Spectrum of quantum KdV hierarchy in the semiclassical limit

TL;DR

This paper calculates the spectrum of quantum KdV charges in the semiclassical limit using $1/c$ expansion, revealing quantum corrections to classical integrable structures and applying results to thermal states and free energy analysis.

Contribution

It provides explicit formulas for quantum KdV charge spectra up to third order in $1/c$, including quantum corrections to classical action variables.

Findings

Derived spectrum of quantum KdV charges up to third order in $1/c$

Identified quantum corrections to classical integrable structures

Applied spectrum results to thermal expectation values and free energy calculations

Abstract

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.