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arXiv:2508.16258 · Journal of High Energy Physics

Modular properties of $$ {\mathcal{W}}_3 $$ generalised Gibbs ensembles

Max Downing, Faisal Karimi, Tanmoy Sengupta, Adarsh Sudhakar, G.M.T. Watts

Published Jun 10, 2026• 0 citations
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Abstract

A bstract In this paper we make a proposal for the solution to a long-standing problem — the asymptotic expansions of the modular S -transform of a generalised Gibbs ensemble (GGE) in a theory with $$ {\mathcal{W}}_3 $$ W 3 symmetry where the GGE includes the first non-trivial charge. Equivalently, we give a proposal for the modular S -transform of traces of arbitrary powers of the zero mode W 0 . We provide evidence in the form of exact results using Zhu’s recursion, results obtained using conjectured results for Verma modules, and exact results for the particular value c = –2. We expect these have generalisations to other symmetry algebras/hierarchies such as the Virasoro algebra/KdV charges, and to GGEs with arbitrary finite sets of charges.

DOI
10.1007/jhep06(2026)114

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We investigate the modular properties of Generalised Gibbs Ensembles (GGEs) in two dimensional conformal field theories. These are obtained by inserting higher spin charges in the expressions for the partition function of the theory. We investigate the particular case where KdV charges are inserted in the GGE. We first determine an asymptotic expression for the transformed GGE. This expression is an expansion in terms of the zero modes of all the quasi-primary fields in the theory, not just the KdV charges. While these charges are non-commuting they can be re-exponentiated to give an asymptotic expression for the transformed GGE in terms of another GGE. As an explicit example we focus on the Lee-Yang model. We use the Thermodynamic Bethe Ansatz in the Lee-Yang model to first replicate the asymptotic results, and then find additional energies that need to be included in the transformed GGE in order to find the exact modular transformation.
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