Generalized Gibbs Ensemble and the Statistics of KdV Charges in 2D CFT

TL;DR

This work analyzes the Generalized Gibbs Ensemble (GGE) for 2D CFTs with an infinite set of quantum KdV charges $I_{2m-1}$ at high temperature, focusing on the large-$c$ limit and the associated saddle-point structure. The authors show that, at leading order, the saddle in the conformal dimension $h$ does not uniquely fix the chemical potentials, implying that finite-$c$ corrections are essential for matching microstates; perturbative finite-$c$ corrections reveal nonuniversal behavior once chemical potentials are turned on. By employing modular transformations, they compute high-temperature correlators of KdV charges from low-temperature data, establishing that connected correlators are suppressed by powers of the temperature and that the KdV charges factorize in the high-temperature regime, with explicit results for low-point functions and $\\log Z_{GGE}$ corrections. In a complementary analysis within a single Verma module, the KdV charge statistics are computed via differential operators on the character; the resulting distributions are sharply peaked, with level statistics aligning with canonical predictions and supporting a level-resolved correspondence. Overall, the paper clarifies how conserved KdV charges shape ensemble equivalence and microstate matching in high-energy 2D CFTs, highlighting the crucial role of modular properties and finite-$c$ effects.

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.