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

Thermal Correlation Functions of KdV Charges in 2D CFT