Movable poles of the solutions of Painleve's equation of the third kind and their relation with mathieu functions

Non-perturbative approaches to the quantum Seiberg-Witten curve

“…It was observed by many authors [63,71,72] that the movable poles of Painlevé III 3 are somehow related to Mathieu functions. In particular in [73], based on [65], it was observed that the zeros of the Painlevé III 3 τ function compute the spectrum of modified Mathieu (with a suitable dictionary).…”

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

“…The situation with the second tau function τ ∞ − (ρ iR , 0, ir) is indeed better. According to (5.40) its vanishing determines the pole of solution, corresponding to the spectral problem for the cosh-Mathieu equation, and this is actually a well-known one-parametric family of solutions, discussed in the literature [19,20]. Moreover, this second tau function τ − can be identified with the spectral determinant from [28,29], giving rise to a Fermi-gas representation for particular PIII 3 tau function and irregular blocks at infinity.…”

“…It has been also discovered that c → ∞ conformal blocks describe the spectra of 2-nd order differential equations [16,17], corresponding to quantum-mechanical version of the Seiberg-Witten integrable system [18], so that exact quantization conditions are written in terms of quasiclassical conformal blocks. Together with relation of c = 1 conformal blocks with 2 × 2 first-order matrix differential equations, arising in the context of auxiliary linear problem by isomonodromy/CFT correspondence, this leads to idea that the blow-up relations for conformal blocks actually arise from the relation between 2-nd order differential equations and 2 × 2 systems, known already for a long time [19]. Inspired by ([21], section 6) we derive the quantization conditions for the quantum cosh-/cos-Mathieu systems 3 as some restrictions on monodromy data of the related 2 × 2 system, supplied with an extra relation on cancellation of apparent singularity, being actually vanishing of the Bäcklundtransformed tau function.…”

“…To do this it is convenient to use the WKB parameterization of monodromies. All definitions and conventions are collected in the appendix A, there is also an elementary overview of the construction, 19 and here we proceed to direct computation of the transition matrices.…”

“…18 In [13] the term 4πνρ − 4πiσ was crucial to solve the difference equations on χ(σ, ν; η). 19 For the rigorous and detailed explanation of exact WKB analysis see [37,38]. WKB parameterization of monodromies is also very well-known in the context of the four-dimensional supersymmetric gauge theories, see [27,39,40] and references therein.…”