A Tau function for q-Painlevé VI as a Fredholm determinant

2026/08/04 by Harini Desiraju, Pieter Roffelsen
Physics and Astronomy · Mathematics · #math-ph #math.CA #math.MP #nlin.SI

paper · pdf

32 pages, 3 figures

arxiv created 2026/08/04 · arxiv updated 2026/08/05

Abstract

We give an analytic construction of a tau function for the q-difference sixth Painlevé equation (qPVI) as a Fredholm determinant through the general Riemann-Hilbert problem associated with it. We show that the tau function is an analytic function on its domain of definition that vanishes at a particular point if and only if the corresponding Riemann-Hilbert problem is point-wise not solvable there. We express the corresponding qPVI transcendents in terms of the tau function as well as three copies of it with some of the parameters shifted. Then the vanishing of any of these four tau functions corresponds to the transcendents taking value in a specific corresponding exceptional line on the initial value space of qPVI. Finally, we derive an asymptotic expansion of the tau function for small times t.

Citations

• Blowup relations and q-Painlevé VI
• Modular transformations of tau functions and conformal blocks on the torus
• Riemann-Hilbert problems, Fredholm determinants, explicit combinatorial expansions, and connection formulas for the general q-Painlevé III₃ tau functions
• Cluster Reductions, Mutations, and q-Painlevé Equations
• On q-Painlevé VI and the geometry of Segre surfaces
• On the monodromy manifold of q-Painlevé VI and its Riemann-Hilbert problem
• The space of monodromy data for the Jimbo-Sakai family of q-difference equations
• On the Riemann-Hilbert problem for a q-difference Painlevé equation
• CFT approach to the q-Painlevé VI equation
• Pure SU(2) gauge theory partition function and generalized Bessel kernel
• The Lattice Structure of Connection Preserving Deformations for q-Painlevé Equations I
• Zeros of the Jimbo, Miwa, Ueno tau function
• A q-anaolg of the sixth Painlevé equation