Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
1997-07-22
J.Geom.Phys. 27 (1998) 113-153
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
47 pages, Plain TeX, no figures, requires AMS font files AMSSYM.DEF and AMSSYM.TEX, minor typos corrected, a reference added
Scientific paper
10.1016/S0393-0440(97)00072-7
We show that 4--dimensional conformal field theory is most naturally formulated on Kulkarni 4--folds, i. e. real 4--folds endowed with an integrable quaternionic structure. This leads to a formalism that parallels very closely that of 2--dimensional conformal field theory on Riemann surfaces. In this framework, the notion of Fueter analyticity, the quaternionic analogue of complex analyticity, plays an essential role. Conformal fields appear as sections of appropriate either harmonic real or Fueter holomorphic quaternionic line bundles. In the free case, the field equations are statements of either harmonicity or Fueter holomorphicity of the relevant conformal fields. We obtain compact quaternionic expressions of such basic objects as the energy-momentum tensor and the gauge currents for some basic models in terms of Kulkarni geometry. We also find a concise expression of the conformal anomaly and a quaternionic 4--dimensional analogue of the Schwarzian derivative describing the covariance of the quantum energy-momentum tensor. Finally, we analyse the operator product expansions of free fields.
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