Generalized Gibbs Ensemble of 2d CFTs at large central charge in the thermodynamic limit

We discuss partition function of 2d CFTs decorated by higher qKdV charges in the thermodynamic limit when the size of the spatial circle goes to infinity. In this limit the saddle point approximation is exact and at infinite central charge generalized partition function can be calculated explicitly. We show that leading 1/c corrections to free energy can be reformulated as a sum over Young tableaux which we calculate for the first two qKdV charges. Next, we compare generalized ensemble with the "eigenstate ensemble" that consists of a single primary state. At infinite central charge the ensembles match at the level of expectation values of local operators for any values of qKdV fugacities. When the central charge is large but finite, for any values of the fugacities the aforementioned ensembles are distinguishable.