WKB periods for higher order ODE and TBA equations
Katsushi Ito (Department of Physics, Tokyo Institute of Technology, Tokyo, 152-8551, Japan); Takayasu Kondo (Department of Physics, Tokyo Institute of Technology, Tokyo, 152-8551, Japan); Kohei Kuroda (Department of Physics, Tokyo Institute of Technology, Tokyo, 152-8551, Japan); Hongfei Shu (Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, Stockholm, SE-106 91, Sweden, Beijing Institute of Mathematical Sciences and Applications (BIMSA), Beijing, 101408, China)
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.