Correlation functions with fusion-channel multiplicity in W3 Toda field theory

Abstract

Current studies of WN Toda field theory focus on correlation functions such that the WN highest-weight representations in the fusion channels are multiplicity-free. In this work, we study W3 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 sl3, and a semi-degenerate primary field with a highest-weight in the fundamental representation of sl3. We show that, when the fusion rules are obeyed, 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 WN theories. We solve this equation, compute its monodromy group, and construct the monodromy-invariant correlation functions.

44 pages, v2 =published version

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• Notes on the solutions of Zamolodchikov-type recursion relations in Virasoro minimal models
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• Ward identities in the ${\mathfrak{sl}}_3$ Toda conformal field theory
• The imaginary Toda field theory
• Example of the $4$-pt non-vacuum ${\mathcal{W}}_3$ classical block
• $W_4$ Toda example as hidden Liouville CFT
• Axion-like particles from soft supersymmetry breaking
• Towards $W_3$ classical blocks with semi-degenerate operators
• Fusion of spin fields in $W_3$ conformal field theories
• Differential equations for classical Virasoro blocks with heavy and light operators