Example of the $4$-pt non-vacuum $\mathcal{W}_3$ classical block

TL;DR

This paper derives the monodromy equations for a specific 4-point non-vacuum classical block in the $ ext{W}_3$ algebra, providing a method to compute these blocks using the heavy-light approximation.

Contribution

It formulates the monodromy problem for the 4-pt non-vacuum $ ext{W}_3$ classical block and computes the block function using the heavy-light approximation.

Findings

Derived monodromy equations within the heavy-light approximation.

Successfully computed the 4-pt non-vacuum $ ext{W}_3$ block function.

Fixed functional arbitrariness using vacuum $ ext{W}_3$ parameters.

Abstract

In this note, we study a special case of the $4$-pt non-vacuum classical block associated with the $\mathcal{W}_3$ algebra. We formulate the monodromy problem for the block and derive monodromy equations within the heavy-light approximation. Fixing the remaining functional arbitrariness using parameters of the $4$-pt vacuum $\mathcal{W}_3$ block, we compute the $4$-pt non-vacuum $\mathcal{W}_3$ block function.