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Riemann-Hilbert problems, Fredholm determinants, explicit combinatorial expansions, and connection formulas for the general $q$-Painlev\'e III$_3$ tau functions

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arxiv 2501.01419 v1 pith:5OD7CRP7 submitted 2025-01-02 math-ph hep-thmath.DSmath.MPnlin.SI

Pavlo Gavrylenko This is my paper
classification math-phhep-thmath.DSmath.MPnlin.SI
keywords fredholmfunctionsproblemriemann-hilbertconnectiondeterminantexplicitpainlev
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abstract

We reformulate the $q$-difference linear system corresponding to the $q$-Painlev\'e equation of type $A_7^{(1)'}$ as a Riemann-Hilbert problem on a circle. Then, we consider the Fredholm determinant built from the jump of this Riemann-Hilbert problem and prove that it satisfies bilinear relations equivalent to $P(A_7^{(1)'})$. We also find the minor expansion of this Fredholm determinant in explicit factorized form and prove that it coincides with the Fourier series in $q$-deformed conformal blocks, or partition functions of the pure $5d$ $\mathcal{N}=1$ $SU(2)$ gauge theory, including the cases with the Chern-Simons term. Finally, we solve the connection problem for these isomonodromic tau functions, finding in this way their global behavior.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A Tau function for $q$-Painlev\'e VI as a Fredholm determinant

    math-ph 2026-08 conditional novelty 8.0 of 10

    The q-Painleve VI tau function is constructed analytically as a Fredholm determinant, with zeros detecting non-solvability of the underlying Riemann-Hilbert problem.

  2. Modular transformations of tau functions and conformal blocks on the torus

    math-ph 2025-08 conditional novelty 8.0 of 10

    The paper derives the modular connection constant for tau functions on the one-punctured torus and obtains an exact closed formula for the c=1 Virasoro modular kernel.

  3. Blowing-up the edge: connection formulae and stability chart of the Lam\'e equation

    hep-th 2025-07 conditional novelty 7.0 of 10

    The paper derives the resummed Nekrasov-Shatashvili free energy from blow-up equations and uses it to compute the band-gap structure, connection formulas, and stability chart of the Lamé equation.

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