Characters of the W3 algebra

TL;DR

This paper advances the understanding of the W3 algebra by computing the first few traces of powers of its zero mode W0 across three calculational avenues: direct Verma-module expansions, null-vector–driven analyses in the 3-state Potts minimal model (c=4/5), and exact all-level results for Verma modules. The authors derive differential equations for traces via null fields, obtain explicit q-series in the Potts case, and present all-level operator expressions for Tr_V(W0 q^{L0}) and Tr_V(W0^2 q^{L0}) whose irreducible-module reductions are then compared with Potts-model results. Across methods, they demonstrate complete agreement up to high orders, reinforcing the consistency of the generalized character framework. The work lays groundwork for extending to other minimal models and higher powers of W0, and it has potential implications for holographic dualities in AdS3/CFT2, where generalized W3 characters relate to higher-spin partition functions and black-hole physics.

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.