidelberg, New York: Springer 1982
Feigin, B.L., Fuchs, D.B.: Funct. Anal. Appl.16, 114 (1982)
Rocha-Caridi, A.: Vacuum vector representations of the Virasoro algebra. In: Vertex operators in mathematics and physics. Lepowsky, J. et al. (eds.). Berlin, Heidelberg, New York: Springer 1984
Moore, G., Witten, E.: Preprint (to appear)
Drinfeld, V.G.: Hamiltonian structures on Lie groups, Lie bialgebras, and the geometric meaning of the classical Yang-Baxter equations. Sov. Math. Doklady32, 68 (1983); Hopf algebras and the quantum Yang-Baxter equation. Sov. Math. Doklady32, 254 (1985); Quantum groups. To appear in the proceedings of the international congress of mathematicians (Berkeley 1986)
Yang, C.N.: Some exact results for the many-body problem in one-dimension with repulsive delta-function interaction. Phys. Rev. Lett.19, 1312 (1967)
Baxter, R.J.: Partition function of the eight-vertex lattice model. Ann. Phys. (NY)70, 193 (1972)
Kulish, P.P., Sklyanin, E.K.: Solutions of the Yang-Baxter equation. Sov. Math. J.19, 1596 (1982)
Belavin, A.A., Drinfeld, V.G.: Triangle equations and simple Lie algebras. Sov. Sci. Rev. Sect. C4, 93 (1984)
Chudnovsky, D.V.: Backlünd transformations and geometric and complex analytic background for construction of completely integrable lattice systems. In: Symmetries in particle physics. Bars, I. et al. (eds.). New York: Plenum 1984
Fadde'ev, L.D., Takhtajan, L.A.: Hamiltonian approach to soliton theory. Berlin, Heidelberg, New York: Springer 1987 (to appear)
Fadde'ev, L.D.: Integrable models in 1 + 1 dimensions. In: The proceedings of the école d'été de physique theorique. Les Houches (1982)
Gelfand, I.M., Dorfman, I.Ya.: Hamiltonian operators and the classical Yang-Baxter equation. Funkts. Anal. Prilozh.16, 1 (1982)
Ooms, A.I.: Commun. Algebra8, (1) 13 (1982)
Eliashvili, A.G.: Frobenius Lie algebras. Funkts. Anal. (1982)
Author information
Authors and Affiliations
Additional information
Communicated by A. Jaffe
Research supported in part by NSF grants PHY 80-19754 and 86-16129
Rights and permissions
About this article
Cite this article
Witten, E. Coadjoint orbits of the Virasoro group. Commun.Math. Phys. 114, 1–53 (1988). https://doi.org/10.1007/BF01218287
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01218287