Rigid Fuchsian Systems in 2-Dimensional Conformal Field Theories

We investigate Fuchsian equations arising in the context of 2-dimensional conformal field theory (CFT) and we apply the Katz theory of Fucshian rigid systems to solve some of these equations. We show that the Katz theory provides a precise mathematical framework to answer the question whether the fusion rules of degenerate primary fields are enough for determining the differential equations satisfied by their correlation functions. We focus on the case of W3 Toda CFT: we argue that the differential equations arising for four-point conformal blocks with one nth level semi-degenerate field and a fully-degenerate one in the fundamental sl(3) representation are associated to Fuchsian rigid systems. We show how to apply Katz theory to determine the explicit form of the differential equations, the integral expression of solutions and the monodromy group representation. The theory of twisted homology is also used in the analysis of the integral expression. The computation of the connection coefficients is done for the first time in the case of a Katz system with multiplicities, thus extending the work done by Oshima in the multiplicity free case. This approach allows us to construct the corresponding fusion matrices and to perform the whole bootstrap program: new explicit factorization of W3 correlation functions as well as shift relations between structure constants for general Toda theories are also provided.