Modular Properties of Generalised Gibbs Ensembles

TL;DR

This paper explores the modular properties of Generalised Gibbs Ensembles in 2D conformal field theories, focusing on KdV charges, their transformations, and explicit examples like the Lee-Yang model using TBA techniques.

Contribution

It derives an asymptotic expression for the modular transform of GGEs with higher spin charges and demonstrates how to incorporate additional energies for exact transformations.

Findings

Asymptotic expansion of transformed GGE in terms of zero modes

Re-exponentiation yields a GGE with modified charges

Explicit analysis of the Lee-Yang model confirms theoretical results

Abstract

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.