Abstract
In Twistor Conformal Field Theory the Riemann surfaces and holomorphic functions of two-dimensional conformal field theory are replaced by “flat” twistor spaces (arising from conformally-flat four-manifolds) and elements of the holomorphic first cohomology. The analogue of a Laurent Series is the expansion of a cohomology element in “elementary states” and we calculate the dimension of the space of these states for twistor spaces of compact hyperbolic manifolds. Our method follows the strategy used in the classical problem of calculating the number of meromorphic functions with prescribed poles on Rieiemann surface. We express the problem globally (in terms of the cohomology of a blown-up twistor space), calculate the holomorphic Euler characteristic of this blown-up space, and then use some vanishing theorems to isolate the first cohomology term.
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References
Bailey, T.N.; Baston, R.J.:Twistors in Mathematics and Physics. Cambridge University Press, 1990.
Baston, R.J.; Eastwood, M.G.: (1989)The Penrose Transform: its interaction with representation theory. Oxford University Press, 1989.
Eastwood, M.G.; Hughston, L.P.: Massless Fields Based on a Line. In: Hughston, L.P.; Ward, R.S. (eds.):Advances in Twistor Theory. Pitman, London 1979.
Eastwood, M.G.; Pilato, A.: On the Density of Twistor Elementary States.Pac. J. Math. 151 (1991), 201.
Eastwood, M.G.; Singer, M.A.: On the Geometry of Twistor Spaces.J. Differ. Geom. 38 (1993), 653–669.
Grauert, H.; Remmert, R.:Theory of Stein Spaces. Springer-Verlag, New York 1979.
Griffiths, P.; Harris, J.:Principles of Algebraic Geometry. Wiley, New York 1978.
Hartshorne, R.:Algebraic Geometry. Springer-Verlag, New York 1983.
Hirzebruch, F.:Topological Methods in Algebraic Geometry. Springer-Verlag, Berlin 1978.
Hitchin, N.J.: Kählerian Twistor Spaces.Proc. Lond. Math. Soc. 43 (1981), 133–150.
Hodges, A.P.; Penrose, R.; Singer, M.A.: A Twistor Conformal Field Theory for Four Space-Time Dimensions.Phys. Lett. B 216 (1989), 48.
Hodges, A.P.: Twistor Diagrams and Feynman Diagrams. In: [1].
Horan, R.E.:Analytic Cohomology on Blown-Up Twistor Space. Ph. D. Thesis, University of Plymouth 1994.
Horan, R.E.: Analytic Cohomology of Blown-Up Twistor Spaces. In: Huggett, S.A. (ed.):Twistor Theory. Marcel Dekker, New York 1995.
Horan, R.E.: A Rigidity Theorem for Quaternionic-Kähler Manifolds. To appear inDiffer. Geom. Appl.
Huggett, S.A.: Cohomology and Twistor Diagrams. In: [1].
Huggett, S.A.: Recent Work in Twistor Field Theory.Class. Quant. Grav. 9 (1992), 127.
Iversen, B.:Comomology of Sheaves. Springer-Verlag, Berlin 1986.
Kodaira, K.:Complex Manifolds and Deformation of Complex Structures. Springer-Verlag, New York 1986.
Singer, M.A.: Flat Twistor Spaces, Conformally Flat Manifolds and Four-Dimensional Field Theory.Commun. Math. Phys. 133 (1990), 75.
Wells, R.O.:Differential Analysis on Complex Manifolds. Springer-Verlag, Berlin 1980.
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Horan, R., Huggett, S. Cohomology of elementary states in twistor conformal field theory. Ann Glob Anal Geom 14, 107–121 (1996). https://doi.org/10.1007/BF00127969
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DOI: https://doi.org/10.1007/BF00127969