938 research outputs found
Monodromy dependence and connection formulae for isomonodromic tau functions
We discuss an extension of the Jimbo-Miwa-Ueno differential 1-form to a form
closed on the full space of extended monodromy data of systems of linear
ordinary differential equations with rational coefficients. This extension is
based on the results of M. Bertola generalizing a previous construction by B.
Malgrange. We show how this 1-form can be used to solve a long-standing problem
of evaluation of the connection formulae for the isomonodromic tau functions
which would include an explicit computation of the relevant constant factors.
We explain how this scheme works for Fuchsian systems and, in particular,
calculate the connection constant for generic Painlev\'e VI tau function. The
result proves the conjectural formula for this constant proposed in
\cite{ILT13}. We also apply the method to non-Fuchsian systems and evaluate
constant factors in the asymptotics of Painlev\'e II tau function.Comment: 54 pages, 6 figures; v4: rewritten Introduction and Subsection 3.3,
added few refs to match published articl
Connection problem for the sine-Gordon/Painlev\'e III tau function and irregular conformal blocks
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 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.Comment: 18 pages, 1 figur
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