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.

65+1 pages, 10 figures, typos corrected, published version

References in corpus (3)

• Pseudo-differential equations, and the Bethe Ansatz for the classical Lie algebras
• WKB periods for higher order ODE and TBA equations
• Quantum periods and TBA equations for $\mathcal{N}=2SU(2)N_f=2$ SQCD with flavor symmetry

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• Thermodynamic Bethe ansatz and wall crossing for deformed supersymmetric quantum mechanics
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• Integrals of motion in $W{E}_6$ CFT and the ODE/IM correspondence
• A short review on TBA equation and scattering amplitude/Wilson loop duality
• Painlevé Kernels and Surface Defects at Strong Coupling