Thermal Correlators and Currents of the $\mathcal{W}_3$ Algebra

TL;DR

This paper constructs local operators in 2D conformal field theories with $ ext{W}_3$ symmetry, linking conserved charges, thermal correlators, and the higher spin algebra to deepen understanding of integrable structures.

Contribution

It introduces explicit local operators corresponding to quantum Boussinesq charges in $ ext{W}_3$ theories, connecting thermal correlators and eigenvalues via the ODE/IM correspondence.

Findings

Constructed local operators for $ ext{W}_3$ charges.

Computed thermal correlators involving stress tensor and spin-3 current.

Derived eigenvalues of charges using ODE/IM correspondence.

Abstract

Two dimensional conformal field theories with the extended $\mathcal{W}_3$ symmetry algebra have an infinite number of mutually commuting conserved charges, which are referred to as the quantum Boussinesq charges. In this work we construct local operators whose zero modes are precisely these conserved charges. For this purpose we study the higher spin conformal field theory on the torus and compute thermal correlators involving the stress tensor and the spin-3 current in a higher spin module of the W3 algebra. In addition we independently obtain the excited state eigenvalues of the quantum Boussinesq charges within the higher spin module via the ODE/IM correspondence. A judicious combination of these data allows us to derive the local operators, whose integrals are the conserved charges of the integrable hierarchy.