Riemann monodromy problem and conformal field theories

A systematic analysis of the use of the Riemann monodromy problem for determining correlators (conformal blocks) on the sphere is presented. The monodromy data is constructed in terms of the braid matrices and give a constraint on the noninteger part of the conformal dimensions of the primary fields. To determine the conformal blocks we need to know the order of singularities. We establish a criterion which tells us when the knowledge of the conformal dimensions of primary fields suffice to determine the blocks. When zero modes of the extended algebra are present the analysis is more difficult. In this case we give a conjecture that works for the SU(2) WZW case.