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We employ semiclassical quantization to calculate spectrum of quantum KdV charges in the limit of large central chargec. Classically, KdV chargesQ_2n−1generate completely integrable dynamics on the co-adjoint orbit of the Virasoro algebra. They can be expressed in terms of action variablesI_k, e.g. as a power series expansion. Quantum-mechanically this series becomes the expansion in 1/c, while action variables become integer-valued quantum numbersn_i. Crucially, classical expression, which is homogeneous inI_k, acquires quantum corrections that include terms of subleading powers inn_k. At first two non-trivial orders in 1/cexpansion these “quantum” terms can be fixed from the analytic form ofQ_2n−1acting on the primary states. In this way we find explicit expression for the spectrum ofQ_2n−1up to first three orders in 1/cexpansion. We apply this result to study thermal expectation values ofQ_2n−1and free energy of the KdV Generalized Gibbs Ensemble.

 

A bstract Studies of Eigenstate Thermalization Hypothesis (ETH) in two-dimensional CFTs call for calculation of the expectation values of local operators in highly excited energy eigenstates. This can be done efficiently by representing zero modes of these operators in terms of the Virasoro algebra generators. In this paper we present a pedagogical introduction explaining how this calculation can be performed analytically or using computer algebra. We illustrate the computation of zero modes by a number of examples and list explicit expressions for all local operators from the vacuum family with the dimension of less or equal than eight. Finally, we derive an explicit expression for the quantum KdV generator Q 7 in terms of the Virasoro algebra generators. The obtained results can be used for quantitative studies of ETH at finite value of central charge.