Abstract
Liouville field theory on a sphere is considered. We explicitly derive a differential equation for four-point correlation functions with one degenerate field . We also introduce and study a class of four-point conformal blocks which can be calculated exactly and represented by finite-dimensional integrals of elliptic theta-functions for an arbitrary intermediate dimension. We also study the bootstrap equations for these conformal blocks and derive integral representations for corresponding four-point correlation functions. A relation between the one-point correlation function of a primary field on a torus and a special four-point correlation function on a sphere is proposed.
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