Correlation functions with fusion-channel multiplicity in W 3 $$ {mathcal{W}}_3 $$ Toda field theory

Current studies of $$ {mathcal{w}}_n $$ toda field theory focus on correlation functions such that the $$ {mathcal{w}}_n $$ highest-weight representations in the fusion channels are multiplicity-free. in this work, we study $$ {mathcal{w}}_3 $$ toda 4-point functions with multiplicity in the fusion channel. the conformal blocks of these 4-point functions involve matrix elements of a fully-degenerate primary field with a highest-weight in the adjoint representation of $$ mathfrak{s}{mathfrak{l}}_3 $$ , and a fully-degenerate primary field with a highest-weight in the fundamental representation of $$ mathfrak{s}{mathfrak{l}}_3 $$ . we show that, when the fusion rules do not involve multiplicities, the matrix elements of the fully-degenerate adjoint field, between two arbitrary descendant states, can be computed explicitly, on equal footing with the matrix elements of the semi-degenerate fundamental field. using null-state conditions, we obtain a fourth-order fuchsian differential equation for the conformal blocks. using okubo theory, we show that, due to the presence of multiplicities, this differential equation belongs to a class of fuchsian equations that is different from those that have appeared so far in $$ {mathcal{w}}_n $$ theories. we solve this equation, compute its monodromy group, and construct the monodromy-invariant correlation functions. this computation shows in detail how the ambiguities that are caused by the presence of multiplicities are fixed by requiring monodromy-invariance.