On Structure Constants of $sl(2)$ Theories

TL;DR

This paper investigates the ratios of structure constants in minimal conformal theories, revealing a simple eigenvector-based expression and clarifying sign conventions, with implications for more complex theories.

Contribution

It provides a new eigenvector-based formula for structure constant ratios and clarifies sign conventions in minimal conformal theories.

Findings

Ratios of structure constants are expressed via eigenvectors of adjacency matrices.

Signs of structure constants are carefully determined and clarified.

Potential extension of identities to more complex theories is discussed.

Abstract

Structure constants of minimal conformal theories are reconsidered. It is shown that {\it ratios} of structure constants of spin zero fields of a non-diagonal theory over the same evaluated in the diagonal theory are given by a simple expression in terms of the components of the eigenvectors of the adjacency matrix of the corresponding Dynkin diagram. This is proved by inspection, which leads us to carefully determine the {\it signs} of the structure constants that had not all appeared in the former works on the subject. We also present a proof relying on the consideration of lattice correlation functions and speculate on the extension of these identities to more complicated theories.