Integrals of motion in conformal field theory with W-symmetry and the ODE/IM correspondence

TL;DR

This paper extends the ODE/IM correspondence to higher-order differential equations linked with affine Toda field theories and W-algebras, providing new methods to compute integrals of motion in conformal field theories.

Contribution

It introduces a generalization of the ODE/IM correspondence to higher-order ODEs associated with affine Lie algebras, and computes integrals of motion for W-algebras on a cylinder.

Findings

Eigenvalues of integrals of motion match period integrals up to sixth order.

WKB expansions correspond to classical conserved currents.

Results enable prediction of higher-order integrals of motion.

Abstract

We study the ODE/IM correspondence between two-dimensional $WA_{r}$/$WD_{r}$-type conformal field theories and the higher-order ordinary differential equations (ODEs) obtained from the affine Toda field theories associated with $A_r^{(1)}/D_r^{(1)}$-type affine Lie algebras. We calculate the period integrals of the WKB solution to the ODE along the Pochhammer contour, where the WKB expansions correspond to the classical conserved currents of the Drinfeld-Sokolov integrable hierarchies. We also compute the integrals of motion for $WA_{r}$($WD_{r}$) algebras on a cylinder. Their eigenvalues on the vacuum state are confirmed to agree with the period integrals up to the sixth order. These results generalize the ODE/IM correspondence to higher-order ODEs and can be used to predict higher-order integrals of motion.