Characters of the W3 algebra

TL;DR

This paper computes traces of powers of the zero mode in the W3 algebra using two methods—null vectors in the 3-state Potts model and Verma modules—and confirms their agreement, contributing to understanding algebra characters.

Contribution

It introduces two novel approaches to calculate traces in the W3 algebra and compares their results for consistency and accuracy.

Findings

Traces computed in the 3-state Potts model using null vectors.

Exact traces obtained for Verma module representations.

Results agree with brute-force diagonalisation at low levels.

Abstract

Traces of powers of the zero mode in the W3 Algebra have recently been found to be of interest, for example in relation to Black Hole thermodynamics, and arise as the terms in an expansion of the full characters of the algebra. We calculate the first few such powers in two cases. Firstly, we find the traces in the 3-state Potts model by using null vectors to derive modular differential equations for the traces. Secondly, we calculate the exact results for Verma module representations. We compare our two methods with each other and the result of brute-force diagonalisation for low levels and find complete agreement.