Theoretical and Mathematical Physics

, Volume 101, Issue 3, pp 1387–1403 | Cite as

Complex projective geometry and quantum projective field theory

  • D. V. Yur'ev


The paper gives a detailed description of the quantum-field analog of complex projective geometry — quantum projective field theory — for the simplest example of minimum dimension: quantum projective [sl(2,C)-invariant] field theory on the Riemann sphere.


Field Theory Projective Geometry Minimum Dimension Riemann Sphere Projective Field 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    D. V. Yur'ev,Teor. Mat. Fiz.,92, 172 (1992);98, 220 (1994);Algebras Groups Geom., Vol. 11 (1994).Google Scholar
  2. 2.
    A. Z. Patashinskii and V. L. Pokrovskii,Fluctuation Theory of Phase Transitions [in Russian], Nauka, Moscow (1982).Google Scholar
  3. 3.
    R. Blumenhagen, M. Flohr, A. Kliem, W. Nahm, A. Recknagel, and R. Varnhagen,Nucl. Phys. B,354, 255 (1991_; W. Nahm,Conformal Quantum Field Theory in Two Dimensions, World Scientific, Singapore (1992).Google Scholar
  4. 4.
    H. G. Kausch and G. M. T. Watts,Nucl. Phys. B,354, 740 (1991).Google Scholar
  5. 5.
    N. N. Bogolyubov, A. A. Logunov, and I. T. Todorov,Introduction to Axiomatic Quantum Field Theory, Benjamin, New York (1975).Google Scholar
  6. 6.
    S. S. Khoruzhi,Introduction to Algebraic Quantum Field Theory [in Russian], Nauka, Moscow (1986).Google Scholar
  7. 7.
    Yu. I. Manin,Gauge Fields and Complex Geometry [in Russian], Nauka, Moscow (1986).Google Scholar
  8. 8.
    A. S. Schwarts,Mathematical Methods of Quantum Field Theory [in Russian], Nauka, Moscow (1975).Google Scholar
  9. 9.
    B. V. Shabat,Introduction to Complex Analysis, Part 1, Nauka, Moscow (1985).Google Scholar
  10. 10.
    I. M. Gel'fand, R. A. Minlos and Z. Ya. Shapiro,Representations of the Rotation and Lorentz Groups and their Applications, Pergamon Press, Oxford (1963); I. M. Gel'fand, M. I. Graev, and N. Ya. Vilenkin,Integral Geometry and Associated Questions of Representation Theory [in Russian], Fizmatgiz, Moscow (1962); N. Ya. Vilenkin,Special Functions and the Theory of Group Representations, AMS Translations of Math. Monogr., Vol. 22, Providence, R.I. (1968).Google Scholar
  11. 11.
    R. E. Borcherds,Proc. Natl. Acad. Sci. U.S.A.,83, 3068 (1986); I. Frenkel, J. Lepowsky, and A. Meurman,Vertex Operator Algebras and the Monster, Academic Press, New York (1988); P. Goddard, in:Infinite-Dimensional Lie Groups and Lie Algebras, ed. V. G. Kac, World Scientific, Singapore (1989), p. 556; S. A. Prevost,Memoires AMS, No. 466 (1992); H. Tsukada,Memoires AMS. No. 444 (1991); I. B. Frenkel, Y.-Z. Huang, and J. Lepowsky,Memoires AMS, No. 494 (1993); I. B. Frenkel and Y. Zhu,Duke Math. J.,66, 123 (1992); C. Dong and J. Lepowsky,Generalized Vertex Algebras and Relative Vertex Operators, Birkhäuser (1993).Google Scholar
  12. 12.
    D. V. Yur'ev,Algebra i Anal.,3, 197 (1991).Google Scholar
  13. 13.
    D. V. Yur'ev,Algebra i Anal.,2, 209 (1990).Google Scholar
  14. 14.
    I. N. Bernshtein, I. M. Gel'fand, and S. I. Gel'fand,Tr. Seminara in I. G. Petrovskii,2, 3 (1976).Google Scholar
  15. 15.
    D. V. Yur'ev,Teor. Mat. Fiz.,86, 338 (1991).Google Scholar
  16. 16.
    D. Juriev,J. Math. Phys.,33, 492 (1992).Google Scholar
  17. 17.
    S. A. Bychkov, S. V. Plotnikov, and D. V. Yur'ev,Usp. Mat. Nauk,47, 153 (1992).Google Scholar
  18. 18.
    A. M. Perelomov,Generalized Coherent States and Their Applications [in Russian], Nauka, Moscow (1987).Google Scholar
  19. 19.
    A. B. Zamolodchikov,Pis'ma Zh. Eksp. Teor. Fiz.,25, 499 (1977); E. K. Sklyanin and L. D. Faddeev,Dokl. Akad. Nauk SSSR,243, 1430 (1978).Google Scholar
  20. 20.
    L. A. Takhtadzhyan and L. D. Faddeev,Usp. Mat. Nauk,34, 13 (1979).Google Scholar
  21. 21.
    N. N. Bogolyubov and V. S. Vladimirov, in:Proceedings of the Congress of Mathematicians at Edinburgh [in Russian], Fizmatgiz, Moscow (1962), p. 27.Google Scholar
  22. 22.
    D. V. Yur'ev,Teor. Mat. Fiz.,93, 32 (1992).Google Scholar
  23. 23.
    D. Juriev,Commun. Math. Phys.,138, 569 (1991);146, 427 (1992).Google Scholar
  24. 24.
    D. Juriev,J. Funct. Anal.,101, 1 (1991).Google Scholar
  25. 25.
    N. Jacobson,Theory of Rings [in Russian], Fizmatgiz, Moscow (1947).Google Scholar
  26. 26.
    D. B. Fuks,Cohomologies of Infinite-Dimensional Lie Algebras [in Russian], Nauka, Moscow (1984).Google Scholar
  27. 27.
    A. Mironov and A. Morozov,Phys. Lett. B,252, 47 (1990); A. Gerasimov et al.,Nucl. Phys. B,357, 565 (1991); Yu. Makeenko et al.,Nucl. Phys. B,356, 574 (1991); S. Kharchev et al.,Nucl. Phys. B,366, 569 (1991); A. Gerasimov et al.,Mod. Phys. Lett. A,6, 3079 (1991); A. Marshakov, A. Mironov, and A. Morozov,Phys. Lett. B,265, 99 (1991); S. Kharchev et al.,Phys. Lett. B,275, 311 (1992); A. Marshakov, A. Mironov, and A. Morozov,Phys. Lett. B,274, 280 (1992); A. Marshakov, A. D. Mironov, and A. Yu. Morozov,Mod. Phys. Lett. A,7, 1345 (1992); A. Mironov and S. Pakulyak,Int. J. Mod. Phys. A,8, 3107 (1993).Google Scholar
  28. 28.
    K. de Vos,Teor. Mat. Fiz.,96, 163 (1993).Google Scholar
  29. 29.
    D. V. Yur'ev and S. A. Bychkov,Usp. Mat. Nauk,46, 161 (1991); S. A. Bychkov,Usp. Mat. Nauk,47, 187 (1992).Google Scholar
  30. 30.
    N. K. Barr,Trigonometric Series [in Russian], Moscow (1961); N. I. Muskhelishvili,Singular Integral Equations, Groningen (1953); V. A. Dishkin and A. N. Prudnikov,Integral Transforms and Operator Calculus [in Russian], Nauka, Moscow (1974).Google Scholar
  31. 31.
    M. A. Semenov-Tyan-Shanskii,Zap. Nauchn. Semin. LOMI. 123, 77 (1983);Funktsional. Analiz i Ego Prilozhen.,17, 17 (1983).Google Scholar
  32. 32.
    P. Goddard and D. Olive,Workshop on Unified String Theories, World Scientific, Singapore (1986), p. 214;Int. J. Mod. Phys. A,1, 303 (1986); Yu. A. Nereshin,Modern Problems of Mathematics. Fundamental Directions, Vol. 22 [in Russian], VINITI, Moscow (1988), p. 163.Google Scholar
  33. 33.
    A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov,Nucl. Phys. B, 241, 333 (1984).Google Scholar
  34. 34.
    J.-L. Gervais, in:Workshop on Unified String Theories, eds. M. Green and D. Gross, World Scientific, Singapore (1986), p. 459; G. Mack, in:Nonperturbative Quantum Field Theory, eds. G. t'Hooft et al., Plenum, New York (1988).Google Scholar
  35. 35.
    M. B. Halpern and E. Kiritsis,Mod. Phys. Lett. A,4, 1373, 1797 (1989); A. Yu. Morozov et al.,Int. J. Mod. Phys. A,5, 803 (1990); M. B. Halpern, “Recent developments in the Virasoro Master Equation”, Berkeley Preprint, UCB-PTH-91/43 (1991).Google Scholar
  36. 36.
    S. Klimek and A. Lesniewski,Commun. Math. Phys.,146, 103 (1992).Google Scholar
  37. 37.
    M. Roček,Phys. Lett. B,255, 554 (1991).Google Scholar
  38. 38.
    Yu. I. Manin,Ann. Inst. Fourier,37, 191 (1987): “Quantum groups and noncommutative geometry”, Publ. CRM 1561, Univ. Montreal (1988); N. Yu. Reshetikhin, L. A. Takhtadzhyan, and L. D. Faddeev,Algebra i Anal.,1, 178 (1989).Google Scholar
  39. 39.
    Yu. I. Manin,Topics in Noncommutative Geometry, Princeton University Press, Princeton (1991).Google Scholar
  40. 40.
    L. C. Biedenharn and J. D. Louck,Angular Momentum in Quantum Mechanics. Theory and Applications, Encycl. Math. Appl. Vol. 8, Addison Wesley (1981);The Racah-Wigner Algebra in Quantum Theory, Encycl. Math. Appl., Vol. 9, Addison Wesley (1981).Google Scholar
  41. 41.
    M. V. Karasev, V. P. Maslov, and V. E. Nazaikinskii,Modern Problems of Mathematics. Recent Advances, Vol. 13 [in Russian], VINITI, Moscow (1979); E. K. Sklyanin,Funktsional. Analiz i Ego Prilozhen.,16, 27 (1982);17, 34 (1983); M. V. Karasev and V. P. Maslov,Nonlinear Poisson Brackets. Geometry and Quantization [in Russian], Nauka, Moscow (1991).Google Scholar
  42. 42.
    D. P. Zhelobenko and A. I. Shtern,Representations of Lie Groups [in Russian], Nauka, Moscow (1983).Google Scholar
  43. 43.
    I. M. Gel'fand and A. V. Zelevinskii,Funktsional. Analiz i Ego Prilozhen.,18, 14 (1984).Google Scholar
  44. 44.
    V. G. Kac,Infinite-Dimensional Lie Algebras, Cambridge University Press, Cambridge (1990); B. L. Feigin and D. B. Fuchs,Representations of the Virasoro Algebra. Representations of Infinite-Dimensional Lie Algebras, Gordon and Breach (1991).Google Scholar
  45. 45.
    E. S. Fradkin and M. Ya. Palchik,J. Geom. Phys.,5, 601 (1988) [reprinted inGeometry and Physics, Essays in Honor of I. M. Gelfand, eds. S. Gindikin and I. M. Singer, Pitagora Editrice, Bologna, and Elsevier Sci. Publ., Amsterdam 1991].Google Scholar
  46. 46.
    V. I. Ogievetskii and E. S. Sokachev,Mathematical Analysis, Vol. 22 [in Russian], VINITI, Moscow (1984), p. 137; A. A. Roslyi, O. M. Khudaverdyan, and A. S. Shvarts,Modern Problems of Mathematics. Fundamental Directions, Vol. 9 [in Russian], VINITI, Moscow (1986), p. 247.Google Scholar
  47. 47.
    D. A. Leites,Modern Problems of Mathematics. Recent Advances, Vol. 25 [in Russian], VINITI, Moscow (1984), p. 3.Google Scholar

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© Plenum Publishing Corporation 1995

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  • D. V. Yur'ev

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