WKB periods for higher order ODE and TBA equations

TL;DR

This paper investigates WKB periods for higher order ODEs derived from affine Toda field equations, establishing a link with TBA equations and confirming the relation through numerical solutions.

Contribution

It introduces a formula connecting WKB periods with Y-functions in the ODE/IM correspondence for higher order equations, supported by numerical verification.

Findings

Derived quantum corrections using Picard-Fuchs operators.

Proposed a formula linking Y-functions and WKB periods.

Confirmed the formula through numerical solutions of TBA equations.

Abstract

We study the WKB periods for the $(r+1)$-th order ordinary differential equation (ODE) which is obtained by the conformal limit of the linear problem associated with the $A_r^{(1)}$ affine Toda field equation. We compute the quantum corrections by using the Picard-Fuchs operators. The ODE/IM correspondence provides a relation between the Wronskians of the solutions and the Y-functions which satisfy the thermodynamic Bethe ansatz (TBA) equation related to the Lie algebra $A_r$. For the quadratic potential, we propose a formula to show the equivalence between the logarithm of the Y-function and the WKB period, which is confirmed by solving the TBA equation numerically.