Spectrum of quantum KdV hierarchy in the semiclassical limit

We employ semiclassical quantization to calculate spectrum of quantum KdV charges in the limit of large central charge c. Classically, KdV charges Q_2n−1 generate completely integrable dynamics on the co-adjoint orbit of the Virasoro algebra. They can be expressed in terms of action variables I_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 numbers n_i. Crucially, classical expression, which is homogeneous in I_k, acquires quantum corrections that include terms of subleading powers in n_k. At first two non-trivial orders in 1/c expansion these “quantum” terms can be fixed from the analytic form of Q_2n−1 acting on the primary states. In this way we find explicit expression for the spectrum of Q_2n−1 up to first three orders in 1/c expansion. We apply this result to study thermal expectation values of Q_2n−1 and free energy of the KdV Generalized Gibbs Ensemble.