## Abstract

Current studies of W_{N} Toda field theory focus on correlation functions such that the W_{N} highest-weight representations in the fusion channels are multiplicity-free. In this work, we study 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 sl_{3}, and a fully-degenerate primary field with a highest-weight in the fundamental representation of sl_{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 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.

Original language | English |
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Article number | 137 |

Journal | Journal of High Energy Physics |

Volume | 2016 |

Issue number | 6 |

DOIs | |

State | Published - 1 Jun 2016 |

Externally published | Yes |

## Keywords

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

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