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Title
Japanese:WKB periods for higher order ODE and TBA equations 
English:WKB periods for higher order ODE and TBA equations 
Author
Japanese: 伊藤克司, 近藤宇泰, 黒田航平, 束紅非.  
English: Katsushi ITO, Takayasu Kondo, Kohei Kuroda, Hongfei Shu.  
Language English 
Journal/Book name
Japanese:Journal of High Energy Physics 
English:Journal of High Energy Physics 
Volume, Number, Page Vol. 2021    No. 10   
Published date Oct. 2021 
Publisher
Japanese: 
English: 
Conference name
Japanese: 
English: 
Conference site
Japanese: 
English: 
Official URL http://dx.doi.org/10.1007/jhep10(2021)167
 
DOI https://doi.org/10.1007/jhep10(2021)167
Abstract <jats:title>A<jats:sc>bstract</jats:sc> </jats:title><jats:p>We study the WKB periods for the (<jats:italic>r</jats:italic> + 1)-th order ordinary differential equation (ODE) which is obtained by the conformal limit of the linear problem associated with the <jats:inline-formula><jats:alternatives><jats:tex-math>$$ {A}_r^{(1)} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>A</mml:mi> <mml:mi>r</mml:mi> <mml:mfenced> <mml:mn>1</mml:mn> </mml:mfenced> </mml:msubsup> </mml:math></jats:alternatives></jats:inline-formula> 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 <jats:italic>A</jats:italic><jats:sub><jats:italic>r</jats:italic></jats:sub>. 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.</jats:p>

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