Isomonodromic Tau Functions on a Torus as Fredholm Determinants, and Charged Partitions

Journal article

Building upon the recent works of Bertola; Fasondini, Olver and Xu, we define a class of orthogonal polynomials on elliptic curves and establish a corresponding Riemann-Hilbert framework. We then focus on the special case, defined by a constant weight function, and use the Riemann-Hilbert problem to derive recurrence relations and differential equations for the orthogonal polynomials. We further show that the sub-class of even polynomials is associated to the elliptic form of Painlev\'e VI, with the tau function given by the Hankel determinant of even moments, up to a scaling factor. The first iteration of these even polynomials relates to the special case of Painlev\'e VI studied by Hitchin in relation to self-dual Einstein metrics.Comment: 31 pages, 6 figure

Authors: Del Monte, F; Desiraju, H; Gavrylenko, P

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer
• Journal: Communications in Mathematical Physics
• Volume: 398
• Issue: 3
• Pages: 1029-1084
• Publication date: 2023-03-01
• DOI: 10.1007/s00220-022-04458-y
• EISSN: 1432-0916
• ISSN: 0010-3616
• Language: English
• Keywords: Computer science; Torus; Mathematics; Complex system; Algebra over a field; Fredholm determinant; Artificial intelligence; Physics; Integral equation; Fredholm integral equation; Pure mathematics; Fredholm theory; Mathematical analysis; Geometry