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Communications in Mathematical Physics

, Volume 120, Issue 3, pp 481–499 | Cite as

The group of local biholomorphisms of ℂ1 and conformal field theory in the operator formalism

  • Robert J. Budzyński
  • Sławomir Klimek
  • Paweł Sadowski
Article

Abstract

Motivated by the operator formulation of conformal field theory on Riemann surfaces, we study the properties of the infinite dimensional group of local biholomorphic transformations (conformal reparametrizations) of ℂ1 and develop elements of its representation theory.

Keywords

Neural Network Statistical Physic Field Theory Complex System Nonlinear Dynamics 
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References

  1. 1.
    Green, M., Schwartz, J.H., Witten, E.: String theory. Cambridge: Cambridge University Press 1987Google Scholar
  2. 2.
    Lepowski, J., Mandelstam, S., Singer, I.: Vertex operators in mathematics and physics. Berlin, Heidelberg, New York: Springer 1985Google Scholar
  3. 3.
    Friedan, D., Martinec, E., Shenker, S.: Conformal invariance, supersymmetry, and string theory. Nucl. Phys. B271, 93 (1986)Google Scholar
  4. 4.
    Verlinde, E., Verlinde, H.: Chiral bosonization, determinants, and the string partition function. Nucl. Phys. B288, 357 (1987)Google Scholar
  5. 5.
    Alvarez-Gaumé, L., Gomez, C., Moore, G., Vafa, C.: Strings in the operator formalism. Nucl. Phys. B303, 455 (1988)Google Scholar
  6. 6.
    Witten, E.: Quantum field theory, Grassmannians, and algebraic curves. Commun. Math. Phys.113, 529 (1988)Google Scholar
  7. 7.
    Milnor, J.: Remarks on infinite dimensional Lie groups. In: Relativity, Groups, and topology. II. Les Houches Session XL. De Witt, B., Stora, R. (eds.) Amsterdam: North-Holland 1984Google Scholar
  8. 8.
    Pressley, A., Segal, G.: Loop groups. Oxford: Oxford University Press 1986Google Scholar
  9. 9.
    Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B241, 333 (1984)Google Scholar
  10. 10.
    Reed, M., Simon, B.: Methods of modern mathematical physics. New York, London: Academic Press 1972, 1975Google Scholar
  11. 11.
    Hamilton, R.: The inverse function theorem of Nash and Moser. Bull. AMS7, 65 (1982)Google Scholar
  12. 12.
    Herman, M.R.: Recent results and some open questions in the Siegel linearization theorem of germs of complex analytic diffeomorphisms ofC n near a fixed point. In: Proc. VIII Int. Congress on Mathematical Physics. Mebkhout, M., Sénéor, R. (eds.). Singapore: World Scientific 1986Google Scholar
  13. 13.
    Feigin, B.L., Fuks, D.P.: Skew-symmetric invariant differential operators on the line and Verma modules over the Virasoro algebra. Funk. Anal. Pril. 16.2, 47 (1982) (in Russian)Google Scholar
  14. 14.
    Friedan, D., Qiu, Z., Shenker, S.: Conformal invariance, unitarity, and critical exponents in two dimensions. Phys. Rev. Lett.52, 1575 (1984)Google Scholar
  15. 15.
    Goddard, P., Olive, D.: Kac-Moody and Virasoro algebras in relation to quantum physics. Int. J. Mod. Phys. A1, 303 (1986)Google Scholar
  16. 16.
    Goodman, R., Wallach, N.R.: Projective unitary positive energy representations of Diff(S 1). J. Funct. Anal.63, 299 (1985)Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • Robert J. Budzyński
    • 1
  • Sławomir Klimek
    • 2
  • Paweł Sadowski
    • 3
  1. 1.Institute of Theoretical PhysicsWarsaw UniversityWarsawPoland
  2. 2.Department of Mathematical Methods of PhysicsWarsaw UniversityWarsawPoland
  3. 3.Institute of MathematicsBiałystok Branch of Warsaw UniversityBiałystokPoland

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