Integrable Structure of Higher Spin CFT and the ODE/IM Correspondence

TL;DR

This paper explores the integrable structure of higher spin conformal field theories with ${\mathcal W}_3$ symmetry, computing eigenvalues of conserved charges and their thermal expectation values using the ODE/IM correspondence and Zhu recursion relations.

Contribution

It introduces a method to compute quantum Boussinesq charges and their thermal averages in higher spin CFTs, linking conserved charges to quasi-modular forms.

Findings

Eigenvalues of quantum Boussinesq charges computed for vacuum and excited states

Thermal correlators involving energy-momentum and spin-3 currents derived

Quantum Boussinesq charges expressed as quasi-modular differential operators

Abstract

We study two dimensional systems with extended conformal symmetry generated by the ${\mathcal W}_3$ algebra. These are expected to have an infinite number of commuting conserved charges, which we refer to as the quantum Boussinesq charges. We compute the eigenvalues of the quantum Boussinesq charges in both the vacuum and first excited states of the higher spin module through the ODE/IM correspondence. By studying the higher spin conformal field theory on the torus, we also calculate thermal correlators involving the energy-momentum tensor and the spin-3 current by making use of the Zhu recursion relations. By combining these results, we show that it is possible to derive the current densities, whose integrals are the quantum Boussinesq charges. We also evaluate the thermal expectation values of the conserved charges, and show that these are quasi-modular differential operators acting on the character of the higher spin module.