Integrals of motion in $WE_6$ CFT and the ODE/IM correspondence

TL;DR

This paper explores the ODE/IM correspondence for an E6-affiliated differential equation, calculating period integrals and integrals of motion, and demonstrating their agreement up to sixth order.

Contribution

It introduces a detailed analysis of the ODE/IM correspondence for E6, including WKB expansion, period integrals, and eigenvalue calculations in W-symmetric CFT.

Findings

Period integrals match integrals of motion eigenvalues up to sixth order.

WKB expansion performed via diagonalization method.

Eigenvalues on highest-weight states agree with period integrals.

Abstract

We study the ODE/IM correspondence for the ordinary differential equation associated with the affine Lie algebra $E_6^{(1)}$. The WKB expansion of the solution of the ODE is performed by the diagonalization method, and the period integrals of the WKB coefficients along the Pochhammer contour are calculated. We also compute the integrals of motion on a cylinder in two-dimensional conformal field theory with W-symmetry associated with $E_6^{(1)}$. Their eigenvalues on the highest-weight state are shown to agree with the period integrals up to the sixth order.