  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 n-th level semi-degenerate field and a
fully-degenerate one in the fundamental sl3 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. This approach
allows to construct the corresponding fusion matrices and to perform the whole
bootstrap program: new explicit factorization of W3 correlation functions as
well shift relations between structure constants are also provided.
