The quaternionic geometry of four-dimensional conformal field theory

We show that four-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 two-dimensional conformal field theory on Riemann surfaces. In this framework, the notion of Fueter analyticity, the quaternionic analog 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 four-dimensional analog of the Schwarzian derivative describing the covariance of the quantum energy-momentum tensor. Finally, we analyze the operator product expansions of free fields. © 1998 Elsevier Science B.V.