Wall-crossing of TBA equations and WKB periods for the third order ODE
Katsushi Ito (Department of Physics, Tokyo Institute of Technology, Tokyo, Japan); Takayasu Kondo (Department of Physics, Tokyo Institute of Technology, Tokyo, Japan); Hongfei Shu (Beijing Institute of Mathematical Sciences and Applications (BIMSA), Beijing, China, Yau Mathematical Sciences Center (YMSC), Tsinghua University, Beijing, China)
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 () 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 and . When the potential is a monomial type, we show the TBA equations obtained from the () and ()-type ODE lead to the and -type TBA equations respectively.
