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Thermal Correlation Functions of KdV Charges in 2D CFT

Alexander Maloney, Gim Seng Ng, Simon F. Ross, Ioannis Tsiares

TL;DR

The paper derives that thermal correlation functions of quantum KdV charges in 2D CFTs are governed by quasi-modular differential operators acting on torus partition functions, with one-point functions transforming as modular forms and higher-point functions acquiring E2-dependent quasi-modularity. It develops two systematic evaluation routes—direct Virasoro commutator cycling and Zhu’s recursion relations—to obtain explicit differential operator forms for various n-point functions and validates them against minimal-model constraints, including vanishing results in (2n+3,2) models via null states and integrability. It also applies these results to extract exact moments of KdV charges at fixed level, showing that distributions become sharply peaked as level grows, and discusses implications for the generalized Gibbs ensemble and ETH in 2D CFTs. Collectively, the work provides a finite-c, finite-temperature framework for KdV charges, clarifies its modular structure, and connects to minimal-model integrability and statistical properties of conserved charges.

Abstract

Two dimensional CFTs have an infinite set of commuting conserved charges, known as the quantum KdV charges, built out of the stress tensor. We compute the thermal correlation functions of the these KdV charges on a circle. We show that these correlation functions are given by quasi-modular differential operators acting on the torus partition function. We determine their modular transformation properties, give explicit expressions in a number of cases, and give a general form which determines an arbitrary correlation function up to a finite number of functions of the central charge. We show that these modular differential operators annihilate the characters of the (2m+1,2) family of non-unitary minimal models. We also show that the distribution of KdV charges becomes sharply peaked at large level.

Thermal Correlation Functions of KdV Charges in 2D CFT

TL;DR

The paper derives that thermal correlation functions of quantum KdV charges in 2D CFTs are governed by quasi-modular differential operators acting on torus partition functions, with one-point functions transforming as modular forms and higher-point functions acquiring E2-dependent quasi-modularity. It develops two systematic evaluation routes—direct Virasoro commutator cycling and Zhu’s recursion relations—to obtain explicit differential operator forms for various n-point functions and validates them against minimal-model constraints, including vanishing results in (2n+3,2) models via null states and integrability. It also applies these results to extract exact moments of KdV charges at fixed level, showing that distributions become sharply peaked as level grows, and discusses implications for the generalized Gibbs ensemble and ETH in 2D CFTs. Collectively, the work provides a finite-c, finite-temperature framework for KdV charges, clarifies its modular structure, and connects to minimal-model integrability and statistical properties of conserved charges.

Abstract

Two dimensional CFTs have an infinite set of commuting conserved charges, known as the quantum KdV charges, built out of the stress tensor. We compute the thermal correlation functions of the these KdV charges on a circle. We show that these correlation functions are given by quasi-modular differential operators acting on the torus partition function. We determine their modular transformation properties, give explicit expressions in a number of cases, and give a general form which determines an arbitrary correlation function up to a finite number of functions of the central charge. We show that these modular differential operators annihilate the characters of the (2m+1,2) family of non-unitary minimal models. We also show that the distribution of KdV charges becomes sharply peaked at large level.

Paper Structure

This paper contains 35 sections, 182 equations, 1 figure.

Table of Contents

  1. Introduction and summary of results
  2. Modular differential form of the thermal expectation values
  3. Differential operator form
  4. Modular transformation properties
  5. Determining thermal expectation values: $q$ expansion and null states
  6. One-point functions
  7. Two-point functions
  8. Higher point functions
  9. Explicit evaluation by commutators
  10. Evaluation using Zhu's recursion relations
  11. One-point functions
  12. Higher-point functions
  13. KdV charges for minimal models
  14. Relation to null states
  15. Relation to integrability
  16. ...and 20 more sections

Figures (1)

  • Figure 1: The mean value ${\overline I_3}(n)$ and the normalized variance $\Delta I_3(n)/ {\overline I_3}(n)$ of the KdV charge $I_3$ as a function of level $n$. These data are plotted for the Verma module with dimension $h=1$ and central charge $c=6$.