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 Ar. 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.
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