Irregular conformal blocks, Painlevé III and the blow-up equations

Abstract

We study the relation of irregular conformal blocks with the Painlevé III3 equation. The functional representation for the quasiclassical irregular block is shown to be consistent with the BPZ equations of conformal field theory and the Hamilton-Jacobi approach to Painlevé III3. It leads immediately to a limiting case of the blow-up equations for dual Nekrasov partition function of 4d pure supersymmetric gauge theory, which can be even treated as a defining system of equations for both c = 1 and c → ∞ conformal blocks. We extend this analysis to the domain of strong-coupling regime where original definition of conformal blocks and Nekrasov functions is not known and apply the results to spectral problem of the Mathieu equations. Finally, we propose a construction of irregular conformal blocks in the strong coupling region by quantization of Painlevé III3 equation, and obtain in this way a general expression, reproducing c = 1 and quasiclassical c → ∞ results as its particular cases. We have also found explicit integral representations for c = 1 and c = −2 irregular blocks at infinity for some special points.

Quantum spectral problems and isomonodromic deformations

“…More recently an alternative proof for the Painlevé VI example was presented in [59,60] based on blowup equation with defects. The interplay between Painlevé and blowup equations appearing in this section is similar to the one of [55,52,56] and does not require any defect.…”

“…for such problem can be found in [52] 4 . More recently, in [55,56,57], a four dimensional version of these relations has been applied in the context of Painlevé equations and spectral theory. We will discuss this further in the main text.…”

“…5 we discuss the compatibility between our results and the Nekrasov-Shatashvili (NS) exact quantization. For the Painlevé III 3 example we essentially follow [56,55] with slight improvement. For the example of the torus these results are new.…”

“…7 we deduce the NS limit of the blowup relations from the regularized action functional and CFT arguments. This is done by following the method developed in [56], which was also inspired by the works of [21,18].…”

“…Connection between the spectrum of modified Mathieu and the poles of Painlevé III 3 have been observed for instance in [16,62,18] at the level of asymptotic expansions as well as numerically. This interplay was recently revisited in [55,56] from the optic of the Ω background and blowup equations.…”