Second level semi-degenerate fields in W3 Toda theory: matrix element and differential equation

TL;DR

This paper derives a sixth-order differential equation for conformal blocks involving semi-degenerate fields in W3 Toda theory, extending previous work on matrix elements with highest-weight fields.

Contribution

It generalizes the analysis of W3 Toda 4-point functions to semi-degenerate fields with null vectors, deriving new differential equations and discussing fusion rules and multiplicities.

Findings

Derived a sixth-order Fuchsian differential equation for semi-degenerate fields.

Analyzed the impact of multiplicities and null vectors on matrix elements.

Extended the understanding of conformal blocks in W3 Toda theory.

Abstract

In a recent study we considered W3 Toda 4-point functions that involve matrix elements of a primary field with the highest-weight in the adjoint representation of sl3. We generalize this result by considering a semi-degenerate primary field, which has one null vector at level two. We obtain a sixth-order Fuchsian differential equation for the conformal blocks. We discuss the presence of multiplicities, the matrix elements and the fusion rules.