WKB periods for higher order ODE and TBA equations

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.

Automated summary

In this study, quantum corrections to the WKB periods of the (r + 1)-th order ordinary differential equation obtained through the conformal limit of the linear problem associated with the A(r)((1)) affine Toda field equation are computed using the Picard-Fuchs operators. The ODE/IM correspondence establishes a relationship between the Wronskians of the solutions and the Y-functions satisfying the thermodynamic Bethe ansatz (TBA) equation related to the Lie algebra A(r). A proposed formula demonstrates the equivalence between the logarithm of the Y-function and the WKB period for the quadratic potential, validated through numerical solutions of the TBA equation.