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

TL;DR

This paper explores 4-point correlation functions in W3 Toda field theory with fusion-channel multiplicities, deriving explicit matrix elements, a novel differential equation, and constructing monodromy-invariant functions, expanding understanding beyond multiplicity-free cases.

Contribution

It introduces the analysis of fusion-channel multiplicities in W3 Toda theory, deriving explicit matrix elements and a new class of Fuchsian differential equations for conformal blocks.

Findings

Explicit matrix elements for fully-degenerate adjoint fields

A fourth-order Fuchsian differential equation with multiplicities

Construction of monodromy-invariant correlation functions

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.