Wall-crossing of TBA equations and WKB periods for the third order ODE

Abstract: We study the WKB periods for the third order ordinary differential equation (ODE) with polynomial potential, which is obtained by the Nekrasov-Shatashvili limit of ($A_2,A_N$) Argyres-Douglas theory in the Omega background. In the minimal chamber of the moduli space, we derive the Y-system and the thermodynamic Bethe ansatz (TBA) equations by using the ODE/IM correspondence. The exact WKB periods are identified with the Y-functions. Varying the moduli parameters of the potential, the wall-crossing of the TBA equations occurs. We study the process of the wall-crossing from the minimal chamber to the maximal chamber for $(A_2,A_2)$ and $(A_2,A_3)$. When the potential is a monomial type, we show the TBA equations obtained from the ($A_2, A_2$) and ($A_2, A_3$)-type ODE lead to the $D_4$ and $E_6$-type TBA equations respectively.