On KdV characters in large c CFTs

TL;DR

This work analyzes the integrable structure of 2D CFTs with Virasoro symmetry in the large central charge regime by linking KdV charges to the quantum Sinh-Gordon theory via a Feigin–Fuchs/Miura framework, valid for $c\ge25$. It develops explicit large-$c$ constructions of the KdV and generalized KdV (GGE) characters, separating vacuum and primary module contributions and deriving leading degeneracy growth through saddles and Hardy–Ramanujan type counts. The study identifies how high- and low-chemical-potential regimes might be connected by nontrivial transformations of the chemical potentials, and it analyzes the constraints on chemical potentials necessary for the large-$c$ approximations to hold. The results illuminate how GGEs capture finer microstructure than thermal states in integrable 2D CFTs and hint at holographic interpretations via multitrace deformations and tau-functions of the KdV hierarchy. Overall, the paper provides building blocks for understanding GGEs in CFTs with an infinite set of conserved KdV charges and opens pathways to finite-$c$ extensions and holographic realizations.

Abstract

Two-dimensional conformal field theories with just Virasoro symmetry are endowed with integrable structure. We review how to construct the integrable charges in a two-dimensional conformal field theory and how to relate them to the charges of quantum Sinh-Gordon theory when c>25. We then explicitly calculate the single charge characters in the large c limit for all charges and thereby reveal how their degeneracies grow within one module. This, in particular, allows us to approximate the characters in the limit of small chemical potential, which source the respective charges. The latter give us insights into possible transformation properties of the characters. We also comment on the full generalized Gibbs ensemble and approximations to pure states.