Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions

Understand

We derive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems with $n$ regular singular points on the Riemann sphere and generic monodromy in $\mathrm{GL}(N,\mathbb C)$.

• The corresponding operator acts in the direct sum of $N(n-3)$ copies of $L^2(S^1)$.
• Its kernel has a block integrable form and is expressed in terms of fundamental solutions of $n-2$ elementary 3-point Fuchsian systems whose monodromy is determined by monodromy of the relevant $n$-point system via a decomposition of the punctured sphere into pairs of pants.
• For $N=2$ these building blocks have hypergeometric representations, the kernel becomes completely explicit and has Cauchy type.