# Correlation functions with fusion-channel multiplicity in \( {\mathcal{W}}_3 \) Toda field theory

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## Abstract

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.

## Keywords

Conformal and W Symmetry Conformal Field Models in String Theory Integrable Field Theories Supersymmetric gauge theory## Notes

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