Thermal correlation functions of KdV charges in 2D CFT

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. Introduction and summary of resultsTwo dimensional conformal field theories (CFTs) have an infinite-dimensional symmetry algebra -the Virasoro algebra of local conformal transformations -which powerfully constrains the structure of the theory. Our goal is to explore the consequences of this symmetry structure for the finite-temperature behaviour of the theory. We will study two dimensional CFTs at finite temperature on the circle. Our interest in this paper is obtaining exact results: we will keep the radius of the circle finite, and will not study the high temperature or large c limits, though these will be discussed in a companion paper [1].Our starting point is the observation that, from the Virasoro algebra, one can define an infinite set of mutually commuting conserved charges [2-4]. In the large c limit these charges generate the KdV hierarchy of differential equations, so they are often referred to as the quantum KdV charges. We will follow the notation of [4] where the charges are denoted I 2m−1 with m = 1, 2, . . . , and the subscript labels the spin of the charge. The first KdV charge I 1 is just L 0 = T (z)dz, the zero mode of the holomorphic part of the stress tensor. The KdV charge I 2m−1 is an m th order polynomial in the Virasoro generators L n .Given this set of mutually commuting charges, one can define a Generalized Gibbs Ensemble (GGE) for two-dimensional CFTs where we introduce a chemical potential for each KdV charge: 1 Z[β, µ 3 , µ 5 , . . .] = Tr [e −βE+µ 3 I 3 +µ 5 I 5 +... ] = Tr [e µ 3 I 3 +µ 5 I 5 +... q L 0 −c/24 ], q ≡ e −β , (1.1)In this equation we have introduced chemical potentials for the left-moving (holomorphic) generators L 0 , I 3 , I 5 , . . . . In general, of course, one should also introduce potentials for the right-moving (anti-holomorphic) generatorsL 0 ,Ī 3 ,Ī 5 , . . . . However, all of our results involve the use just of the Virasoro algebra so the left-and right-moving sectors completely factorize.1 See [5-9] for some recent work on GGEs and [10,11] for recent experimental realisations. This ensemble was studied from a holographic perspective in [12]; see also [13].These KdV charges act within each Virasoro module, so can be simultaneously diagonalized level by level with...

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