Abstract
By the use of the Klein method instead of the theta-function method of Jacobi we are able to relate a conformal quantum theory or Riemann surfaces to the corresponding flat-space field theory and its Virasoro algebra. Physical positivity holds on a distinguished real subset in the manifold with nonrivial Hausdorff dimension which in the general case g > 1 cannot be shifted by a hamiltonian. Our picture of obtaining curved two-dimensional quantum field theories by applying a special diffeomorphism to flat ones resembles that of the Hawking-Unruh effect.
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