Connection problem for the sine-Gordon/Painlevé III tau function and irregular conformal blocks
A. Its, O. Lisovyy, Yu. Tykhyy
TL;DR
This work addresses the connection problem for the tau function of the radial sine-Gordon/Painlevé III3 equation by exploiting isomonodromic deformations and conformal-block representations. It establishes convergence for the short-distance tau expansion in irregular c=1 blocks and proposes a structured long-distance expansion with a monodromy-dependent prefactor. The authors derive an explicit conjectural formula for the connection coefficient χ in terms of Barnes G-functions and a dilogarithm-based generating function, tying two canonical monodromy-coordinate systems together via a generating function. They validate consistency through symmetry properties and numerical checks, extending the conformal-block/gauge-theory viewpoint to Painlevé III3 and contributing to the broader constant-term problem for Painlevé tau-functions.
Abstract
The short-distance expansion of the tau function of the radial sine-Gordon/Painlevé III equation is given by a convergent series which involves irregular $c=1$ conformal blocks and possesses certain periodicity properties with respect to monodromy data. The long-distance irregular expansion exhibits a similar periodicity with respect to a different pair of coordinates on the monodromy manifold. This observation is used to conjecture an exact expression for the connection constant providing relative normalization of the two series. Up to an elementary prefactor, it is given by the generating function of the canonical transformation between the two sets of coordinates.