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.
Highlights
A W-algebra will typically have a larger set of commuting zero-modes for which one can try to extend the reduced character to a full character
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 find the traces in the 3-state Potts model by using null vectors to derive modular differential equations for the traces
Summary
The ultimate aim is to calculate the full character of both the Verma module and irreducible representations, TrV (qL0−c/24yW0 ) = TrV (qL0−c/24e2πizW0 ). In this paper we calculate the first few terms in the expansion of this second form for several classes of expressions. To summarise the content of the rest of the paper: In section 3, we obtain explicit results up to level 6 for any highest weight Verma module for the traces TrV W0qL0 through TrV W05qL0 by direct calculation. We conclude with some open questions and directions for future research
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