Connection problem for the sine-Gordon/Painlev\'e III tau function and irregular conformal blocks

Understand

The short-distance expansion of the tau function of the radial sine-Gordon/Painlev\'e 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.