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Flat twistor spaces, conformally flat manifolds and four-dimensional field theory

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A definition is proposed of “four-dimensional conformal field theory” in which the Riemann surfaces of two-dimensional CFT are replaced by (Riemannian) conformally flat four-manifolds and the holomorphic functions are replaced by solutions of the Dirac equation. The definition is investigated from the point of view of twistor theory, allowing methods from complex analysis to be employed. The paper fills in the main mathematical details omitted from the preliminary announcement [15].

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References

  1. Andreotti, A., Grauert, H.: Théorêmes de finitude pour la cohomologie des espaces complexes. Bull. Soc. Math. France90, 193–259 (1962)

    Google Scholar 

  2. Aronszajn, N.: A unique continuation theorem for solutions of elleptic partial differential equations or inequalities of the second order. J. Math. Pures Appl.36, 235–249 (1957)

    Google Scholar 

  3. Atiyah, M. F., Hitchin, N. J., Singer, I. M.: Self-duality in four-dimensional Riemannian geometry. Proc. R. Soc. Lond.A362, 425–461 (1978)

    Google Scholar 

  4. Atiyah, M. F.: Geometry of Yang-Mills fields. Lezioni Fermiane, Accademia Nazionale dei Lincei & Scuola Normale Superiore, Pisca (1979)

    Google Scholar 

  5. Bailey, T. N., Ehrenpreis, L., Wells, R. O., Jr.: Weak solutions of the massless field equations. Proc. R. Soc. Lond.A384, 403–425 (1982)

    Google Scholar 

  6. Baston, R. J., Eastwood, M. G.: The Penrose transform: Its interaction with representation theory. Oxford Mathematical Monographs, Oxford: O.U.P. 1989

    Google Scholar 

  7. Bismut, J.-M., Gillet, H., Soulé: Analytic torsion and holomorphic determinant bundles I–III. Commun Math. Phys.115, 49–78, 79–126 and 301–351 (1988)

    Google Scholar 

  8. Cartan, H., Serre, J.-P.: Un théorème de finitude concernant les variétés analytiques compacts. C. R. Acad. Sc.237, 128–130 (1953)

    Google Scholar 

  9. Donaldson, S. K.: Infinite determinants, stable bundles and curvature. Duke Math. J.54, 231–247 (1987)

    Google Scholar 

  10. Eastwood, M. G., Penrose, R., Wells, R. O., Jr.: Cohomology and massless fields. Commun. Math. Phys.78, 305–351 (1981).

    Google Scholar 

  11. Eastwood, M. G., Pilato, A. M.: The density of twistor elementary states. In: Mason, L. J., Hughston, L. P. (eds.). Further advances in twistor theory. Oxford: Pitman 1990

    Google Scholar 

  12. Folland, G. B., Kohn, J. J.: The Neumann problem for the Cauchy-Riemann complex. Ann. of Math. Studies vol.75. Princeton: P.U.P. 1972

    Google Scholar 

  13. Hitchin, N. J.: Linear field equations on self-dual spaces. Proc. R. Soc. Lond.A370, 173–191 (1980).

    Google Scholar 

  14. Hitchin, N. J.: Kählerian twistor spaces. Proc. Lond. Math. Soc.43, 133–150 (1981)

    Google Scholar 

  15. Hodges, A. P., Penrose, R., Singer, M. A.: A twistor conformal field theory for four space-time dimensions. Phys. Lett.B216, 48–52 (1989)

    Google Scholar 

  16. Mickelsson, J., Rajeev, S. G.: Current algebras ind+1 dimensions and determinant bundles over infinite-dimensional Grassmannians. Commun. Math. Phys.116, 365–400 (1988)

    Google Scholar 

  17. Penrose, R.: Nonlinear gravitons and curved twistor theory. Gen. Rel. Grav.7, 31–52 (1976)

    Google Scholar 

  18. Penrose, R., Rindler, W.: Spinors and space-time, vol. 2. Cambridge: C. U. P. 1986

    Google Scholar 

  19. Penrose, R.: Pretzel twistor spaces. In: Mason, L. J., Hughston, L. P. (eds.). Further advances in twistor theory. Oxford: Pitman 1990

    Google Scholar 

  20. Pressley, A., Segal, G. B.: Lomp groups. Oxford Mathematical Monograph. Oxford: O.U.P. 1986

    Google Scholar 

  21. Quillen, D. G.: Determinants of Cauchy-Riemann operators over a Riemann surface. Funct. Anal. Appl.19, 31–4 (English) (1985)

    Google Scholar 

  22. Schoen, R., Yau, S.-T.: Conformally flat manifolds, Kleinian groups and scalar curvature. Invent. Math.92, 47–71 (1988)

    Google Scholar 

  23. Segal, G. B.: The definition of conformal field theory (to appear, 1990)

  24. Singer, M. A.: Twistors and conformal field theory. In: Mason, L. J. (ed.). Further advances in twistor theory. Oxford: Pitman 1990

    Google Scholar 

  25. Witten, E.: Quantum field theory, grassmannians and algebraic curves. Commun. Math. Phys.113, 529–600 (1988)

    Google Scholar 

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  1. Mathematical Institute, OX1 3LB, Oxford, UK

    M. A. Singer

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  1. M. A. Singer
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Communicated by A. Connes

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Singer, M.A. Flat twistor spaces, conformally flat manifolds and four-dimensional field theory. Commun.Math. Phys. 133, 75–90 (1990). https://doi.org/10.1007/BF02096555

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  • DOI: https://doi.org/10.1007/BF02096555

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