The paper derives the resummed Nekrasov-Shatashvili free energy from blow-up equations and uses it to compute the band-gap structure, connection formulas, and stability chart of the Lamé equation.
Conformal blocks and Painlev\'e functions
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abstract
This paper is based on my presentation at RIMS workshop on "Theory of Integrable Systems and Its Applications in Various Fields" held in Kyoto on 19--21, August 2015. The aim of the present paper is to give a short account of recent studies on relations between conformal blocks in the two-dimensional conformal field theory and Painlev\'e functions. In addition, we present a conjecture on a combinatorial expansion formula of the three-point irregular conformal block at an irregular singular point, with two regular singular points and one irregular singular point. Our conjectural expansion formula is written in terms of pairs of skew Young diagrams, while the four-point regular conformal block, by AGT correspondence, is written in terms of pairs of Young diagrams.
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Blowing-up the edge: connection formulae and stability chart of the Lam\'e equation
The paper derives the resummed Nekrasov-Shatashvili free energy from blow-up equations and uses it to compute the band-gap structure, connection formulas, and stability chart of the Lamé equation.