Abstract
It is shown that the global charges of a gauge theory may yield a nontrivial central extension of the asymptotic symmetry algebra already at the classical level. This is done by studying three dimensional gravity with a negative cosmological constant. The asymptotic symmetry group in that case is eitherR×SO(2) or the pseudo-conformal group in two dimensions, depending on the boundary conditions adopted at spatial infinity. In the latter situation, a nontrivial central charge appears in the algebra of the canonical generators, which turns out to be just the Virasoro central charge.
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Abbott, L. F., Deser, S.: Stability of gravity with a cosmological constant. Nucl. Phys.B195, 76 (1982)
Abbott, L. F., Deser, S.: Charge definition in non-Abelian gauge theories, Phys. Lett.116B, 259 (1982)
Geroch, R.: Asymptotic structure of space-time. In: Asymptotic structure of space-time, Esposito, F. P., Witten, L. (eds.) New York: Plenum Press 1977
Ashtekar, A.: Asymptotic structure of the gravitational field at spatial infinity. In: General relativity and gravitation: One hundred years after the birth of Albert Einstein, vol. 2 Held, A. (ed.). New York: Plenum Press 1980. See also his contribution in Proceedings of the Oregon conference on mass and asymptotic structure of space-time. Flaherty, F. (ed.). Berlin, Heidelberg, New York: Springer 1984
Nelson, P., Manohar, A.: Global color is not always defined. Phys. Rev. Lett.50, 943 (1983); Balachandran, A. P., Marmo, G., Mukunda, N., Nilsson, J. S., Sudarshan, E. C. G., Zaccaria, F.: Monopole topology and the problem of color. Phys. Rev. Lett.50, 1553 (1983)
Henneaux, M.: Energy-momentum, angular momentum, and supercharge in 2 + 1 supergravity. Phys. Rev.D29, 2766 (1984); Deser, S.: ‘Breakdown of asymptotic Poincaré invariance inD = 3 Einstein gravity.’ Class. Quantum Grav.2, 489 (1985)
Arnold, V.: Mathematical methods of classical mechanics. Berlin, Heidelberg, New York: Springer 1978
Regge, T., Teitelboim, C.: Role of surface integrals in the Hamiltonian formulation of general relativity. Ann. Phys. (N.Y.)88, 286 (1974)
See for instance Scherk, J.: An introduction to the theory of dual models and strings. Rev. Mod. Phys.47, 123 (1975)
Brown, J. D., Henneaux, M.: to appear, J. Math. Phys.
Jackiw, R.: Introduction to the Yang-Mills quantum theory. Rev. Mod. Phys.52, 661 (1980)
Deser, S., Jackiw, R., 't Hooft, G.: Three-dimensional Einstein gravity: Dynamics of flat space. Ann. Phys.152, 220 (1984)
Deser, S., Jackiw, R.: Three-dimensional cosmological gravity: Dynamics of constant curvature. Ann. Phys.153, 405 (1984)
See for example, Dirac, P. A. M.: The theory of gravitation in Hamiltonian form. Proc. Roy. Soc.A246, 333 (1958); Arnowitt, R., Deser, S., Misner, C. W.: In: Gravitation: An introduction to current research. Witten L. (ed.). New York: Wiley 1962
Henneaux, M., Teitelboim, C.: Asymptotically anti-de Sitter spaces. Commun. Math. Phys.98, 391 (1985)
Benguria, R., Cordero, P., Teitelboim, C.: Aspects of the Hamiltonian dynamics of interacting gravitational gauge and Higgs fields with applications to spherical symmetry. Nucl. Phys.B122, 61 (1977)
Penrose, R.: In: Relativity, groups, and topology. Dewitt C., De Witt B. (eds.). New York: Gordon and Breach 1964
Hanson, A., Regge, T., Teitelboim, C.: Constrained Hamiltonian systems. Acc. Naz. dei Lincei, Rome 1976
Teitelboim, C.: Commutators of constraints reflect the spacetime structure. Ann. Phys. (N.Y.)79, 542 (1973)
This has been recognized independently by A. Ashtekar and A. Magnon-Ashtekar (private communication)
See for instance, Faddeev, L. D.: Operator anomaly for the Gauss law. Phys. Lett.145B, 81 (1984); Alvarez, O., Singer, I. M., Zumino, B.: Gravitational anomalies and the family's index theorem. Commun. Math. Phys.96, 409 (1984); and references therein
See for instance, Jackiw, R.: Three-cocycle in mathematics and physics. Phys. Rev. Lett.54, 159 (1985); Grossman, B.: A 3-cocycle in quantum mechanics. Phys. Lett.152B, 93 (1985); Wu, Y. S., Zee, A.: Cocycles and magnetic monopole. Phys. Lett.152B, 98 (1985); Boulware, D. G., Deser, S., Zumino, B.: Absence of 3-cocycles in the Dirac monopole problem. Phys. Lett.153B, 307 (1985)
For example, in the problem of spacetime symmetries of gauge fields, see Henneaux, M.: Remarks on spacetime symmetries and nonabelian gauge fields. J. Math. Phys.23, 830 (1982)
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Communicated by S. W. Hawking
Chercheur qualifié du Fonds National Belge de la Recherche Scientifique
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Brown, J.D., Henneaux, M. Central charges in the canonical realization of asymptotic symmetries: An example from three dimensional gravity. Commun.Math. Phys. 104, 207–226 (1986). https://doi.org/10.1007/BF01211590
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DOI: https://doi.org/10.1007/BF01211590