Thermal Correlation Functions of KdV Charges in 2D CFT

TL;DR

This paper computes thermal correlation functions of quantum KdV charges in 2D CFTs, revealing their modular properties and behavior in minimal models and at large levels.

Contribution

It introduces a method to express KdV charge correlations as quasi-modular differential operators acting on the torus partition function, with explicit formulas and modular transformation analysis.

Findings

Correlation functions are given by quasi-modular differential operators.

Modular properties of these functions are explicitly determined.

Distribution of KdV charges becomes sharply peaked at large level.

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.