Differential equation for 4-point correlation function
in Liouville field theory and elliptic 4-point conformal blocks
Andre Neveu
The Liouville field theory on a sphere is considered.
We explicitly derive a differential equation for
5-point correlation functions with one degenerate field
Vm/2.
From it, we calculate exactly and study a class of four-point conformal blocks which can be represented by finite dimensional integrals of elliptic theta-functions for arbitrary central charge and intermediate dimension. We study the bootstrap equations for these conformal blocks and derive integral representations for the corresponding 4-point correlation functions. A curious relation between the 1-point correlation function of a primary field on a torus and a special 4-point correlation function on a sphere is also proposed. Finally, we present a kind of
Backlund transformation which enlarges this class of conformal blocks
by shifting in various ways the values of the external dimensions.