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Abstract |
I am going to explain the explicit construction of general solutions to isomonodromy equations, with the main
focus on the Painlevé VI equation. I will start by deriving a Fredholm déterminant representation of the
Painlevé VI tau function. The corresponding integral operator acts in the direct sum of two copies of
$L^2(S^1)$. Its kernel is expressed in terms of hypergeometric fundamental solutions of two auxiliary 3-point
Fuchsian systems whose monodromy is determined by the monodromy of the associated linear problem via a
decomposition of the 4-punctured sphere into two pairs of pants. In the Fourier basis, this kernel is given by
an infinite Cauchy matrix. I will explain how the principal minor expansion of the Fredholm determinant yields
a combinatorial series representation for the general solution to Painlevé VI in the form of a sum over pairs of
Young diagrams. The latter series coincides with the dual Nekrasov partition function of the $\mathcal N=2$
$N_f=2$ $SU(2)$ gauge theory in the self-dual $\Omega$-background. |
