Abstract
Two-dimensional CFTs have an infinite set of commuting conserved charges, known as the quantum KdV charges. We study the Generalized Gibbs Ensemble with chemical potentials for these charges at high temperature. In a large central charge limit, the partition function can be computed in a saddle-point approximation. We compare the ensemble values of the KdV charges to the values in a microstate, and find that they match irrespective of the values of the chemical potentials. We study the partition function at finite central charge perturbatively in the chemical potentials, and find that this degeneracy is broken. We also study the statistics of the KdV charges at high level within a Virasoro representation, and find that they are sharply peaked.
Highlights
JHEP03(2019)075 is to explore the relation between this ensemble and an individual microstate of the theory, in the limit of large energies
We have studied the structure of the Generalised Gibbs ensemble with chemical potentials for the KdV charges in the high temperature limit, and the comparison of the expectation values of the KdV charges in this ensemble to their values in a particular microstate
We found that in the large central charge limit, the ensemble partition function could be obtained from a saddle-point approximation, and the expectation values of the KdV charges match those of a primary state
Summary
We will consider the GGE in the limit of large temperature and large central charge, where we can compute the partition function directly by a saddle-point analysis. At leading order at large c, the KdV charges take approximately the same value on the descendents contributing to the GGE partition function. In the final expression we have expanded the result near μ3 = 0, and see that at small μ3 the saddle reduces to the usual thermal saddle with h = 1 as expected Substituting this back into f (h) to compute the GGE partition function gives log ZGGE (μ3 ). At subleading orders in k the values of the KdV charges for a typical microstate are not determined by the energy; we need to consider contributions from the non-zero. As noted in [23], solving for the chemical potentials to match the values of the KdV charges in a particular microstate would require us to solve an infinite system of equations. We will be able to determine many features of the finite k ensemble quite explicitly
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