Abstract
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.
Highlights
Number of particles, its thermal properties are described by the grand canonical ensemble, with the effective number of particles controlled by the value of chemical potential [20]
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
We studied Generalized Gibbs Ensemble (1.5) of 2d CFTs in the limit when the size of the spatial circle goes to infinity
Summary
The crucial simplification of thermodynamic limit → ∞ is that saddle point approximation becomes exact. Ln is the conventional Virasoro algebra generator related to the stress tensor on the plane,. Parametrized by the dimension of Virasoro primary ∆ and sets {mi}, k i=1 mi n, arranged in the dominance order. The sum ∆ goes over all Virasoro primaries including possible multiplicities, and P (n) is the number of integer partitions — Young tableaux consisting of n elements. The sum over n can be substituted by an integral. The sum over ∆ in (2.4) can be substituted by an integral multiplied by the density of primaries given by Cardy formula [34]. It is easy to see that in the limit → ∞ the saddle point approximation is exact, 2n β
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